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Electrophysiology of the Action Potential
and its Propagation
James T. Fulton
Neural Concepts
Abstract: The action potential is defined rigorously for the first time, based
on its theoretical form. The action potential is the most obvious signal in the
brain but not the most important. Less than 5% of the neurons in the brain
generate action potentials; the other 95% or greater number of neurons generate
analog output signals that are much more difficult to characterize. Because of
this lopsided ratio, the neural system is best characterized as an analog computer.
The action potential is created by a conventional monopulse relaxation oscillator,
that can be driven or free running. The monopulse action potential does not
involve two stable values; it is not a binary signal. The action potential is always
a positive-going signal at its source, the axon of a three terminal neuron. It has a
nominal amplitude at its source of 100 millivoltsand a pulse width in endothermic
animals of between 1.0 and 2.0 milliseconds. In addition, there is a negative-
going waveform, of nominal 20 millivolts and the same width, associated with the
podaplasm of the three-terminal Activa within the stage 3A neuron. The
oscillator uses a single Activa within a single stage 3A neuron. The stage 3 action
potential is used throughout the chordate phylum to transmit information over
distances of greater than two millimeters. The analog signals from stage 2, 4 & 5
circuits are encoded by stage 3A relaxation oscillators into a series of action
potentials using the same phase code as used by the Inter-Range Instrumentation
Group, IRIG. The IRIG code was used extensively in rocketry prior to the 1970's.
Its phase character is to be note; the first pulse is a time of occurrence marker,
followed by a series of pulses identifying the amplitude of the analog information.
For long distances, the action potentials are regenerated by stage 3B Nodes of
Ranvier at a nominal two millimeter spacing. The analog information propagated
by the action potentials is decoded by a simple diode detector within a stage 3C
neuron. The stage 3 action potential is delineated from analog generator potential
of stage 1 external sensory neurons. The generator potential is defined by the
Excitation/De-excitation, E/D, Equation also developed in this work and presented
in closed form. Section 6.3.6 Discusses the requirement to compensate the probe
used in patch-clamp experiments to avoid the introduction of artifacts into the
data.
Keywords:Action potential, Generator potential, theory, relaxation
oscillator, IRIG code,
Purpose: This material is in support of several other paper on this site.
This material is in draft form but includes all of the science and conceptual
framework for understanding the physiology of the neural system of animals.
This material is in draft form because I (who am now 87 yrs old) may not get the
opportunity to work it into final (preprint) form and post it on ResearchGate.
2 Neurons & the Nervous System
The NEURONS and
NEURAL SYSTEM:
a 21st CENTURY PARADIGM
This material is excerpted from the full -version of the text. The final
printed version will be more concise due to further editing and
economical constraints.
A Table of Contents and an index are located at the end of this
paper.
A few citations have yet to be defined and are indicated by “xxx.”
James T. Fulton
Neural Concepts
jtfulton@neuronresearch.net
December 15, 2022
Copyright 2011 James T. Fulton
2 Neurons & the Nervous System
1Released: 14 March 2004
2Bockris, J. & Reddy, A. (1970) Modern Electrochemistry, vol. 1. NY: Plenum Chapter 5
6. Electrophysiology of Action Potential
Propagation1
6.3 Reformulating the concept of signal transmission in neuroscience
This section will begin to define the electrophysiology of the neural system. Subsequent chapters
and sections will provide more in depth analyses. This section will begin with a more
comprehensive development of the methods of signal transmission available within biology. It
will then combine the modes of transmission with other mechanisms required to achieve such
transmission efficiently (and largely error free). These mechanisms are found in the various types
of conexuses found in biology. The primary goal is a broad understanding of the electrophysiology
of the neural system.
The actual operation of the neural system differs significantly from the putative operation found
in the current neuroscience literature. In fact, it involves many techniques not discussed in the
current literature. The method of propagation used to transmit neural signals over distances
greater than one millimeter is entirely absent from the literature. To overcome these problems,
this section will, of necessity, present a range of material at an academic level. Of specific
importance will be the mathematics associated with the General Wave Equation (GWE) of
Maxwell. It provides the framework for discussing all means of signal transmission within the
neural system. The various forms of the GWE will be developed as appropriate to the signaling
function in Section 9.1.1. This discussion will highlight the role of Thompson (Lord Kelvin),
Hermann, Cole, Rall, Hodgkin & Huxley, Tasaki, Taylor and others in the description of neural
signaling.
An important distinction must be made between four related terms, diffusion, transmission,
conduction and propagation. The last one has been overlooked in neuroscience.
1. Diffusion will be used in the conventional sense used in physical chemistry. However, it is
important to recognize that the diffusion coefficients of a given material may differ considerably
based on the state of matter involved. The liquid, liquid crystalline, crystalline solid, amorphous
solid states are particularly relevant. Diffusion relates to the movement of physical matter within
one or more of these states.
2. Conduction as used in electronics and electrolytic chemistry relates to the diffusion of
electrical charges from one location to another. Again, the state of matter involved is critical. In
some (most) states, the transfer of charge does not involve the physical motion of heavy ions. The
dominant carrier is usually a negatively charged particle, the conventional electron, except in
some liquid or solid crystalline materials. In those cases, the dominant carrier appears to be the
positive equivalent of the electron, known as a hole. As noted in Section 1.3.2.2.3, the hole has
the mass of an electron but a different mobility, because it is in fact due to an electron moving in a
salutatory manner through such a crystalline material. Bockris & Reddy discuss the motion of
holes in great detail without recognizing it is the negative charge that is actually moving2. They
speak of it as the non-conforming ion.
3. Transmission is used to describe the transfer of signals between two points. It exists in two
principle forms, conduction and propagation. Transmission by conduction involves the physical
transport of some material or charge from one point to another. The direction is usually
described with respect to a field gradient that causes the movement. The direction of this field
can usually be described by a local vector. However, conduction is a scalar phenomenon.
Transmission by conduction is intimately related to the physical properties of the intervening
medium (Section 6.3.4).
Electrophysiology 6- 3
3Piccolino, M. (1998) Animal electricity and the birth of electrophysiology: the legacy of Luigi Galvani Brain
Res Bull vol. 46, no. 5, pp 381-407
4Taylor, R. (1963) Cable Theory in Nastuk, W. Ed. Physical Techniques in Biological Research, Vol. 6. NY:
Academic Press
5Goldman, L. & Albus, J. (1968) Computation of impulse conduction in myelinated fibers; theoretical basis of
the velocity-diameter relationship Biophys Journal vol. 8, pp 596-607
4. Propagation is a distinctly different form of transmission. Propagation is capable of
transferring a signal, including energy, between two points largely irrespective of the physical
properties of the intervening medium. Propagation depends on a separate set of electrical
parameters. Propagation is a vector based mechanism. This mechanism involves two orthogonal
quantities, an electrical field and a magnetic field. The vectorial product of these two fields
generates a Poynting vector (the phonetic pronunciation of the name of the man honored by this
term is ironic). This vector indicates the direction the signal (energy) is traveling largely
independent of the medium through which it travels. However, the direction is significantly
affected by the electrical and magnetic environment through which, and near which, it travels.
Propagation in the technical sense defined here plays a major
undocumented role in the operation of the neural system of all animals.
Propagation occurs in two distinctly different forms in the animal kingdom. The first form
involves the transmission of signals between two distant points as efficiently as possible (Section
6.3 5). The second form involves the special case of an unmyelinated large diameter (giant) axon
(Section 6.3.7). This axon is optimized as a tapped delay line that can support the
synchronization of a large number of otherwise independent muscles. When used in pairs, such
tapped delay line axons can provide great programming flexibility.
To achieve the goal of a broad understanding, it is first necessary to review the current framework
proposed in the literature. With this review, the reader is better able to discern where the actual
operation differs from this baseline.
6.3.1 General Background
The discipline of electrophysiology is not uniquely defined in the literature. The On-line Medical
Dictionary defines it as: That branch of physiology that is concerned with the electric
phenomena associated with living bodies and involved in their functional activity.
Physiology is in turn defined as: The study of how living organisms function. Dowben further
defines a sub-discipline of systems physiology--study of functional integration. This Chapter is
primarily concerned with the electrophysiology of the neural system as a total system. The
electrophysiology of other cell types in other systems is not of concern. Even the electrophysiology
of the soma of the neuron, and its internal constituents, is not of concern here.
Piccolino has provided a recent historical review of the field of electrophysiology from its
beginning with the famous experiments by Galvani (1791) on the legs of the frog3.
Many investigators have performed experiment to quantify the speed of signal transmission
within the neural system. However, no example of an investigator providing a comprehensive
model of the process has been found in the literature. Taylor provided a review of cable theory as
known in the biological community as of 19634. Goldman & Albus followed with a similar
mathematical derivation in 19685. Like Taylor, they relied upon the Equation of Heat rather
than the General Wave Equation, although they did introduce “isopotential patches of excitatory
membrane” to model Nodes of Ranvier. Like Hodgkin & Huxley, they could not solve rigorously
the equations they developed and relied upon dimensional analysis and numerical methods. The
velocities presented by Goldman & Albus are calculated using parameters chosen to match the
4 Neurons & the Nervous System
average velocities found in the literature (8-45 meters/sec) and not the phase velocities. The value
of r0, that was “set arbitrarily” could not be found in the paper. The condition noted by Taylor
appears to still be true. No contemporary textbook has appeared that addresses signal
transmission within the neural system comprehensively. Before Taylor, the previous broad work
dates from 1939. This sparsity of texts may be due to a situation described in Section 6.3.1.1.1.
It appears that most biological investigators have not been exposed to the second order
differential equation needed to discuss signal transmission in detail. In addition, it appears no
investigator has previously differentiated between axons transmitting signals over short distances
for purposes of signal processing, and those transmitting over long distances for purposes of
interconnection. As will be shown, no previous work has addressed the propagation of neural
signals consisting of action potentials at the theoretical level. This section will address the
mathematics of neural signal transmission in more detail than in other sections in order to
overcome this problem. It will require considering second order ordinary differential equations as
defined in mathematics and engineering, but differing from the common nomenclature of
chemistry.
The following sections will offer a comprehensive neural signaling framework. Subsequent
paragraphs will emphasize the importance of thinking in terms of the time delay between two
points rather than just the velocity of signals between the same two points. Developing the time
delay between two points in increments is preferred because it allows the easy introduction of the
significant time delays associated with the regeneration process that is used so effectively within
stage 3 signal projection circuits.
When investigating the transfer of signals, all of the methods available must be considered if the
correct model is to be obtained. Theoretically, the transfer of a signal can involve conduction
(diffusion) through a medium, convection (circulation) within a medium, and propagation
(radiation) through (or along) a medium. The latter is a technique not previously considered in
the literature with respect to neurons.
The transmission of neural signals involves a variety of different approaches and mechanisms.
These approaches generally relate to the character of the signals found within the various stages
of the system defined earlier (Section 10.1.6). The mechanisms are quite varied and relate more
to the local environment than to the stage where they are employed. Within the Activa, the
mechanism is quantum-mechanical transport within a liquid-crystalline semiconductor
environment. Within the conexus, outside of the Activa, the mechanism is generally one of charge
diffusion. However, there appear to be two alternatives available, the transfer of charge through
a viscous, possibly liquid-crystalline electrolytic medium (bulk diffusion) and the transfer of
charge along the surface formed by the boundary layer between the bulk plasma and the
enclosing membrane. Outside of the conexus in the conduits interconnecting conexuses, several
alternatives are also found. The one most often considered relates to diffusion via either of the
mechanisms just discussed. However, there is a second mechanism that is used extensively, and
possibly exclusively, in Chordata. It involves electromagnetic propagation (similar to free space
radiation) over an essentially loss-free transmission line formed by a cylindrical section of
myelinated axon. Myelination of a neuron reduces the capacitance per unit length of the axon by
more than an order of magnitude. Since the velocity of electromagnetic propagation is a function
of the capacitance of the medium, it will be shown that this electromagnetic propagation is orders
of magnitude faster than propagation by electrolytic diffusion. This mode of signal transmission
is critical to the development of large animals with a highly centralized nervous system, i. e., large
mammals.
It is important to differentiate between the processes of conduction (the
Electrophysiology 6- 5
6Ramo, S. & Whinnery, J. (1953) & subsequent additions. Fields and Waves in Modern Radio. NY: John
Wiley & Sons
2
2
2
2
1
VV
t
diffusion of electrical charges in the presence of a static electrical field),
and the process of signal propagation (due to the dynamic interaction
between electrical and magnetic fields).
Stage 1, 2 & 4 neurons process electrotonic signals via conduction in the
presence of essentially static electrical fields.
Stage 3 neurons generate action potentials and propagate them using
interacting, dynamic, electrical and magnetic fields.
Propagation does not require the flow of any electrons or ions along the
length of the axon or axon segment.
Due to historical precedent, and the greater familiarity of most biologists with diffusion, the
subject of propagation of neural signals has not been recognized by the neuroscience community
(see Section 6.3.3 ). This is in spite of the availability of significant measurements supporting
the mode of propagation within the system. The biologists have continued to describe the
transmission of action potentials within a conduction mode framework. This section will discuss
the conduction and the propagation of neural signals in separate subdivisions.
6.3.1.1 Technical background and the introduction of Maxwell’s
Equations
It is difficult to review the history of neuroscience concerned with signal transmission in the
presence of our current knowledge of Maxwell”s Equations. James Clerk Maxwell (a
contemporary and associate of Hermann) developed a set of equations in 1863 that can be used to
describe a very broad range of physical and electrical problems6. One of the solutions to these
equations, the general wave equation, predicted the advent of radio transmission and other
propagation related phenomena (such as the transmission of heat by radiation as an alternative to
conduction). The equations have been expressed in integral, differential and matrix form. This
makes their application easy in the hands of an experienced investigator. However, they are
frequently daunting in the eyes of the novice.
Maxwell’s General Wave Equation is of interest here. In its most compact matrix form, it can be
expressed as;
Where the inverted delta is a vector operator but not technically a vector itself. Capital V is a
vector and nu, is a scaler (that may be a complex mathematical function). This equation applies to any
problem involving up to the 2nd differential of the variable, V, with respect to any coordinate system required to
adequately express the variable.
In differential form, it is usually expressed in Cartesian coordinates as;
6 Neurons & the Nervous System
2
2
2
2
2
2 2
2
2
1
V
x
V
y
V
z
V
t
( ) ( )
A
tk A c
n
The equation describes the sum of the 2nd order differentials of the variable with respect to the
three spatial coordinates as equal to the 2n d order differential of the variable with respect to time.
In many applications, the equation simplifies significantly when some of the terms in the
expanded form of the equation are equal to zero.
1. If the value of the right hand term equals zero, the equation is known as Laplaces’s Equation.
This form is important in the study of electrostatics and many other problems involving static
fields.
2.If the value of the right hand term is a constant, the equation is known as Poisson’s Equation.
It applies to more complex electrical field problems that contain a localized concentration of
charge, and many other problems.
3. If the value of the right hand term is proportional to the time and the left hand term only
involves the first order differential with respect to position, the equation is known as Kelvin’s
Equation or more often, the Equation of Heat Flow. The equation represents the transfer of heat
between a stationary source and a sink by conduction.
4. The general condition is the condition shown in Cartesian coordinates. The right hand term is
a second derivative with respect to time. This is the form that introduced the dynamic interaction
between the variable with respect to both distance and time. This is the form that predicted the
propagation of radio and light waves through free space (independent of any aether or other
medium). This is also the form used to describe the operation of electrical filters of arbitrary
complexity. Transmission lines are a special class of electrical filters. The coaxial cable is a
specific type of transmission line that is of particular interest here.
6.3.1.1 Limit of mathematics used in chemistry
A cursory scan of Physical Chemistry texts quickly confirms a relevant fact. The field of
chemistry does not usually employ any variant of the General Wave Equation except for Poissan’s
Equation (a term other than zero on the right side). The kinetics of chemical reactions seldom if
ever oscillate between two states. Lacking involvement with the second order differential, they
tend to use the term order to describe the highest exponent of the dependent variable in the
equation. This term is described as first order in a unimolecular reaction, second order in a
bimolecular reaction and third order in a termolecular reaction. In its simplest form, the equation
used in chemistry is given by;
where n is the highest exponent of the
concentration, A.
This can cause confusion in interdisciplinary matters (although it does allow easier verbal
expression among chemists). It would be preferred if the chemist described the equation for what
it was in the language of a mathematician. It is a first order ordinary differential equation with
Electrophysiology 6- 7
7Ludimar Hermann, Untersuchungen über den Stoffwechsel der Muskeln ausgehend von des Gaswechsel
derselben (Berlin, 1867). The results reported in this book are extended and summarized in the revised
third edition of Hermann's textbook, Grundriss der Physiologie des Menschen (Berlin, 1870), esp. 233-239.
The Grundriss was published first in 1863 .
8Models and Instruments in the Development of Electrophysiology, 1945-1912 Historical Studies in the
Physical Sciences, Vol. 17, Berkeley: University of California Press, 1987: 1-54. Also available on the web
in 2003 http://www.stanford.edu/dept/HPST/TimLenoir/Publications/Lenoir_ModelsInstruments.pdf.
the variable present in a specified power. Thus a quadratic first order ordinary differential
equation would have the dependent variable present in the second power (regardless of whether
the power applies to the variable or the differential of the variable). Some texts describe a zero-
order, or pseudo-zero-order, chemical reaction. In this case, the equation contains no differential
of the dependent variable. It is not an ordinary differential equation. The controlling equation
does contain the concentration of the dependent variable, generally in the first power.
In the suggested notation, a unimolecular reaction is a first power reaction. A bimolecular
reaction is a second power reaction, etc. These definitions will be used here. All equations of
Kinetics are first order ordinary differential equations in various powers of the dependent
variable.
This nomenclature quickly surfaces a problem in neuroscience. The majority of the investigators
were not exposed to second order differential equations during their training. This leaves them
poorly prepared to discuss the second order partial differential equations that apply to signal
transmission within the neural system. This is particularly obvious when preparing the analog of
a chemical reaction using electrical terminology. There is no component equivalent to the
inductance of electricity. Only relationships expressible in terms of resistance and capacitance
are needed.
6.3.1.2 Background from the literature–Hermann vs Maxwell
Hermann began an extensive investigation into the responses of nerve using the most primitive
of tools (using sulfuric acid as an analog of a biological plasma). He initially reported his work in
18637. While drawing some useful conclusions concerning the chemistry related to neurons, his
electrical model was quite primitive. Lenoir has provided a historical account of the early work of
Hermann on the electrophysiology of the nerve8. His figure12 and 15 are very illustrative of
Hermann’s proposals based strictly on diffusion. Figure 6.3.1-1 shows these two figures as
transcribed by Lenoir from the original Hermann figures. The top frame shows his interpretation
of nerve transmission and communication of action wave in muscles under his alteration theory.
Quoting Lenoir, “In a stimulated region, E, of the nerve or muscle, chemical changes take place
analogous to those in rigor mortis; these render E temporarily electronegative with respect to
neighboring nerve or muscles’s tissue. The flow of current from the neighboring regions makes
them electronegative to regions more remote, and so on. The initial stimulus rapidly travels down
the length of the nerve or muscle fiber to successive positions, such as A.” Two much should not
be read into a conceptualization. The curved lines appear to represent electrical field lines
representing the paths followed by charges moving between the potential dipole created by the
tissue. The symmetry of the lines are a bit unusual. They would suggest the source within the
tissue was distributed with its two positive terminals near the centerline but on both sides of the
hatched area. Conversely, the negative poles would be found at both ends of the vertical bar.
Hermann shows no field lines from the structure at A to that at E. On the other hand, he seems
to imply the field pattern at E has moved to the location A. No magnetic field lines are shown. In
the lower frame, Hermann’s cable model is shown. Quoting the caption provided by Lenoir, “LL
8 Neurons & the Nervous System
Figure 6.3.1-1 Hermann’s conception of
the electrophysiology of a nerve. Top;
nerve transmission under his alteration
theory. Bottom; his cable model. Both as
transcribed by Lenoir from the original by
Hermann. See text.
2
Tc
k
T
t
P
k
represents a conducting wire bathed in an electrolytic fluid. Current is applied at A, conducted by
the fluid to the wire, through the wire, and via the fluid to K. The effect is attenuated by
increasing the distance between A and K. The loops on the exterior of the cable indicate the
direction of current a galvanometer would register if placed in a metallic circuit coincident with
the loops. Points in or on the cable right of K are progressively positive with respect to K, and
point left of A are progressively negative with respect to A.” Assuming Lenoir’s representation is
faithful, this description is awkward. The remark concerning attenuation suggests that the
conductor, LL, is resistive. The description suggests there is a preexisting condition where the
cable to the left of A is connected to the negative terminal, and the cable to the right of K is
connected to the positive terminal, of a separate battery. In the absence of such a battery , all
points in the figure must be less positive than point A. Points to the left of A would all assume a
potential at or less than the potential of the wire at A (less the potential drop within the fluid
between A and the closest point of the wire). All points to the right of K would take on a potential
at or greater than the potential at K. That is all points to the right of K must be at a potential
between A and K. As a result, the situation can be stated more simply. All points within the fluid
and the wire are less positive than A and more positive than K.
William Thomson (Lord Kelvin) was working
on the problem of heat conduction in the
1850's. This work was used in the
“marketing” of the first undersea telegraph
cables. The cables did not perform as
expected based on the application of his
equations for heat flow to the problem of
electrical conduction. This obviously led him
into debates with the Maxwellian School
that arose a decade or two later. While his
accomplishments were many, his best
remembered quotations from later in his life
include: “Radio has no future" and X-rays
are a hoax.” He obviously did not consider
the general solution of Maxwell’s Equations
too important.
As noted above, his work in heat led to a
mathematical formula that would later be
shown to be a simplified form of the general
wave equation. It can be written:
where P equals the rate of heat production
per unit volume. At a point where no heat is
being introduced into the system, the last term on the right vanishes. The resulting equation is
known as the Equation of Heat Flow. The second differential of T with respect to time is zero in
this equation. The operation described by this equation have been interpreted by many. “It
contains, however, a first rather than a second time derivative, and this results in solutions which
are exponentially damped, like a particle with resistance but no restoring force, rather than
oscillating solutions. The particular case where the temperature is independent of the time, the
Electrophysiology 6- 9
9Slater, J. & Frank, N. (1933) Introduction to Theoretical Physics. NY: McGraw-Hill pg 198
Figure 6.3.1-2 The electrical and magnetic
fields associated with a coaxial
transmission line
steady state, leads simply to Laplace’s equation, the term in time vanishing9.”
The early investigators attempted to describe this operation under more dynamic conditions by
solving for the temperature under two separate steady state conditions. However, these solutions
did not properly account for the transient condition that necessarily occurred between these two
terminal conditions. The same situation occurs when using the equation to describe the electrical
potential along a cable. As the time interval between the two steady state situation becomes less,
it is mandatory that the general solution to the General Wave Equation be considered. Said
another way, whenever the operating interval includes a change in the temperature (or the
electrical potential), this change must be accounted for in the mathematical model. The
appropriate equation is no longer that of Laplace or Poisson. If the change only involves the first
order derivative of temperature or potential with respect to time, it requires Kelvin’s Equation.
However, if the change involves both a first and second order derivative with respect to time (such
as for a sinewave), the General Wave Equation of Maxwell must be evaluated.
6.3.1.2.1 The contribution of Maxwell–The General Wave Equation
Of primary importance is the fact that a current that is changing with time induces a magnetic
field. And of critical importance as Maxwell first deduced, the same changing magnetic field
induces a changing electrical field. Because of this interaction, the general solution to the
General Wave Equation is a representation of two traveling waves, traveling in opposite
directions.
Figure 6.3.1-2 describes the fields encountered in a coaxial cable under dynamic conditions. The
left views represent a conventional man-made cable where the resistance per unit length of the
cable is negligible compared to the inductance. In this case, the vertical electric field lines are
perpendicular to the surface of the conductors at all intersections and all current travels along the
surface of the conductor nearest the dielectric. The magnetic field lines are shown as circular in
the top view and going in (closed dots) and out (open dots) of the paper in the lower view. Note
the arrows representing current are facing each other. There is no net current flow along either
the inner or outer conductors of the cable even as the energy associated with the fields is moving
along the cable. The right views represent the case of an axon where the inner and outer
conductors are formed of plasma of significant resistivity. In this case, the electric fields still
intersect the inner and outer conductors perpendicularly in the upper view. However, the
electrical fields may intersect these surfaces
at an angle in the lower view. In addition,
the electric field lines in the lower view are
shown extending into the two plasmas. The
electric field lines are more easily shown to
be continuous in this view, just as are the
magnetic field lines. One point should be
noted. Taylor assumed the resistivity of the
outer plasma was negligible in the biological
case because of its infinite extent away from
the dielectric. As seen in this view, that
assumption is not appropriate. It is the
resistance of the plasma in the immediate
vicinity (both surfaces ) of the dielectric that
is of paramount importance.
In comparing the figures from Hermann and
the modern representation of a lossy coaxial
cable shown in this figure, several points are
noteworthy. The description accompanying
the Hermann figures are basically
conceptual. They date from an era that
preceded modern electrical transmission
theory and cell cytology. The upper figure
does not explain why the signal travels along
10 Neurons & the Nervous System
10Rushton, W. (1951) A theory of the effects of fibre size in medullated nerve. J. Physiol. (London) vol. 115,
pp. 101-122
11Ritchie, J. (1995) Physiology of axons In Waxman, S. Kocsis, J. & Stys, P. The Axon. NY: Oxford University
Press pp 68-73
the nerve (although the text of Lenoir does give a verbal account that fits the heat flow scenario).
Both figures apply only to the static electrical condition. They do not address the actual
electromagnetic conditions present in an axon acting as a coaxial cable and conveying a signal.
In reviewing the contribution of Lord Kelvin, it is also clear his work preceded, and in fact
surfaced the need for, modern transmission line theory. His equation only applies to electrical
conduction (diffusion of charge) under temporally static conditions. The original undersea cable
highlighted the limitations of his theoretical approach and confirmed the broader framework
introduced by Maxwell.
6.3.1.2.2 Rationalizing the Hermann & Maxwell approaches--The concept
of the local circuit
The 1800's were years of great discovery accompanied by frequent changes of concepts as new
investigative and mathematical tools became available. The early exploratory work of Hermann
can be interpreted in more than one way based on current knowledge. His 1863 paper presented
a figure reproduced in [Figure 6.3.1-1(top)]. While the text describes the flow of the signal along
the axon in accordance with the Heat Equation of Thompson (more properly the Laplace Equation
because of the presence of concentrations of charge), the figure can be interpreted as showing two
zones of electrical polarization separated by a finite space. Hermann and James Clerk Maxwell
were associates at the time. It appears that Hermann’s figure could show the struggles he and
Maxwell were facing in explaining signal propagation at that time. While the figure does not
suggest the presence of any magnetic field, Maxwell published his famous equations in the same
year, 1863. Thus, it would not be difficult to conclude that Hermann’s figure would quickly
mature to include the interaction of dynamic electrical and magnetic fields as the mechanism
supporting the separation of these distinct charge concentrations. This would have caused
Hermann to abandon the “first order” Heat Equation of Thompson in favor of the “second order”
General Wave Equation of Maxwell.
In 1951, Rushton presented a series of calculations using dimensional analysis as a tool. The
calculations began with his interpretation of the work of Hermann. Rushton described signal
transmission along an axon using Hermann’s model but assigning it the name “local circuit
theory.” Piccolino (pg 393) has pointed out that the term local circuit theory does not appear in
Hermann’s writings. Rushton apparently interpreted the points A and E in the above figure as
locations of distinct charge congregation and redrew the figure to resemble the lower left of [
Figure 6.3.1-2] but without the magnetic field lines10. In his concept, the charge packets
maintained their charge distribution profiles while they moved along the axon at a finite velocity.
While conceptually attractive, the laws of electrostatics do not allow charge concentrations to be
maintained within a conducting medium in the absence of an appropriate static electrical field
(whether moving uniformly or stationary). The charge would diffuse until it was evenly
distributed in accordance with the geometry of the conducting medium. The so-called local circuit
of Rushton cannot be maintained under static conditions, even if the relevant region is moving
uniformly through space.
Ritchie reviewed his concept of the local circuit in Waxman, et. al. in 199511. While using the term
propagation in the vernacular, he made no distinction between conduction and propagation in the
technical sense. Nor did he differentiate functionally between the types of neurons represented in
his caricatures (figure 4.1). His semantics in the caption to this figure is particularly awkward.
While the average velocity of signal transmission in both conduction and propagation may be
similar (within an order of magnitude), the peak velocities are grossly different. More defined
caricatures will be presented in Sections 6.3.4 & 6.3.5. His directions of current flow near a
Node of Ranvier do not represent the actual situation when examined as a function of time (see
Section 10.5).
Electrophysiology 6- 11
“Local circuit theory,” a concept devoid of a magnetic field component, is unknown in the electrical
engineering literature. The idea of a loop of current, where the current is necessarily the same
amplitude all along the loop is only found in a DC circuit under steady state conditions. Such a
local circuit can convey no information. However, the simile between the ideas of Hermann and
the caricatures of Rushton with what Maxwell presented (as described in [Figure 6.3.1-2]) is
clear. The introduction of competing electrical and magnetic fields provides the mechanism for
concentrating local regions of charge that appear to move along the conducting medium. In fact,
the charge is like the longitudinal waves on the ocean (in the absence of white caps). The energy
associated with the waves moves along the surface but the individual particles of water remain in
a localized region. The change from the magnetism free situation to the electromagnetic situation
is of fundamental importance in understanding the operation of an axon. When the magnetic
field component is negligible, transmission is by conduction. When the magnetic and electric field
strengths are comparable, transmission is by propagation.
To understand the operation of the neural system, it is necessary to move beyond a single concept
applicable to all neurons. Different conditions related to unmyelinated, myelinated and pseudo-
myelinated axons result in different underlying mechanisms. Different mathematical solutions
describe these mechanisms. Demyelination results in yet another situation that is clearly
pathological.
By closely examining the operation of neurons, in the context of the Wave Equation, it is possible
to define two fundamentally different operating modes. One deals with the conduction of
electrotonic signals (energy) through a bulk medium. The second deals with the propagation of
phasic signals (energy) largely independent of the medium present (except it may be guided by
major discontinuities in the electrical properties of the medium or the surroundings).
Only the ummyelinated neuron is properly represented by a first order differential equation such
as the Heat Equation.
When Maxwell’s equations are applied to the myelinated axon situation, they call for the
interaction of electrical and magnetic fields. These fields can be related to two primary
parameters, a resistive component and a reactive component. The reactive component is
represented by the imaginary terms in the equations (see Section 6.3.2). This component is
present in two forms. In physical systems, the positive imaginary component is associated with
the inductance of the system. The negative imaginary component is associated with the
capacitance of the system. In transitioning from a heat flow analog to an actual electromagnetic
application, the inductance of the axon measured by Cole (Section 9.1.2.4.2 ) becomes quite
understandable.
It is unfortunate that other investigators did not challenge the work of Rushton and suggest a
reinterpretation of his views of the model provided by Hermann. Replacing the static “local
circuit” by the dynamic “local interaction” of electromagnetic fields would have led to an entirely
different perspective on neural signal transmission.
6.3.1.2.3 Problems in the recent literature relying upon the old Hermann
concept
For some unknown reason, the biological community adopted the Hermann cable concept of
conduction but never moved forward to accept the more complete Maxwellian Theory of
Propagation. As noted above, Lord Kelvin fought the Maxwellian concept of propagation until he
passed from the scene.
Hodgkin & Huxley ran into complications in analyzing their data acquired on the basis of lumped
parameter measurements. A similar problem was encountered by the competing Cole team.
Their solution was to re-frame their operating concept and consider the axon a transmission line.
They made this conversion relying upon the Hermann Cable approach involving only resistors
and capacitors. This leaky transmission line does not describe the unmyelinated axon of Loligo or
the mammalian myelinated axon.
In switching conceptual approaches, they introduced their equation 27 (pg 522) without providing
any reference or discussion of its constraints. The equation is commonly known within biology as
the cable equation as originally promoted by Hermann based on the earlier work of Lord Kelvin.
12 Neurons & the Nervous System
12Albert, A. (1934) Electrical Communications. NY: J. Wiley & Sons. pp 192-214 in the 3rd edition of 1950
13Cole, K. (1968) Membranes, Ions & Impulses. Berkeley, CA: University of California Press pp 77-87
14Taylor, R. (1963) Cable Theory in Nastuk, W. Ed. Physical Techniques in Biological Research. NY:
Academic Press
The equation has a number of constraints that are addressed indirectly in both Cole (pages 60 and
212) and Taylor. H&H did not address the impact of their change in concept from a lumped
constant axon to an axon supporting a traveling wave. It is difficult to follow the brief discussion
in H&H related to their transition from a lumped constant model to the transition to a cable
solution based on an action potential present as a traveling wave. Cole did address the change, at
least in terms of its impact on the construction of his axial probe. The larger problem relates to
the introduction of the “general wave equation” and a propagation velocity, . They use the 1st order
cable equation, that is lossy, as a baseline but substitute into it expressions from the 2nd order general wave
equation that only applies to a loss-free line. They appear to have done this to keep a real number (as opposed
to a complex number). It is not clear they were aware of this subtlety.
The above is typified by their use of the equation Vm = f(x0) = Vm1 = f(x1-vt) for a periodic traveling wave. If the
line is loss free, this equation holds and the velocity of transmission, v =( x1 -x0)/t where x1 - x0 =, the
wavelength of propagation. However, the condition that the line is loss free prohibits the introduction of a series
resistance, R, and a capacitance, C, into the cable equation. The presence of R requires the attenuation constant,
, not be identical to 1.00. The presence of R and C requires the phase constant, , not be identical to 1.00. If
and are not equal to 1.00, then amplitude of f(x0) cannot equal the amplitude of f(x1-vt) at any
frequency. The correct solution of the general wave equation to a lossy coaxial line was well
known by the time of Cole and Hodgkin12. This author was exposed to it during sophomore year
in 1955. Specifically, the dispersion in the signal is due to the difference in velocity of the signal
components as given by the equation, velocity = .
Hodgkin and Huxley were aware of the complex plane plots of the impedance of the squid axon
produced by Cole13. These plots clearly demonstrate that the squid axon involved a 2nd order
differential equation with both inductive and capacitive components. An axon exhibiting both
inductance and capacitance is not compatible with the 1st order differential equation Hodgkin and
Huxley used. Nor is it compatible with the Hermann Cable. While Hodgkin & Huxley recognized
the claim of Cole that an inductance was present, and they calculated a value for it from one of
their relaxation curves, all of their equations and calculations involved real (as opposed to
complex) numbers. No imaginary terms appear anywhere in their equations. The phase shift
associated with an inductance was never addressed (pg 540).
Taylor attempted to rationalize some of the many models of cables in the literature up through
that of Hodgkin & Huxley. Taylor presented a thorough mathematical review of “Cable Theory,”
as restricted to an RC cable, in detail14. He includes a section on the giant axon with an axial wire
introduced into it. Taylor does note the following crucial fact ten years after the work of Hodgkin
& Huxley. “Since the 1939 papers of Tasaki, in which he demonstrated directly that only the
nodes of Ranvier in mammalian myelinated nerve are excitable it has been abundantly shown and
generally conceded that the ‘salutatory’ theory as propounded by Lillie is correct.” This position
may have been more controversial than he suggests. However, it remains correct today. He
references Tasaki’s 1953 paper and provides a caricature of the myelinated axon with Nodes of
Ranvier. Why Taylor did not interpret the complex plane data of Cole as calling for a second
order differential equation containing both inductance and capacitance is not clear.
Taylor then makes a set of assumptions. “We make the assumptions that the radial currents in
the core and external region are to be ignored, that the myelin sheath between nodes has an
infinite resistance and negligible capacitance and that the node width is negligibly small but with
finite impedance.” This statement defines a loss-free line and not the Hermann Cable he
continued to use in his analyses.
While the review by Taylor in 1963 provided a more modern interpretation of the work of
Hermann and Thomson, He chose to “consider only the steady state in time, and only the direct
Electrophysiology 6- 13
15Rushton, W. (1951) A theory of the effects of fibre size in medullated nerve. J. Physiol. (London) vol. 115,
pp. 101-122
16Hille, B. (1984) Ionic channels in excitable membranes. Sunderland MA: Sinauer Associates pp. 27
17Ritchie, J. (1995) Physiology of axons. In The Axon. Waxman, S. Kocsis, J. & Stys, P. Ed. NY: Oxford
Univ. Press pg. 84
18Fitzhugh, R. (1962) Computation of impulse initiation and saltatory conduction in a myelinated nerve fiber
Biophysical Journal vol 2, pp 11-20
19Frankenhaeuser, B. & Huxley, A. (1964) The action potential in the myelinated nerve fibre of Xenopus Laevis
as computed on the basis of voltage clamp data. J. Physiol. (London) vol. 171, pp. 302-315
current resistance and conductance concepts.” By also ignoring any radial electrical fields within
the plasmas, he further restricted his analysis to the Equation of Heat Flow. He posited that if
the membrane was represented by only a parallel combination of a resistance and a capacitance,
his core conductor model represented an “ideal submarine cable,” the differential equation for
which has been derived a number of times since first treated by Lord Kelvin. This approach
essentially takes the field back in time to that of Kelvin and Hermann and ignores what was
learned about a real submarine cable from the in-place test results.
6.3.1.3 Background from the literature–Rushton & others, 1950's
A large amount of material has accumulated in the literature attempting to prove a strong link
between the so-called conduction velocity, i. e. the average velocity of the neural system, to the
diameter of the axon fiber. It begins with the calculations of Rushton15 in 1951, based on what he
describes as “local circuit theory.” Local circuit theory, unknown in the electrical engineering
literature, apparently arose from the writings of Hermann in the late 1800's16. However, that
expression is not found in his writings (Piccolino, pg 393). A figure similar to that used to describe
“Local circuit theory” does appear in the electrical engineering literature. However, it is
conceptually quite different (see Section 6.3.1.2.2 ) Rushton’s calculations showed that the
conduction velocity in an axon should be related to the square root of the axon diameter. The
equations were only solved by dimensional analysis.
Rushton modeled the axon as a simple circuit consisting of a leaky insulating surface with current
moving in opposite directions along its two sides. He did not incorporate capacitance into his
calculations and he did not solve the partial differential equations he presented. Instead, he
performed a “dimensional analysis” to show that if the surface were a cylinder and certain other
conditions were met, the velocity of charge movement along that cylinder should be proportional
to the square root of the diameter of that cylinder. One of the conditions was that the ratio of the
length to diameter of a given class of neurons was fixed. Ritchie17 points out that this prediction
has not proven accurate. Until the introduction of the concept of salutatory conduction in nerve
fibers in the late 1950's, Rushton’s approach was not challenged within the community. In fact,
Ritchie presented it again in 1995 in parallel with other material taking a different approach.
Rushton’s analysis did not address the Node of Ranvier at all. In hindsight, he was calculating
the end to end or average velocity of a signal along an uninterrupted axon without considering the
effect of any series inductance or shunt capacitance. His solution does not apply to the
propagation of a pulse over a transmission line at all. His calculation applies to the transmission
of a continuous, D.C., current along a solid conductor inside a thin insulator with a thin external
conducting coating.
In 1962, Fitzhugh18 presented a quite different analysis of the problem. He assumed saltatory
conduction and models the axon as consisting of an excitable membrane, using the Hodgkin-
Huxley membrane model, at each Node of Ranvier with a section of cable between them. The
cable consisted of only resistors and capacitors. The membrane was the conventional two
terminal network of the time. Although Fitzhugh spoke of a delay at each node, he did not
incorporate such delay in his model. The equations presented contained more than twenty
adjustable parameters, six of which were of unknown physical significance19. Four of the six were
14 Neurons & the Nervous System
20Frankenhaeuser, B. & Huxley, A. (1964) Op. Cit.
21Goldman, L. & Albus, J. (1968) Computation of impulse conduction in myelinated fibers; theoretical basis
of the velocity-diameter relation. Biophysical Journal, vol. 8, pp. 596-607
complex algebraic functions of time. Fitzhugh was unable to solve the equations and resorted to
numerical calculation based on circuit element values given by Tasaki in 1955. The stimulus was
a square current pulse of 0.01 msec. duration applied to the membrane at Node(0). His figures 2
and 3 are still presented today. Figure 2 actually shows the action potential growing in amplitude
as it progresses along a section of the transmission line. Figure 3 on the other hand shows the
action potential maturing in shape as it progresses along the line but there is no attenuation
between nodes or between points within one cable segment. Although Fitzhugh’s results are
technically unsatisfying, and he only reports a node to node average velocity on the order of 12
meters/sec, he does discuss the fact that the time to traverse a given internode segment, 0.015
msec, is only about 10% of the total node to node delay of 0.168 msec,
“Hodgkin and Huxley's standard temperature of 6.3 C was used.”
“The peak time increases only 0.015 msec. along one internode, while the node-to-node conduction time
is 0.168 msec. (These times should be divided by 2 or 2.5 to correct for the difference of temperature.)”
Frankenhaeuser & Huxley20 presented a paper using the same mathematical framework for the excitable
membrane as Fitzhugh and listed 15 caveats applying to their evaluation of the same differential equations. They
did not solve the equations but converted them to the difference form to accommodate a digital computer.
A frequently referenced paper followed in 1968 by Goldman & Albus21. This paper adopted the same set of
partial differential equations as used by both Fitzhugh and by Frankenhaeuser & Huxley. It did not incorporate
the notion of saltatory transmission by projection neurons. In fact it treated the conseg in the nodal gap as an
isopotential patch of excitable membrane. It continued the previous approach of depending on dimensional
analysis to accredit the equations and then proceeded to convert them to numerical difference equations for
evaluation via a general purpose digital computer. They included two crucial comments. “This treatment of
course, produces no information as to the exact form of the solution nor can any actual conduction velocity
values be obtained.” Second, “Since L (the internode length) is proportional to D (the outside diameter of the
myelin enclosing the axon), (the impulse conduction velocity) will be proportional to D if the internode delay
time is constant with changing diameter.” Their conclusion is that the above stipulations “are found to be
essential to produce proportionality between fiber diameter and conduction velocity.” Of course! However, the
actual conduction (average) velocity is determined almost entirely by the length of the internode (axon segment)
divided by the regeneration delay of each node. To justify the relationship between the average velocity and the
diameter of the axon segment, it must be shown that either L is a constant function of D or the regenerator delay
is inversely proportional to the diameter of the axon.
These four analyses must be considered exploratory by current standards. They did not adopt equations for a real
transmission line and they did not treat the Node of Ranvier as an active, three terminal, site. A transmission line
of only resistors and capacitors must exhibit considerable attenuation, not shown in the above evaluations, and
also severe phase distortion for pulses traveling along the line, also not shown in the evaluations. Lacking a
solution to their simultaneous differential equations, there is no way to prove their equations have been
formulated properly and are represented by the proper initial conditions.
6.3.2 Technical background related to the action potential
There are a variety of descriptions of the action potential in the literature. Some relate the
potential to a specific neuron morphology and some take a more generic view. Some recognize the
salutatory nature of longer neurons and some do not. Lacking precision, these descriptions result
in confusion in the literature. In this work, an action potential is defined as the output of a stage
3 monopulse oscillator at its collector (axon) terminal. Any other similar waveform should be
described using adjectives to maintain traceability. This is particularly true when probing the
soma of a neuron. The recorded waveform is frequently not that associated with the axon but that
Electrophysiology 6- 15
22Segev, I. & London, M. (1999) A theoretical view of passive and active dendrites In Stuart, G. Spruston, N.
& Hausser, M. Dendrites. NY: Oxford University Press pp 205-230
23Shepherd, G. (1991) Foundations of Neuron Doctrine. Pg. 277
24Matthews, G.(1991) Cellular physiology of nerve and muscle. Boston, MA: Blackwell Scientific Publications,
pp. 64-65
25Steriade, M. Jones, E. & McCormick, D. (1997) Thalamus vol. 1, NY: Elsevier, pg 73
associated with the dendroplasm of the monopulse oscillator circuit. Segev & London22 discuss
this briefly on their page 207. Another problem arises when a differential probe circuit is used to
record a waveform. The result is frequently the derivative of the underlying waveform. These
variants will be discussed below in detail.
Action potentials can be generated by any conexus exhibiting a net internal
feedback gain greater than 1.00. Thus any internal conexus, any Node of Ranvier
and virtually any synapse can be made to generate an action potential if its
parameters are changed artificially. This is a particular problem in some
experimental configurations where sufficient capacitance is added to the circuit by
the probe to cause oscillation.
Contrary to the common view, reiterated by Shepherd23 in 1991, the action potential is not a
membrane-based event. The membrane of the axon alone is an entirely passive structure. The
action potential is the result of a change in the charge contained within the axolemma.
This work shows the origin of the action potential is due to the operation of a typical electrical
circuit containing a monopulse oscillator. The underlying mechanism is “transistor action” within
an Activa found within a conexus. This Activa is formed by a unique junction between two
lemma. The potential between the axoplasm and the surrounding environment is shared
between the axolemma and other circuit element depending on location. These other elements
include any myelin present and any electrostenolytic process located on the surface of the
axolemma. The leading edge of the action potential is due to transistor action associated with the
Activa. The trailing edge is due to the electrostenolytic process replacing the charge within the
axoplasm.
Matthews gives a particularly clear list of the properties of an action potential24 . However, he
does not define his electrical coordinate system, does not differentiate the conduits of a neuron,
and does not recognize the salutatory process. This section will address the action potential in
more detail. It will present an alternate set of action potential properties for comparison with
those of Matthews.
It is particularly important to separate the response of a passive circuit to a step
input from the response of an active circuit to a short duration input. The action
potential is the response of an active oscillatory circuit to a short pulse exceeding a
threshold. The action potential never remains at a high positive potential.
Similarly, its peak amplitude is always more than 0.1 msec later than the peak in
the exciting pulse.
It is also important to review the error introduced into electricity by Benjamin
Franklin and his contemporaries. They postulated that the conventional electrical
current flowed from the positive terminal to the negative terminal of a battery.
We now know the actual material involved is the negatively charged electron and
that it (the electron current) flows from the negative terminal to the positive
terminal of a battery.
The remainder of this section will focus on the action potentials, and ganglion cells generating
them, associated with the retina, the optic nerve and other neural paths outside of the CNS.
These neurons are known to exhibit a maximum pulse frequency of between 100 and 150 Hz.
There are substantial reports of action potentials occurring within the CNS at clock rates of 800-
1000 Hz25. It is assumed these high clock rates are achieved over short path lengths in order to
achieve required processing time targets. The power consumption associated with these clock
16 Neurons & the Nervous System
26Kandel, E. (1977) Handbook of physiology–Section 1, The nervous system–Vol 1, Cellular Biology of
Neurons, Bethesda, MR: American Physiological Society, pg. 100
27Waxman, S. Kocsis, J. & Stys, P. (1995) The axon. NY: Oxford University Press. Chap. 13 thru 18
28Bowe, C. Kocsis, J. & Waxman, S. (1985) Differences between mammalian ventral and dorsal spinal roots
in response to blockade of potassium channels during maturation. Proc. R. Soc. Lond. B Vol. 224, pp. 355-366
rates and long axon lengths would be substantial. These special cases will be addressed in
Section 14.6.3.5 and Chapter 15.
There are a number of similar waveforms that are often confused with the true action potential,
these will be discussed in Section 6.3.2.4.
6.3.2.1 The electrical characteristics associated with the action potential
(as generated)
There are many actual and purported action potential waveforms in the literature. It is
important to discriminate between them. Most of the actual action potentials exhibit a duration
of about 0.5 to 1.5 msec. depending on the temperature and measurement criteria26. Waxman, et.
al27. provides a mine of experimental data, including some comparisons between action potentials
in the same specimen at different temperatures. Many of these action potentials appear to have a
much shorter rising time constant and a more constant time constant in the falling waveform
than that shown in Bowe, et. al. Waxman’s figure 13-3 is particularly clear in this respect for a
human at an unspecified but assumed nominal temperature. The rising waveform clearly starts
as a first order exponential after the threshold is exceeded by the 0.5 msec. depolarizing
waveform.
Great care should be taken in differentiating between action potentials, that are the result of an
oscillatory process after the cessation of excitation, and various forced responses of circuits to step
changes in excitation that are labeled action potentials. It is also important to differentiate
between waveforms obtained by penetration of a lemma (contact with a plasma) and those
collected by a probe located within a surrounding medium (field measurements). The latter are
frequently composite waveforms only indirectly related to a single action potential.
Great care must also be taken to discriminate between waveforms obtained by accessing the soma
of a neuron. These waveforms are almost always pseudo-action potentials related to the
dendroplasm or the podaplasm of the neuron. It is only when the axoplasm found within the
hillock is accessed that a true action potential is obtained from within the soma. Virtually no
waveforms are available from accessing the ramified dendritic or poditic trees because of their
small dimensions.
In contact measurements, the probe penetrates the lemma and senses the voltage of the plasma.
In field measurements, the waveform is obtained by sensing the voltage in the current field of the
interneural matrix near the axon. Field measurements are critically dependent on the precise
location of the test probe and the conductivity of the matrix. This sensitivity is even more
pronounced in physically congested areas such as the nodal gaps. The recorded field potential is
typically lower than the recorded contact measurement. Lacking a detailed circuit model relating
to the area probed, it becomes very difficult to interpret the recorded waveform. Contact
measurements are less sensitive to the characteristics of the plasma but are sensitive to the
location along the length of the axon.
6.3.2.1.1 The graphical description of an action potential
Figure 6.3.2-1(A) shows the intra-axonal recording of the action potential of an eight-week-old
rat ventral root fiber to artificial stimulation before and after superfusion with 4-aminopyridine,
(4-AP)from Bowe, et. al28. They claim the superfusion of the nerve with 4-AP causes a delay in
action potential repolarization. They relate this phenomenon to the effect of the 4-AP on the
Electrophysiology 6- 17
blockage of putative “fast K+ channels.” The precise nature of this chemical was not found in the
article. Figure 6.3.2-1(B) displays the same waveforms expanded in time by a factor of five. The
potentials on the right have been added by this author. Bowe, et. al. did not indicate the
quiescent potential prior to the initiation of the excitation. VT indicates the presumed threshold of
the input circuit as it appears reflected into the axoplasm circuit. It is within a few millivolts of
the resting potential of the axoplasma. VINM represents the nominal potential of the surrounding
medium. VSAT represents the saturation voltage of the Activa within the neuron. The voltage
difference between VINM and VSAT approximates the potential of the base (poditic terminal) of the
Activa during saturation. There is no component of the waveform more negative than VT other
than the brief transient due to capacitive coupling near time zero.
The theory of this work allows a different interpretation of these waveforms from that of Bowe, et.
al. First, there is no delay, in fact no change, associated with the rising phase of the waveform
due to 4-AP. Second, there is no delay, in the mathematical sense associated with the falling
waveforms. Only the recharging that occurs following the peak in the response is affected by the
4-AP. It is suggested that the 4-AP interferes more with the reactants controlling the recharging
of the axoplasm than it does with the membrane of the cell. It appears that the second waveform
is the result of a change in the time constant of this recharging process.
In the absence of any delay related to the peak of the waveform, no significant delay in the overall
operation of the neural system would be expected from application of 4-AP to the in-vivo system.
Third, note the distinctly linear character of the sloping parts of these graphs. The reasons for
these features will be discussed below.
The fundamental action potential is a monopolar monopulse waveform beginning at the resting
potential of the axoplasma of a stage 3 neuron. The pulse rises monotonically until it reaches the
saturation potential of the Activa within the neuron. At that peak in the waveform, the trailing
edge falls monotonically until it returns to the resting potential of the axoplasma. The duration of
the resultant pulse is temperature dependent and is typically 0.2 to 5 milliseconds within the
biological temperature range. The peak typically occurs within 0.1 to 1.5 milliseconds of the start
of the pulse. Variations of the fundamental action potential encountered in practice are discussed
below.
6.3.2.1.2 Potential monopulse oscillators for generating action potentials
To determine the specific form of the monopulse oscillator used in biological neurons, two major
categories must be explored. First is the operating mode of typical monopulse oscillators. Second
is the means of achieving a net positive feedback gain of greater than 1.00 in such circuits.
A stage 3 monopulse oscillator can operate in at least two distinct modes. The analyses applicable
to these two modes is different. The simplest operating mode considers the oscillatory circuit to
contain a simple switch. In this model, the Activa is forced to operate outside of its normal range.
It operates as a switch. It exhibits zero impedance between its collector and emitter circuits when
the switch is closed and an infinite impedance between these two circuits when it is open. In this
case, the rising portion of the action potential would be described by the initial potentials of the
two circuits and the time constant of the combined circuit elements of the dendroplasm and
axoplasm circuits. The expected initial response would be exponential. After the switch is
opened, the falling portion of the response would describe the time constant of the collector (axon)
circuit alone. It would also be expected to be exponential in the first order.
In the second mode, the stage 3 monopulse oscillator operates within its large signal operating
range. In this mode, it can operate as a constant current source over much of its dynamic range.
This can be observed by measuring the collector (axon) potential. As the emitter (dendroplasm)to
base (podaplasm) potential becomes more positive than the cutoff potential, current will begin to
pass through the collector circuit of the Activa. This will cause the potential of the axoplasm to
become more positive and the potential of the podaplasm to become more negative. The effect is
to cause the emitter to base potential to become more positive and thereby cause more current to
flow in the collector circuit. This positive feedback quickly causes the Activa to achieve its
maximum collector current condition. As a result, the capacitance of the collector (axon) circuit is
discharged by a constant current source until, the collector potential falls to nearly the potential
of the base (podaplasm). The collector potential changes linearly in such a circuit during the
18 Neurons & the Nervous System
rising portion of the action potential. When the collector potential reaches saturation, the total
feedback gain of the circuit falls below 1.00 and the transfer characteristic of the Activa becomes
constant as a function of applied input voltage. As a result, the potential of the collector begins to
become more negative due to its power supply recharging the axoplasm. The podaplasm potential
begins to become more negative due to its power supply returning this circuit to its quiescent
condition. This change in podaplasm potential causes the emitter to base potential to become
more negative which in turn further reduces the current through the collector circuit of the
Activa. As a result, the collector potential falls rapidly following the peak in the action potential.
The fall is controlled by a combination of the current through the collector of the Activa and the
recharging capability of its power supply.
The common base configuration of the Activa found in stage 3 neurons are particularly susceptible
to positive feedback. This feedback encourages their oscillation when large. Any significant
impedance in the base lead of the circuit introduces positive internal feedback. Any significant
capacitive impedance between the collector and the emitter of the Activa introduces positive
external feedback. When the net feedback gain associated with these circuits reaches a value
greater than 1.00, the circuit will commence oscillation. It appears different combinations of
positive internal and external feedback are used in different neural circuits. This variation will be
discussed below.
Figure 6.3.2-1(C) displays the theoretical action potential waveforms based on the above
discussions. The heavy solid exponential lines describe the operation of the monopulse oscillator
under nominal switching conditions. The rising waveform is shown becoming horizontal in the
absence of switching. The falling waveform is shown by the solid line in the absence of
pharmacological interference. The light solid line represents the expected rising portion of the
potential under nominal large signal conditions and a positive feedback gain exceeding 1.00. This
line also becomes horizontal when the collector reaches the saturation potential of the Activa. The
falling portion of the action potential would be marginally straighter than the exponential
response shown during its early fall due to the internal feedback within the Activa circuit that
tends to reduce the current through the Activa. However, it will eventually become exponential
as the Activa ceased conducting.
In the switching mode, the effect of temperature is distinctly different between the rising and
falling portions of the waveform since these portions are dominated by different mechanisms. In
the large signal case, the difference is less clear because both the Activa and the recharging circuit
are active during the falling portion of the waveform. The effect of temperature is discussed
below.
The dashed falling line represents the affect of intervention by pharmacological or other means.
Several points should be noted:
+ The action potential waveform is the summation of two separate waveforms, one
created during the discharge phase of the regenerative repeater circuit and the
second during the recharging phase of the same circuit.
Electrophysiology 6- 19
Figure 6.3.2-1 Comparison of recorded
and calculated action potentials. (A) and
(B) are action potentials obtained by
intra-axonal recording by Bowe, et. al.
(C) is the first order action potential
calculated for the same neuron based on
this work (fixed time constants). Straight
lines suggest second order (current
limited) condition in Activa and charging
circuit. All voltages are relative to the
local INM. See text for details.
+ The positive going or discharge
phase is dominated by the “inward
current” traditionally attributed to
the flow of sodium into the axoplasm
through the axon membrane. The
current is actually due to the flow of
electrons out of the volume of the
axolemma through the Activa.
+ The negative going or recharging
phase is dominated by the “outward
current” traditionally attributed to
the flow of potassium out of the
axolemma volume through the
axolemma. It is actually due to the
flow of electrons into the volume of
the axolemma due to the
electrostenolytic process powering
the neuron.
+ While the currents through the
Activa and the recharging circuits
are dominant at different times, they
do overlap at any time that the
potentials of the circuit are not at
their quiescent values. These
overlaps are correctly represented,
although misinterpreted, in the
empirical equations of Hodgkin and
Huxley.
+ There is no necessity for ions to
flow through the membrane of the
axon if the electronic model of the
neuron proposed in this work is
adopted.
+ Typically, the time constant of the
rising or discharge waveform is
independent of minor
pharmacological intervention.
However, the falling or recharging
waveforms are directly affected by
anything that affects the current capability of the power supplies supporting the
regenerator circuit
To precisely match the theoretical action potential to a given experimentally obtained action
potential, it is necessary to have knowledge of several voltages related to the individual intrinsic
membrane potentials of a given regenerator. It is also necessary to have knowledge of the
capacitance and the saturation current capability of the Activa and electrostenolytic sources.
With this knowledge in hand, the theoretical waveforms can be matched to the experimental
waveforms to any degree of accuracy required.
The rising and falling portions of the measured action potential in frames A and B are sufficiently
different to suggest they are controlled by different mechanisms, with different time constants.
The proposed monopulse circuit does propose different time constants in these two portions and a
switching phenomenon occurring near the peak of the response. More sophisticated
measurements should confirm a discontinuity in the overall waveform near its peak. Such a
discontinuity would confirm the circuit model proposed by this theory.
6.3.2.1.3 The equations for a switching mode monopulse oscillator
20 Neurons & the Nervous System
29Stampli, R. (1981) Overview of studies on the physiology of conduction in myelinated nerve fibers. In
Demyelinating disease: basic and clinical electrophysiology, Waxman, S & Ritchie, J. Ed. NY: Raven Press.
pg. 14.
30Schwarz, J. & Eikhof, G. (1987) Na currents and action potentials in rat myelinated nerve fibres at 20 and 37
Celsius. Pflugers Arch. vol. 409, pp. 569-577
31Hodgkin, A. & Katz, B. (1949) The effect of temperature on the electrical activity of the giant axon of the
squid J Physiol vol 109, pg 240+
Figure 6.3.2-2 Measured action potentials
as a function of temperature for rat motor
neurons. The 37C response was elicited
by a 30 sec current pulse. The 20C
response was elicited by a 100 sec
current pulse. From Schwarz & Eikhof,
1987
The closed form equations for a switching mode type monopulse oscillator are developed in
Section 9.2.5.
Note that the analysis does not assume or require the axoplasm to become positive relative to the
INM. Such an overshoot is frequently encountered in the literature for three reasons.
1. Inadequate documentation, such as relying on a value for the quiescent axoplasm potential
measured in a different experiment can lead to a putative overshoot.
2. In-appropriate biasing of the input circuit of an in-vitro experiment can result in measurements
that would suggest an overshoot in the original in-vivo circuit29.
3. An overshoot may also be due to poor equalization of the probe circuit used to acquire the action
potential(see Appendix ZE).
6.3.2.2 The action potential versus temperature
The effect of temperature on the action potential aids in understanding the underlying
mechanisms.
Figure 6.3.2-2 presents a replot of measured action potential data as a function of temperature
from figure 1 of Schwarz & Eikhof30. They did not measure the absolute potentials associated
with the action potentials but estimated them based on other experiments. Their paper also
includes a current-voltage characteristic obtained under clamp conditions in excellent agreement
with the theory of this work (See Section 10.8.4.5).
The fact that the rising time constant of the
putative action potential of Loligo is largely
independent of temperature while the falling
time constant is temperature dependent is
also documented in Hodgkin & Katz31. A
virtually identical measured response to that
shown (also at 20 C) is provided by
Frankenhaeuser and Huxley for the toad,
Electrophysiology 6- 21
32Frankenhaeuser, B. & Huxley, A. (1964) The action potential in the myelinated nerve fibre of Xenopus laevis
as computed on the basis of voltage clamp data. J Physiol vol. 171, pp 302-315
Xenopus laevis32. Unfortunately, their precision is not supported with data points. Neither
recorded waveform shows any undershoot in the action potential. On the assumption that the
quiescent potential was –150 mV, neither waveform shows any polarity reversal relative to the
INM.
There are a number of significant features in this figure. First, the threshold level of 18.5 mV
(referred to the axoplasm) remains essentially unchanged with temperature. This is
understandable since this value is the difference between two temperature sensitive voltages, the
dendritic and poditic supplies. The threshold is reached well after the termination of the
excitation pulse in both cases. This may reflect a shortcoming in the test set, a transit delay due
to the dendroplasm or the fact that the current pulses generated a potential at the location of the
Activa that was only slightly super-threshold. Both the rising and falling components of the
action potentials are a function of temperature. This is probably indicative of the sensitivity of
the source impedance of both the Activa and the electrostenolytic supply to temperature. The
falling potential returns to the quiescent value and not the threshold value. The maximum
amplitude of the pulses is only changed marginally with temperature and reflects the intrinsic
electrostenolytic potential of the axoplasm . Schwarz & Eikhof indicate the zero voltage level of
the figure varied with temperature. It is likely that the components of the threshold value also
varied although the threshold itself may have remained nearly constant. The zero value was -152
mV at 20C with reference to the INM. It was -144 mV at 37C. The threshold value they show as
18.5 mV is then between -125 and -133 mV relative to the INM. Using these numbers, neither of
the waveforms became positive with respect to the INM as claimed in the conventional wisdom.
There appear to be a significant delays between the time when the axoplasm first reaches the
equivalent threshold level and the time the oscillatory process actually begins. A test probe in the
dendroplasm and or podaplasm would be useful in understanding this effect.
The following values applicable to the first order equivalent circuit at a temperature of 37 Celsius
can be derived from their data.
For VT = 18.5 mV; VSAT = VT + 95 mV, VS =VT + 94 mV, tR = 0.012 msec, tS = 0.075 msec & tF =
0.25, Temp. 37 Celsius.
All potentials are with reference to the quiescent axoplasm potential.
Their data at 20 Celsius for the same animal exhibits time constants similar to those of Bowe, et.
al. Because of the shape of the rising waveform near its peak, more experimentation is needed to
refine these numbers. The waveform is quite sensitive to the ratios of VM to VS and tR to tS. The
complete second order circuit should explain why the falling exponential drops to the baseline
quicker than a simple exponential in Schwarz & Eikhof while that of Bowe, et. al. drops to the
baseline more slowly.
The currents present in the regenerator circuit, and related to the two portions of the action
potential, are also exponential in character. The observed current associated with the rising
waveform is maximum at time zero. The measurement of this current peak may be subject to
limitations in test instrumentation. This current was described as the discharge current in
[Figure 10.5.4-2]. The current is related to the transfer of stored charge from the post Activa
conduit capacitor to the pre-Activa conduit capacitor during the interval the Activa acts as a
closed switch. It may or may not be measurable in a given situation. It is particularly difficult to
measure at a Node of Ranvier since most of this current circulates entirely within the confines of
the nodal recesses. Only a portion of this current passes through the nodal gap. This portion is of
the same magnitude as the current injected into the next axon segment.
Based on this data, changes in temperature can have a significant impact on both the circuit delay
associated with individual action potential regenerators and the overall average velocity of neural
signals.
Rinberg et al. have recently provided useful data on the stability of neural oscillators as a function
22 Neurons & the Nervous System
33Rinberg, A. Taylor, A. & Marder, E. (2013) The Effects of Temperature on the Stability of a Neuronal
Oscillator http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002857
of temperature33.
Electrophysiology 6- 23
34Piccolino, M. (1998) Animal electricity and the birth of electrophysiology Brain Res Bul vol. 46, no. 5, pg
393
6.3.2.3 Overshoot, undershoot and polarity reversal
In the above figures, and that of Frankenhaeuser and Huxley, the entire action potential is
monphasic with reference to the quiescent condition. This shows that there is no intrinsic
overshoot in the waveform with respect to either the rising or falling part of the waveform.
However, many putative waveforms in the literature are interpreted to exhibit overshoot or
undershoot (a crossing of the quiescent potential associated with a falling waveform). There are
also comments in the literature that the plasma of a neuron may become positive with respect to
the INM for a short interval during the generation of an action potential. Several factors account
for these observations and interpretations. Many are due to the use of an AC coupled test
configuration. Others are due to the observation of a pseudo-action potential related to the
dendroplasm instead of a true action potential arising in the axoplasm.
A key factor in the assertion that an action potential can cause the plasma of a neuron to become
positive with respect to the INM concerns the measured quiescent voltage. Whenever the
quiescent voltage is less than –100mV, it is highly likely that the observed voltage is actually
associated with the dendroplasm, instead of the axoplasm of a neuron. Under this condition, the
observed waveform is a pseudo-action potential by definition.
This section will discuss and define each of the terms overshoot, undershoot and polarization
reversal. It will rely upon Section 9.3.1 where the theoretical waveforms found in the different
plasmas of a stage 3 monopulse oscillator were developed.
6.3.2.3.1 Precise definitions
In electrical engineering, a polarity reversal is a formalism. It describes the change in the
electrical polarity of a signal with respect to an arbitrary reference.
An action potential is defined as the monopulse output of a conexus within a stage 3 neuron as
measured between the axoplasm of such neuron and the adjacent inter-neural matrix, INM. Such
a waveform is always positive going and monophasic relative to the quiescent potential of the
axoplasm of the neuron (typically –150 mV relative to the INM). Because of the saturation effect
within the Activa of that neuron, it is impossible for the potential of the axoplasm to become
positive with respect to the adjacent INM. The only exception to this statement is if the
podaplasm of the Activa is more than 20 mV positive with respect to the INM. Such a condition is
pathological.
The term overshoot is used in electrical engineering to describe the short-term rise of the leading
edge of a pulse waveform to a potential that is higher than the steady state value during the
pulse. This definition will be used here. The definition does not apply to an action potential since
it does not exhibit a steady state value like that associated with a long pulse waveform. A natural
action potential is a monopulse of short duration.
The term undershoot is used in electrical engineering to describe the polarity reversal associated
with a falling waveform, following a forced steady state condition, that crosses the quiescent
condition momentarily before finally returning to the quiescent condition. An undershoot can be
considered the overshoot of a falling waveform as it returns to its quiescent condition.
6.3.2.3.2 Discussion
The term overshoot in physiology has had a tenuous history. It was originally
applied by Bernstein to the fact that the current measured during the active phase
of an action potential exceeded the level of the resting current measured between
an intact and an injured segment of nerve34. “Bernstein’s observation on the
possible ‘overshoot’ by the nerve signal of the injury current was afterwards
largely forgotten. It reappeared in the main path of the science only in 1939 with
24 Neurons & the Nervous System
35Matthews, G. (1991) Op. Cit. pg. 64-65
the studies of Hodgkin and Huxley.” In fact, it reappeared in an entirely different
context, that of a positive going voltage rather than an increased current.
As shown in Section 9.3.1, the action potential measured in the axoplasm of a neuron (near the
collector terminal of the Activa, is always a very simple waveform. It is an impulse type positive
going voltage waveform relative to the typically –150 mV quiescent potential of the axoplasm
relative to the INM. The action potential typically has an amplitude of about +100 to +120 mV
relative to its quiescent value. However, its potential relative to the INM remains negative at all
times. Normally, the most positive (but still negative) potential of the axoplasm is described by
the saturation potential, Vcsat, of the Activa in that circuit. The true action potential exhibits no
intrinsic undershoot.
When attempting to measure the action potentials of a stage 3 neuron, it is common to probe the
soma because of its physical size relative to the remainder of the neuron. However, both the axon
and poditic portions of the soma are very small. Generally, the investigator first obtains a
recognizable signal when he contacts the dendroplasm of the neuron.
The voltage waveforms measured in the podaplasm and the dendroplasm are much more complex
than in the axoplasm. This is partly because of their impedance level, typically higher than that
of the axoplasm, partly due to the finite travel time of signals traveling between the boutons of the
dendrites and the emitter terminal of the Activa, and partly due to the summation of currents at
these locations due to multiple mechanisms. The precise form of the podaplasm and dendroplasm
waveforms depends on the time constants of their individual power supplies and the relative
importance of the feedback used in the overall conexus. In the absence of capacitive feedback, the
dendroplasm potential tends to be an integral of the input excitation of the neuron. However,
capacitive feedback is frequently observed between the axoplasm and the dendroplasm. This
feedback introduces a scaled version of the action potential into the input circuit along with the
otherwise present input excitation. The summing of these two potentials is the source of the most
commonly measured pseudo-action potential reported in the literature. Because the dendroplasm
potential (just before the beginning of oscillation) is typically –20 to –30 mV with respect to the
INM, the introduction of a scaled version of the positive going action potential frequently causes
the dendroplasm to become positive relative to the INM. This phenomenon is frequently
described as an overshoot in the literature. While it does constitute a polarity reversal relative to
the INM, it does not constitute an overshoot as defined here.
It is common to observe an undershoot when recording the pseudo-action potential associated
with the dendroplasm of a neuron. This is most frequently due to the limited low frequency
response of the test set. A low-pass filter is frequently introduced into the test set to minimize
background noise when operating at low signal levels. This filter may be introduced using
hardware or software. The use of a bandpass filter in the test instrumentation automatically
introduces what is known as droop in the test channel. This droop is most easily recognizable as
an undershoot following any pulse type waveform.
6.3.2.4 Summary characteristics of the action potential (as generated)
The action potential as generated can now be described in considerable detail. Its character as it
travels along a real dispersive transmission line (an axon) will be discussed in Section 6.3.3.4.3.
The characteristics of the action potential can be summarized in a format similar to that of
Matthews35. However, the following represents a more detailed characterization and there is no
requirement for any changes in the permeability of the axolemma. Thus it represents a more
fundamental research oriented presentation than that of Matthews.
1a. Action potentials are triggered (in Nodes of Ranvier and the ganglion cells of the
luminance channels) whenever the emitter to base potential of an Activa contained within
a pulse oscillator circuit of a neuron is exceeded. The potential of the axolemma is not
changed except through transistor action.
Electrophysiology 6- 25
36Nicholls, J. & Baylor, D. (1968) Specific modalities and receptive fields of sensory neurons in CNS of the
leech J Neurophys vol. 31, pp 740-756, fig 9
1b. Action potentials are triggered (in the chrominance channels) continuously due to the
quiescent emitter to base potential of the Activa contained within the pulse oscillator
circuit of the neuron. The pulse interval in this circuit is changed by changing the emitter to
base potential. The potential of the axolemma is not changed except through transistor action.
2. The emitter to INM threshold level for initiation of an action potential is typically
10–20 mV more positive than the quiescent emitter to INM potential. The threshold level is
reached when sufficient charge is injected rapidly into the dendroplasm of the cell.
3. Individual action potentials are all or nothing events. The action potential is the result
of the stimulation of a monopulse oscillator. Until the stimulus threshold is reached, the only
change in the axolemma potential is due to the sub-threshold amplification factor of the circuit.
In the sub-threshold region, the short term potential of the axolemma faithfully reproduces the
potential of the dendroplasm.
4. An action potential degrades as it propagates along a neural path and is regenerated
every few millimeters by a Node of Ranvier. The result is a signal waveform that is
salutatory. The action potential waveform at the end of the neural path associated with the
propagation stage of a neural path is very similar to that at the start of the stage. However, the
precise waveform at any point along the neural path is determined by the parameters of the last
Node of Ranvier and the subsequent length of transmission line traversed prior to regeneration.
The signal degradation along the transmission line is due to dispersion of the signal energy in
time and not to the dissipation of the energy in resistors.
5. After a Node of Ranvier or other monopulse oscillator circuit generates an action
potential, there is a brief period, called the absolute refractory period, during which it is
impossible to cause the circuit to generate another action potential. During this period,
the emitter to base voltage of the circuit returns to its pre-oscillation quiescent value in
accordance to the diode-capacitance time constant of the dendritic portion of the Activa circuit.
This period is on the order of one millisecond for a typical monopulse circuit in a neuron.
6. The pseudo-action potential of the dendroplasm frequently exhibits an overshoot and
an undershoot as a result of the positive external capacitive feedback used to improve
the switching time of both encoding and regenerative stage 3 neurons. The intrinsic
membrane potential, the intrinsic electrostenolytic potential, and the maximum amplitude of the
true action potential are not affected by this process.
6.3.2.4.1 Difference between action potential vs its derivative
Figure 6.3.2-3 presents a series of waveforms obtained in a single experiment36. The skin of a
leech was stimulated with a piezoelectrically driven stylus (as represented by the signal labeled
S). This stimulus may have occurred simultaneously with the artifacts introduced into the left
portion of each waveform for timing purposes. However, the authors did not state this as a fact.
Two responses, R1 & R2, were obtained using pairs of wires supporting individual nerve bundles
and acting as extracellular probes. One response, R3, was obtained using an intracellular probe.
The difference in the signal obtained by these two different techniques are obvious. While R3
obtains a direct absolute measurement of the measured waveform, R1 & R2 measure the
differential voltage between two points in the external solution bathing the nerve. This indirect
measurement represents the derivative of the instantaneous waveform recorded by R3 but does
not sense the slowly varying component of R3. The reasons for this are also clear. The waveforms
recorded as R3 are actually pseudo-action potentials recorded by contacting the dendroplasm of
the stage 3 neuron within the ganglia. The waveforms measured as R1 & R2 are derivatives of
the true action potentials propagated along the stage 3 axons. Current was introduced into the
dendroplasm via the intracellular probe while a short duration stimulus was applied to the skin,
in the waveforms on the right. While the current was applied, the potential of the dendroplasm
was driven more negative. As a result, the signal received from the root nerve, as recorder by R1,
does not drive the dendroplasm sufficiently positive to achieve unity internal positive feedback.
No action potential is generated within the ganglion neuron under this condition. R2 continually
26 Neurons & the Nervous System
Figure 6.3.2-3 Interpretation of experimental stage 3 waveforms generated by the
tactile sensors of the leech, Hirudo medicinalis. S is the signal representing the
mechanical stimulation applied to the skin of the subject. This stimulus may have
been introduced coincidently with the timing marks (on the left of each waveform)
that were introduced into each recording channel for timing purposes. Note the
difference in scales on the right. The R1 & R2 signals are differential electrical
signals obtained from the voltage difference in the solution surrounding each
root. R3 is a pseudo-action potential obtained through direct electrical contact
with the dendroplasm of the neuron in the ganglia. It is 1000 times larger than the
indirect signals. See text. The figure is modified from Nicholls & Baylor, 1968.
reports a true record of the signal leakage from the axon of the cell penetrated by R3. The only
signal recorded by R2 on the right is noise from other cells and the timing artifact.
While Nicholls and Baylor speak of overshoot in their waveforms, there is no significant overshoot
in the action potentials and pseudo-action potentials illustrated, only in their derivatives recorded
as R1 and R2. The derivative recorded as R1 suggests an undershoot (an overshoot of the trailing
edge of the original action potential). Note the increasing time delays between the stimulus, S,
and each of the recorded signals, R1, R3 and R2. These reflect the finite signal transmission times
involved.
Electrophysiology 6- 27
37Dowling, J (1992) Neurons and Networks. Cambridge, MA: Harvard University Press pg 58
38Dowling, J. (1992) Op. Cit. pg 80
6.3.2.4.2 Difference between action potential vs generator potential
The action potential and the generator potential are fundamentally different in form and
underlying mechanism. The action potential is a discontinuous waveform generated by the
switching monopulse oscillation of a stage 3 neuron. Action potential have a constant amplitude
in the absence of fatigue effects and regardless of the illumination level. The generator potential
is a continuous waveform. resulting from the photoexcitation/de-excitation process and appears
initially at the pedicle of a photoreceptor cell. Within the mesotopic region, the amplitude of the
generator potential is a direct function of the illumination level. Dowling37 has presented a table
describing additional functional differences between action potentials and generator potentials.
Examining the derivative of a phasic waveform can provide rapid identification of the source of
that waveform. A high quality recording of an action potential from a probe near the hillock or
the post-nodal area of a Node of Ranvier will show three definitive characteristics. The waveform
will show a discontinuity near its peak associated with the switching character of the source
oscillator. Second, the leading and falling edges of the waveform will show nearly constant time
constants (constant slopes for the waveform when plotted against a logarithmic vertical axis)
compared to the constantly changing derivative of the generator waveform. The time constants
change abruptly at the switching time. Third, the time constant of the falling portion of the
waveform (but not the leading portion) is a strong function of specimen temperature (See Schwarz
& Einkof above). These characteristics are drastically different from those of the generator
potential. They also appear to be drastically different from the waveforms used by Hodgkin &
Huxley to derive their conceptual equations.
The generator potential is a continuous waveform represented by the mathematical difference
between two exponential functions. As such the slope of the waveform is continuously changing.
The amplitude of its derivative confirms this fact. It also changes continuously. The derivative
also passes through zero without a discontinuity. Temperature affects the time constants of the
leading and trailing portions of the waveform equally.
6.3.2.4.3 Difference between action potential vs pseudo-action potentials
By reviewing the above material, it is possible to recognize the differences between the true action
potential, associated with the axoplasm or the plasma in a subsequent axon segment, and the
pseudo action potentials found in the dendroplasm and podaplasm of stage 3A neurons. The true
axoplasm is positive going but does not become positive relative to the INM. The waveform
associated with the podaplasm is negative going and exhibits a shorter duration than does the
true action potential. The waveform associated with the dendroplasm is positive going and
exhibits a shorter positive duration, like the podaplasm, than the true action potential. The
dendroplasm waveform may also become positive, relative to the INM, due to capacitive coupling.
Most of the waveforms found in the literature are actually pseudo-action potentials measured by
probing the dendroplasm instead of the axoplasm. These waveforms are easily identified by at
least one of two features. They frequently have a resting potential more positive than –100 mV.
They also frequently show a polarity reversal relative to the INM. It is much easier to probe for a
pseudo-action potential in the dendroplasm than it is to electrically contact the much smaller
diameter axoplasm of the same cell. Dowling38 caricatures a pseudo-action potential without
providing data points. Note the negative excursions associated with his negative going current
pulses. These excursions would be observable in the dendroplasm but would not be observable in
the potential of the axoplasm since the Activa would already be at its most negative potential
(cutoff) value. As interpreted here, the inward current associated with a true action potential is
due to current through the Activa. The outward current in the same situation is due to the
electrostenolytic process recharging the plasma. In the pseudo-action potential, the “inward
current” (for potentials above threshold) is due to current through the Activa augmented by
positive capacitive feedback during action potential generation. The outward current in this
situation consists of two components. It is due to the positive capacitive feedback (occurring
during the collapse of the oscillation) as well as any restoration current through the
28 Neurons & the Nervous System
electrostenolytic process associated with the dendroplasm.
6.3.3 Mathematics of signal conduction and propagation in neural
circuits
The neuroscience literature has not previously differentiated between the signal transmission
associated with the signal processing neurons of stage 1, 2 & 4 portions of the neural system and
the signal projection neurons of stage 3. This differentiation must be made. It is frequently
associated with the presence or absence of myelination. The more clear differentiation is between
those neurons processing action potentials and those that do not. As noted earlier (page 4), over
95% of the neurons in a given neural system do not process action potentials. It will be shown
that these neurons transmit signals, over distances of less than one millimeter, by conduction (the
diffusion of charge) along the axons in general agreement with the theory outlined by Thomson,
Hermann and Taylor. It will also be shown that the transmission of signals over stage 3 neural
paths involves a fundamentally different mechanism not previously addressed in the literature.
This mode of transmission is called propagation. It is nearly 100 times faster than conduction and
its existence is well documented by Smith, Bostock & Hall (1982, also reproduced in Bostock,
1993). Propagation is used to transmit neural signals over distances exceeding one millimeter.
This mode does not involve the net physical movement of charge along the axon. Smith, et. al.
make one of the few references to the difference in velocity related to myelinated and
unmyelinated neurons in their remyelination studies. The said, “impulses could indeed traverse
whole demyelinated internodes (i.e. axon segments) by changing from their normal saltatory mode
of conduction (i.e. propagation) to a slow, continuous mode.” This statement will receive more
attention below.
A conexus within a stage 3 neuron acts as a source of action potentials. This source generates an
action potential at a specific impedance level. To effectively transmit that action potential over a
finite distance, it is necessary to arrange the impedance of the transmission element so that
energy is transferred to that transmission element from the source. Matching a source to a
transmission element (and the matching of a transmission element to a receiving element) is a
well understood electrical design task. Failure to achieve a satisfactory match leads to a very
inefficient transmission system. Attempting to transmit a phasic signal over a demyelinated axon
is a typical pathological situation of significant interest. A good match can transfer more than
95% of the available energy to the transmission line.
To achieve a good match between sources, transmission media and receivers, the neural system
employs a variety of recognized electrical techniques. The framework for applying these
techniques is presented in [Figure 6.3.5-1]. It can be expanded here by further overlaying the
figure with the electrical equivalents of the matching sections between the lumped circuit
elements of the conexuses and the distributed circuitry of the axon segment as a transmission
line.
See the discussion of the role of the hillock in Section 5.2.3.2 of the three-terminal neuron.
Four distinct modes of signal transmission are used within the neural system. Two are used
within the conexus and two relate to the conduits.
1. The quantum-mechanical transfer of charge between the emitter and the collector of the Activa
requires the least time. The distance involved is measured in Angstrom and the time delay is
estimated at 10–8 seconds or less. This quantum-mechanical delay is associated with all of the
conexuses in the system. This very short delay explains why there have been few creditable
measurements of the delay associated with a synapse.
2. A longer delay is associated with the transfer of a signal from the input structure of the conexus
to the output structure. This delay is most obvious in the action potential regeneration process
associated with stage 3 neurons. It is dominated by the time required for the regenerated pulse to
reach peak amplitude after initial excitation. This regenerative time delay is nominally 0.19
millisecond (Section 9.1.1.5.4). It is directly relatable to the discharge time constant of the
Electrophysiology 6- 29
39Smith, K. Bostock, H. & Hall, S. (1982) Saltatory conduction precedes remyelination in axons demyelinated
with lysophophatidyl choline J Neurol Sci vol. 54, pp 13+
individual regenerative conexus.
3. Several time delays are found within the system that are related to the conduction of charge
(diffusion of charge in response to static electrical fields) over short distances. These conduction
delays are associated primarily with the neurites but are most easily measured with respect to
electrotonic axons. In most signal processing neurons, the dimensions of these elements are so
small that the individual delays are measured in fractions of a millisecond.
4. A largely overlooked delay, because it is so short, is associated with the stage 3 neurons used to
transmit signals over long distances. This propagation delay is electro-magnetic in origin. It is
uniquely different from the conduction delay and does not involve the net movement of charge
within the neuron. In the free space, propagation travels near the speed of light. However, it is
considerably slower in the neural system, with a nominal velocity of about 4400 meters/sec (See
measured data of Smith et al39., Section 9.1.1.5.4) . This velocity corresponds to about 0.45 sec
for each 2 mm of axon length. The delay is the result of the finite velocity of signal transmission due to the finite
due to the impedance of the axon segment when operating as a coaxial cable (Section 6.3.3.4).
As will be demonstrated in Section 9.1.1.6, the one millisecond regeneration delay associated with each
functional unit of each stage 3 neuron is the longest identifiable fixed delay in the system. When summed, these
delays contribute to a delay approaching one second between stimulation of a nerve in the toe of a two meter tall
man and his awareness of the stimulation. Fortunately, the delays associated with vision and hearing involve
much shorter propagation paths and smaller total delays. In vision, the delay is on the order of three or four
milliseconds between excitation by photons and the arrival of a signal reporting that excitation at the CNS.
Because the impact of the quantum-mechanical delay associated with a synapse is negligible in the neural system,
it will not be discussed in detail below. The best available data on the delay associated with a type 3 conexus, a
synapse, is found in Section 9.5. Similarly, the delay associated with a type 1 conexus is hard to evaluate with
precision because it depends on the actual electrical circuit elements found within that conexus. The type 2
conexus appears to be reasonably standardized within the neural system of animals. Its performance is discussed
in Section 9.4.
An interesting observation can be made relative to the electric fields present in the above situations. In situation
1, the quantum-mechanical forces present within the Activa over-ride any static electrical fields present. In
situations 2 and 3, the static electrical fields present control the movement of the currents present. In situation 4,
the dynamic interaction of the electrical and magnetic fields essentially ignore the presence of any static electrical
field.
6.3.3.1 Importation of the morphological model into electrophysiology
Figure 6.3.3-1 provides a series of caricatures of how a neural system evolves from an electrical
perspective. As in the previous sections of PART C (specifically Section 10.1.6), it begins with
the simple serial connection of a series of alternating electrical conduits and junctions. However,
the morphologically invisible options become more numerous. Initially, the goal is to transmit a
signal from one end of the neural chain to the other as shown in (A). Traditionally, it has been
considered impossible to do this electrically because the space between adjacent sections of
conduit has been assumed to be an insulator, thereby introducing an electrical interruption at
each side of each conduit. However, this assumption is not defendable. The larger dotted box will
be shown to represent the portion of the overall signal path associated with a single neuron cell.
The smaller dotted portion will be seen to represent the minimum functional block of the signal
path. This fact was discussed without supporting comment in Section 10.1.1.1 in the context of
the Neuron Doctrine.
The smaller dotted box. shows a junction associated with a portion of a pre- and post junction
section of an axon segment. The portion of (A) within the smaller dotted box has been extracted
and reproduced as (B). Look first at (B) from an electrical perspective. If there was a current
flowing along the reticulum of the leftmost conduit, iin, the object would be to generate the same
level of current flowing in the same direction in the rightmost conduit, i out. Assuming the lemma
of each axon segment was made of a very high quality insulating biological material, this would
30 Neurons & the Nervous System
40Kriebel, R. (1975) Neurons of the dorsal lateral geniculate nucleus of the albino rat J Comp Neurol vol. 159,
pp 45-68
appear impossible. Let the lemma be specialized in the region of the junction. Let the two
bilayers of each lemma be asymmetrical at the molecular level in such a way as to form an
electrical diode. Further, let them be oriented and biased so that the interior of each axon
segment is maintained at a negative voltage with respect to the exterior interneural matrix, INM,
without leakage through the diode. If the material of the junction between these two lemmas, the
100 Angstrom thick hydronium crystal, is now placed at an electrical potential such that the
leftmost diode is forward biased and the rightmost diode is reverse biased, a unique situation
arises. More than 99% of whatever current is applied to the leftmost diode will appear at the
output of the rightmost diode by transistor action. This situation is shown in (C) using standard
electrical notation. Note the circuit in (C) has three independent terminals; the input or emitter
terminal, the output or the collector terminal and the base terminal. Note that any current
flowing into the emitter terminal must flow through the base terminal as well. Note also, that
any current flowing through the collector terminal must also flow through the base terminal.
These two currents normally flow in opposite directions and the net current through the base
terminal is very low. The electrical bias is shown applied to the base terminal of the transistor.
The current found flowing in the bias connection to the base, consisting of the hydronium crystal,
is normally less than one percent of the current flowing into the emitter, or the collector.
(D), (E) & (F) shows three potential biasing situations applicable to the Activa within a conexus.
(G) and (H) show where these circuits are found in practice. (G) shows the typical electrotonic
situation associated with stage 1, 2 and 4 neurons. (H) shows the typical phasic situation
associated with stage 3 neurons. While not documented here, it is likely the conexus shown in (E)
is found in the memory circuits of the stage 4 neurons, e.g., primarily within the CNS.
6.3.3.1.1 Packaging of functional neurons based on morphology
The neuroscience community has struggled a long time to define neurons based on their
morphology. It has obviously been impossible to do this in an orderly fashion. Kriebel has
provided a recent synopsis of his designations compared with those of many others based
primarily on light microscopy40. These designations only apply to the dorsal neurons of the lateral
geniculate nuclei. Kriebel noted his inability to identify the axons in two of his three types. He
associated this problem with the lack of dye absorption by these structures in his experiments.
Other investigators have made similar attempts at classification for various portions of the neural
system. They have usually encountered similar limitations. Until, the investigators associate the
morphological structures of the neuron with their functional performance (probably associated
with electron microscopy), this situation is not liable to change.
There are only a few fundamental morphological forms of neurons. All neurons exhibit at least
one output conduit called historically an axon. This conduit may bifurcate a few times as it leaves
the vicinity of the soma. If it is part of a stage 3 projection neuron, it will be interrupted at
regular intervals by Nodes of Ranvier. These interruptions subdivide the axon into axon
segments. Axon segments have frequently been labeled interneurons in the literature. However,
this is poor practice and leads to confusion with complete neurons labeled interneurons within the
stage 3 signal projection system of chordates. All neurons exhibit at least two input conduits, one
of which is associated with the signal inversion function and is labeled a podite. The podite
conduit may or may not be ramified depending on the role of the neuron. If it is acting primarily
as an electrotonic relay neuron (generally labeled a bipolar neuron) within stage 1,2 or 4, the
podite conduit may be used only as a route to provide electrical contact between the Activa and
the surrounding INM. In that case, its non-signaling role does not require ramification. When
acting as a lateral neuron, the podite terminal is typically ramified in order to collect signals to be
delivered to the inverting input terminal (the base) of the Activa. All neurons contain at least
one conduit associated with the non-inverting signal terminal of the neuron. This conduit delivers
the signals it collects to the non-inverting terminal (the emitter) of the Activa. The dendritic
conduit frequently bifurcates before reaching the exterior surface of the soma. As a result,
multiple dendrites are found to extend from the soma. While these dendrites ramify individually,
they eventually coalesce into one conduit within the soma. Only a few sensory neurons, such as
the photoreceptor cells, are known to deviate from these fundamental morphologies.
Electrophysiology 6- 31
41Bennett, M. Nakajima, Y. & Pappas, G. (1967) Physiology and ultrastructure of electrotonic junctions. III.
J. Neurophysiol. vol. 30, pg. 216
The fact that much of the surface of the morphologically defined soma of a cell may support
terminals associated with the dendritic and poditic conduits is of little consequence. Functionally,
the lemma of the conduits associated with the dendrites and podites may form part of the outer
plasma membrane of the complete cell. Only electron microscopy or other very sophisticated dye
techniques can define the perimeters of the dendrites and neurites relative to the soma of a
neuron. Alternately, electrical probing can map the potentials associated with these two neurites
within the soma.
Finally, the degree and precise form of the ramified neurites is controlled by the need to form
conexuses with a wide number of spatially dispersed axons. As a result of the above
considerations, the detailed external morphology of each neuron is controlled by the physical
space available to it. In this sense, the form of each neuron is controlled by its function and the
available topography.
6.3.3.1.2 Neural signals are orthodromic (unidirectional)
There have been many experiments exploring antidromic, or reverse, signal flow through
projection neurons. It is generally accepted now that artificially induced action potentials may
propagate antidromically within an axon segment. However, propagation stops at the first
encounter with a synapse, Node of Ranvier or hillock41 . These conclusions are consistent with the
presence of an active unidirectional electrical amplifier at each of these sites.
The term transconductance, one form of transimpedance, is used to describe the ratio between the
current out of a general network divided by the voltage applied to the input of that same network.
Here, the transconductance between the input structure and the output structure will be given
very exactly by the input conductance of the leftmost diode. A change in the output current,
typically in picoamperes for a typical neuron, will be controlled by a change in input voltage,
typically of a few millivolts.
32 Neurons & the Nervous System
Figure 6.3.3-1 The basic electronic forms of a fundamental neural path. (A); the
fundamental neural signal path showing the region incorporated into a single
neuron and the smaller region forming the fundamental building block of the
neural system. (B) & (C); the fundamental building block of the neural system
shown using morphological and electrical symbology. (D), (E) and (F); the
building block shown in electrical symbology with various modifications to the
circuit. (G); the morphological symbology for two electrotonic neurons connected
in series. (H); a single phasic neuron shown consisting of multiple fundamental
building blocks. See text for details
Electrophysiology 6- 33
Note that under these bias conditions, and ignoring the third terminal associated with biasing the
hydronium crystal, the overall circuit appears to be a simple diode. It is not. If the bias to the
hydronium crystal is removed, the circuit is now represented by two diodes connected back to
back. The transconductance will now be very low, essentially an open circuit. If the bias
conditions are reversed, the direction of current flow will be reversed without making any other
changes to the circuit. It will be as if the “rectifier” has been turned around. When the neuron is
properly biased, the resulting transistor action is the key to the operation of both the electrotonic
and phasic circuits of the neural system. Referring to (A), as many sections of conduit can be
connected in series as needed, subject to satisfying the biasing requirements. In practice, it
becomes difficult to satisfy the biasing requirements in a long series of conduits. Good practice in
man made circuits is to limit the number of junctions to less than five in what are called direct
coupled circuits. As will be seen below, this criterion has also been applied in neural systems.
6.3.3.1.3 Electrophysiological/histological model of a neuron
The internal operation of a neuron, or Node of Ranvier with modification in biasing, is shown in
Figure 6.3.3-2, reproduced from Section 2.7.5.
34 Neurons & the Nervous System
Figure 6.3.3-2 Fully implemented electrophysiological/histological Neuron,
showing charge entering and leaving a cylindrical axon. All current flow is via
electrons. The capacitance of the axon, CA, is indicated at lower right. The
capacitance of the denrite, CD, is shown on left. Other notation defined in Section
2.7.5. There is no need for an ephemeral sodium-pump.
6.3.3.1.4 Myelin as a critical element in signal propagation
The extreme thinness of the lemma of a neuron is a distinct impediment to efficient signal
transmission within a neural system. To overcome this problem, Nature has introduced a variety
of techniques to effectively increase the distance between the various neural plasmas and the
surrounding INM. In the higher animals, the use of a specialized cell to provide a multilayer
sheath isolating individual neurons has become popular. This multilayer sheath has come to be
known as myelin. It actually consists of hundreds of wraps of the BLM of the protective cell
around the individual neuron. The same technique is found in less developed form in the lower
animals along with an alternate technique. A pseudo-myelin can be formed by surrounding a
Electrophysiology 6- 35
42Morell, P. (1977) Myelin NY: Plenum Press
43Pethig, R. (1979) Op. Cit. pg 208
specific neuron with a large number of closely packed smaller neurons. The effect is largely the
same with respect to the primary neuron. The capacitance between the axoplasm of the neuron
and the surrounding INM is reduced.
The physical properties of myelin have been studied extensively. However, the electrical
properties are hard to find in the literature. Morell has provided an entire book on the physical
properties42.
Morell has provided a tabulation of the composition of CNS myelin and brain for a variety of
species. Human myelin is given as 70% total lipids and 30% protein (dry weight). 43% of the total
human lipids are phospholipids (wet weight) and two-thirds of this material is ethanolamine
phosphatides (PE). This is as expected. Pethig gives slightly different values of 81% lipid and
19% protein for nerve axon myelin43. The individual membranes of Myelin are in fact plasma
membrane of Schwann cells. It is likely the portion of the membrane wrapped around the
individual axon segment is primarily type 1 BLM and consisting of PE. The mixed phosphatides
are most likely associated with type 3 BLM supporting the manufacture and distribution of
lactate and pyruvate to the neurons. These cells are believed to act as an alternate source of these
two chemicals, thereby relieving the need to transport them from the soma to the extremities of a
propagation neuron (that might be one to two meters long).
6.3.3.1.5 The transversal filter and time dispersion in the neural system
As investigators begin to adopt multi-probe data collection techniques, it is important that they
recognize the importance of time dispersion within the various sensory modalities, particularly
vision and hearing. The rate of signal transmission within the neural system is quite low relative
to conventional electronic circuit where it typically is higher than six-tenths of the speed of light.
The faster neural circuits transmit signals at about 44 meters/sec compared to 200,000 meters/sec
in a typical electronic circuit. The speed of acoustic signals within the cochlea is even slower,
typically 3-6 meters/sec.
As a result, signals from different regions of the retina and different locations along the cochlea
are not time coherent when they appear in the electrolytic signal processing domain. In essence,
the neural system avails itself of, and takes advantage of, transversal filter techniques.
Figure 6.3.3-3 shows the general plan of a transversal filter as used in hearing. The acoustic
signal, x(t) passes along the Organ of Corti where it is dispersed, based on frequency into the
vertical channels. The elements labeled wn, correspond to the sensory neurons. The middle
horizontal bar represents any of a number of intermediate signal processing engines (stages 2 or
4), used to extract information from the raw neural signals from stage 1. Some of the vertical
signal paths may not be processed by a specific signal processing engine. Because the delays
along the top horizontal path are significant, it is generally necessary to recreate a temporally
coherent copy of x(t) as z(t) for introduction into the saliency map of stage 5. Thus additional
delays are introduced into the overall system. Note the introduction of these subsequent delays
means that much of the signal processing within the neural system occurs before the animal is
able to perceive the event as documented in the saliency map and accessed by the stage 5
cognition circuits.
These delays, and those due to the path lengths involved in stage 3 circuits, explain why the
knee jerks before you are aware of the Doctor’s hammer contacting your kneecap or why the
baseball player hits the ball before he is totally aware that he has.
36 Neurons & the Nervous System
44Terman, F. (1943) Radio Engineers’ Handbook. NY: McGraw-Hill pp 770-780
A similar transversal filter can be defined in vision. In that system, the lower delay lines are
formed by what the anatomist calls Meyer’s loops within the CNS. The upper delays are
introduced by the lengths of the stage 3 neurons originating in different regions of the retina.
This leaves the LGN effectively processing the dispersed y(t). This may make change detection
much simpler.
The signals within the cochlear nucleus of hearing and the geniculate nucleus of both vision and
hearing are good examples of the intermediate signals represented by y(t) in the figure. The
signals within these bodies can be mapped back to the input signals but when they are mapped,
they are not time coherent with the input signals.
6.3.3.2 Introduction to Modern Cable Theory
The title of this section is intentionally relatable to the paper discussed above by Taylor. This
discussion will begin from the perspective of Maxwell’s Equations and specifically the General
Wave Equation. This equation will be shown to describe the movement of signals within the
neural system much more completely than does a first order differential equation of potential with
respect to time.
When discussing the transmission of phasic signals, the method of coupling of the signal
generator to the signal transmission medium must be considered. The output of any oscillator is
capable of being detected at a distance based on the electrical and magnetic fields it generates.
However, the achievable distance depends greatly on the efficiency of coupling between the
oscillator circuit and the intervening medium44. If the output impedance of the oscillator is
matched to the characteristic impedance of the medium, propagation is achieved efficiently. The
signal can be transferred very efficiently within the medium by the interaction of electrical and
Figure 6.3.3-3 General plan of a transversal filter. The signal, y(t) does not
represent a coherent copy of the input, x(t) but can be very useful. The signal, z(t),
may or may not be a coherent copy of x(t). See text.
Electrophysiology 6- 37
45Lafontaine, S. Rasminsky, M. Saida, T. & Sumner A. (1982) Conduction block in rat myelinated fibres
following acute exposure to anti-galactocerebroside serum. J Physiol (Lond.) vol 323, pp 287-306
magnetic fields (and without the net movement of electrical charge through the medium). If the
output impedance of the oscillator is not matched to the characteristic impedance of the medium,
some signal transmission can still be achieved over much shorter distances. This method is
similar to the inductive transfer of a signal within a transformer or the capacitive transfer
between impedances on each side of a capacitor of random size. This type of transmission is
known as inductive transmission and is exemplified by the conduction of a signal through a
electrolytic medium. The actual mode of transmission achieved depends greatly on the specific
parameters of the application. Frequently, lumped circuit elements are used between the
oscillator and the medium to insure efficient coupling between the two. These elements are
described as matching elements or (as a group) a matching filter.
6.3.3.2.1 Efficient transmission over long distances–propagation
In propagation, the electric and magnetic fields are proportional to each other and in phase at a
given point along the propagation path. In the electromagnetics of an axon, the signal is
propagated by the interaction of these two fields without the net movement of charge along the
axon. This is the basis of all radio, TV and Radar transmission.
A concept not found in the chemical theory of the neuron is that of impedance matching at
junctions employing propagation in either part of the pre-junction or post junction element, to
avoid reflections due to impednance mismatches (Section 6.3.5.2). This technique is very
important in stage 3 projection neurons where the distance between NoR are maximized when
reflections at the cable interface are minimized.
6.3.3.2.2 Adequate transmission over short distances–conduction
In conduction, the electric and magnetic fields are not proportional to each other and are
generally not in phase. At a given point, one type of field may dominate to a significant degree.
This situation is optimized in transformers where only the magnetic field is used.
Bostock provided a scarce value on page 112 and Fig. 5-6 (citing Rasminsky & Sears, 1972) for the
average conduction velocity in a demyelinated rat ventral root fiber in-vitro at 30°C over an axon
segment and a following NoR before the NoR failed. Failure probably occurred when the NoR did
not reach threshold at its input due to paranodal demyelination; and propagation beyond that
NoR ceased. Whereas they noted a typical value of 26 microseconds (0.026 milliseconds) in the
myelinated case (including the NoR regeneration time of 0.015 to 0.019 milliseconds), they
recorded 600 microseconds in the demyelinated case when the signal was interrupted. Taking the
normal average velocity in their neurons as 36 m/sec, ratioing these values would suggest a
demyelinated average velocity of only 1.2 m/s. This is within the commonly given range for
conduction by diffusion within unmyelinated neurons. It could be smaller if the NoR delay was
removed from both the numerator and denominator of the ratio.
Quoting Bostock,
“Lafontaine et al45. made a more extensive investigation of demyelination, indicating a 11-
fold in membrane capacitance and a smaller increase in conductance (averaging 5-fold).”
The wording is ambiguous. The intent was probably to indicate a net reduction in the
conductance. See Section 6.3.8.5.1 for additional discussion of Lafontaine et al.
6.3.3.3 Propagation over the ideal coaxial cable
The axon, like the real submarine cable is not represented by only resistive and capacitive
elements. Any coaxial transmission line exhibits an inductance under transient conditions. The
value of this inductance per unit length is a function of the geometry of the cable and the
permeability of the dielectric medium. The equation for this inductance is (B) in the following set
38 Neurons & the Nervous System
46Chie, S. & Ritchie, J. (1980) Potassium channels in nodal and internodal axonal membrane of mammalian
myelinated fibers Nature vol. 284, pp 170-171
Figure 6.3.3-4 Ideal cable Equations based
on inner/outer radii of dielectric
(myelination, r = 3.0, = 1.0 in this case).
where ro is the outside radius and ri is the inside radius of the dielectric, Figure 6.3.3-4 .
The equation in (B) is also known as the external inductance of a coaxial cable to differentiate it
from the internal inductance that is related to the inductance of a straight solid conductor. The
total inductance of a cable is the sum of the external (or mutual) inductance and the internal (or
self inductance) of the cable. The internal inductance of an axon could be very large because of its
small diameter if the axoplasm was a good conductor like copper. However, the resistivity of the
axoplasm is many orders of magnitude higher than that of copper. It may be several orders
higher than that of sea water due to its liquid crystalline (gel-like) character. The total
inductance of the axon appears to be dominated by the external inductance.
The presence of an inductance associated with the giant axon of Loligo was extensively
documented by Cole. However, he did not associate that inductance with a cable. Instead, he
associated two inductances with the h and n parameters and an additional capacitance with the m
parameter of the equations and equivalent circuit of Hodgkin & Huxley. He diagramed them as
parallel circuits in their shunt circuit diagram (pg 299). Cole notes that: “Every element of the
circuit, except the membrane capacity C, changes with the potential difference across the
membrane.” As a result, the symbols used should not be interpreted as associated with fixed
circuit elements.
Any coaxial transmission line also exhibits a capacitance under transient conditions. The value of
this capacitance per unit length is a function of the geometry of the cable and the permeability of
the dielectric medium. It is not found by using the flat plate equivalent area of the dielectric.
The equation for the capacitance per unit length of an axon is (A) in the following set of equations
where is the permittivity of the dielectric.
Figure 6.3.3-5 presents the calculated value for
these impedances for various size axons. The
values are for both unmyelinated and myelinated
neurons based on the membrane dimensions and for
a permittivity assumed to be equal to 3.0 for
discussion. As calculated, 100 layers of membrane
causes a change in capacitance of about 12:1. Note
the significant change in these values as a function
of axon radius. The unmyelinated values shown
intersecting the gray bar are typical of the values to
be expected in the experiments of Hodgkin &
Huxley. These values are clearly not typical of the
1-5 micron radius applicable to stage 3 projection
neurons in Chordata.
Chiu & Ritchie measured the capacitance of normal
and unmyelinated axonal segments in mammals
(rabbits)46. Although they did not provide the
radius or length of their segments, they described an
increase of 20 +/– 2.6 times in the capacitance after
demyelination from an initial total value of 3 +/–
1.2 pF. The capacitance ratio of 20:1 would
suggest a myelination consisting of about 150–200
BLM layers in their specimen.
Electrophysiology 6- 39
Figure 6.3.3-5 The capacitance and
inductance of an ideal axon segment. R =
G = 0. The vertical gray bar represents
the radius of the typical Loligo axon
without surrounding tissue.
40 Neurons & the Nervous System
47Ramo, S. & Whinnery, J. (1953) Op. Cit. pg 41
Figure 6.3.3-6 The characteristic
impedance and phase velocity along an
axon. Top, an ideal axon segment. R = G
= 0. The phase velocity does not vary
significantly for typical biological
dielectrics. The estimated values are for
a real axon where R is finite but G = 0. The
vertical gray bar represents the radius of
the typical giant axon of Loligo,
suggesting a very low phase velocity.
Figure 6.3.3-6 shows the characteristic impedance, Z0, of an ideal axon segment based on
equation (D). A major problem with using the giant axon of Loligo is seen immediately. The
characteristic impedance of that size axon is very low. As a result, it is extremely difficult for the
conexus of the neuron to excite it. While it can support the transmission of a signal by diffusion, it
may not be able to support it by propagation. While the giant neuron of Loligo makes a good
signal processing (stage 2) neuron, it makes a very poor stage 3 signal propagation neuron. The
giant axon of Loligo is not analogous to the stage 3 propagation neurons found in
Chordata. The propagation velocities shown in the figure represent ideal transmission lines
with various values of dielectric constant at the top and estimates associated with lossy lines such
as a real axon. The latter will be discussed in greater detail below.
Equation 2(C) shows that an ideal axon with
a dielectric constant of 3.00 has a
transmission velocity of (1/3)0.5 that of the
speed of light or about 17 x 108 meters/sec.
This extremely high value is not compatible
with the values found in the literature for
signals flowing from one end of a neuron to
the other, or even along a single axon
segment. This fact suggests that the axon
segment is not an ideal axon and that its
series resistance and shunt conductance
must be considered more carefully.
However, the form of equation 2(C) explains
why a signal impressed in the middle of an
axon segment is propagated in both the
positive and negative directions along the
segment. There are two values for the
function. One travels to the right and one
travels to the left. It also shows a significant
variation in the velocity of propagation as a
function of frequency. Like a simple RC
filter network, the ideal axon exhibits a
significant variation in the velocity of a sine
wave traveling along an axon as a function
of frequency.
The equations for the velocity of a wave
traveling through a lossy axon segment
(coaxial cable) are quite complex but readily
available in text books47. They show that the
velocity is slowed considerably as a function
of the frequency and of the series resistance and shunt conductance. In the case of the real axon,
the specific resistance, e.g. resistivity, of both the axoplasm and the INM are reported to be on the
order of 110 Ohm-cm. This compares with a value of 1.7 x 10-6 Ohm-cm for copper wire. Such a
high resistivity has a significant effect on the movement of charge within the electrolyte adjacent
to the dielectric. Figure 6.3.3-7 shows the typical situation for both an ideal axon (left) and a real
axon (right). The left figures are similar to those found in many biology texts, except they have
been expanded. They now show the electric field lines between the charges and the magnetic field
lines surrounding the fields due to the charge on the inner and outer conductors. With conductor
resistivities very small relative to the inductances, all charge is located very close to the walls of
the dielectric and it can move along these walls very rapidly. As a result, the propagation velocity
of a wave along this cable is a fraction of the speed of light. For a real axon as shown on the right,
Electrophysiology 6- 41
Figure 6.3.3-7 Electric and magnetic fields within an axon. Left; as conventionally
described in the biological literature, the magnetic lines (solid and hollow circles)
are omitted and the conductors are considered ideal. Right, the real case of an
axon with a nearly ideal dielectric but a lossy axoplasm and surrounding inter
neural matrix.
the situation is different. The resistivity of the plasmas is now far from zero and significant when
compared to the inductance of the structure. The electric field lines now enter the plasmas a
significant distance and the speed of electrons along these field lines is considerably reduced. As a
result, the position of the electrons at any instant is hard to define and the propagation velocity of
the wave is considerably reduced. Note the electrical field lines are closed in both the ideal and
real axons. There is no net current flow in the direction of propagation. As shown in Section
10.5.7, the nominal propagation (phase) velocity for the wave traveling along an axon segment is
4400 meters/sec (10–5 times the speed of light).
A phase velocity of 4400 meters/sec is 300 to 1000 times faster than the speeds calculated by
Taylor and Hodgkin. For they were trying to make their Kelvin—>Hermann based approach
agree with the average velocity of a neural signal traveling between two identical points in a Node
of Ranvier and axon segment, instead of the phase velocity of an axon segment. The modern
models of Taylor and of Hodgkin do not predict the actual phase velocity of an axon
segment.
42 Neurons & the Nervous System
Figure 6.3.3-8 The electrical circuits used to describe an axon (a coaxial cable).
(A); the general circuit in unbalanced lumped-parameter form. (B); the same
circuit drawn in balanced lumped-parameter form. In efficient cables, the
conductive currents passing through the resistive elements are negligible
compared to the displacement currents through the inductive and capacitive
elements. As a result, a simpler notation is available. (C); the simplified
distributed network based on the dominant displacement currents. It forms an
efficient propagation medium. (D); the half-section used to terminate either end
of the line in (C) based on the component values in (A).
The presence of the inductance requires the adoption of a model of a coaxial cable
including this inductance. Such a model is given in Figure 6.3.3-8. (A) is such a model in an
unbalanced form, the usual form for discussing a coaxial cable. (C) is the balanced form often
found in the biological literature (without the inductances). (C) shows the unbalanced form in its
distributed form. To properly connect to the form in (C), a half-section matching filter is used of
the type shown in (D). A uniform distributed line terminated at both ends with the appropriate
matching section can achieve 98% transfer efficiency over considerable distances if the required
bandwidth is not to great.
The introduction of an inductance into the model negates all of the equations found in
the biological literature for the coaxial cable representing the biological axon in a
stage 3 projection neuron (including the Rall model and earlier variants). The
mathematical description of a real axon requires solution of the General Wave Equation of
Maxwell. The solution of this equation also introduces a concept not found in the lower order
Laplace and Poisson equations, the characteristic impedance of the axon (or coaxial line). The
value of this impedance plays a crucial role in the efficiency of the axon as a transmission line.
Electrophysiology 6- 43
Figure 6.3.3-9 Measured impedance
(inductive & capacitive) of a real axon
after unspecified preparation procedures.
Replotted from Cole, 1968, and annotated.
The solution for the one-dimensional transmission of energy over an axon is well known and is
given as 44m/sec at a presumed temperature of 32 C (including Nodes of Ranvier at an average
spacing of 2.0 mm). See Section 6.3.5.4.2.
6.3.3.4 Propagation over a lossy coaxial cable–real axon
Cole was the first to report impedance measurements on a variety of neural and muscular tissue.
Figure 6.3.3-9 shows actual impedance measurements by Cole on the squid axon after undefined
preparation procedures. Caution should be observed here. It is likely that the preparation
changed the effective conductance of the fluid within the axon. The plasma was frequently
replaced by sea water to simplify the instrumentation procedure. Such a change would have a
significant change in the conductance of the internal fluid. He describes the empirical derivation
of the expected impedance on pages 36-37 (without considering any potential inductance) and
then notes the considerable consternation in the community when the axon was found to exhibit
considerable net positive reactance at low frequencies. Pages 78-79 describe conceptual
discussions aimed at avoiding the obvious. The coaxial axon exhibits significant inductance.
Cole presented the data in a non-standard
form in his original work. By using
conventional filter theory and complex plane
plotting conventions (positive reactance at
the top), the revised figure can be used to
evaluate the nature of the impedance
measured on a lossy axon based on
conventional filter theory.
The following equations replace equations
(C) and (D) in the previous set when the
values of the resistance and conductance
associated with the coaxial cable or axon are
not negligible compared to the inductance
and capacitance. Two new equations have
been added in this set. An approximate
equation for the attenuation along the line
and an approximate equation for the phase
constant along the line. The exact equations
are much more complicated and are not
needed here.
44 Neurons & the Nervous System
p
R j L G j C A
ZR j L
G j C B
R
Z
G Z C
LC RG
LC
G
C
R
LD
1
2 2
14 8 8
0
0
0
2
2
2 2
2
2 2
( ) ( ) ( )
( )
( )
( )
(5) Equations of a lossy coaxial
cable applicable to an axon
The interpretation of these equations in a specific application is aided by experience. As a general
rule, equation (A) is not used. It is re-written in the form of equations (C) and (D) which are
easier to interpret. (C) describes the attenuation constant applicable to the cable. (D) describes
the phase shift constant applicable to the cable. A brief theoretical interpretation is provided
below. The comparison of the theoretical and measured values for the axon will be presented in
Section 6.3.5.
6.3.3.4.1 The theoretical intrinsic propagation velocity and attenuation
on a lossy line
Equation (A) gives the phase velocity of the signal when the coaxial line exhibits significant
resistivity and conductivity. The propagation velocity is dependent on four parameters in a very
complex manner. In cases where the resistivity and conductivity dominate, propagation is slowed
considerably below the value for the ideal cable. The velocity is given by p = (R•G)– 0.5 for this case
(which always occurs at low frequencies). At high frequencies, the velocity is usually given by p = (L•C)– 0.5.
At intermediate frequencies, equation (A) is generally written in a different form that allows the
separation of the attenuation constant for the cable (C) and the phase constant for the cable (D).
The phase constant describes the velocity of propagation as a function of wavelength or frequency.
This velocity varies with the wavelength (or frequency) of each Fourier component in the signal.
The variation leads to the dispersion of the energy in the original signal with distance along the
cable. Dispersion appears to be the principle cause of deterioration in the neural system.
The intrinsic propagation (phase) velocity of even a lossy axon is much higher than the average
velocity of propagation. The average velocity is limited by the significant delay associated with
the periodic signal regeneration process (involving Nodes of Ranvier, see Section 9.4).
6.3.3.4.2 The theoretical impedance of a lossy cable
Equation (B) in this set of equations describes the characteristic impedance, Z0, of a lossy coaxial
cable. This equation can be useful in interpreting the complex impedance plane measurements of
Cole. However, it must be noted first that the equation only applies to a transmission line that is
terminated at each of its ends using an impedance equal to this characteristic impedance. If the
cable is not properly terminated, measurements such as Cole made will depend on the precise
position along the axon where they are made. The measured values will reflect this inadequate
termination by exhibiting a voltage standing wave ratio, VSWR, along the line. This parameter is
an indicator of the error in termination.
In the case of a real axon, the adequacy of the termination is currently unknown. However, a
matching filter is usually required between the myelinated portion of the axon and the conexuses
at each end of each axon segment. Such a matching filter is usually formed by the unmyelinated
section of the axon segment.
The Cole figure above can be annotated using equation (B). At zero angular frequency, the value of the
impedance is given by the square root of the ratio of R divided by G. This impedance is purely resistive and is
about 10.4 kilohms in the figure. Conversely, at very high frequencies, the value of the impedance is given by the
Electrophysiology 6- 45
square root of the ratio of L divided by C. This impedance is also purely resistive and is about 6.6 kilohms in the
figure. The value of 11.7 kilohms at 260 Hertz is also descriptive of the cable. It is a purely resistive impedance.
This resistive value will move toward the characteristic impedance as the terminations are adjusted to optimum.
At frequencies below 260 Hz, the cable exhibits a positive reactance. It can be considered to be inductively
loaded at this location. At frequencies above 260 Hz, the cable exhibits a negative reactance. It can be
considered capacitively loaded at this location.
6.3.3.4.3 Intrinsic pulse dispersion along a lossy line
Equations (C) and (D) of the above set are also instructive. They are approximations of the attenuation constant,
, and the phase constant, , of a lossy cable. The describe the distortions in a pulse transmitted along a lossy
cable. For the distances of interest in neuroscience, it will be shown that the attenuation constant is not
significant. The amplitude of the energy in the Fourier components of the pulse does not diminish significantly
over distances of a few millimeters. However, the phase constant is a direct function of the frequency of the
Fourier component. This term causes each Fourier component to exhibit a different phase velocity when
propagating over a lossy cable. The effect is to smear the shape of the action potential pulse, eventually to the
point of unrecognizability. To effect useful signal propagation, the pulse shape must be restored before it reaches
this point. This restoration through regeneration is the precise role of the Nodes of Ranvier. They must be
placed at sufficiently frequent intervals along an axon to sense the signal accurately and reliably before it falls
below the threshold amplitude of the conexus within the Node of Ranvier.
The phase constant, , associated with a real axon, and represented by a lossy cable, makes one of
the several assumptions of Taylor in his analysis of a cable untenable. He made the following
assumption (page 226). “If the open circuit potential V0 does not vary with position on the
membrane, we may use as our variable of voltage Vm - V0 = V(t).” This is clearly not the case
within the individual groups of measurements by Smith, et. al. Taylor assumed that the velocity
of all components of a pulse waveform traveled at the same velocity in a lossy line. This was also
the failing in the original analysis of the undersea cable by Lord Kelvin.
6.3.3.4.4 The mobility coefficients of typical electrolytes & semiconductor
materials
A drastic difference exists between the mobility of charges in liquid crystalline (and other
semiconductor materials which rely upon quantum-mechanical phenomenon) and in electrolytes
(which rely upon charge conduction by diffusion). In the case of solid state materials, the
mobilities of the carriers are typically in the range shown in the upper portion of Table 6.3.3 The
lower portion of the table shows simlar values for typical ions in dilute solution. All of these
values are quite slow for reasonable voltages relative to the speed of light. However, the distances
involved are typically measured in nanometers within the liquid crystalline materials of interest
(primarily hydronium). The transit times over these short distances remain very small for the
voltages encountered in the neural system. In the case of electrolytes, the situation is different.
The mobilities are quite low and the distances involved can range from millimeters to meters.
46 Neurons & the Nervous System
48Millman, J. & Halkias, C. (1972) Integrated Circuits NY: McGraw-Hill pg. 29
49Adamson, A. (1973) A textbook of physical chemistry NY: Academic Press pg. 506
50Adamson, A. ibid. pp. 506-508
51Ochs, S. (1971) General properties of nerve. in Physiology, 3rd ed. Selkurt, E. editor Boston MA: Little
Brown & Co. pg. 45
52Berthold, C-H. (1978) Morphology of normal peripheral axons, In Waxman, S. Ed. Physiology and
Pathobiology of Axons. NY: Raven Press pg 4
Table 6.3.3
Mobilities of selected charge carriers
In Semiconductors 48 at 300 K negative charges positive charges
Germanium 3,800 cm2/V-s 1,800 cm2/V-s
Silicon 1,300 500
In Electrolytes49 at 25 Celsius & infinite dilution in units of cm2/V-s x 104
OH–- 20.5 36.30* H+
F–- 5.7 4.01 Li+
Cl–- 7.91 5.19 Na+
Br— 8.13 7.62 K+
I— 7.95 6.16 Ca++
* The mobility given is believed to be for a hydrated form of the hydrogen ion. However, the
question of how the hydrogen ion moves in an electrolyte is a very difficult one. There is
considerable reason to believe that hydrogen is represented in an electrolyte much as a hole is in a
solid state material; i. e., there may be little physical motion but a transfer of bonds within a
matrix of molecular water “clusters”50. These clusters may exhibit a longer life than normal
because of the concentration level of the solution. If this is the actual situation, the analogy
between the semiconductor of the solid state and the semiconductor of the liquid crystalline state
would be even stronger. In ice, the mobility of the positive charge is 1,000 times higher than in
water at 25 degrees C.
Clearly, the speed of signal propagation by conduction through an electrolyte is much slower than
in solid state, metallic or electromagnetic systems.
As a simple example, in an axon that is 100 microns long with a potential
difference between its input and its output of 10 millivolts, the transit velocity of
OH- is about 2,000meters/sec. If the actual effective voltage was this high, the
resulting velocity of 2 meters per millisecond would result in negligible transit
delay within the retina.
This estimated propagation velocity of an electrotonic signal conducted through a
plasma is similar to the velocity of an action potential51. Action potentials are
propagated as electromagnetic waves and typically travel at a phase velocity along
an axon segment of 4400 meters/sec. However, they are slowed to an effective 44
meters/sec by the regenerative process required frequently at Nodes of Ranvier.
(See measured data of Smith et al., Section 9.1.1.5.4). The “conduction velocities”
given in Table 1-1 of Afifi & Bergman and fig 1 of Berthold52 are actually the
effective (average) propagation velocities along propagation neurons of unspecified
length. The number of NoR present in their measurements of velocity makes a
large difference in the average value measured.
It is important to state the morphological and cytological value of the 1978 book edited
Electrophysiology 6- 47
by Waxman. It includes a large number of electron micrographs at various
magnifications. While they are all interpreted under the chemical theory of the neuron
prevalent at the time, they can be reinterpreted quite easily under the framework of The
Electrolytic Theory of the Neuron.
6.3.4 Transmission by conduction of electrotonic signals within Stage 1,2
& 4 neurons
Conduction is the fundamental mode of signal transmission within the signal sensing and signal
processing neurons. It is the dominant mode of signal transmission in Chordata based on neuron
count and total equivalent distance over which signals are transmitted. However, the role of
conduction is often overlooked because of the ease of measuring the velocity of signals using the
propagation mode in the peripheral nervous system.
Only concepts related to neural conduction have been found in the literature. The chemical
theory of the neuron has not provided a framework with which to develop a theory of conduction.
The Electrolytic Theory of the Neuron provides a much more detailed framework and points to the
need for additional specific tests.
In the few cases where electrotonic signals are transmitted over appreciable distances (hundreds
of microns in the retina), their velocity has not been widely reported. It appears to be slow. Less
than one meter per second.
6.3.4.1 Conduction of signals within axons of stage 1,2 & 4 neurons
Most signal processing neurons are compact with axon lengths of less than a few hundred
microns. These neurons are best modeled as lumped parameter circuits, as has been done in
earlier sections. However, some signal detection and signal processing neurons have longer axons
that are best modeled as distributed parameter circuits. These include the photoreceptor cells
and several of the bipolar and lateral cell types in the retina. Figure 6.3.4-1 presents several
potential models of the conduction of electrotonic signals within unmyelinated signal detection
and signal processing neurons. Because the axolemma is not myelinated, the capacitance per unit
length of the axon far exceeds the inductance per unit length. The inductance can be considered
mathematically negligible. Recall the resistive symbols represent impedances based on
electrolytes. These resistive values exhibit significant time delay due to the limited mobility of
the ionized components within the medium. The model in (A) applies to neurons
48 Neurons & the Nervous System
Figure 6.3.4-1 Models of signal conduction within unmyelinated stage 1, 2 & 4
neurons. A; lumped parameter case. The impedance of the power source creates
the voltage at V1. R1 is small relative to the output impedances 1 & 2. B & C;
Distributed parameter cases. Optimum location of the power source depends on
value of R1 and impedances at 1 & 2. D; a simple analog of the circuit in A that can
be used for analysis. See text.
with short axons. The conductive delay associated with the axoplasm is negligible and the total
axon capacitance is essentially that associated with the conexus itself. (B) & (C) show two
alternatives for neurons with long axons. In (B), the axon power source is located near the
Electrophysiology 6- 49
53Reddick, H. & Miller, F. (1955) Advanced Mathematics for Engineers, 3rd ed, NY: John Wiley & Sons
conexus within the neuron. The load impedance associated with the power source creates the
voltage V1. Negligible delay is introduced between the voltage at V1 and the input signal to the
conexus. However, the RC networks shown between V1 and V2 and the load impedances form a
voltage divider network. In addition, each of the series resistances introduces a measure of delay.
Therefore, the signal at V2 is reduced in amplitude and delayed relative to that at V1. Each
additional increment in axon length also introduces an additional RC low pass filter section. The
shunt resistance is very high compared with the series resistance and each incremental circuit
can be considered a series resistance and a shunt capacitance. These reduce the frequency
response of the overall neuron incrementally. (C) places the load impedance associated with the
power source at the node associated with the output impedances. Whether (B) or (C) provides the
maximum circuit performance depends on the specific values of the parameters. In the case of the
photoreceptor cell, the work of Vardi, et. al. suggests that configuration (C) is used.
6.3.4.1.1 Calculation of the signal conducted along an axon
As indicated earlier, Hermann described his concept of signal transmission within a neuron as
analogous to the transmission of heat, although he settled on a poor graphical analogy combining
pieces of our modern understanding of conduction and propagation. The mode of transmission
used in electrotonic neurons is that of conduction, the diffusion of charge, along an electrolytic
pathway. The mathematics of conduction in a less than ideal medium, where the travel velocity is
low relative to the distances involved, is complicated and generally cannot be described in closed
form. Reddick & Miller describe the situation in detail and provide several examples53. Their
articles 7(d) and 70 describe one-dimensional heat flow as generally needed to describe neural
signaling. Article 7(d) describes the basic concepts of heat flow and defines the simple terms used
in the differential equations describing a given situation. Article 70 derives the partial
differential equation of one-dimensional heat flow as originally conceptualized by William
Thompson (Lord Kelvin). They then solve a typical problem in three steps. First, they define the
differential equation. Second, they determine the initial conditions (or quiescent conditions).
Third, they determine the boundary conditions. They then separate the solution into two parts, a
steady state solution (corresponding to the boundary conditions) and a transient solution. It is the
transient solution that is of most interest in neural signaling. However, the complete solution is
needed to most easily interpret the results.
Defining the boundary conditions related to a real in-vivo axon is difficult in the neural problem
because of the mixed technologies used to form(transistor action), transmit (simple electrolyte)
and receive (quantum mechanical diode) the signal. A two-step approximation can be used to set
up a manageable example. Consider the case where the power source is adjacent to the Activa, as
in frame (B) of the above figure. Then simplify the axon and its synapse as shown in (D). (D)
describes a single long axon with the right end grounded for AC signals and at the same voltage
as the conexus, V1, with respect to its DC potential. The axon is shown as bent and an output
connection sampling the voltage, V2, of the axoplasm. The voltage at V2 is the result of the two
parts of the axon acting as a voltage divider. Let the source of the voltage change at the conexus
be ideal. Let the output impedance of the conexus equal zero. A step change in the source voltage
will result in a change in the potential along the conduit that is described by the one-dimensional
heat equation.
The transient solution for one-dimensional heat flow is given by:
uK
nExp n
Ltn x
L
t
n
142
2 2 2
2
1
sin
where K is proportional to the height of the step excitation at time zero. L is the length of the
conduit and alpha equals the quotient of the thermal conductivity of the medium divided by the
product of the specific heat capacitance and the density of the medium. At first glance, this
equation goes to zero rapidly as n increases. However, on closer examination, it is the overall
coefficient in the exponential term that determines the size of the individual terms and the
number of terms in the series that are significant. An arbitrary value for alpha of 5.0 and an
50 Neurons & the Nervous System
54Juusola, M. et. al. (1996) Graded-potential transmission through tonically active synapses Trends in Neurosci
vol 19(7), pp 292-297
Figure 6.3.4-2 Net voltage versus distance
along a conduit with time as a parameter.
The top dashed line is the steady state
voltage at any point along the conduit.
This curve is the boundary condition for
the differential equation. The horizontal
axis forms the initial condition for the
equation. The solid curves show the
voltage at any point after different
intervals of time. The lower dashed line
shows the half-amplitude criteria. See
text.
arbitrary conduit length of five units was used to calculate the voltage along a conduit.
Stimulation consisted of a 20 mV step that lasted indefinitely. Figure 6.3.4-2 plots the complete
solution as a function of distance with time as a parameter. The solution required eight terms in
the summation to obtain realistic results.
The curve representing the signal moves up
and to the right only as long as the stimulus
remains at a constant 20 mV. For a short
stimulus that terminates within 20 units of
time, like an action potential, no signal will
reach points beyond 3.0 in this figure.
Removal of the stimulus will cause the
energy along the conduit to redistribute
according to a different equation with much
of the energy within the conduit returning to
the source. The potential will eventually
equal zero at all points along the line under
this condition. This situation is
fundamentally different from the situation
related to propagation that will be discussed
below. Hermann did not differentiate
between this conduction mode of
transmission and the propagation mode in
his early caricatures.
As an analog describing the transmission of
energy, conduction acts like the “bow wave”
ahead of a modern submarine on the surface.
Propagation acts like the wave associated
with the sides of a boat. The “bow wave”
dissipates quickly if the submarine stops or
slows. However, the propagation wave, once
launched, continues to move outward at
nearly constant amplitude and velocity
regardless of the motion of the boat.
Juusola, et. al54. have recently explored some aspects of graded-potential conduction along
electrotonic signal paths, including electrotonic synapses. Table 1 compares the stage 3 synapse
with the synapses found in analog synapses under the chemical theory of the neuron. Their figure
4B shows a rare parameter, the low frequency performance of a stage 3 synapse. Their figure 4C
presents data on the various quiescent potentials (resting potentials) found by different
investigators for both pre and post synaptic plasmas. Unfortunately, the different data points on
the pre- and post synaptic side of the junction are not paired.
6.3.4.2 Delay associated with conduction of signals within stage 1,2 & 4
neurons
It takes more than 500 units of time for the response to reach the steady stage value all along the
conduit (the upper dashed line). It takes only one unit of time for the response to reach 50%
amplitude at a distance of 0.2 units. This 50% response level is shown by the lower dashed line.
It takes 22 units of time for the signal to reach 50% amplitude at 1.0 units of distance. After 100
units of time, the signal reaches 50% amplitude at 3.0 units of distance. Note, there is nearly no
response at 3.0 units of distance after 20 units of time. The voltage at any point along the conduit
as a function of time is easily calculated from the complete mathematical solution. This
calculation allows the “latency” of the signal to be determined based on any amplitude criteria
Electrophysiology 6- 51
55Pannese, E. (1994) Op. Cit. pg 150
56Rall, W. (1977) Core conductor theory and cable properties of neurons. Chapter 3 In Handbook of
Physiology, Section 1, Vol. I, Kandel, E. ed. pg. 60
desired.
Little specific data describing the velocity of conduction of electrotonic signals along an axon was
found in the literature. The best available data relates to the velocity of signals between the
adaptation amplifier and the pedicle of the photoreceptor cells in the retina. Unfortunately, the
delay associated with this velocity is combined with the delay due to the P/D process and the
operation of the adaptation amplifier. Section 17.5.6 suggests this velocity is about 7 mm/sec for
exothermic animals. New experiments need to be performed to obtain optimum values for this
velocity.
In addressing the delay within the conexuses of stage 1, 2 & 4 neurons, it is necessary to
differentiate between small signal and large signal conditions. Under small signal conditions, the
delay within the Activa of the conexus is negligible in biological situations (typically measured in
small fractions of a microsecond). Similarly, the lumped parameter electrical elements associated
with the Activa charge very rapidly. No latency is associated with such a circuit. However, if
large signals are involved, the lumped parameters may require considerable current to pass
through the Activa. The finite internal impedance of the output circuit of the Activa and the
lumped electrical elements result in a limit on the slew rate, the rate the voltage at the collector
can change. This limitation may be recognized as a latency if the time to reach a certain
percentage of maximum is used as a performance criteria.
The above paragraph applies to both the conexuses within neurons and those within a synapse
between neurons.
6.3.5 Transmission by propagation of phasic signals within Stage 3
neurons
Recently, Pannese55 has provided a brief historical comment concerning his interpretation of the
difference between signal transmission in myelinated versus unmyelinated axons. It is
conceptual and appears to be shared by many others. It is compatible with, but less specific than,
the more technical aspects of propagation presented here. He speaks of the polarization in
unmyelinated axon segments moving through the bulk of the axoplasm, whereas for myelinated
axon segments, the polarization is “confined to the region of the nodal axolemma, so the current
flows longitudinally node to node without depolarizing the internodal” axoplasm. “According to
this theory, therefore, the nodes of Ranvier act as current generators.” This last statement is
Pannese’s recognition of The Electrolytic Theory of the Neuron & the application of the
GWE to the myelinated axon in 1994! The GWE is described in Section 6.3.
6.3.5.1 Background–the electrical circuit of the axon as a transmission
line
As discussed in Section 10.4.3.1, many authors have discussed the axon (and the axon segments
in a long axon), such as found in a projection neuron, in the context of a transmission line.
Section 6.3.1 has presented the technical and historical reasons why the neurological literature
has overlooked the propagation mode of signal transmission. Since 1850, it has attempted to treat
neural signal transmission as strictly conductive. This treatment is incompatible with the
operation of a wide temporal bandwidth transmission line. When examined from the perspective
of an electrical transmission line, the axon is seen to be a coaxial in form. Such a structure of
necessity exhibits both capacitance and inductance. It may or may not exhibit significant
resistance, although it will always e60xhibit a resistive component in its input characteristic.
Rall has provided a short mathematical rationale between signal propagation by diffusion versus
electromagnetic waves56. However, he does not differentiate between projection neurons involving
action potentials and signal processing and other neurons. The matter of myelination is a crucial
factor in discussing these two major classes of neurons. Without recognizing the difference
52 Neurons & the Nervous System
57Kaplan, S. & Trujillo, D. (1970) Numerical studies of the partial differential equations governing nerve
impulse conduction: the effect of Lieberstein’s inductance term. Math. Biosci. vol 7, pp. 379-401
between these classes, it is unrealistic to draw conclusions as is done at the end of the above
rationale. The premise of this work is that myelinated signal projection neurons employ
electromagnetic wave propagation while the considerably shorter unmyelinated signal processing
neurons employ a parabolic partial differential equation as found in mechanism employing
diffusion. In the case of electromagnetic wave propagation, the partial differential equation is
hyperbolic. Rall also provides a simple definition of the Neuron Doctrine, based on morphology,
that dates from Cajol. Although accepting the part of the doctrine that suggests the signal path is
not continuous but contiguous with interruptions at (at least) the synapses, it discounts the idea
of the neuron as the basic building block in favor of the Activa-conduit pair. He also presents a
brief definition of the core conductor concept of an axon which is considered too simple for
purposes of this work. His table of parameters for different types of axons is useful.
Rall references Kaplan & Trujillo57 who considered the transition between these two functional
environments. They used the physical model of an axon found in Rall and the empirical
differential equations of Hodgkin & Huxley. The physical model was a a simple core of axoplasm
surrounded by an infinite volume of INM. Their computations involved numerical solution by
digital computer. They adopted Hodgkin & Huxley’s lossy axolemma in their analysis and
assumed physical transport of charge along the axon in the form of ionic flow. They did not
discuss how ionic flow was achieved in a viscous, possibly liquid crystalline material. They
discuss a “left-over” term in their equation fitting their model and use an impulse function to
simulate the leading edge of an action potential. Unfortunately, they adopt an average velocity of
12.3 m/sec as a reference value for the velocity of neural signals common to all neurons instead of
introducing the much higher phase velocity, of 4400 m/sec, when addressing projection neurons.
Scott reviews the Kaplan & Trujillo paper but finds it wanting in several respects while indicating
his analysis is “a drastic simplifications of the properties of a real active membrane.” He
introduces the terms magnetic inductance and inertial inductance, computes the axon capacitance
based on the properties of a flat sheet capacitor and proceeds to compute a negative inductance for
a simple linear circuit. His one sentence conclusion is primarily political. However, he does
suggest that the omission of any inductance by Hodgkin & Huxley in their equivalent circuit was
based on “good intuition.” He is then forced to introduce the analogy of “little green boys” (pages
45–46) to explain the variability in the membrane of Hodgkin & Huxley. Good intuition must
continually be re-confirmed by up-to-date experiments. The Scott work will not be referenced
here.
It is important to note that there is no requirement that a transmission line provide a conductive
path for electrons or ions along its length. It is only necessary that energy can be continually
shared between infinitesimal segments of a capacitor and an inductor. The process is similar to
the propagation of radio signals through free space. No ether is required. The phase velocity of
signals along such a line is determined by the product of the inductance and the capacitance per
unit length. It does not depend on the conductivity of any plasma within the conduit.
6.3.5.1.1 The physiological model for a signal projection neuron
Figure 6.3.5-1 presents the electrical circuit of the typical axon within the larger context of the
neural system. The figure is meant to be generic and apply to both the simple neuron and the
more physically extended neuron consisting of one or more Nodes of Ranvier. In this context, the
two titles above the figure are dual and the last Activa on the right may be either a Node of
Ranvier or a generic synapse. By referring to the individual zones of the axon, it is possible to
avoid confusion. For compatibility with other figures, the Activa on the left is shown being driven
by a voltage source, Ve. The conduit in the center of the figure is the transmission line of interest.
It is partially myelinated. The electrostenolytic voltage sources are shown as Vcc and Vee. The
dashed line within the conduit represents the reticulum of the axon (or axon segment). The solid
lines from the Activas to the reticulum are shown symbolically for convenience. In fact these
conductors are formed by the physical concentration of the reticulum as it approaches the Activas.
The region labeled x constitutes a lumped constant capacitance associated with the collector
circuit of the first Activa. This capacitance is large due to the extremely short distance between
the reticulum and the electrolyte surrounding the axon. Within the area labeled y, the
capacitance of the transmission line is distributed and much smaller due to the increased distance
Electrophysiology 6- 53
58Stampfli, R. (1981) Overview of studies on the physiology of conduction in myelinated nerve fibers. In
Demyelinating disease: basic and clinical electrophysiology. Waxman, S. & Ritchie, J. ed. NY: Raven Press
pg. 16
59Hermann, L. (1905) Lehrbuch der Physiologie, 13th ed. Berlin. See also Hille, pg. 27
60Kraus, J. (1953) Electromagnetics. NY: McGraw-Hill pp. 417-444
Figure 6.3.5-1 Generic neurological transmission line; the electrical circuit of the
axon in the context of the overall neural system. X; the postsynaptic transition
(matching) section. Z; the presynaptic transition (matching ) section. Y; the
distributed parameter transmission section.
between the inner and outer electrolytes. The area labeled z constitutes the lumped capacitance
of the unmyelinated part of the conduit in the vicinity of the input to the second Activa.
A coaxial transmission line is an unbalanced transmission line. Such a line does not require a
conductive path between its two terminals. However, if energy is extracted from the terminal, a
conductive path back to the source must generally be provided. In man-made circuits, this is
typically through a “ground” connection which is independent of the actual transmission line. In
many high performance transmission lines, the conductive path associated with at least one of the
conductors is intentionally broken to avoid undesirable extraneous loop currents.
Little is known about the detailed electrical properties of the individual membranes and
electrolytes forming the conduit. This is partly due to the oversimplified model of the conduit
used previously. Stampfli says the specific resistance of the INM is similar to that of Ringer’s
solution (90 ohm-cm at 20 Celsius)58. However, because of the high quality of the insulator
represented by the axolemma and the myelination, little error is introduced by making the above
assumption. Under this assumption, the capacitance and inductance of the transmission line and
the two lumped capacitors can be calculated and compared to the measured values in the
literature. In general, only lumped capacitance values are found in the literature.
6.3.5.1.2 The electrical transmission line model of a signal projection
neuron EDIT
There has been a general misconception in the biological literature, dating from Hermann in the
late 1800's59, that an electrical transmission line of adequate bandwidth can be composed of only
resistors and capacitors. This position is not founded on electrical engineering principles60 nor
supported by subsequent physiological experiments. The axon of a neuron can not be
54 Neurons & the Nervous System
61Cole, K. (1968) Membranes, Ions and Impulses: A Chapter of Classical Biophysics. Los Angeles, CA:
University of California Press, pp 77-103
62Schwan, H. (1963) Determination of biological impedances In Physical Techniques in Biological Research
NY: Academic Press pp 323-406
properly modeled as a Hermann Cable. Cole61 discussed this fact extensively in 1968. He
showed that the typical axon exhibited a large inductive component through measurement. While
he did not provide a satisfactory explanation for this inductance, it was clearly present, and is
well documented. As Cole describes it, “The suggestion of an inductive reactance anywhere in the
(neural) system was shocking to the point of being unbelievable.” To this day, the fact that the
axon of a projection neuron exhibits considerable inductance has not been accepted in academia
and pedagogy. However, the tide was turning at the research level even then. Schwan62 also
began discussing the inductance of axon in 1963. As seen in this work, it is the combination of
inductance and capacitance that determines the very high signal propagation velocity of the
individual axon segment when operating in the signal propagation mode (where it is usually
myelinated). This velocity is much higher than the velocity achieved by diffusion in axons
associated with stage 1, 2 and 4 neural operations.
Hermann adopted a simple RC network to represent what he understood to be “a leaky telegraph
cable.” It must also have been a short leaky telegraph cable. An RC configuration suffers from
significant temporal dispersion in the frequency components of the signal with distance. A useful
transmission line always consists of capacitors, inductors and resistors. Technically, the
resistor is optional. A transmission line is an electrical circuit designed to emulate free space but
to direct the signal energy to a more limited target location. Free space has a resistive
characteristic impedance of 376.7 Ohms although it is completely empty and exhibits no resistive
element. An electromagnetic wave travels through free space at the speed of light. Except in a
few novel situations, a passive transmission line always exhibits a characteristic impedance less
than that of free space. The velocity of energy propagation along a transmission line is always
lower than the speed of light.
[ Figure 6.3.4-4] illustrates the equivalent circuits representing a coaxial axon. (A) shows the
typical lumped-parameter representation. In the ideal case, and within the operating range of the
axon, the values of the resistive elements are such that the currents through them can be
ignored. The dominant currents are reactive currents associated with the inductive and
capacitive elements. The resulting lumped constant network is show by the heavier lines. To
more accurately represent the physical length of a real axon, it is more appropriate to describe it
by using the heavier network on the left replicated on an incremental basis. The resulting
distributed model is shown in (C) This simple network describes the electrical performance of the
real axon over the distance between its terminal components.
The network in (C) has a characteristic impedance that is seldom well matched to the driving
source. This leads to electrical in-efficiency. To avoid this problem, good design practice calls for
the introduction of a “matching filter” section between the transmission line and the source and
between the transmission line and the following sink. These matching sections are lumped
parameter networks similar to that in (D). These filters are similar to one half of the filter shown
in (A) but with slightly different parameter values.
Several significant features should be noted for such an ideal circuit.
1. No resistive elements appear in the distributed equivalent circuit of the axon. The circuit is
not dissipative of electrical power.
2. An actual axon consists of two portions, the extended middle section described by the
distributed equivalent circuit, and the “matching” end segments which are best modeled using a
circuit similar to the lumped constant model.
3. The signal propagation velocity along the axon is determined entirely by the ratio of the
inductive to capacitive components per unit length.
4. The loss in signal amplitude along the axon is very low in the ideal case.
Electrophysiology 6- 55
In a real myelinated stage 3 signal propagation axon. The shunt conductance is negligible but the
series resistance (representing the conductivity of the plasmas on each side of the myelinated
axolemma, is not. It is the primary parameter limiting the propagation velocity of the axon. It
also dissipates some power. This resistance introduces both attenuation and phase distortion into
the transmission line. As will be discussed in the data section below, it appears phase distortion
is a greater problem in real axons than is attenuation.
These features of a real axon differ considerably from the Hermann Cable model.
A transmission line exhibits a series of important parameters. Four of these parameters are
basic;
+ the series inductance per unit length of the line, L/m
+ the series resistance per unit length of the line, R/m
+ the shunt capacitance per unit length of the line, C/m
+ the shunt conductance per unit length of the line, G/m
The series impedance of the line is given by Z/m = L/m + R/m. The shunt admittance is given by
G/m + C/m.
Three additional parameters are calculated from the above;
+ a characteristic impedance--it is important in how a separate circuit drives and how a
receiving circuit terminates the transmission line.
+ the velocity of signal propagation along the transmission line
+ the propagation constant of the transmission line. It is a measure of the quality of a
transmission line.
To understand, derive, and evaluate expressions for these parameters, it is necessary that the
investigator be familiar and comfortable with complex algebra. The characteristic impedance, Z0,
is defined as the square root of the series impedance divided by the shunt admittance determined
from the four basic parameters. The propagation constant, , is defined as the square root of the
product of the series impedance and the shunt admittance. These two quantities are usually
defined using complex algebra, where they are the real and imaginary parts. The real part of the
propagation constant describes the attenuation constant, . The attenuation factor for a line is
given by the expression e-x, the decrease in signal amplitude with distance along the line. The
imaginary part describes the phase constant, . The propagation velocity is given by the
frequency of the signal divided by the phase constant. In transmitting a pulse waveform over a
transmission line, it is important that all Fourier components of the pulse travel at the same
velocity. Otherwise, the line is considered dispersive and the pulse is distorted as it travels along
the line. If is a not a linear function of frequency, the velocity of propagation of the
transmission line will be a function of frequency and the pulse will become distorted as it travels
along the line.
There are only a few special cases among these equations. A lossless line does exhibit a
characteristic impedance given by a pure resistance, as does the lossy line where G/C = R/L. In
general, a lossy line does not satisfy this ratio. Any other lossy transmission line exhibits a
complex characteristic impedance and all lossy transmission lines exhibit a phase constant that is
a function of frequency--resulting in waveform distortion. The phase constant associated with a
transmission line lacking in inductance, i. e. made up of only resistors and capacitors, is highly
dispersive and such a line is not appropriate for pulse transmission. Fortunately, all coaxial
transmission lines, such as used in neurons, exhibit considerable inductance.
6.3.5.1.3 The cylindrical transmission line
The axon segment of a stage 3 signal projection neuron is a cylindrical transmission line. Its
performance is significantly increased if it is myelinated to reduce the effective capacitance of the
axon segment per unit length. In this case, the input impedance of the axon segment is reduced
56 Neurons & the Nervous System
63Schwan, H. (1963) Determination of biological impedances In Physical Techniques in Biological Research
NY: Academic Press pp 323-406
considerably and the propagation velocity is increase significantly (to the values measured in the
laboratory by Smith et al.).
The biological community has generally calculated the capacitance of an axon by calculating the
apparent area of the axon and considering it equivalent to a flat plate capacitor. For precise work,
it is necessary to recognize and use the correct formula for the capacitance of a concentric
cylindrical structure. This structure exhibits both a capacitive and an inductive electrical
component. The formulas are given in the previous reference to Kraus. The result is that any
axon transmitting a waveform of complicated shape with respect to time consists of all of the
circuit elements discussed above. It is important that the waveform does not encounter phase
dispersion due to the axon.
As noted by Schwan63,. the in-vivo capacitance of a myelinated axon, due to the thick myelination
(higher dielectric material) that normally surrounds it, may exhit a very low capacitance per unit
length. This variation may aid in rationalizing some of the capacitance values found in the
literature. The theoretical value is shown in Section 6.3.3.3
6.3.5.2 The interface between a distributed line and an Activa
Filter theory, within the field of Electrical engineering, has frequently found it useful to introduce
a transition section between a coaxial transmission line and the amplifier circuit driving that line.
By using this “half-section,”a better impedance match is obtained between the two circuits. The
same situation occurs in the neural system and it appears a similar optimization procedure has
been used. The lumped parameter values associated with the conexus of a projection neuron
appear to act as a half-section filter connecting the Activa output to the distributed parameter
values of the transmission line. A similar half-section filter approach appears to be used at the
dendritic interface with the Activa in the orthodromic conexus. The result of using matching
sections is higher electrical efficiency, leading to longer distances between axon segments.
The subject of matching the impedance of an axon segment to a synapse or NoR is not
addressed in the chemical theory of the neuron.
6.3.5.2.1 Importance of impedance matching at junctions
Figure 6.3.5-2 illustrates a boundary between two media, such as at the input junction between
an axon segment and a NoR. Any reflection at this boundary reduces the amplitude of the signal
at the Activaforming the NoR. A similar mismatch at the output junction between the NoR and
the following axon segment. boundary.
Within the first medium, the vectorial sum of the incident and reflected wave forms a standing
wave pattern described by the voltage standing wave ratio, VWSR; the ratio between the
maximum and minimum amplitude of this standing wave.
Any VWSR greater than 1.00 is undesirable. To overcome a high VWSR, steps should be taken to
mediate this condition. The most common method of adjusting the impedance in man-made
transmission lineis to introduce a “balun,” effectively a gradual impedance change over a
relatively short distance.
Electrophysiology 6- 57
Figure 6.3.5-2 Importance of impedance
matching at a boundary between medium
1, a transmission line, and medium 2, a
medium of different impedance. Any
reflected wave reduces the amplitude of
the transmitted wave. From Kraus, 1953.
6.3.5.2.2 Half-section filter design
Matching filter design is beyond the scope of
this work. However, the concept is straight
forward. It is usually taught in graduate
school to those specializing in analog filter
design. When constructing a typical “T” or
“” filter, the designer has the flexibility to
match the external input and output driving
impedances with a variety of internal circuit
values. If these internal values are selected
to match some other impedance level, the
filter can be split into two symmetrical half-
sections and turned back-to-back. A
distributed transmission line can then be
inserted between the two original driving
points. If it has the same driving impedance
as the filter as designed, a very efficient
configuration results. The two new external
terminals were designed specifically to
match the lumped parameter networks
associated with the conexuses. Thus, the
overall network now exhibits a very good
impedance match from end-to-end. Transfer
efficiencies greater than 95% can be
expected for this overall circuit, subject to
any loss component in the distributed line.
This is the technique implemented in the
transition sections associated with the
typical Node of Ranvier.
6.3.5.2.3 The impedance of the terminal portions of an axon segment
The axons of stage 3 signal propagation neurons generally consist of a multitude of individual
axon segments (including the initial axon segment). For a given axon, these segments tend to be
nearly identical in size and performance. Each segment has two terminal portions as illustrated
above. They can be defined electrically as electrical filter “matching half-sections.” The
parameters for these half-sections are difficult to calculate because of the complex shapes involved
(Section 10.5.2). However, their role is important. They typically consist of a high lumped
capacitance (because of the lack of myelination), negligible inductance, and one or more diode
impedances associated with the electrostenolytic power supply and the Activa within the conexus.
This latter impedance is either the collector source impedance or the emitter input impedance.
The capacitance of the two lumped capacitors is usually calculated in the literature assuming a
flat plane analogy. The calculation based on this analogy gives an acceptable but less than precise
value. If the terminal structure is reasonably circular, a better approximation is obtained from
Eq 6.3.5.-1 of the next section. However, the form of the terminal section of the axon segment is
frequently not circular. A precise calculation of the capacitance requires the use of spatial
transform algebra or the graphical techniques of electrostatics. For the termination, the
capacitance per unit length of the unmyelinated portion of the axon is to be multiplied by the
length of the capacitor, either x or z. is the permittivity of the material between the axoplasm
and the electrolyte surrounding the conduit. “b” represent the radius of the outer surface of the
lemma and “a” represents the radius of the inner surface. Since the ratio is very nearly 1.0 in the
above formula, the natural logarithm of that ratio is very nearly zero. The resulting capacitance
is quite high. Because of the high capacitance of the termination, it is unnecessary to calculate
the inductance at these locations.
The resistive impedance associated with this half-section will be inferred below from test data.
6.3.5.3 The electrolytics/electrostatics of the axon segment
58 Neurons & the Nervous System
In the middle of the 19th Century, the state of knowledge concerning coaxial cables did not exist
and the fact that a myelinated axcon was not understood to be a coaxial cable. In the case of the
stage 3 axon or axon segment, the myelinated axolemma is the insulator, and the axoplasm and
the INM constitute the two conductors of a coaxial cable.
6.3.5.3.1 The geometry and electrostatics of the axon segment
Figure 6.3.5-3 describes the detailed geometry of a typical axon segment. It will be used to
discuss both the electrostatics and the electrodynamics associated with the axon segment. Frame
A has been reproduced earlier and provides a reference. See Section 6.3.3.1.2. Frame B is a first
expansion to illustrate and stress that a double layer is formed around the interior perimeter of
the lemma of the axon whenever its voltage differs from that of the surrounding medium. This
double layer is a feature defined by electrostatics and is a conventional feature of
electrochemistry. It is discussed in more detail in Wikipedia. The thickness of this layer depends
on the molar concentration of the axoplasm. As noted in Section 10.1.5, the thickness of a
axolemma would be less than 0.002 microns containing an axoplasm having the propertiesof sea-
water except the concentration of NaCl is less than that of sea water. This is the generally
accepted case. The significance of this lemma will be discussed below. As in earlier figures
(Section 2.1.4), the lemma of an axon is formed entirely of type 1 (symmetrical) BLM except in
the areas associated with the solid black boxes (where it is type 2) and a potential small type 3
area used to transport ions through the lemma for purposes of metabolism. The type 3 area is
generally found between the volume of the soma and the axoplasm. The type 1 and 2 portions of
the lemma are entirely impervious to the flow of ions (even more so if it is sheathed by myelin as
is the routine case of stage 3 projection neurons). This discussion will converge around
discussions of the stage 3 projection neuron.
Electrophysiology 6- 59
Figure 6.3.5-3 The geometry and electrostatics of the initial axon segment of a
neuron. A; a completer neuron for reference. B; an expanded view of the axon
segment stressing the continuity of the electrostatic “double layer” around the
entire inner perimeter of the axolemma. This region will be discussed in terms of
the “electrolytic skin effect.” C; a more expanded view of part of the axon
segment showing various field lines and electron flow paths. Note the
dimensions on the right. Small box at lower right; an arbitrary path of integration
enclosing part of the axoplasm and part of the myelin surrounding the axon
segment. The shaded areas are electrically neutral during the entire lifetime of the
neuron. See text.
Frame C is a further expansion of only a portion of an axon segment. It shows the location and
operation of the Activa (the black vertical box on the left) and the electrostenolytic process
providing power to the axoplasm (horizontal black box at the top). This frame will be discussed
separately for both the static and dynamic condition of the axon segment. The first feature to
note is that there is no requirement to transport ions across the lemma of the axon (or any other
cell lemma for purposes of signaling). Electrons can flow out of the interior of the lemma through
the Activa and they can flow into the interior of the lemma through the electrostenolytic process.
The actual flow of electrons out of the interior through the Activa corresponds to the conceptual
“inward sodium current” proposed by Hodgkin and Huxley. The actual inward flow of electrons
through the electrostenolytic process corresponds to the conceptual “outward potassium current”
proposed by them. See Section 6.3.3.2.2 and the figure in Section 2.7.5.
This expanded view provides more detail concerning how the excess electrons within the lemma
are constrained to the double layer area of classical electrochemistry. This area is immediately
adjacent to the lemma. It can be looked at, withn the context of electrolytic chemistry as
60 Neurons & the Nervous System
supporting an “electrolytic skin effect.” This skin effect causes the axon segment to exhibit
several special features. First, no net current related to signaling or any other function can flow
through the shaded area. This is due to the high mobility of the ions dissolved in the axoplasm
solution. If there is no net flow through this area, due to the compensating ability of the free ions
in solution, It is axiomatic that there are no lines of magnetic flux in this area. Therefore, any self
inductance of the axoplasm can only be due to the flow of electrons within the electrolytic skin
effect region. As will be discussed further below, the flow of current along the inside of the lemma
is matched precisely by the flow of current in the opposite direction along the exterior of the
lemma. As a result, there is no net flow along the axon at any point. This causes the net self
inductance associated with the lemma of an axon to equal zero. A similar finding can be made
concerning the net self inductance of the Inter neural matrix, INM, in the immediate area of the
lemma (or if myelin is present, the immediate area of the myelin surface in contact with the INM).
The above conclusions can be supported using the small box shown at lower right. The perimeter
of this box represents an integration path. The fundamental law of electrostatics (and
electrodynamics) like so many other laws is named after Gauss. This law says the net charge
within such an integration path must be zero. The reader is free to draw a similar box anywhere
in the figure. The net charge within the box must always be zero. The application of Gauss’s
Law to the transmission of energy through space was what caused Maxwell to include the
displacement current term in his General Wave Equation. It is the displacement current that
eventually formed the theoretical foundation explaining the propagation of electromagnetic
energy through space.
It is important to note that the interior of a lemma can support an excess of electrons while the
number of positive and negative ions within the lemma remains in electrical balance. The lemma
forms the dielectric of a capacitor. A charge can be introduced onto one surface of the lemma and
an equal and opposite “static” charge will be induced on the opposite surface. Although present at
any dielectric surface, this effect will be explored further when discussing the myelin sheath in
Section 6.3.5.
6.3.5.3.2 The electrodynamics of the axon segment
The axon segment has an awkward role to play its input end needs to pretnd it is biased at a DC
voltage similar to a dendrite, but its output terminal must be biased at a DC voltage like an axon.
Fortunately, the signal is transferred from input to output by propagation according to the GWE
mechanisms that are not effected by the DC voltages of the axon segment
6.3.5.3.3 The impact of the electrolytic skin effect on the axon segment
The fact that all of the current flowing within an axoplasm is constrained by the laws of
electrolytic chemistry to the double layer adjacent to the lemma has a profound effect on the
operation of the axon segment. First, the effective series resistance of the axoplasm is computed
using only the resistivity of the material within the double layer and the cross section of that
double layer. This makes the series resistance much higher than calculated previously and
accounts for the slow propagation velocity of the action potential within the axon, compared to the
speed of light. This velocity is the phase velocity of the action potential which is about 100 times
faster than the average velocity measured between two or more Nodes of Ranvier (Section
6.3.5.4.2). The net electron current flowing within the double layer within the axolemma is
matched precisely by a similar current flowing along the opposite side of the lemma (if the lemma
is assumed to have negligible thickness and be unmyelinated).
If the axon is myelinated, the situation is a bit more complicated as indicated in the lower right.
Myelin is an excellent insulator and supports no measurable flow of current. However, any
dielectric will support a displacement current equal to that required to satisfy Gauss’s Law. The
result is the formation of two displacement currents within the dielectric matching the currents
flowing outside of the dielectric as discussed above. These displacement currents are shown by
dashed arrows in the figure. Note the presence of the myelin moves the current flowing in the
INM further away from its counterpart flowing in the axoplasm (solid arrows). This is reflected in
the calculated value of the capacitance per unit length of the axon when myelinated.
Electrophysiology 6- 61
64Bockris, J. & Reddy, A. (1970 & continuing) Modern Electrochemistry, vol. 1. NY: Plenum Press
65Yeager, E. Bockris, J. Conway, B. & Sarangapani, S. (Continuing) Comprehensive Treatise of
Electrochemistry NY: Plenum Press
66Bauer, H. ( 1972) Electrodics. Stuttgart, Germany: Thieme
6.3.5.3.4 The electrodynamics of the axon associated with propagation
Note the paired arrows moving in opposite directions a the lower left of frame C. These represent
local charges moving in opposite direction in close proximity to each other. From the perspective
of gauss’s Law, they neutralize each other. However from the perspective of Maxwell’s Laws, they
constitute a moving couple. Such a couple generates a local magnetic line of flux. This line of flux
influences the motion of other charges and couples nearby. This is the essence of propagation
suggested in Figure 6.3.3-4 in Section 6.3.3.3. The local movement of charges due to electric
fields generates a magnetic flux. This magnetic flux causes adjacent charges to move, thereby
generating a change in the electrostatic field. The mechanism is perpetuated in the form of
propagation. This propagation does not require the presence of a conducting medium and
operates perfectly well in the vacuum of space. The velocity of propagation is given by the
equations for free-space or a lossy medium as required by the environment.
6.3.5.4 The impedances of the distributed part of the axon
A myelinated axon forms a very highly optimized transmission line. To describe such a
transmission line, it is necessary to determine all four of the relevant electrical parameters of
such a line. The shunt conductivity of the dielectric formed by both the BLM and the myelin
sheath is taken as zero for practical purposes. The shunt capacitance of the dielectric and the so-
called external inductance related to the internal plasma and the surrounding plasma, INM, are
also easily calculated. The most difficult to calculate is the effective series resistances related to
these plasmas. This is because of the limited resistivity of the plasmas and the displacement
currents associated with the plasmas under dynamic conditions. Because of the unique
configuration of the axon compared to other interesting electrolytic circuits (primarily involving a
metal-electrolyte interface), it is difficult to find relevant calculations and data in the literature.
As a result, the best values of the effective series resistance of an axon will be derived from the
propagation velocity measurements reported in Section 6.3.5.4.2.
In this section only type 1 and type 2 membranes will be considered. It is a premise of this work
that type 3 membrane is not found along the distributed portion of any stage 3 axon. Thus, no
pores, gates or other passages through the dielectric formed by the lemma is proposed, needed or
allowed in developing the propagation mode capability of an axon. Any currents within either the
axoplasm or the surrounding INM may approach the surface of the dielectric formed by the
lemma and/or the myelin sheathed lemma or travel parallel to that surface. In approaching the
surface, they are not allowed to penetrate it.
The following material related to, and largely drawn from, electrochemistry relies heavily on the
massive materials at least coauthored by J. O’M Bockris. These include the series’, Modern
Electrochemistry64 and Comprehensive Treatise of Electrochemistry65. Unfortunately, the field of
electrolytic chemistry applied to a biological organism (electrochemistry without the presence of a
metallic electrode, and without any redox processes or mechanisms at such electrodes) is less well
understood and documented. The only redox mechanisms known to be involved in the operation
of the neural system are related to the generation of electrical biases through electrostenolysis of
organic molecules.
Precise definitions play a key role in understanding all areas of electrochemistry. The most
concise set of definitions for the terms of electrochemistry appears in Bauer66. See Section 6.3.3.3
for the equations expressed in terms of the thickness of the myelination.
The data in the literature is marginally adequate for evaluating these expressions. The value of
the phase velocity can be determined from the graph of Figure 6.3.3-6(A). It is approximately
4,400meters per second. This velocity is much higher than the average velocity of the signal
reported in the literature for a long projection neuron. The average velocity is a calculated value
using a distance interval divided by a time interval. The time interval includes an additional time
62 Neurons & the Nervous System
67Brismar, T. (1983) Nodal function of pathological nerve fibers, Experientia, vol. 39, pp. 946-952
delay related to the regeneration time delay associated with each Node of Ranvier.
6.3.5.4.1 The axolemma-myelin junction
The difficulty of introducing a foreign agent into the space between the axolemma and the plasma
membrane of the myelin along most of the length of the axon is significant. From a functional and
electrical perspective of the axon, this is extremely important. As will soon appear obvious, a de-
myelinated neuron is a functionally different neuron. A primary purpose of the myelination is to
limit the capacitance between the axoplasm and the low resistivity electrolyte of the surrounding
interneural matrix. Although the dielectric constant of the myelination, typically a Schwann cell,
is approximately five, the total capacitance per unit length of sections of the axolemma "protected"
by the myelination is significantly reduced because of the thickness of the protection.
There is considerable data in the literature on the capacitance of the axon. However, the values
quoted for the same situation vary over more than a magnitude. Brismar67 speaks of the
capacitance of a 200-micron section of myelinated internode [sic] being in parallel with the node
but contributing <0.3 pF to the measured capacitance of >1-2 pF.
Whereas the dielectric constant of the myelin may be five times higher than the axolemma, the
thickness of the region consisting of both the axolemma and the myelin is considerably larger
than just the axolemma. The result is a reduction of about an order of magnitude in the
capacitance per unit length per 100 layers of myelin regadless of axon diameter (Section 6.3.3.3).
On the other hand, where the surrounding fluid is in direct contact with the axolemma, the
capacitance per unit length can be greatly increased. If electrically conductive fluid could
penetrate the space between the axolemma and the Schwann cell, the capacitance between the
axoplasm and the surrounding fluid would be as high as for a non-myelinated neuron of the same
diameter axon. As shown elsewhere, the higher capacitance of the paranodes is absolutely critical
to the electrical regeneration of the action potentials at the Nodes of Ranvier (and elsewhere in
the neuron).
From the perspective of hydraulics, the rings seen in electron micrographs of the axolemma of
axon segments as they approach the Node of Ranvier have a logical foundation. It is proposed
that these rings are formed by the tight banding of the myelin layers as each terminates near the
node. The resulting labyrinth seal is a very effective device for preventing material under
pressure from passing the seal. This prevents fluid from reaching the exterior surface of the
cylindrical structure contained within the seal. The dielectric integrity and conductive isolation of
the periaxonal space are protected in this way.
As is very clear in the clinical literature, anything that causes the loss or
separation of the myelin sheath from a projection neuron causes a
serious pathological and neurological condition. The cause is a major
reduction in the impedance of the axon-INM interface. This reduction in
impedance can cause the Activa in the nearby Node of Ranvier to fail to
regenerate the action potential delivered to it by the previous section of
healthy axon. The onset of such failure can be correlated with the
increased refractory period displayed by the affected Node.
6.3.5.4.2 Capacitance of the distributed part of an axon
There is general recognition in the literature that in projection neurons, 90% of the capacitance
between the axoplasm and the interneural matrix is concentrated at the synaptic end of the axon
(or at the paranodal terminals of an internodal segment) of a projection neuron. However,
accurate measurements are quite difficult using a point type, as opposed to either a ring or
Faraday cage type, probe. This is partly due to the variable resistivity of the plasmas and matrix
materials surrounding the membranes of plasma membrane, consisting of the axolemma, myelin
sheath of the associated Schwann cell (oligodendrocyte within the CNS)..
Electrophysiology 6- 63
68Taylor, R. (1963) Cable Theory in Nastuk, W. Ed. Physical Techniques in Biological Research. NY:
Academic Press pg 224
Electrostatics allows the calculation of the capacitance of virtually any configuration of dielectric
and conducting material. For many simple configurations, such as a coaxial cylindrical capacitor,
simple formulas are available.
Electrolytic materials are known for their very high capacitance. In an earlier time, electrolytic
capacitors were a mainstay of electrical devices. They were also known to be asymmetrical with
applied voltage. The thin structures employed actually consisted of a very large area
semiconductor diode that was reverse biased. The bias voltage caused a very high capacitance to
be exhibited at the terminals of the device.
Taylor notes the variability of the capacitance of a BLM as a function of voltage and of the
frequency used to make the test68. He provides several references. Unfortunately, he did not
recognize the inductance of the BLM.
6.3.5.4.3 The capacitance of an unmyelinated axon
Since most impedance values in the literature for the axon are lumped values, care must be taken
with them. It must be understood how they were obtained experimentally.
Unmyelinated neuron--A sample calculation for the capacitance of the unmyelinated portion of
the conduit gives a value of 4.4 x 10-10 Farads/meter. This is obtained for an axolemma thickness
of 80 Angstrom (8 nm.), an inner axolemma radius of 4 microns, and a relative permittivity of the
axolemma equal to 5.0. The range of relative dielectric constants available 2-5 unless based on
space charge This value corresponds to 44 picofarads per meter or 44 micro-picofarads per micron
of axon length (a very small value which probably can not be measured directly). However, the
lump sum value for a section of unmyelinated axon that is two mm long would be about 0.088
picofarads, a reasonable value. For a value this large, the circuit would probably be oscillatory.
The Activa would almost certainly oscillate if the additional capacitance of a probe were
added to its collector circuit.
Myelinated--If a better capacitance per unit length for the myelinated portion of the axon were
available and the permittivity were known more accurately, the inductance of the axon could be
calculated based on the data for the phase velocity noted above. This would also allow the
characteristic impedance (Section 6.3.3.3) and driving point impedance of the line to be
calculated .
6.3.5.4.4 The inductance of the distributed part of an axon
The total inductance of a coaxial structure such as an axon consists of three components, the so-
called external inductance related to the magnetic flux found outside of the two conducting
elements and the internal inductances related to the magnetic flux within the axoplasm and the
magnetic flux within the INM.
The external inductance is the major component of the total inductance in a practical axon (or any
transmission line). This inductance is given by a simple equation when the dielectric of the
lemma and myelin are well behaved. This is the case for type 1 membrane and is approximately
the case for type 2 membrane. The case for type 3 membrane is highly dependent on the
particular membrane. The premise is that the type 3 membrane is of little concern along the
distributed portion of the axon.
In reviewing the literature, two important points quickly appear. First, the use of stored energy
calculations in the electrical domain are very similar to the kinetic equations used in chemistry.
They both provide simple answers without clearly defining a variety of important parameters.
The results of such calculations cannot be accepted without very careful analysis of the underlying
conditions and assumptions. This is typified by the simple equation for the low frequency
inductance of a solid round wire given by L = /8 in Henries/meter. While correct, it depends on
the current density of a “well behaved conductor” being uniform across its cross section. A well
behaved conductor is typically a metal where all of the atomic nuclei within it are frozen in
position. This is not the case in an electrolyte where the ions are free to move about.
Furthermore, the current density within an axoplasm related to propagation has not been shown
64 Neurons & the Nervous System
69Bockris, J. & Reddy, A. (1970) Modern Electrochemistry, vol. 1. NY: Plenum Press pp 399-440
to be uniform.
In general, the current density within the axoplasm will be zero after sufficient time has elapsed
for the charge distribution within the electrolyte to redistribute in accordance with the laws of
static electricity. Similarly, the charge distribution within the INM will also redistribute in
accordance with the laws of static electricity. These redistributions will tend to leave the interior
of the axoplasm and the bulk of the INM free of conductivity currents and paired magnetic fields.
As a result, the internal inductances related to both the axoplasm and the surrounding INM will
approach zero based on the time constant of ionic motion within these electrolytes (to be
determined in the next section). These time constants can be described in terms of a relaxation
effect for electrolytes.
6.3.5.4.5 The effective resistance of the distributed part of an axon
The effective series resistances associated with both the axoplasm of a neuron and the
surrounding INM control the performance of the axon in propagation. These resistances are
strongly affected by a variety of fundamental parameters related to the electrolytes that are not
easily determined from the literature. Electrolytic chemistry and electrochemistry involve many
nonlinear mechanisms that make them difficult disciplines. Electrochemistry is defined as the
interface between one or more metallic electrodes and an electrolyte. A majority of the literature
relates to aqueous solutions of low concentrations. Adapting this literature to the liquid
crystalline or gel-like state does not appear to be straight forward. Electrolytic chemistry is
defined here to involve the electrochemistry of electrolytes in the absence of metallic conductors.
Whereas most electrochemistry involves continuous DC currents flowing through the electrolyte,
the case of interest in the electrolytic chemistry of the neural system is to determine the
movement of charge within an electrolyte in response to an electrostatic charge on an adjacent
dielectric. No direct current flow is involved. This difference makes extrapolation from the
literature difficult.
A potentially important condition has surfaced when discussing the mobility of carriers in an
electrolyte. It addresses the longstanding question of why the ratio of sodium to potassium ions is
different within many plasmas than within the surrounding medium. The subject of
interdependence of ionic drifts and potential decay time constants, is discussed in Bockris and
Reddy, vol. 169. The material cannot be explored here in detail.
Bockris & Reddy note the conventional wisdom (page 400) that if there are equal numbers of
various ions in a solution, those which have higher mobility contribute proportionally more to the
total current through the solution. However, one finding is that in an aqueous solution of both
sodium chloride and potassium chloride, the mobility of each ion is controlled by the concentration
of all other species. By an appropriate ratio of sodium to potassium to hydrogen, the current
carrying capacity of the highest mobility ion (hydrogen is about five times more mobile than either
sodium or potassium) is greatly reduced (as much as 200:1). In the simple case of a solution of
HCl and KCl, they make the following statement. “In fact, the transport number of the H+ ions
under such circumstances is virtually zero, . . .”(page 402). In the case of a mixture of NaCl and
KCl, it appears the control of the ratio of sodium to potassium can control the overall resistivity
and decay time constant of the overall electrolyte. By varying this ratio in different cells of the
organism, different electrical performance can be obtained.
The mechanics involved in determining the time constant associated with these ions and an
external field are also discussed in Bockris and Reddy (367-399).
The finding that the performance of a neural system is dependent on the ratio of
sodium to potassium suggests additional laboratory investigations are needed
under very controlled conditions (including temperature control), to document this
ratio in axoplasm of the different neurons found in different species.
Electrophysiology 6- 65
70Fried, I. Rutishauser, U. Cerf, M. & Kreiman, G. (2014) Single Neuron Studies of Human Brain. Cambridge,
MA: MIT Press
6.3.5.4 The conduction of signals in a demyelinated, or stripped, axon
When a normally myelinated or otherwise electrically isolated neuron is stripped of its electrical
isolation, its capacitance per unit length increases significantly. This causes the characteristic
impedance of the axon to also change significantly. The result is an abrupt change in the
impedance level as a function of position along the axon. This results in a significant reflection at
the location of the change and a much lower transmission efficiency. It is also likely that a
significant change in the phase constant of the circuit will be introduced.
The material reviewed in Section 6.3.3, particularly that of Kaplan & Trujillo, can be
readdressed using different parametric values to determine the signal velocity along an
unmyelinated axon. The parameters of such an axon can cause it to exhibit signal transmission
by conduction rather than propagation. Alternately, the changes can result in propagation but
under conditions of significant phase (and therefore pulse) distortion. Lafontaine et al. have also
reported in this area. See Section 6.3.8.5.1.
The clinical community has made considerable progress in understanding and treating
demyelination.
6.3.6 Importance of compensated probes to avoid artifacts
The literature is littered with recordings of waveforms in neuroscience that show artifacts, such
as overshoots and undershoots due to poor understanding of the importance of compensating the
probes leading to the input impedance of their oscilloscopes and other instrumentation. This lack
of compensation is the primary source of positive-going and negative-going over shoots that occur
without the presence of any potential source capable of supporting these phenomenon.
This section will illustrate how these overshoots can be encountered due strictly capacitive
differences that are not matched to their associated resistive differences.
Fried et al70. wrote a book on “single neuron studies.” After the first four chapters, their focus was
on multi-probe techniques. The text is written for entry level students of clinical medicine. It
does not address multi-neuron configurations explicitly, although chapter six is devoted to multi-
neuron recording (figure 6.7). The text does not address analog waveforms explicitly, and
generally considers the analog signal background as noise to be eliminated (page 68-81). They
note,
“Several different types of signals are often referred to as ‘LFPs’.” While investigators have
used the term LFP to discuss scalp electroencephalographic recordings or intracranial
recordings . . , those signals are less ‘local’ than the ones to be considered here.”
The authors are clearly not aware that greater than 95% of all neurons operate in the LFP domain
that they do not clearly define (Section 2.5 of this work); their focus is on the less than 5% of the
neurons related to action potentials, which they describe routinely as spikes. They suggest
limiting the lower passband of test set configurations to above 100 Hertz to eliminate the
background analog signals when recording spikes (page 69). This recommendation also distorts
the shape of the recorded spikes and makes the delays, associated with neural operation, difficult
to measure precisely (Section 9.2.6).
6.3.6.1 Background regarding Probes EMPTY
6.3.6.1.1 Single wire Probes, extraneural & Intraneural EMPTY
6.3.6.1.2 Multi wire Probes
66 Neurons & the Nervous System
Figure 6.3.6-1 Assortment of multi wire probes. Probe A is a multi wire probe
where the wires are of equal length, as is probe B. Probe C and possibly other
was developed to be inserted in the cochlea of the ear to actively stimulate the
hair cells. Each wire terminates at one of the dark areas. The size and insulation
of each wire is described in the caption of the original figure. From Fried et al.,
2014.
Figure 6.3.6-1, from Fried et al. shows an assortment of probe configurations. Probe C was
designed to be inserted into the cochlea of a human ear for long periods to allow the stimulation of
groups of hair cells along its length. The end of the probe was to stimulate the lower audio
frequencies. The others were designed for more general application. The functionality of the
large diameter of probe D was not indicated.
Figure 6.3.6-2 exhibits extraneuronal recording with one of these probes. The amount of cross
talk between channels appears negligible. The caption is reproduced verbatim with annotation
added. The lower limit of the bandpass filter in each channel was not specified, but probably at
100 kHz to supress the summation of the analog signals. Note the small amplitude of the
extraneural signals roughly 1/1000 of the intraneural axon signals. The action potentials are
riding on a summated analog signal of similar amplitude.
Electrophysiology 6- 67
Figure 6.3.6-2 “Spontaneous extracellular unit activity recorded from a
microelectrode containing eight microwires (numbered 1-8; impedance 100-300
kHz [sic, probably kilohms] measured at 1 kHz) with tips spaced at 500 micron
intervals positioned in entorhinal cortex. Continuous data were sampled at 27.8
kHz per channel (16-bit resolution; 1 Hz to 6 kHz; Neuralynx, Inc. Bozeman, MT)
and recorded with respect to adjacent uninsulated microwire used as a local
reference ( 1-3 kHz [sic, probably kilohms] measured at 1 kHz). Negative polarity
of signals is oriented down.” From Fried et al., 2014.
6.3.6.2 Probes used in patch-clamp investigations
There are two different cases to be examined; the patch-clamp to a neuron to record the natural
occurring action potential and the patch-clamp where the patch-clamp is used to cause an action
potential in any neuron be inserting a charge into the selected plasma (generally using the same
probe) to cause the generation of an action potential.
The probe used in patch-clamp measurements is much different than the simple wires discussed
above. The probe is usually liquid-filled and therefore introduces a serial resistance, due to the
liquid, in parallel with a capacitance. There may also be a capacitance in shunt with the signal
path due to the container holding the liquid. This combination must be compensated, when
connected to the measurement oscilloscope to insure a transient artifact is not introduced into the
recorded data.
Figure 6.3.6-3 illustrate the situation when the patch-clamp probe is not properly compensated.
The subscripts, P, refer to the probe, the subscripts, O, refer to the oscilloscope input impedances.
The ratio of the real and imaginary parts of the parallel impedances (within the
bandpass of the test instrument) must be the same to insure no artifact is introduced
into the recorded data.
68 Neurons & the Nervous System
Figure 6.3.6-3 Effect of probe compensation on the recorded oscilloscope
waveform using a one kiloHertz square wave. Left; typical probe/oscillograph
input circuit with probe and oscillscope impedance shown. Center middle;
requirement for the sought response at right middle. Center top; equation
resulting in overshoot condition at right top. Center bottom; equation resulting
in the undershoot condition at right bottom.
Most of the literature attempts to insure a maximum value for RP without any regard to CP; the
investigators then calibrate the steady state amplitude of the oscillscope display based on the
ratio of RP and RO. Ignoring the ratio Of CP and CO resulty in the two transient condition shown
in the figure. One overshoots the true amplitude of the subsequent action potential (that may be
interpreted, erroniously as the action potential going positive). the other condition results in the
recorded action potentail as being less in amplitude and somwhat broader in time than it actually
is. It is necessary for the center middle equation to be satisfied if accurate results are
to be obtained.
6.3.6.2.1 Background relating to the patch-clamp techniques
The patch-clamp technique is a very good development; however its operational characteristics
are poorly understood. First, under the three chamber concept of The Electrolytic Theory of the
Neuron, it is possible to access all three plasmas of a specific neuron, Second, the impedance
level in these plasmas differ greatly. Third, there is an active electrolytic device within the
neuron accessed, whose state of biasing must be considered. Fourth, the impedance of the source
determining the clamp level is usually orders of magnitude lower than the intrinsic levels of the
plasmas. These characteristics must be understood if the patch-clamp technique is to be used
effectively and proper conclusions drawn based on the data recorded. A primary task is
determining what stage the neuron is participating in; whether it is a stage 3A neuron or
performing in a different stage not associated witg action potential generation. Any neuron is
Electrophysiology 6- 69
capable of generating an action potential if its bias arrangement is changed to
appropriate values.
It is extremely important to make the measurements under patch-clamp
conditions in the DC mode, rather than the AC mode. Otherwise, the plasm being
interrogated cannot be explicitly determined. The DC mode also insures an additional
capacitance has not been added in shunt with the input resistance of the oscilloscope used
after the probe has been properly compensated.
6.3.6.2.2 Patch-clamp technique used with a stage 3A neuron
The recording of an action potential within a stage 3A neuron ( or at the axon of a stage 3B
Node of Ranvier) is a simple process. First, it is important to understand which plasma has been
accessed, by determining the quiescent DC voltage of the plasma compared to the nearby
extraneuronal voltage. There are two diabolically different conditions, whether the neuron is of
stage 3A and normally generating action potential, OR not of stage 3A, and not normally
generating action potentials.
In the absence of creating an action potential, the quiescent axoplasm should be near --110 to --
140 mV. In the absence of creating an action potential, the quiescent podaplasm should be less
than -- 20 mV. In the absence of creating an action potential, the quiescent dendroplasm should
be more negative than the podaplasm and generally less than -- 20 mV.
During action potential generation, the axoplasm should become more positive by nominally 100
to 110 mV the podaplasm should become more negative by about 20 mV and the dendroplasm
should become more positive by a fraction of the axoplasm positive rise. These values can only be
measured with a test set with an impedance an order of magnitude higher than the plasma to be
measured. The high impedance dendroplasm is probably most difficult, and the low impedance of
the podaplasm the easiest, to measure in this regard.
The use of an external source to set a plasma to a specific voltage (Section 6.3.6.2.3) may also
interfere with the voltage or current measurements. This is particularly important if the source is
a low impedance source and the probe impedance is relatively low compared to the plasma.
6.3.6.2.3 Patch-clamp technique used with a NON stage 3A neuron
EMPTY
As indicated above, any neuron can be made to generate an action potential; it is only necessary to
make the dendroplasm go positive relative to the podaplasm when the poditic terminal impedance
is not purely resistive. This can generally be accomplished by injecting a charge into the plasma
of the podaplasm or extracting charge from the dendroplasm. It can also be accomplished by
extracting a larger charge from the axoplasm (due to fact the dendroplasm and axoplasm are
capacitively coupled. Any of these methods must be appropriately labeled “parasitic stimulation
generating an action potential.”
The circuit for generating a parasitic action potential is more complex than shown in the above
figure as shown in Figure 6.3.6-4. This circuit diagram assumes a common probe is used
for both stimulation and recording the resulting action potential. In practice the presence
of the curret pulse in the oscillograph output is usually suppressed, except for the occurrence of
the leading edge. A similar requirement to probe compensation is described in the caption for the
compensation of the capacitance of the current source to avoid introducing a delay in action
potential generation.
70 Neurons & the Nervous System
71Coggeshall, R. & Fawcett, D. (1964) The fine structure of the central nervous system of the leaech, Hirudo
medicinalis. J Neurophys vol. 27, pp 229-289
72Nicholls, J. & Baylor, D. (1968) Specific modalities and receptive fields of sensory neurons in the CNS of
the Leech J Neurophys vol. 31, pp 740-756
73Kristan, W. et. al. (1974) Neuronal control of swimming in the medicinal leech (three papers) J Comp Physiol
vol 94, pp 97-119, 121-154, 155-176
74Frank, E. Jansen, J. & Rinvik, E. (1975) A multisomatic axon in the central nervous system of the leech. J
Comp Neurol vol. 159, pp 1-14
75Masino, M. & Calabrese, R. (2002) (2 papers) Phase relationships between segmentally organized oscillators
in the leech heartbeat pattern generating network J Neurophysiol vol 87, pp 1572-1585 & 1586-1602
Figure 6.3.6-4 Current driven circuit, to
generate a parasitic action potential. The
circuit is now more complex. CS/CP must
now equal RP/RS to ensure the current
pulse entering the neuron under test is
not distorted, particularly in time.
6.3.7 Computational anatomy as
a flexible tool in the neural
system
6.3.7.1 Delay–based
computational anatomy between
the retina and the visual cortex
In texts discussing the morphology of the
overall visual system, Meyer’s Loops are
usually described without any reference to
their function. These are two fan shaped
bundles of myelinated stage 3 neurons
projecting to the Occipital Cortex from the
Lateral Geniculate Nuclei of the midbrain.
They are fan shaped for the particular
purpose of introducing different time delays
in the transmission paths. What is not
generally known is that these delays are
introduced to compensate for a previously
introduced set of delays.
The LGN is assigned a spatial correlation task. However, correlation of information with respect
to space that is all received in time correlated “frames” is difficult. To alleviate this problem, the
signals from the retina are transmitted along paths requiring different times based on their
originating points in the retina. These spatially related time delays are accentuated by the fact
the axons leaving the ganglion cells are not myelinated until they reach the lamina cribosa, the
start of the optic nerve. Thus a set of dispersion based delays are introduced that are
complimentary to those of Meyer’s Loops. This fanning of the signals from the ganglion will be
labeled Reyem’s Loops because of this feature.
The availability of this feature changes the computational strategy of the LGN greatly. The
signals it receives are no longer spatially correlated. The spatial location of scene elements,
relative to the line of fixation, can now be determined readily by using simple time correlation
techniques. This subject is addressed in greater detail in Section 15.6.5.
6.3.7.2 The special case of a syncytium formed by the ganglia of the leech
The relative simplicity of the neural system of the leech, Hirudo, makes it particularly attractive
to morphologists. Significant exploratory data was published during the 1960's and 1970's related
to the medicinal leech, Hirudo medicinalis71,72,73,7 4. More recently, Masino and Calabrese have
used more modern techniques to provide details related to the heartbeat pattern of the leech75.
Electrophysiology 6- 71
Using the Electrolytic Theory of the Neuron, this material can be correlated into a single more
comprehensive view of the neural system of this animal.
Members of Annelida display a bilateral architecture, a primitive heart, eye spots, and two closely
aligned nerve cords. Each of these cords appears to contain a “giant fiber” that extends the length
of the cord. Some texts suggest that only one giant fiber is packaged separately in Faivre’s nerve
that is found between the two nerve cords. Annelids are the most highly organized animals
capable of complete regeneration.
6.3.7.2.1 Background
Kristan, et. al. offered an overview and extensive experimentally supported analysis of the neural
system of Hirudo medicinalis along with new segment and neuron numbering systems. These
systems is more compatible with physiological experiments than the previous morphologically
based system. Masino and Calabrese have also provided an extensive experimentally supported
analysis. They appear to have used the old terminology since they were not concerned with the
physical motions of concern to Kristan, et. al. Most of the tests reported by both Kristan, et. al.
and Masino and Calabrese were limited to AC responses due to their bandpass electrical
configuration. This fact is evident in their figures and supported by their notes on methods.
Kristan, et. al. describe the nervous system of the leech as consisting of 34 interconnected ganglia.
They make an intersting assertion.
“Fused masses of 13 of these ganglia form two brains: a 6-ganglion head brain and a 7-
ganglion tail brain.”
“Head and tail brains are linked by a ventral nerve cord consisting of the remaining 21
ganglia and their connectives.”
While Masino and Calabrese describe the electrical circuits required to operate a heart as being
formed within the first four ganglia counting from the “head” brain, they did not note a similar
structure associated with the putative tail brain. However, for the two portions of a bisected leech
to continue normal swimming motions for “many hours” as claimed by Kristan, et. al., it would
appear the animal should have two hearts and two brains.
Kristan, et. al. develop the electrophysiology of the efferent neural system of the leech when
swimming in considerable detail. They note many important features of the topology, and
operating characteristics, of the neural system They also echo the work of another investigator
that the characteristics of the oscillators controlling the swimming motion are those of relaxation
oscillators.
Nicholls and Baylor develop the electrophysiology of the afferent neural system of the leech, when
the skin is abraided, in considerable detail. Their experiments and findings are clearly
exploratory and limited to the phasic (and presumed stage 3) neurons of both the peripheral and
central nervous system of the leech. However, contrary to the suggestion in the title of their
paper, they did not explore the electrotonic signals to be expected to be found in stage 1 sensory
neurons. While it may be the sensory neurons are multifunctional in these simple animals, and
are capable of generating action potentials like the stage 2 eccentric cells in Limulus, this feature
was not demonstrated. Their recordings were a mixture of AC and DC responses due to the form
of their instrumentation. This fact was not developed in the description of their results.
However, it is clearly evident in their figures (see Section 6.3.2.1.2 ). The extracellular
recordings are AC based and the intracellular recordings are DC based.
Their photomicrographs clearly show the bilateral character of the longitudinal connectives and
the root nerves projecting laterally from the individual ganglia. One (fig 4A) suggests that certain
nerves (axons) pass straight through the individual ganglion. They also provide specifics
concerning some of the less descriptive neuron names found in this literature (e.g .Retzius cells).
Their own cell descriptors are not functionally oriented (pg 746). The photographs also show the
inherent symmetry of the ganglia described by Kristan, et. al. [Nicholls and Baylor also use the
singular and plural of ganglion in interesting ways.] Apparently, many morphologists and
physiologists treat one of multiple ganglia along a nerve cord as a ganglion although it contains a
great many stage 3 neurons. Others treat a group of neurons found along any nerve as a ganglia.
In vision, it has been traditional to speak of each of the stage 3 neurons in the retina as a ganglion
cell, although they are not grouped at all (See Glossary).
72 Neurons & the Nervous System
Frank, et. al. have explored a particularly large axon in the medial bundle that extends along the
entire length of the nerve cord in the Leech (possibly Hirudo medicinalis, and about 10-12 cm long
based on their source). Their paper does not contain details concerning the location or topology of
the ganglia they worked with. They claim this axon supports the bidirectional transmission of
signals. However, it is unclear whether this happens in-vivo or only under pathological conditions
associated with their test configuration. As suggested by the title of their paper, they
conceptualized a single axon associated with multiple soma. The caricature in their figure 1
suggests the axon is interrupted by multiple soma. However, their electron micrograph of figure
3(A–C) appears compatible with an alternate morphology. While they did not define the locations
of interconnections between the axon and each soma, they did claim the arrangement required
either direct cytoplasmic connection or synapses between the elements that could transmit signals
bidirectionally. They offered no mechanism for accomplishing the latter feat.
They examined what they describe as a syncytium Such arrangements are frequently described
in the literature as containing a “giant axon.” In this case, giant refers to an axon diameter of less
than 5 microns in a small animal of undefined dimensions. Their caricature shows five neurons
arrayed along a conduit that appears to be bidirectional. Their timing diagrams are very
informative and clearly illustrate action potential waveforms acquired from axons associated with
the neurons along the main axon. They define these as S-cells because the output signal at their
axon can be described as a “spike.” Other waveforms are obviously pseudo-action potentials
acquired from the dendroplasm of similar neurons (see below). The dendrite waveforms of these
neurons clearly show the integrating process leading to positive internal feedback while the axon
signals show well defined waveforms with no overshoot, undershoot or integration.
They did not describe their test configuration in detail. However, they did note a speed of signal
transmission of 1.5 meters/sec at 16 C that apparently applies to the giant axon. While the
excitation used in their figure 1 is clearly a short duration pulse, the excitation used in figure 2a
appears to be a long pulse (~50 ms). It is the length of this second pulse that accounts for the
integration in the dendritic waveforms of this figure. The integration leads to as many as three
action potentials during a single long pulse. Under short pulse excitation, only one action
potential is generated at each individual stage 3 neuron.
These authors specifically noted several important points. They found no axon segments (and
presumably no Nodes of Ranvier) along the axons of these animals. Dyes traveled the length of
the main axon without observable points of concentration or dilution. They also introduced long
pulses into the axon that were not comparable to action potentials. Such long pulses were not
reported in-vivo by Kristan, et. al. (pg 171).
6.3.7.2.2 An explicit configuration used in swimming
An alternate configuration to that of Frank, et. al. is suggested by similar circuitry frequently
used in man-made radar systems. Such a system would use a single continuous axon as a delay
line that was “tapped” at intervals by individual orthodromic neurons. Figure 6.3.7-1 presents
such an alternate configuration based on the Electrolytic Theory proposed here and consistent
with figure 3A in their paper. The new caricature adds additional detail related to the
electrophysiology of each neuron and further annotates the experimental results. Note the axon
skirts the target cell in figure 3A and in this figure. Certain optimizations related to each of the
five projection neurons to cause them to generate action potentials have not been shown. The
dendrite of each of the five projection neurons synapse with the axon as shown. Since the
dendrite is not ramified in this situation, it may appear morphologically that the synapse is
between the axon and the soma. However, functionally, it is between the axon and the
dendroplasm.. An interesting feature, compatible with the test data of Frank, et. al. is the ability
of the axon to support signals traveling in opposite directions simultaneously over the single axon.
This mode of operation would suggest the signals are propagated and not conducted. Such
propagation may account for the large size of the axons encountered. The large size, and their
pseudo-myelination, allows them to propagate signals a sufficient distance without requiring
regeneration by Nodes of Ranvier.
Although not required for this discussion, a third probe is shown contacting the dendroplasm of
the fifth projection neuron. In several of the waveforms provided by Frank, et. al., particularly
those showing an integration prior to initiation of oscillation, it is clear that their waveform was
representative of the dendroplasm potential. This waveform is sometimes labeled a pseudo-action
Electrophysiology 6- 73
potential in this work.
Frank, et. al. only reported on the performance of two out of the five motor command neurons
served by a single giant axon (shaded area). Frank defines these motor neurons as S-cells and
indicates the giant axon is associated with Faivre’s nerve. He describes Faivre’s nerve as located
between the two nerve cords forming the connectives between the ganglia. It is proposed that the
giant axon is intrinsically a propagation mode axon that can be excited from either of its end
terminals. It can be thought of as a phasic axon segment that can be served by two conexuses. As
shown, a conexus on the left can send an action potential, in response to S1, traveling to the right
along the axon. Similarly, the conexus on the right can send an action potential traveling to the
left. Only impulse type signals will be considered here. Frank, et. al. also sent longer
(pathological) pulses along the conduit as shown in their figure 2(A) and pairs of pulses from S2 as
shown in figure 1(C). This annotated figure is consistent with all of the data in Frank, et. al.
However, the time delays recorded by those authors were skewed due to the placement of their
probes. Simple calculations (e. g. no conexus and similar delays) demonstrate several points.
Their two projection neurons were separated, at their synapses, by 5.8 ms of conduction travel
time along the giant axon. The probe points on the two neurons varied in position by enough to
give a propagation travel time difference of 0.6 ms due to this factor alone. As shown in this
figure, their probe labeled R1 was farther from the conexus (X = 3.0) of the third projection neuron
than was their probe labeled R2 (Y = 2.4) from the fourth projection neuron. The calculated
lengths along the giant axon, and shown below it, would suggest that the projection neurons were
not equally spaced along the giant axon as shown in the drawings.
74 Neurons & the Nervous System
Figure 6.3.7-1 Cartoon of the syncytium in Leech based on the Electrolytic Theory
of the Neuron. Five stage 3 (monopulse oscillator) neurons are shown equally
spaced along a stage 2 (propagation mode) giant axon. The giant axon can be
excited from either the left, right or both conexuses shown. The time scale on the
left shows when the leading edge of the action potential of each neuron occurs
when excited by the left conexus (when the baseline of each waveform is
extended to the time axis). The time scale on the right shows when each action
potential occurs if excited by the right conexus. The shaded area shows the area
explored by Frank, et. al. The dotted box contains the putative second conexus
supplying excitation to the giant axon that is antidromic to that supplied from the
conexus at S1. See text.
In the configuration shown, there is no requirement for any bidirectional synapses with the giant
axon.
Two requirements do exist that cannot be addressed under the Chemical Theory of the Neuron.
The signals, as they are propagated along the giant axon, must remain of sufficient amplitude to
cause the last projection neuron to generate an action potential. This causes the amplitude of the
signal arriving at the distant end of the giant axon to be relatively large and this could result in a
reflection at the terminal point if the terminal impedance does not match the characteristic
impedance of the giant axon well. Negligible signal will be reflected back along the giant axon if a
satisfactory match, probably involving matching filter sections, is provided at each conexus.
Electrophysiology 6- 75
6.3.7.2.3 Summary of the neural system of Hirudo medicinalis
The efferent neural system of Hirudo is understood to a remarkable level at this time. By combining the work in
the papers referenced, it is possible to draw a number of conclusion. First, as every young child quickly learns,
the bisected parts of members of Annelida are able to survive for many hours. The segments appear to continue
to carry out a variety of normal functions. The neural system defined by morphology supports the concept of two
brains located at the opposite extremes of a ventral nerve cord. It is likely that the ventral cord also contains two
simple heart beat generators located near the two brains. At least a very few of the ~7500 neurons associated
with the CNS appear able to support bidirectional transmission of neural signals over the length of the specimen.
At least over distances of five ganglia, there are no Nodes of Ranvier to interrupt the diffusion of dyes or to
enhance the transmission of signals. It appears that the normal body motions, such as those used
in swimming, are supported by a set of gated relaxation oscillators essentially like those found in
stage 3 of all other neural systems. It is the method of gating and the fact the giant axon operates
as a delay line that differentiates the neural system of Hirudo and other members of Annelida
from more complex systems.
The afferent neural system of Hirudo is also well understood based on the work of Nicholls and
Baylor. They mapped the cutaneous receptors associated with each annulus. However, they did
not isolate the sensory neurons of the animal. More work is needed to determine the location of
the conexus(es) between the dendritic structure of the cutaneous sensory (stage 1) neurons and
the recorded action potentials along the dorsal nerve paths. Whether the cutaneous signals are
propagated to both brains of the animal prior to a traumatic bisection was not explored.
It appears the giant axon(s) are optimized to allow the animal to survive following bisection.
Lacking any (directionally restrictive) Nodes of Ranvier, the axon can be excited by the neurons
associated with the tail brain when necessary. This excitation would allow the bisected portion of
the animal lacking the head brain to survive long enough to escape and regenerate the missing
portions of its anatomy.
Further study of the papers of Kirstan et. al. and Frank, et. al. can lead to a more complete
understanding of the operation of this neural complex. The giant neuron relies upon its size, and
possibly pseudo-myelination, to propagate signals over a significant distance without
regeneration. The signals are propagated at a very low velocity due to the size of the axon. The
associated neurons of the syncytium appear to operate as threshold detecting stage 3 neurons. As
a result, the stage 3 neurons fire in synchronism based on their physical location along the length
of the giant axon. These action potentials would be used to drive motor neurons.
A prominent question at this time concerns the control of the bursts of action potential associated
with the contraction of the muscles of each segment. While Frank, et. al. showed that isolated
segments of the giant axon can support long pulses in a conductive mode, and thereby cause his S-
cells to generate bursts of action potentials, Kirstan, et. al. did not show that the natural
excitation of the giant axon involved long pulses. It appears that a different topology is needed
from that suggested by Frank, et. al. There are a variety of simple signaling topologies that could
cause bursts to be generated by stage 3 neurons based on individual (stage 3) action potentials
propagating along the giant axon(s). Most of these would call for a stage 2 neuron (or at least a
stage 3 decoding neuron) between the giant axon and the S-cells to act as a latch. Such a latching
circuit has not been isolated to date.
The giant axon provides a simple functional circuit for generating a sequential series of muscle motions that are
reasonably fixed in time between responses (addressed in the Kirstan, et. al. series of papers). When combined
with other features of computational anatomy, such a sequence is particularly useful in swimming, flying, walking
(in multi-legged species) and the type of locomotion used by the sidewinder snake. It may also be used to
control the timing in multi-chamber hearts.
6.3.7.3 The special case of the giant axons of the 3rd stellate ganglia in the squid
The literature suggests there are three stellate ganglia in the squid and what is called the third stellate ganglia
actually representa a pair of neurons. The fact the electrical probe significantly changed the electrical fields
within the axon, the replacement of the plasma significantly changed the conductivity of the plasma and the
physical removal of the pseudo-myelin provided by the other smaller axons, significantly changed the impedance
of the axon relative to the surrounding medium.
The “giant axon” of the dwarf squid, Loligo forbesi, has gained eternal fame (or notoriety) because of its easy
76 Neurons & the Nervous System
76Purves, D. Augustine, G. Katz, L. LaMantia, A-S. & McNamara, J. (1997) Neuroscience Sunderland, MA:
Sinauer Associates, Inc. pg. 46
Figure 6.3.7-2 The nature of the giant axon of Loligo forbesi and other features of
its nervous system. The smaller axons surrounding the giant axon may provide
the same electrical function as myelin does when associated with a chordate
neuron. The length of the giant axon typically used by Hodgkin & Huxley was
30–50 mm. The diameter of the giant axon was typically 500–700 microns.
accessibility using relatively crude electrical probes. However, it is seldom discussed in terms of its functional
and morphological arrangement. The giant axons found in Arthropoda and Mollusca are not typical axons and
should not be considered prototypical. They are examples of specialized structures used in computational
anatomy. They are designed to function as tapped delay lines. Purves, et. al. have provided several figures
defining this special configuration and other features of the squid neural system76. They use the term stellate
loosely to describe the star-like morphology of the neuron rather than to describe its location or function. For the
dwarf squid, it appears that the individual axons closely packed around the giant axon provide the mollusc’s
adaptation to the problem of reducing the high capacitance of the extremely thin axolemma. Isolated chambers of
axoplasm form very effective dielectric insulation (see Section 10.2.1.1.4). Figure 6.3.7-2 defines the relevant
parameters. Purvis, et. al. appear to compare apples and oranges. They compare a typical unmyelinated
mammalian neuron of 2 microns diameter with the outer dimension for the giant axon and its surround of smaller
axons at 800 microns. It may be more appropriate to consider a myelinated neuron at 8-15 microns.
If the axons surrounding the giant axon are closely packed over a considerable portion of their length, they can
effectively isolate the giant axon from the surrounding INM in the same way myelin does. Although probably
less effective than myelin in chordates, due to the displacement currents that could occur in each individual small
axon, the large difference in diameter between the axoplasm o f the giant axon and the surrounding INM would
reduce the capacitance associated with the giant axon considerably. Note the thickness of the actual membrane
wall of the giant axon would not be visible on a figure at this scale. The thickness of the smaller axons as a group
relative to the membrane thickness of the giant axon should be compared to the thickness of the myelin relative to
the lemma thickness in myelinated neurons of Chordata.
Electrophysiology 6- 77
77Frankenhaeuser, B. & Hodgkin, A. (1957) The action of calcium on the electrical properties of squid axons
J Physiol vol 137, pp 218-244 referenced in Abbott, N. et. al. ed. (1995) pg 232
78Waxman S. & Wood, S. (1983) Impulse conduction in inhomogeneous axons: (Calculated) effects of variation
in voltage-sensitive ionic conductances. Brain Res. vol. 294, pp. 111-122
Figure 6.3.7-3 Records of the action
potentials in squid, Loligo forbesi, lasting
about one second (note break in record).
(a); the overshoots. (b); the undershoots.
Temperature of 19.8° C. 10mM of K+ in
external bath. From Frankenhaeuser &
Hodgkin, 1957.
6.3.7.3.1 Measured and calculated action potentials of the squid
Figure 6.3.7-3 documents one of the best examples of action potentials in the literature from the squid, Loligo
forbesi. The figure is from Frankenhaeuser and Hodgkin77 but did not appear in the published paper (see
footnote). The experiments employed in the 1940's test set of Hodgkin & Huxley with a mutilated axon as in the
1940's experiments. There are three problems with the indirect data. They did not make any direct
measurements of sodium or potassium ions moving through a membrane. It is not easy to interpret the DC
potentials associated with these waveforms. It is not definite that the waveforms are not pseudo-action potentials
from the dendroplasm of the neuron. Frankenhaeuser and Hodgkin did not explicitly state that their test
instrumentation was adjusted for a “flat response” as a function of temporal frequency.
It is probable that these are pseudo-action potentials measured at the dendroplasm of the giant nerve of Loligo
forbesi and that the reduction in waveform height with a time constant of about 0.3 seconds is due to the time
constant of the electrostenolytic supply of the axon.
Under this interpretation, the resting potential of
the dendroplasm is a reasonable –52 mV. Assume
the oscillator threshold is about –30 mV and there
is considerable capacitive feedback between the
axoplasm and the dendroplasm (but no stray
capacitance in the test set. The resulting action
potential at the axon terminal is about 120 mV and
about 80 mV of that 120 mV is coupled into the
dendroplasm to reduce the rise time of the leading
edge of the action potential. This raises the
potential of the bulk dendroplasm to about +50 mV
(although it is not clear the actual emitter of the
Activa within the conexus goes more positive than
the base potential of about +20 mV. After
achieving its peak amplitude, the action potential of
the axoplasm returns to its baseline of –150 mV.
About two-thirds of this amplitude is capacitively
coupled back into the dendroplasm. This drives the
dendroplasm down to about –65 mV for a short
time. This is a significant undershoot in the
pseudo-action potential at the dendroplasm that is
unrelated to the actual action potential at the
axoplasm. In this scenario, the actual action
potential at the axoplasm does not go positive and it
exhibits no polarity reversal with reference to the
INM nor does it exhibit any undershoot as it returns
to its quiescent potential of –150 mV.
Waxman & Wood provide a relatively unique article in that it presents both the voltage profile and the current
profile at different locations along a neuron several centimeters long78. Unfortunately, all of the data is calculated
via computer simulation using the unsolved partial differential equations of Hodgkin & Huxley (1952). They do
incorporate a set of transition regions between the nodal region and the bulk of the axon-segment to improve their
simulation. These regions would be labeled matching sections in standard electrical filter design. These sections
minimize the reflection of the electrical signals at the terminus of the axon segment. In comparing those
published computed waveforms, Figure 6.3.7-4, it is clear that the model presented here is compatible with the
computed results. However, more precise measured values for these voltage and current parameters as a function
of time are needed. It is proposed that such data would demonstrate a discontinuity in the mathematical
description of the waveform at the peak of the voltage waveform not predicted by the Hodgkin & Huxley
78 Neurons & the Nervous System
79Huxley, A. & Stampfli, R. (1949 to 1951) References in Stampfli, R., Overview of studies on the physiology
of conduction in myelinated nerve fibers. In Waxman, S. & Ritchie, J. Demyelinating Disease. NY: Raven
Press. pp 11-23
Figure 6.3.7-4 The calculated voltage and
current profiles of the same action
potentials using the empirical equations
of Hodgkin & Huxley..Inward current is
upward. The polarity of the voltage is not
specified. The first three waveforms in
each group are distorted by the artificial
stimulus. The subsequent current
waveforms show a distinct change in
character as they pass through zero. The
waveforms (after the first three in each
group) are compatible with this theory.
From Waxman & Wood (1983)
equations. Huxley & Stampfli have provided some additional data of this type79.
Whereas the Hodgkin & Huxley rely upon the inward and outward flow of metallic ions through a membrane,
the same results can be obtained by conventional current being balanced between the charging and discharging
current paths of the neural circuit. The current flowing through an impedance in one direction to charge a
capacitor can cause the rising portion of the action potential waveform. When flowing through a different
impedance in the opposite direction to discharge a capacitor, the current can generate the falling portion of the
waveform. If the shape of the waveforms were known more precisely, it is possible the discontinuity near the
peak of the waveforms could be seen. In that case, the rising portion of the waveform would be given by the
current through the regenerator Activa discharging the capacitance of the axon and the falling portion would be
given by the time constant of the recharging circuit of that neuron, as illustrated in Section 2.7.5.
6.3.7.3.2 A reinterpretation of the
Hodgkin & Huxley waveforms
Hodgkin & Huxley recorded waveforms in
their famous experiments of the late 1930's
and 1940's, reported in 1952. They
essentially emasculated their “giant axon of
the pygmy squid, loligo forbesi,” by their
cleaning process; stripping away the pseudo
myelination provided by the surrounding
neurons (Section 6.3.7.3) and removing all
extraneous tissue, that included the
dendritic and poditic terminals of the
neuron, and inserting a conductor along the
axis of the remaining axon. They then
proceeded to stimulate the axon
parametrically. The resulting recorded
waveforms do not have the characteristics of
Electrophysiology 6- 79
80Hodgkin, A. & Huxley, A. (1952) A quantitative description of membrane current and its application to
conduction and excitation in nerve. J. Physiol. vol. 117, pp 500-544, pg 525
Figure 6.3.7-5 Tracings from the test configuration of Hodgkin & Huxley purported
to be action potentials. Axon 17 measured at 6 degrees Celsius. Numbers next
to curves give the “shock strength” in mcoulomb/cm2. The significant delay
between the stimulus and the response is not found in normal action potentials
(nominally 0.019 msec) or generator potentials of sensory neurons (range of 0.2
to 4.0 msec but a distinctly different shape). Horizontal line at 110 mV is a
calibration artifact. From Hodgkin & Huxley, 1952.
action potentials or generator potentials80. Their waveforms, reproduced in Figure 6.3.7-5,
do not appear to be exponential in character,
differ significantly from those of Bowe, et. al. and of Schwarz & Eiknof shown above,
differ significantly from the pedagogical description of an action potential in Section 9.3.2,
differ significantly in time scale from the same data published in Hodgkin, Huxley & Katz
(Section 2.6.1.2.1).
differ significantly from the theoretical action potential waveforms of this work (Figure 2.5.3-4
in Section 2.5.3.5).
The peak of the stimulus causing the creation of an action potential always occurs within less
than one millisecond of the start of the action potential. There is never an excursion in the
opposite direction between the stimulus and the generated waveform in an action potential. The
amplitude of the stimulation, resulting in creation of an action potential, is always less than the
amplitude of the action potential.
Hodgkin & Huxley did not attempt a theoretical mathematical solution describing their
waveforms (page 506). They noted the difficulty in fitting their equation #26 to the measured
waveforms. Four distinctions were noted and discussed on their page 542-543. They used
equation #31 as a baseline later in the paper. However, it did not reflect the falling phase of their
waveforms. Their equation #33 was used to account for this portion of their solution. Each of
their equations was linear and involved four empirically derived terms in a primary equation.
This equation was supported by as many as nine auxiliary equations. The linear equations
reflected their assumption of a linear membrane model (although with variable resistance
elements). The parameters in these equations were determined by curve fitting. This is a
sufficiently large number of parameters to fit any conceivable waveform.
A modern review of their waveforms and their test set would suggest a totally different
80 Neurons & the Nervous System
81Bostock, H. (1993) Impulse propagation in experimental neuropathy. In Peripheral Neuropathy, 3rd. Dyck,
P. & Thomas, P. Ed. Philadelphia: W. B. Saunders pp. 109-120
explanation for their waveforms. The responses are clearly the impulse response of their
membrane combined with some leakage of their excitation impulse into their recordings (at time
intervals from zero to about 0.1 milliseconds. The remainder of the impulse response shows
significant rectification of a sinusoidal response. The rectification introduces significant harmonic
content into the output. This is the expected response from a circuit (asymmetrical biological
membrane) that contains a diode and resistance in series shunted by a capacitor. Under this
interpretation, the waveforms cannot go below the baseline because of the diode.
While their efforts were monumental for the time period, the equations of Hodgkin & Huxley have
been optimized to emulate a waveform that is not an action potential but an impulse response
function due to parametric stimulation. This impulse response function reflects the parameters of
both their biological membrane and their test set. It is not representative of an action potential as
documented using modern experimental techniques.
The protocol used by Hodgkin & Huxley was suitable for an exploratory research
program but not an applied research program producing parameters that could be
relied upon.
6.3.8 Correlation of the transmission theory of axons with the measured
data
This section will rationalize the theory above with the measured data from the literature.
6.3.8.1 Rigor required when transitioning from exploratory to exploitive
research
The literature of the action potential is not known for its rigor. Much of the data was collected
before a clear distinction was developed between the generator potential and the action potential.
Most of the data referring to the actual action potential was collected in stand-alone experiments
without all parameters well controlled. In some cases, this makes a rigorous comparison between
the measured data as published, and the theory quite difficult.
6.3.8.1.1 The special case of the giant axons of the 3rd stellate ganglia in
the squid
The above material has defined three fundamental types of axons. The axons of stage 1, 2 and 4 neurons are
designed to conduct electrotonic signals. They are generally short and unmyelinated and in most cases can be
successfully modeled using lumped-parameter models. The axons of stage 3 neurons are designed to propagate
phasic signals efficiently over relatively long distances efficiently. They are most appropriately modeled as
distributed cables (with significant inductance per unit length) that are terminated by half-section matching filters.
The special case of the giant axons of the 3rd stellate ganglia in the squid is worthy of consideration; this special
class of axons are those used to create tapped delay lines in support of anatomical computation. While these
operate as stage 3 axons, they exhibit several special features. They are unmyelinated (or pseudo-myelinated)
and very large in order to achieve an efficient axon with a very low propagation velocity. Their length can be
measured in centimeters but they are not interrupted by Nodes of Ranvier (indicating very low attenuation per
millimeter). To optimally explore the available data, the differences between these types of axon must be
recognized.
Bostock81 also presents a series of membrane current contour maps obtained by probing the interneural matrix
near the axon at 100 micron intervals. However, as pointed out in his text, the widths of the contours reflect the
diameter of the probe more than they do the source of the currents. Although the contours extrapolate the
location of the Nodes of Ranvier to within about 50 microns, the Nodes are typically only 10 microns wide.
The 2005 edition of Dyck Thomas do not contain the current contour maps of the 1993 edition. The 2005
edition is in two volumes to accommodate considerably more clinical information. In the 2005 edition,
Electrophysiology 6- 81
82Bennett, M. Nakajima, Y. & Pappas, G. (1966) Op. Cit. pg. 175
83Paintal, A. (1978) Conduction properties of normal peripheral mammalian axons. In Physiology and
Pathobiology of axons, Waxman, S. Ed. pp.131-143
84Cole, K. & Baker, R. (1941) Longitudinal impedance of the squid giant axon J Gen Physiol vol 24(6), pp
771–788 https://doi.org/10.1085/jgp.24.6.771
Bostock is joined by two other authors. On page 115 of volume 1, they show an ephemeral sodium ion
pump powered conceptually by ATP. The need for a sodium ion pump disappears under “The
Electrolytic Theory of the Neuron” (Section 2.7.5).
The contour maps and the assertion by Huxley and Stampfli that only the Nodes of Ranvier generated inward
membrane [conventional] current agree with the circuit presented above if the electron flow is actually leaving
the axoplasm via the Activa.
In agreement with the previous statement above, and the observation of Bennett, et. al.82, the current waveforms
in Bostock are not precise enough to determine the electrical characteristics of any delay line formed by the axon
segment. It appears in his figure 5-3 (presented and discussed further in Section 10.6.3) that the peak amplitude
of the current pulse falls about 50% in the internodal distance of about 0.9 mm. However, the total integrated
current arriving at the node, an indication of the level of dispersion, cannot be determined from the figure with
sufficient accuracy. Without an accurate number for the dispersion, the level of energy dissipation due to
resistance in the delay line cannot be determined.
Many of the test sets enumerated above have been used in conjunction with pharmacological studies of nerve
conduction. Bostock has provided some additional material in this area. Most of the data has been acquired by
mapping the response to single pulses. The resulting data does show the recharging performance of the
electrostenolytic supply at each node under a variety of conditions, the reflection of the action potential at the
pre-node under pathological conditions, and many other useful pieces of information. Paintal83 has provided
considerable information concerning the pulse characteristics of the action potential under a variety of conditions.
All of this data would be of greater value if interpreted in terms of a more precise electrophysiological model of
the neuron as presented here (Section 6.3.3.1.3 ). Such a reinterpretation would move much of this data from
the realm of exploratory research to a higher level of understanding.
It is important to state the morphological and cytological value of the 1978 book edited by Waxman. It
includes a large number of electron micrographs at various magnifications. While they are all interpreted
under the chemical theory of the neuron prevalent at the time, they can be reinterpreted quite easily under the
framework of The Electrolytic Theory of the Neuron. The other parameters must also be reinterpreted under the
The Electrolytic Theory of the Neuron ; otherwise they are frequently archaic. As indicated above, Paintal’s
values must be reinterpreted. Many of his values and figures, and those of other authors in the book, contain
uncontrolled variables.
6.3.8.1.2 The equivalent circuit of the giant axon-Cole
Cole84 has provided a lumped-parameter circuit model of the giant axon of the squid based on his
impedance measurements; his figures 6 & 7 are combined in Figure 6.3.8-1.
82 Neurons & the Nervous System
Figure 6.3.8-1 The RLC character of a giant axon of squid. Left; An approximate
equivalent membrane circuit for the squid giant axon, consisting of capacity, C,
resistance, R, and inductance, L. Right; Longitudinal impedance locus, series
resistance, Rs versus series reactance, Xs, for squid giant axon. Negative, or
capacitative, reactances are plotted above the resistance axis. Frequencies
indicated are in kilocycles. From Cole & Baker, 1941.
After some discussion and further mathematical analysis, Cole concludes,
“Taking a value of 1.1 uf /cm2, for the membrane capacity, this leads to a value for the
membrane inductance of 0.21 henry cm2.”
“For the axon of Figs. 3 and 6 the membrane resistance, R4, is 290 ohm cm2, while the other
axons gave resistances from 260 ohm cm2, to 420 ohm cm2, with an average of 350 ohm
cm2.”
“If then X , is positive, T - U and, consequently, T must be greater than zero. Since this is a
kinetic energy which is associated only with inductance, it follows that an inductance is
necessary.”
“the inductance is to be found in the membrane.”
Measurements by the Cole team and the Hodgkin team found speeds of 10-40 meters/sec in
various in-vitro giant axons of Loligo that had been physically modified. The modification
frequently involved the replacement of the plasma by sea-water without showing that the
resistivity (and electronic mobility) of these two fluids was the same. This is also strange because
of their basic assumption that the concentration of the axoplasm was deficient in sodium
compared to sea-water.
6.3.8.2 Node of Ranvier test circuits
In exploring the voltage and current characteristics of the Nodes of Ranvier and the axon
segments, a variety of test circuits have been used. All of them have suffered serious limitations.
Bostock has summarized five different test configurations and provided references to each. Only
two of the five involve direct current measurements related to the assumed continuous current
flowing through the axon. These circuits involve physical intervention into the electrical circuit of
the axon. The impact of this intervention is not well understood. The other three utilize probes in
the interneural matrix near the axon attempting to measure either displacement currents through
Electrophysiology 6- 83
85Ritchie, J. (1984) Physiological basis of conduction in myelinated nerve fibers. In Myelin, 2nd Ed., Morrel,
P. Ed. NY: Plenum Press pp. 117-145
86Nicholls, J. Martin, A. & Wallace, B. (1992) From neuron to brain, Sunderland MA: Sinauer Associates, Inc.
Pg. 136.
87Barrett, E. & Barrett, J. (1982) Intracellular recording from vertebrate myelinated axons, J. Physiol. (London)
vol. 323, pp. 117-144
88Cahalan, M. (1978) voltage clamp studies on the Node of Ranvier In Physiology and Pathobiology of Axons,
Waxman, S. Ed. NY: Raven Press
the myelin sheath or conduction currents through the nodal gap at a Node of Ranvier. In the
latter case, the experiments were performed without detailed knowledge of the underlying
circuits. By probing the interneural matrix, the measurements necessarily involve a spatial
integration of the potentials or currents encountered. Bostock presents some membrane current
data from Huxley and Stampfli. Most of the current data presented is the result of an integration
to average out unspecified noise sources. Typically, these are test set related noise sources
because of the low level of the individual neuron currents and the extremely high impedance
levels present.
All of the voltage data presented in Bostock is calculated data based on an ambitious integration
regimen or a computer model. The fact that the potentials presented in this paper were not
measured is not clearly stated in the caption supporting the data. However, the text does say that
these potentials were a result of a spatial integration based on a series of assumptions.
Ritchie85 presents a similar set of test circuit diagrams to those of Bostock attributed to the same
individual investigators. Nicholls, et. al86. has presented an expanded diagram after Tasaki.
Barrett & Barrett87 have provided an even more detailed test circuit diagram to explore the
various difficulties and tradeoffs in developing a defendable test strategy. They included the
recesses associated with the nodal gap in their graphic for purposes of orientation but did not
discuss its purpose. Cahalan88 has presented a schematic addressing the physical aspects of
testing more completely.
The overwhelming problem with all of the above test circuit configurations is their fundamental
assumption that the neuron consists of only passive linear electrical components. A second major
problem is their general assumption that there is a common ground point for all of the different
sub-circuits associated with a single neuron. A third problem, shared with much photoreceptor
research, is the assumption that a probe of unspecified electrical potential brought into proximity
to one side of a generally cylindrically symmetrical circuit element immersed in an electrically
leaky medium provides a signal that is representative of the total current into or out of that
circuit. At the impedance levels found in biological tissue, a simple electrostatic field plot would
show the unlikelihood that this technique gives unequivocal results. This problem is addressed
pedagogically by many authors in Waxman. These pedagogical assertions frequently require
reinterpretations under “The Electrolytic Theory of the Neuron.” A fourth problem is insuring the
electrical well being of the in-vitro neuron during the complete test interval. It is common to find
the test duration to be limited due to deteriorating performance of the neuron during the test
sequence.
As indicated earlier, there are essentially no resistors (as both commonly and technically defined)
in the neural system of an animal. There are many diodes present that exhibit a resistive (though
non dissipative) component. There are also a variety of types of Activa. The Activa, like any
active semiconductor device is highly nonlinear and exhibits parameters that vary dramatically
with the biases applied to it as well as with the other components connected to it. In this respect,
the voltage clamp technique can be extremely valuable, but only if one has a detailed
understanding of what one is clamping in an overall circuit.
6.3.8.3 Redefinition of axon dynamics
In establishing a new base line for research into the dynamics of neurons containing Nodes of
Ranvier, refining the terminology used is necessary. This more precise terminology will also be
applied in later sections on other neuron cases. It is conventional to speak of the “conduction
84 Neurons & the Nervous System
89Ramo, S. & Whinnery, J. (1953) Modern Radio NY: John Wiley & Sons pp. 23-85
90Huxley, A. & Stampfli, R. (1949) Evidence for saltatory conduction in peripheral myelinated nerve fibres. J.
Physiol. Vol. 108, pp. 315-339. [Also at Stampfli, R. (1981) Overview of studies on the physiology of
conduction in myelinated nerve fibers. In Demyelinating Disease: basic and clinical electrophysiology,
Waxman, S. & Ritchie, J. ed. NY: Raven Press pp. 11-23]
velocity” of a nerve fiber, the diameter of that fiber and the distance between nodes. Many papers
have attempted to relate the conduction velocity to the diameter of the fiber, and the conduction
time to the conduction velocity and internode length. However, these are only first order
approximations. To understand the operation of the axon, it is necessary to differentiate the
intrinsic phase velocities and time intervals associated with each electrical element or circuit in
the axon. The conventional models used in electrical engineering can aid significantly in this
area; specifically the models associated with a transmission line and the delay associated with a
pulse signal repeater89.
6.3.8.3.1 Specialized glossary
With these models in mind, the following terminology will be adopted.
Attenuation coefficient-- R/2Z0) + (GZ 0/2)--The loss in signal amplitude as a function of
distance along a transmission line.
Average velocity– The velocity of the leading edge of a signal waveform between two distant
terminals of an active network containing many individual types of delay.
Dispersion coefficient--A factor describing the differential velocity along a transmission line as
a function of frequency. Controls the rate of degradation (typically seen as broadening) of a pulse
on such a line.
Group velocity--The velocity of the leading edge of the signal waveform between terminals of a
passive network. This velocity is usually measured relative to a given location on the leading
edge of a pulse signal. See also average velocity and phase velocity.
Phase coefficient----The change in phase of a sinusoidal waveform with position along a transmission line
Phase velocity--The instantaneous velocity of the current along a specific section of an overall network. For a
high quality distributed element transmission line, this velocity is typically given by the formula: vP = 1/ (LC)½
Characteristic impedance of a transmission line--Zo = (L/C)½ , a term that appears
frequently in transmission line analysis
Capacitance of a coaxial line— C = 2/ln(b/a) in Farads/meter where b is the inside dimension
of the outer conducting cylinder and a is the outside diameter of the inner conducting cylinder.
is the dielectric constant of the medium between the cylinders.
Inductance of a coaxial line— L = (/2) x ln(b/a) in Henries/meter where b is the inside
dimension of the outer conducting cylinder and a is the outside diameter of the inner conducting
cylinder. is the permeability of the medium between the cylinders.
The fundamental electrical values are given in per unit length terms in the above expressions. In
complex circuits, there is always a degree of indeterminacy in the definition of measurement
locations within the circuits. For the Node of Ranvier, the definitions can be related to the
morphology reasonably well.
6.3.8.3.2 Transmission of the Action Potential
Section 10.6.3 presented data supporting the proposition of this model that the Node of Ranvier
is an electrical regeneration circuit within a neuron. Figure 6.3.8-2 from Stampfli & Huxley90
presents the Node of Ranvier in a broader context. The left frame describes the longitudinal
Electrophysiology 6- 85
91Rasminsky, M. & Sears, T. (1972) Internodal conduction in undissected demyelinated nerve fibres J Physiol
(London) vol. 227, pp 323+
Figure 6.3.8-2 The currents measured in
the return paths (the INM) related to the
signals propagating along the axon of a
projection neuron. From Huxley &
Stampfli (1949) in Stampfli (1981).
signal currents (action potentials) within the axon of a projection neuron by collecting and
measuring the total current flowing in the return path external to the neuron. The signal is
proceeding along the fiber, shown on the right, from Node R1 to R5. They indicate the poor shape
of the last two waveforms was due to the knot of hair (possibly from Huxley’s infant daughter,
Sarah, or nylon used to constrain the fiber during testing. The caption of the right frame says the
waveforms describe the difference in the above currents in the return path external to the neuron
at adjacent points separated by 0.75 mm. This waveform is also not a signal waveform. It is
described as a transverse current in the original caption.
Note how the lowest waveform and the last
waveform of each group of three in the left
frame is of poorer quality than the previous
waveforms. They are slightly broader and
reduced in amplitude. This is the result of
passive transmission along the neuron. As
discussed elsewhere, the action potential
only deteriorates slightly before it is
regenerated. This insures good fidelity with
respect to the timing of the pulse. Note also
the very short time interval between the
waveforms representing the signals within a
given axon segment. This is indicative of
the high phase velocity of the signal through
the axon segment. The time between the
last waveform of each group of three and the
first waveform of the next group is
indicative of the regenerator delay at the
intervening Node of Ranvier. Although not
as accurately calculable as in the figure of
Section 6.3.3, the curves are consistent
with the phase velocity of 4,400 m/sec and
the regenerator delay of 0.019 msec.
found there. The waveforms are all
monophasic.
In the right frame, the difference signal clearly displays the shape and amplitude of the degraded
action potential arriving at the node as well as the characteristics of the regenerated waveform. It
is also clear that the node near 7 mm. did not regenerate adequately and the node near 9 mm.
received a low amplitude signal which it may not have regenerated at all. In the original caption,
the tick marks above each waveform were described as occurring at the peak of the voltage
waveform associated with each of these waveforms. It was stated that the potential spread was
continuous in contrast to the current spread in the left frame. However, looking along the row of
ticks shows they are not in a rigorously straight line and do display the same discontinuity as in
the left frame.
Considerable information is presented in Figure 6.3.8-3 from Bostock. It is reproduced here with a series of
dashed lines and a heavy vertical arrow added by this author. The specimen was a rat spinal root axon in-vivo.
(A) was obtained by launching a pulse along an axon using the test configuration of Rasminsky & Sears91 and
measuring the voltage profile at 0.1 mm intervals using a probe next to the outside of the Schwann cell
surrounding the axon. Usually, the detected signal was due to displacement currents. The leakage current
component of the interneuron where it is myelinated is exceedingly low. It is possible the waveforms contained a
conductive component for the probe positioned in the perinodal space. The signal levels were so low that
considerable data processing was used to obtain these waveforms, typically averaging 128 individual traces at
each location. (B) was obtained by taking the trace to trace differences after averaging. This process
emphasizes the major changes occurring in the areas of the Nodes of Ranvier. These waveforms were then used
to construct the contour map in (C). The focus of each contour group represents the best estimate of the location
of the Node of Ranvier (along the horizontal axis). It also represents the time of occurrence of the peak
86 Neurons & the Nervous System
amplitude of the signal at that location relative to the signal at the previous measurement point. Note that the best
estimate of the node position is in fractions of a millimeter in this presentation whereas the actual axial node
dimension is in microns. The probes used had a nominal diameter of 100 microns and effected the precision of
the phase velocity measurement. The contour map is a particularly compact form for interpreting the information
obtained. However, it obscures some features of the raw data. The grouping of the waveforms in (A) is very
important. In (C), overlying a sloping line through the peaks of the contours to obtain a average velocity for the
neural signal over a distance of several nodes is possible. The overlay shown corresponds to an average velocity
of 44 meters/sec., a number consistent with the literature for a myelinated axon of about eight microns overall
diameter. The peaks of the contours represent the peak of the newly regenerated action potential at each node. A
second dashed line can be drawn slightly below this line and with the same slope. This second line is shown as
intersecting the skirt of the contour at approximately the 10% amplitude point. It is meant to represent the
average velocity line measured at the point where the threshold level of the regenerator circuit is exceeded. This
is the point where the integrated current from the transmission line of the previous axon segment has resulted in a
voltage exceeding the threshold voltage.
Looking again at frame (A), one sees a quite different picture. A similar dashed line drawn through the peaks of
each of the individual groups of measurements is very nearly horizontal. At the scale shown, this overlay shows
the phase velocity of the signal within a given axon segment is immensely faster. At the scale of the drawing, the
phase velocity is clearly at least 30 times faster and probably 100 times faster than the average velocity in (C).
The dashed lines in (A) are drawn at 100 times the average velocity slope in (C), e. g. 4400 m/sec. Although
the lines appear horizontal, they are not. Pending further experimentation, the phase velocity of the signal within
an individual axon segment of a myelinated neuron of about eight microns overall diameter will be taken as
4,400 meters/sec. This measured value is at least a factor of 100 times the accepted transmission velocity
(average velocity) associated with a myelinated neuron of this size. If dashed lines, again apparently horizontal,
representing this phase velocity are introduced into (C) as the velocity between nodes, there is a considerable
delay left unaccounted for at each node. This delay is the pulse regenerator delay associated with the charging
time of the “one shot” oscillator circuit that reproduces the action potential. The delay derived from the graph is
approximately 0.19 msec. at each node. This delay will be taken to be 0.019 msec even though Fitzhugh
mentions 0.015 under different conditions. The fundamental conclusion is obvious. The average velocity in a
myelinated neuron is not related to the phase velocity within the axon segments. The delay associated with the
conduction velocity within each axon segment is negligible. The average velocity is calculated from the average
axon segment length divided by the pulse regenerator delay at the associated Node of Ranvier.
Electrophysiology 6- 87
Figure 6.3.8-3 Saltatory conduction in a normal rat ventral root fiber: longitudinal
current, derived membrane currents and membrane current contour map, plotted
to same distance and time scales, with dashed lines and arrowheads added by
this author. A, Averaged longitudinal currents (n=128) recorded at 100 micron
intervals along the root (calibration bar: 4 microamps.). B, Membrane current
records calculated by subtracting adjacent records of (A). C, Membrane currents
re-plotted as a contour map. Inward currents are indicated by continuous lines,
outward currents by dotted lines. Contour interval: 0.5 nA/100 micron electrode
displacement, 37C. Dashed lines relate to signal velocities in the neuron, see text.
Modified from Smith, et. al. (1982)
88 Neurons & the Nervous System
The average velocity, the value normally presented in the literature as the conduction velocity, is not related
directly to the diameter of the axon. It is related to the regenerator delay at the node. This delay may be related
to an axon diameter from a topography perspective. The topography affects both the amount of the bioenergetic
material available to power the pulse generator and the capacitance present. This capacitance determines the rate
of voltage rise of the action potential produced by the generator. According to this hypothesis, the conduction
velocity is only an indirect function of the axon diameter. There is no primary causal relationship and no simple
mathematical equation linking the two. Only a “link” based on data correlation has been described in the
literature. Whereas all of the data in the above figure was measured, it must be pointed out that the voltage
waveforms in other figures in the Bostock paper were not measured. They were created using a computer model
of the axonal signaling process.
6.3.8.3.3 Derivation of the average transmission velocity
Saltatory conduction implies regeneration at each node and regeneration implies a finite delay
associated with the regeneration amplifier. To demonstrate the problem, the average velocity will
be derived using elementary mathematics, to show the relationship of this velocity to the phase
velocity and the average length, not the diameter of the axon segment. If the internode is modeled
as a transmission line, the inductance of that line must be considered along with its resistance
and capacitance if serious distortion due to dispersion (differential phase shift with frequency) of
the transmitted pulse is to be avoided.
Consider an axon consisting of multiple segments called internodes separated by Nodes of
Ranvier. The signal velocity along each individual segment can be calculated based on very
simple measurements; the time required for the peak signal to traverse a measured distance
along that segment. This calculation gives the phase velocity of the electrical signal along that
segment. If the Nodes of Ranvier incorporate an active amplifier, there will be a finite time delay
between the arrival of the peak of the input signal and the departure of the peak of the output
signal. By summing the delays associated with the traverse of each segment of the axon and the
delay associated with each Node, the total delay associated with the axon can be determined. If
the length of the axon is divided by this total delay, the average velocity can be determined for
that axon as follows:
Let VA = The average signal velocity along the axon of total measured length, L
TA = The time required for the signal to traverse the length, L.
TI = The time required for the signal to traverse the physical length of the axon
segments.
TR = The average signal delay associated with each signal regenerator in the Nodes of
Ranvier
VP = The phase velocity of an axon segment given by the average length of an axon
segment, LI, divided by the average time, TA, required for the signal to transit an
axon segment
N = number of node-axon segment pairs included in the length L
and assume that the nodes are of negligible length compared to the axon segments.
Then
VA = (total length of axon)/(total delay through the axon) Eq. 10.6.3-1
VA = L/(N x TR + N x TA) Eq. 10.6.3-2
VA = L/(N x TR + N x( LI/VP)) Eq. 10.6.3-3
but N x LI = L therefore,
VA = (N x LI) /(N x TR + (N x LI)/VP) Eq. 10.6.3-4
by rearrangement,
VA =( LI/TR) x ( VP /(VP + (LI /TR))) Eq. 10.6.3-5
Electrophysiology 6- 89
92Coppin, C. & Jack J. (1972) Internodal length and conduction velocity of cat afferent nerve fibres J Phsiol
vol 222, pp 91-93 Reference #74 in Waxman S. ed. Physiology and Pathobiology of Axons. NY: Raven Press
page 18
93Zagoren, J. Fedoroff, S. (1984) The Node of Ranvier NY: Academic Press pg. 97
94Paintal, A. (1978) Conduction properties of normal peripheral mammalian axons. In Waxman, S. Ed.
Physiology and pathobiology of axons. NY: Raven Press pp. 131-143
VA = ( LI/TR) x (1 /(1 + ( LI/TR)/VP)) Eq. 10.6.3-6
Based on the data from the above figure, VP is at least 100 times larger than ( LI/TR). Therefore,
0.99( LI/TR) < VA < ( LI/TR) Eq. 10.6.3-7
The average signal velocity of a single axon is given almost exclusively by the ratio of the average
axon segment length and the average regenerator delay. Notice this equation for the average
signal velocity does not contain the diameter of the axon in any form. It does contain the
average length of the axon segments. If a relationship can be established between the average
length, LI of an axon segment and its diameter, D, then an appropriate substitution can be made
in the equation. The LI /D ratio is reported in the literature to be species dependent and imprecise
within a species. Rushton said it varied between 100 and 200 in man based on review of scatter
diagrams.
Coppin & Jack92 have suggested that the axon segment (internode) length is linear as a function of
the outer diameter of a myelinated neuron on a semilogarithmic plot but he did not specify the slope.
Berthold, writing on the same page 18 in Waxman (1978) suggests the number of layers of the
myelination sheath is a semilogarithmic function of the diameter of the axon (to diameters of the
axon extending to at least 17 microns).
Zagoren & Fedoroff93 present statistics on the axon segment lengths for cultured fetal mouse and
additional references. Their data supports a standard deviation in segment lengths of about 0.20.
Although there are many statements that the segments of a given axon are of equal diameter,
Berthold gives actual statistical data for several fibers in cat. The standard deviation becomes quite
large for axons of less than three microns, 0.28 to 0.48. This variation may be due largely to
experimental technique. For larger axons, the standard deviation in diameter ranges from 0.06 to
0.225. A standard deviation of 0.225 does not justify a determination that the axon segments
of a neuron are of equal length.
Until a stronger relationship between the length and diameter of an axon segment within a species
can be determined, only a casual link can be assumed between the diameter of an axon and its
average signal velocity, e.g. the conduction velocity. The diameter of an axon may well influence the
phase velocity of a signal, as Rushton derived. However, the phase velocity is largely irrelevant in
calculating the average velocity in a real axon, and more specifically in a real myelinated projection
axon. The amount of myelination is a significant factor (Section 6.3.3.3). Because of this fact,
discussions concerning whether the average velocity is proportional to the first power or the one-half
power of the axon diameter are of little importance.
Paintal94 has provided extensive data on the relationship between the action potential width, rise
time, and absolute refractory period versus the “conduction velocity” of an axon from a cat. He also
presents data on the maximum transmissible frequency versus this conduction velocity and
temperature. All of his temporal data is represented well by a hyperbolic function with respect to the
conduction velocity. If this conduction velocity is in fact the average velocity derived above, the
relationship should be hyperbolic since each of his temporal measures is directly related to the
regenerator delay in Eq. 10.6.3-7. His frequency data introduces the interesting idea that the
conduction velocity is a function of the position of a pulse in a pulse train, leading to dispersion of the
pulses with axon length. He also presents data on the maximum action potential regeneration rate
at a node as a function of temperature. This data appears to show the change with temperature
occurs over a limited biological range and not the typical range expected in chemistry.
90 Neurons & the Nervous System
95Fukada, Y. Motokawa, K. Norton, A. & Tasaki, K. (1966) Functional significance of conduction velocity in
the transfer of flicker information in the optic nerve of the cat. J. Neurophysiol. Vol. 29, pp. 698-714
96Huxley, A. & Stampfli, R (1949) Saltatory transmission of the nervous impulse. Arch. Sci. Physiol., III, pp.
435-448
97Waxman, S. Kocsis, J. & Stys, P. (1995) The Axon. NY: Oxford University Press pg 69
It is possible to compute the phase velocity of the action potential along the axon segment. However,
the nature of the propagation must be known. Historically, it has been assumed to be by conduction,
either within the plasma or along the lemma of the conduit. As discussed in Section 8.2.6, the
conductive velocity of ions within even a dilute electrolyte is quite low compared to the measured
value of action potentials. The transit velocity of the action potential is actually calculated from the
electromagnetic properties of the conduit considered as a loss free coaxial transmission line (See
Section 6.3.5.4).
Much of the electrophysiology of vision has used an even broader definition of average velocity, which
has usually been called “conduction velocity.” This conduction velocity has included the time delay
associated with the photoexcitation/de-excitation process, with the signal processing process and the
delays associated with the regeneration amplifiers of the projection neurons. This expression of
conduction velocity contains several unspecified variables since the delay associated with the P/D
process is a state variable dependent on temperature, illumination, and illumination history.
Furthermore, it is clearly an average velocity. Fukada, et. al95. give a average velocity under these
conditions for a cat (a warm-blooded animal) of 10 to 70 m/sec when the measured travel distance
ranged from 34.5 to 38.9 mm (average of 37.3 mm).
Three important conclusions;
The label conduction velocity has been applied to an algebraic calculation for “speed,”
an overall distance divided by an overall time to cover that distance. This notion is
called a average velocity in this work in analogy to electrical usage.
The instantaneous velocity along a neuron is much higher and corresponds to a phase
velocity in analogy to a telephone or similar system, and based on the calculus. It can
be properly called the transit velocity of the electromagnetic wave associated with the
action potential.
The diameter of an axon may exhibit a casual relationship but does not exhibit a
causal relationship to the conduction velocity (average velocity) or the phase velocity
of a neuron.
6.3.8.3.4 The intrinsic propagation velocity on a lossy line
Computing the signal velocity along an extended neural conduit is difficult because of the number of
variables. Reference is made to the table in Section 6.3.3.4.4. Equation (A) of Section 6.3.3.4
describes the velocity of transmission of a pulse over a lossy coaxial cable. The velocity of propagation
over a loss-free line was found to be a large fraction of the speed of light and determined only by the
permittivity and permeability of the dielectric. For a lossy cable, this velocity is reduced considerably
by the values of R and G. Little data is available in the literature describing the phase velocity of
action potentials traveling over real axons. Huxley and Stampfli provided the earliest data96. The
principle graphic is reproduced by Ritchie writing in Waxman, et. al97. Smith, et. al. have provided
Electrophysiology 6- 91
98Smith, K. Bostock, H. & Hall, S. (1982) Saltatory conduction precedes remyelination in axons demyelinated
with lysophophatidyl choline J Neurol Sci vol. 54, pp 13+
99Bostock, H. (1982) Conduction changes in mammalian axons following experimental demyelination, in Culp,
W. & Ochoa, J. Abnormal Nerves and Muscles as Impulse Generators, NY: Oxford University Press pp 236-
252
100Bostock, H. (1993) Impulse propagation in experimental neuropathy. In Peripheral Neuropathy, 3rd. Dyck,
P. & Thomas, P. Ed. Philadelphia: W. B. Saunders pp. 109-120
101Rasminsky, M. & Sears, T. (1972) Internodal conduction in undissected demyelinated nerve fibres J Physiol
(London) vol. 227, pp 323+
some data that updates the Huxley and Stampfli results98. Bostock has presented additional data99,100.
The first three figures of Smith, et. al. describe real test configurations and real results. However,
the reader is cautions, the mathematical models used to create figures 5-4 and beyond do not
represent the propagation of a real signal along a real axon. No phase constant was introduced into
their equations. In the absence of any phase constant, no dispersion occurred. Similarly the constant
amplitude in the absence of a phase constant suggests no attenuation constant was introduced either.
Although the experiments of Smith, et. al. were not optimized for measuring the velocity of action
potential propagation, an estimate can be made from their data. Figure 6.3.8-4 provides an
annotated version of their data. Frame (A) of their data shows the arrival time of an action potential
propagating along an axon segment. The individual groups of wavy lines indicate the profile recorded
at different locations. These groups are separated by a larger time interval associated with the
regeneration time at an Node of Ranvier. The horizontal scale suggests the axon segments were
approximately one millimeter long. The time scale suggests the pulses move one millimeter in less
than a small fraction of a millisecond. By connecting the peaks of the waveforms, a line describing
the velocity of the action potentials is obtained. The four lines shown all suggest a phase velocity on
the order of 4400 meters/second. While this value is far below the speed of light, it is far above the
velocities of signal propagation normally found in the biological literature. The obvious reason is the
values usually given in the literature refer to the average velocity of signal transmission. This
average value includes the delays inherent in the signal regeneration mechanism associated with
each Node of Ranvier. Smith, et. al. calculated this average value using the data in frame (C), as 45
meters/sec (essentially the same as the 44 meters/sec calculated by overlaying a straight line on their
figure). The precise spacing of the Nodes of Ranvier can be obtained from their differential traces in
frame (B). These were obtained by differencing the pairs of traces in (A) after indexing them in time
to account for the delay due to the phase velocity. The average value is about 930 microns for the
ventral root fiber selected.
It is not clear why both Huxley & Stampfli and Smith, et. al. calculated an average velocity for their
data but neither group addressed the obviously much higher phase velocity in their data.
Rasminsky and Sears101 have provided similar data for a demyelinated ventral root nerve. Even in
this condition, their data shows propagation velocities of 500 to 600 meters/sec or higher (in some
cases, much higher by scaling).
The nominal value of 4400 meters/second for the phase velocity of an action potential is initially
surprising. This value is approximately three times the speed of sound in fresh water (1435
meters/second) and salt water (1500 meters/second). Since particle diffusion within a electrolyte could
not possibly travel at a velocity exceeding the speed of sound in the medium, the idea of ion transport
as the mechanism of action potential propagation is completely ruled out by this data. While electron
or hole transport through the electrolyte, or along the surface of the membrane remain possibilities,
propagation by electromagnetic means is clearly the most logical explanation for the mechanism of
signal (action potential) propagation along an axon segment. This data also suggests the description
of the pathology of a demyelinated neuron should use the term propagation failure and not conduction
failure.
6.3.8.3.5 Performance of the pulse signal repeater
A parameter of major interest is the time delay associated with the regenerator circuit of a projection
92 Neurons & the Nervous System
102Waxman, S. Kocsis, J. & Stys, P. (1995) The Axon. NY: Oxford University Press
103Waxman, S. Kocsis, J. & Stys, P. (1995) The axon. NY: Oxford University Press. Chap. 13 thru 18
neuron. This time delay is most easily calculated as the interval between the peaks of two waveforms.
The first waveform is measured at the input circuit of the regenerator. The second waveform is
measured at the output of the regenerator. The time delay between these two waveforms is typically
0.019 msec. at 37 Celsius.
If the regenerated action potential pulse is reasonably symmetrical, this would imply the action
potential would have a pulse width near the 10% amplitude points of about 0.38 msec. Paintal has
provided data at 37C for the action potential pulse width as a function of conduction (average) velocity
in myelinated neurons. The value for 44 m/sec. is 0.4 msec. The action potential is frequently
assymmetrical under both normal and pathological conditions.
Reviewing the material in Section 6.3.5.4.2, the expected regenerator delay associated with each
Activa circuit at a Node of Ranvier in a projection neuron can be seen to be about 0.019 milliseconds
at a 32 C. The exact delay is determined by the time constants of the oscillator circuit and to a much
smaller extent the threshold setting of the amplifier. These are the same time constants that
determine the shape of the action potential generated. For the nominal action potential generator
used in this work, and derived from figure 11-11A in Waxman et al102 .
Waxman, et. al103. provides a mine of electron micrographs and experimental data. The volume
includes comparisons between action potentials in the same specimen at different temperatures
and, unfortunately, with a variety of uncontrolled variable.
The time constants of a warm blooded animal (the rat) at 37°C are available in Section 9.2.6.2 and
the resulting nominal delay is in agreement with the above 19 microseconds. (Section 6.3.8.3.2). With this
number in hand, plotting another dashed line in is possible (C) that is parallel to the average velocity
line through the contour peaks. This line would be drawn through the leading edge of the contours
at the level of threshold exceedance. From a signaling theory perspective, this line would represent
a more precise point separating the arrival of the signal at each node. It is the point where sufficient
charge has been integrated onto the input capacitance of the repeater to cause oscillation of the
circuit. As will be discussed below, this point is subject to distortion by pharmacological factors
affecting the Node of Ranvier, specifically the electrostenolysis of the pre-nodal axon segment.
With a detailed circuit diagram of the regenerator amplifier in a Node of Ranvier, it is possible to
correlate the measured data for the absolute refractory period, ARP, of Paintal with the calculated
values. Recognizing that the average signal velocity is dominated by the inverse of the regenerator
delay interval, it is easy to see “the obvious inverse relation between the conduction (average) velocity
and the ARP . . . ”
6.3.8.3.6 Performance of the axon segment
Investigators have struggled for a long time to determine the electrical performance parameters of the individual axon
segment. Their efforts have invariably assumed a simple resistor-capacitor transmission line, with leakage through
the axolemma (or the combination axolemma-myelin-Schwann cell) frequently assumed. Since their work has lacked
a definitive model and been largely exploratory, the measurements have usually involved more than just the axon
segment. In this work, the transmission line and its phase velocity are proposed to be quite different from this earlier
work. The phase velocity of the signal traveling between the two paranodes of an axon segment is at least 100 times
faster than assumed in the previous literature. There are no resistors in the conventional sense in the proposed
electrical model of the axon segment and the phase distortion of the neuron is quite low. Because of these facts, the
applicability of a simple transmission line, consisting of resistors and capacitors and frequently used in the literature,
is questionable.
If one compares the pulse waveforms associated with one of the axon segments in the figure, the attenuation
coefficient and the dispersion coefficient of an equivalent transmission line can be estimated. If one looks at the
shape of the typical action potential and the greatest length of time between action potentials, an estimate can be made
of the harmonic content of the overall waveform. This was done for the 1st and 5th waveforms in the second group.
The 5th waveform is 60% of the height of the 1st. There is no appreciable pulse broadening. There may actually be
some pulse sharpening. Comparing a typical action pulse width of about 1.5 msec to a maximum pulse interval of
Electrophysiology 6- 93
more than 50 msec suggests that the signaling channel must be quite broadband. The situation can be compared to
a train of cosine-squared pulses. The ratio of the period to the half pulse width is about 66 and the signal returns
quickly to the baseline following the pulse. This type of performance requires a signaling bandwidth for the internode
that is flat with respect to frequency over at least 100 times the interpulse frequency of 20 Hz. These facts suggest
that;
+ any transmission line used to represent the axon segment must be of the resistor-inductor-capacitor, RLC, type such
that the phase velocity is constant with frequency across the Fourier bandwidth of the action potential,
+ the attenuation coefficient is 0.12 per 100 microns and the loss is quite probably related to dielectric heating in
the capacitor formed by the axolemma and myelin sheath of the axon segment.
No values could be found in the biological literature for the inductance associated with an axon. The values found
for the resistance of an axon vary over orders of magnitude due to lack of a model of the circuit being measured and
limited instrumentation. Usually, the measurements do not separate the series impedance of the conduit formed by
the reticulum from the impedance of the membrane at each end of the conduit. The impedance of the membrane at
each end of the conduit is that of a diode. Unless care is taken, the experimenter is liable to measure the axon by
measuring two back to back diodes and the intervening conduit resistance without knowing it. The impedance of two
back to back diodes is state sensitive, generally high, and essentially indeterminate. On the other hand, the impedance
of an electrolyte of less than a few millimeters is quite low. Measurement of the shunt capacitance of the axon, both
within the STIN of an axon segment and the paranodes, is quite difficult because of their size and frequently poor
experiment design.
Because of the frequent use of a small probe in electrophysiological experiments, the capacitance values in the
literature appear to relate more to the probe than the actual neuron. These probes are usually immersed in an
interneural matrix, or a man-made equivalent. They are frequently mounted so they can be easily moved along the
length of the axon. Therefore, the nature of the circuit between the actual neuron and the probe is quite indeterminate.
This makes determination of the fundamental electrical characteristics of the axon segment with this apparatus very
difficult.
The capacitance between a cylindrical structure such as an axolemma immersed in a surrounding fluid is easy to
calculate. The difficulty lies in determining the exact form of the electrostatic fields present. For a highly conductive
external medium and a highly conductive internal conductor, the problem devolves to that of a simple coaxial cable.
Moving one step closer to the real situation, the interneural matrix appears adequately conductive to allow the coaxial
cable equation to be used. However, there is some concern about the nature of the plasma within the dielectric
formed by the lemma. The question relates to the material outside the reticulum. Is it conductive or should it be
considered a dielectric? The following calculation will consider this material sufficiently conductive that its dielectric
properties are inconsequential.
Using the equation for the capacitance of a coaxial cylinder, the capacitance between the two surfaces of an axon is
easily calculated. Using an axolemma diameter of 8 microns, a relative dielectric constant of 5.0, and a membrane
thickness of 8.0 nm for an unmyelinated axon, a typical capacitance per unit length of the axon would be expected
to be near 1.4x 10-6 Farads/meter. For a myelinated axon of the same dimensions but with a combined membrane-
myelin thickness of 3.5 microns, a capacitance of 4.4 x 10-10 would be expected. The impact of the myelin on the
axon shunt capacitance is striking--reducing it by a factor of 3,200:1. Most lipid tissue has a relative dielectric
constant of 5.0 or less.
The above calculation applies to both static charges and charges moving along the length of the dielectric. However,
if the charge is moving along the dielectric, it is necessary to calculate the inductance associated with this same
structure. This inductance can be used to determine how the signal travels along the cylinder and the input impedance
of the overall structure to excitation at a given temporal frequency. No calculation of the inductance or input
impedance of a neuron could be found in the literature. The inductance of a coaxial transmission line can be
calculated using a formula similar to that for the capacitance. However, there is an additional complication if the
current along the conductors is not confined to the surface of the two conductors. The general effect is to increase
the inductance if the current is able to flow through the entire cross section of the inner conductor. If the inner
conductor is made in a helical form, the additional inductance can be great enough to change the phase velocity of
the signal along the transmission line. Thus, it may be necessary to consider the configuration of the microtubules
within the reticulum if precise results are important. Using the equation for a coaxial inductor, an axolemma diameter
of 8 microns, an absolute relative permeability of 4 x 10-7 henries/meter (reative permeability 0f = 1.0), a
membrane thickness of 8.0 nm for an unmyelinated axon, and a membrane-myelin thickness of 3.5 micron for a
myelinated axon, typical inductances per unit length of each axon type would be expected to be near 4 x 10-11 and
1.25 x 10-7 Henries/meter respectively. The impact of the myelin is again significant.
It is important to note that if any of the conductive fluid associated with the interneural matrix
94 Neurons & the Nervous System
104Schnapp, B. Peracchia, C. & Mugnaini, E. (1976) The paranodal axo-glial junction in the central nervous
system studied with thin sections and freeze-fracture. Neuroscience, vol. 1, pp 181-190
105Ritchie, J. (1984) Physiological basis of conduction in myelinated nerve fibers. In Myelin, Morell, P. Ed.
pg. 123
should enter the space between any of the layers of the myelin coat or the periaxonal space between
the axolemma and the myelin, the capacitance calculated above for the myelinated axon would be
greatly reduced. In the same situation, the inductance would be greatly increased. It is extremely
important that the myelin layers form tight seals between themselves and with the axolemma. This
is the primary purpose of the “barley corn” morphology of the myelin near a Node of Ranvier and
of the repeating ridges and grooves found surrounding the paranode of each axon segment. These
ridges and grooves are formed by the constricting of the axon by the myelin layers at their point
of termination. This structure constitutes a labyrinth seal; one of the most effective seals against
fluid leakage. Although the purpose of these structures is not obvious from morphology104, they are
clearly hydraulic features designed to maintain the electrical integrity of the axolemma-myelin
interface.
For the above myelinated internode that is 2 mm long and has an unmyelinated portion within the conseg at each end
of 5 microns length, the capacitance of the myelinated portion would be 8.8 x 10-13 Farads or 0.88 pF. The
capacitance of each conseg portion would be 7.0 x 10-12 Farads or 7.0 pF. These numbers are very significant. The
capacitance of the very small portion within the conseg is about ten times larger than the capacitance of the entire
length of the myelinated portion.
Using the pairs of capacitances and inductances for the unmyelinated and myelinated case, a
maximum phase velocity and a characteristic impedance for the axon segment as a transmission line
can be calculated. See Section 6.3.3.3 .
Ritchie105 has provided a set of consistent electrical values (except for the inductance) for a myelinated nerve fiber.
For a axon segment that is 1.38 mm. long and an axonal diameter of 10.5 microns (a total external diameter of 15
microns), he gives a node capacitance of 1.6 pF for each paranode and 2.2 pF for the intervening STIN. The value
for the STIN would correspond to approximately 1.6x10-6 Farads/Meter. This value is within 15% of that calculated
above. Ritchie did not provide a length for the paranode in his presentation. His paranode capacitance is lower by
a factor of five than that calculated above.
By accumulating the values in previous paragraphs, estimating the missing parameters in the axon model of this
theory is possible. For the axon segment as a transmission line with a phase velocity equal to or exceeding 4,400
m/sec and a capacitance per meter of 1.6 x 10-6 F/m, the inductance required of the model is less than or equal to 3.22
x 10-2 Henries/Meter. For a two micron long axon segment, the series inductance of the transmission line equivalent
would be less than 6.4 x 10-8 Henries.
Using the inductance and capacitance of the line, calculating the characteristic impedance of the line is possible. For
C = 4.4 x 10-10 F/m and L = 3.22 x 10-2 H/m, the characteristic impedance is 8.5 x 103 Ohms. Using this value and
the attenuation coefficient of the line, computing the dissipative losses along the line is possible. The relevant
equation is:
= (R/2Z0) + (GZ0/2) Eq. 10.6.3-8
At present, there is little information available concerning the relationship between the series resistance and the shunt
conductance of the axon. The equation can be evaluated based on only one element being present. The result will
provide two worst cases. Using = 0.12 per 100 microns, and Z0 = 8.5 x 103 Ohms,
R = 2 x x Z0 = 2040 Ohms/ 100 microns For G=0 Eq. 10.6.3-9
G = (2 x )/Z0 = 2.8 x 10-5 Mhos/100 microns For R=0 Eq. 10.6.3-10
Because this is a myelinated axon, and the myelination forms an excellent insulator, the assumption will be made that
the insulator conductance is zero and the losses in the transmission line are due to the series resistance, R. By this
analysis, the series resistance of the transmission line representing a two micron long axon segment is very small,
typically 41 Ohms for a signal phase velocity of 4,400 m/sec. , compared with the values in the literature. The value
of 41 Ohms is quite large for the series resistivity of a simple electrolyte. This value may represent the sum of the
Electrophysiology 6- 95
resistance in the reticulum electrolyte and the resistive component of the diode formed by the axolemma wall(s) at
the Node of Ranvier. Here, the impedance of the two walls would be represented by the transimpedance of the Activa
within the Node of Ranvier when properly biased. It is not obvious how many of the values in the literature were
obtained. Values in the megohms for the series resistance of a signaling medium do not appear compatible with a
signal channel bandwidth of 1000 Hz and low driving power. Most of the published values are highly test
configuration and test environment dependent and most of them were obtained in-vitro. If the specimen were not
operating under in-vivo bias conditions, the predicted measurement is indeterminate and almost any resistance value
could be measured during a given test procedure.
It should be recalled that this signal transmission process is taking place in a regime where quantum level interfaces
may be significant within the liquid crystalline environment of the reticulum. In addition, there is presently no
assurance that a transmission line is the proper model for the axon segment. A simple model based on hole transport
may be more appropriate.
6.3.8.4 Effect of pharmacological agents on the Action Potential
The observed current associated with the falling portion of the action potential is also shown in
[Figure 10.5.4-2]. There are two distinct current paths within the regenerator circuit related to
recharging. The recharging current associated with the emitter circuit is almost entirely internal but
may be observable by instrumentation monitoring the plasma of the pre-Activa conduit. The more
commonly observed recharging current is that associated with the collector terminal of the Activa
since it is observable by instrumentation monitoring the post Activa conduit plasma. Here again,
most of the current flows within the nodal membrane and the recesses of the nodal gap. However,
a portion is used to drive the post Activa conduit plasma. This portion results in a current passing
through the nodal gap.
Based on the proposed electrical structure of this neuron, superfusion of a nerve with 4-
aminopyridine, (4-AP) can and did affect the recorded waveform. However, it probably did so by a
different mechanism than described by Bowe, et. al. There are many ways 4-AP could affect the
observable operation of a neural regenerator circuit. The simplest is probably to change the
conductivity of the electrolyte surrounding the neuron. The material could also disturb the
electrostenolytic process supporting the collector circuit of the Activa. Such a disturbance could
change the intrinsic electrostenolytic potential supporting the plasma, the intrinsic impedance of the
membrane, or possibly the effective capacitance of the membrane. The last two possibilities would
cause a change in the recharging time constant that would be observed as a change in the time
constant of the falling portion of the action potential.
An alternate mechanism impacting performance would be by changing the nature of the material
coating the outside of the neuron near the poda, the third electrical terminal of the neuron. The state
of this material amounts to an uncontrolled variable that has not been controlled in most past
experiments. The intrinsic electrostenolytic potentials developed in conjunction with this material
and the podal membrane affect the quiescent operating point of the Activa. Furthermore, this
electrostenolytic process may affect the impedance of the podal power supply. This impedance
primarily affects the discharge (rising) portion of the action potential. However, this impedance is
in series with the collector power supply that is believed to dominate. The near identity of the two
rising waveforms in Bowe, et. al. would support this assumption.
It the podal power supply is affected sufficiently to affect the emitter to base voltage of the Activa, the
regenerator circuit could be transformed into a gated free running oscillator. This appears to have
occurred in other waveforms presented by Bowe, et. al. In those cases, the shape of the action
potential remains the same but the interval between action potentials is now controlled by two
parameters. These are the steady state bias voltage between the emitter and base, and the recharge
time constant of the emitter circuit.
By affecting the podaplasm potential, 4-AP can also impact whether the Activa generates more than
one action potential as Bowe, et. al. indicates frequently happens in dorsal root fibers from similar
animals of the same age. However, the primary impact of 4-AP appears to be on the impedance in
the collector circuit of the regenerator. It probably impacts the impedance of the electrostenolytic
process at that location and thereby causes a lengthening of the recharging cycle. This is observed
as a lengthening of the falling time constant of the action potential.
6.3.8.5 The pathological condition of transmission over a demyelinated
96 Neurons & the Nervous System
106Shrager, P. & Rubinstein, C. (1990) Optical Measurement of Conduction in Single Demyelinated Axons
J Gen Physiol vol 95, pp 867-890
107Lafontaine, S. Rasminsky, M. Saida, T. & Sumner A. (1982) Conduction block in rat myelinated fibres
following acute exposure to anti-galactocerebroside serum. J Physiol (Lond.) vol 323, pp 287-306
axon
6.3.8.5.1 Chemical demyelination & regrowth of frog sciatic nerves
Shrager & Rubinstein106 have provided detailed information on the physiology and pathology of frog,
Xenopus, sciatic nerves with demyelination and the regrowth of myelination in a series of papers,
including mammalian subjects, cataloged in the 1990 paper. In the 1990 paper, they find the type
2 lemma extends under the normally myelinated regions of the axon segment. They employ an RC
model of their axolemma and the Schwann cells, myelin, when present. They do not treat the axon
as a coaxial cable with an inductive component. Table 1 gives parameters for the elements of their
cable. Their capacitance values per unit area appear quite low; they do not give a value for their
typical myelinated neurons.
“All experiments were run at room temperature.”
They used a voltage-clamp configuration to establish a resting potential of -65 mV. They also
provided electron micrographs of the neurons during the life of their experiments.
The demyelinated axon typically propagated an action potential at 1.1 m/sec or less over a distance
of 2.58 mm. Figure 8A shows considerable phase distortion along the length of a demyelinated axon
segment. They report a normal myelinated neron propagating at an average velocity of 15 m/s (page
873).
Lafontaine et al107. made similar measurements to those of Shrager using a rat.
Lafontaine et al. relied upon a simple RC cable for their model and the chemical theory of the neuron.
When they speak of sodium, Na+, passing into the axoplasm of a neuron, this is equivalent to electrons
escaping from the axoplasm via the Activa conducting charges; when they speak of K+ leaving the
axoplasm via the external lemma, this is equivalent to the axoplasm being recharged by electrons
from the electrostenolytic source on the surface of the outer surface of the exterior lemma. The need
for a sodium ion pump disappears under The Electrolytic Theory of the Neuron (Section 2.7.5).
Lafontaine et al do not provide any velocities for their transmission along a demyelated axon.
6.3.8.5.2 Matching a Node of Ranvier to a cable
The many figures in the cited paper by Bostock (1993) provide valuable information concerning the
performance of the regenerative amplifier at a Node of Ranvier. Their interpretation leads to many
suggestions for expanding this work. The contour profile of a given node is tear drop shaped at the
scales shown. The contours are quite symmetrical horizontally because of the dominance of the probe
size. They tend to be symmetrical vertically because they are dominated by the shape of the action
potential. However, under a variety of pathological conditions, the decay characteristic of the action
potential may be altered.
If the vertical contour profile of a given node varies over the short term, it is generally associated with
poor performance of the circuit power supply or a significant increase in the circuit capacitance.
These parameters both affect the driving impedance of the following axon segment transmission line.
In either case, the result is an abnormally slow recovery of the “one shot” regenerative oscillator. This
is usually reflected in an extended “refractory period” for the oscillator and a lower maximum pulse
frequency for the signaling circuit. The large vertical arrow in the annotated Smith et al. Figure
9.1.1-12, (Section 9.1.1.5.4) shows the tendency of the contour to become elongated under these
circumstances. Contour plots tend to show a considerable variation in the performance of adjacent
and nearby regenerative amplifiers. Many of these changes have been observed to be caused by de-
myelination of one or more axon segments. In other experiments, they have been caused to occur via
Electrophysiology 6- 97
pharmacology. Under severe pathological conditions, the Activa may fail to oscillate and the
regeneration process may be stopped or exhibit signaling fibrillation.
Bostock’s figure 5-10 shows a situation that he interprets as a signal reflection, due to
pharmacological intervention, along the axon involving several axon segments. Section 6.3.5.2
offers a totally different interpretation based on an impedance mismatch, that may be aggravated by
pharmacological intervention. No model was presented in that paper that supported two way signal
communication through a Node of Ranvier. It is suggested, based on this theory, that a different
interpretation is needed.
Reflection is generated at an impedance mismatch (Section 6.3.5.2).
Reflection at a given pre-nodal terminal of a single axon segment is conceivable if the impedance of
the node is grossly changed compared with the impedance of the associated transmission line. This
could be easily explained by pharmacological intervention. However, reflection back through a series
of nodes is not supported by this theory. The question does raise the suggestion that similar contour
maps be prepared using multiple pulse excitation of the neuron where the interpulse interval is a
variable. The result would be two nominal average velocity loci separated by the interpulse interval
on a single graph. The data of Paintal would suggest this interpulse interval should decrease toward
the range of 1.0 to 1.6 msec. for an eight micron myelinated neuron. Many additional effects related
to fatigue in the regeneration amplifiers may become apparent.
As better data becomes available, it would be interesting to learn whether the fluting of axon
segments (Fig. 2.6.3-2B in Section 2.6.3.2) was employed to match the terminal impedance of the
NoR to the characteristic impedance of the axon segment transmission line.
The subject of matching the impedance of an axon segment to a NoR is not addressed in
the chemical theory of the neuron. See Section 6.3.5.2 for more details.
6.3.9 The time delays relevant to the neural system
The term latency, as used in physiology and psychology, is usually defined in terms of the time of
occurrence of an event at a point in the neural system relative to an external stimulus. On the other
hand a delay is usually related to the time difference between related events occurring at two
different points within a system, One or more identifiable delays frequently contribute to a latency.
A given delay is usually defined in terms of a distance a signal travels divided by the average velocity
with which it travels. The range of the delays found within the neural system is remarkably large
because of the range of distances, and the drastically different velocities of the mechanisms, involved.
Distances range from tens of Angstroms (nanometers) in the thickness of the base region in an Activa,
through microns in the diameters of neural cells, to ten meters for the longest neurons in a whale.
There are total time delays and differential time delays amounting to milliseconds within the visual
system of warm blooded animals. The delays may approach a second in cold blooded animals at
temperatures near the freezing point of water. Documentation of these delays for a specific species
is straightforward but beyond the scope of this work.
6.3.9.1 Sources of signal delay in the external sensory neurons
There are four fundamentally different types of signal delay within the neural system. They are
based on different mechanisms and exhibit different characteristics. The four primary types of signal
delay are:
1. Delays based on quantum-mechanical mechanisms. The E/D Equation of the external sensory
neurons is a primary source of this type of delay. The delay is dominated by the transduction
mechanism transferring the signal to the input of the 1st Activa within the neuron. This delay is a
function of the intensity of the stimulus and therefore a differential delay as a function of intensity
(Section 8.10 ).
2. The delay due to the finite velocity of fundamental charges and ions within the electrolytes of the
neural conduits. This velocity is highly dependent on the precise concentration of these electrolytes
and whether they support charge transmission within a lattice associated with a liquid crystalline
state of matter or merely within a fluid state of matter. The delay is typically a small fraction of a
98 Neurons & the Nervous System
millisecond. It is encountered in all stage 1, 2, 4 & 5 neurons.
3. The delay related to the finite velocity of propagation of electrical signals in a non-ideal medium.
This delay is typically a small fraction of a millisecond for each millimeter of distance. It is
encountered in all stage 3 & 6 neurons.
4. The delay related to the average velocity of action potentials within the signal propagation stage
of the visual process. This group delay is in itself made up of two different delay mechanisms. The
first mechanism is the finite delay at each monopulse oscillator (typically a Node of Ranvier). This
delay is between the peak amplitude of the input excitation waveform and the peak amplitude of the
subsequent output waveform. The second delay mechanism is the finite phase velocity of the
electromagnetic transmission of the signal information along the transmission line formed by the
axon. The total delay is dominated by the regeneration delay and is usually a significant fraction of
a millisecond.
Each type of delay is the result of a signal of finite velocity traversing a different physical distance.
These three fundamentally different kinds of signal delay are based on a total of four different
mechanisms. These mechanisms are the exciton velocity within a quantum-mechanical energy level
of a crystal, the charge velocity within an electrolyte, the rate of charge buildup on a capacitor as a
result of electrical switching by an Activa, and the energy velocity along an electromagnetic
transmission path. The properties and extent of each of these types of signal delay must be
interpreted correctly to understand their role in the visual process.
There is a fifth type of signal delay involved in each of the active devices of vision. This is the velocity
of the majority charge carriers within the device. This velocity is so high and the distance involved
so small that this type of delay appears to be negligible within the neural system. It is usually
measured in nanoseconds in similar man-made devices.
6.3.9.2 Effects of temperature on each source of signal delay
The sensitivity of each significant type of signal delay to temperature is in the order they appear
above. The sensitivity of each is unique. Evaluating the parameters of the P/D equation based on
the experimental literature has demonstrated that the temperature coefficient associated with exciton
velocity is clearly not that associated with a conventional Arrhenius Equation with Absolute
temperature as a variable. The sensitivity of charge velocity within the electrolytes is even more
complex because of the changes in state that may occur with changes in temperature. The sensitivity
of energy velocity associated with pulse transmission along an axon appears to be relatively
independent of temperature. The major change appears to be related to the expansion coefficient of
the material forming the dielectric wall of the axolemma.
6.3.9.3 Primary uses and effects of signal delay within the visual system
Each type of signal delay discussed above has a different impact on the visual system. These impacts
are largely related to the relative distances each type of signal must travel within the visual system.
The delay associated with exciton velocity is dependent primarily on the dimensions of the Outer
Segment in relation to the base of the adaptation amplifiers of the photoreceptor cells. In general,
these dimensions do not vary with spectral sensitivity. Therefore, there is little difference in delay
between the different spectral channels due to the effects of exciton velocity and geometric distance.
There is a delay in each spectral channel associated with the signal level. However, for normal scene
content (small signal conditions), this difference in delay between spectral channels remains low.
Under laboratory conditions where large signal conditions may be imposed, this difference may be
measured. It can be milliseconds for warm blooded animals and major fractions of a second for
animals at zero to eight degrees Celsius. This delay is discussed in detail in Section 12.7.2.
The delay associated with charge velocity is important in the operation of many areas of signal
processing. It is particularly important in the signal processing associated with the 2nd lateral
processing matrix of the retina. Whereas the literature has generally explored the differencing of
signals between nearby signal paths as if they were synchronous with each other, this is not the case.
The mere length of the neuritic structures associated with a single amercine, or other lateral cell, can
introduce a significant temporal delay into the signal processing. This delay is very valuable in
Electrophysiology 6- 99
108Segev, I. & London, M. (1999) A theoretical view of passive and active dendrites In Stuart, G. Spruston, N.
& Hausser, M. Dendrites. NY: Oxford University Press pp 205-230
109Rall, W. (1977) Core conductor theory and cable properties of neurons. In Kandell, E. ed. Handbook of
physiology. The Nervous System: Cellular Biology of Neurons. Bethesda, MR: American Physiology Society.
pp 39-97
forming spatial filters, of the “comb filter” type, with respect to object space. This process is discussed
more fully in Section 9.2.2.3.
The delay associated with the transmission of action potentials has only recently achieved prominence
with the discovery and documentation of visual signals in the higher levels of the cortex that occur
before any signals have appeared in the so-called “primary visual cortex.” Clearly, these signals have
not passed through the primary visual cortex (area 17). It is proposed that they entered via the
Pulvinar pathway. It is also proposed that these are the signals uniquely related to the foveola and
critical to the fine discrimination associated with reading and other functions in man.
Although the inherent phase velocity of action potentials is quite high relevant to other velocities, the
considerable delay associated with the regeneration of the action potentials at each Node of Ranvier
and the physical distances involved between the retina and the cortex make these delays associated
with the action potentials quite important.
6.3.10 The complex electrical structure of the ramified neurites
The neurites of neurons are found in four general configurations:
1. Podites of negligible length making contact with the INM immediately adjacent to the neuron.
2. Dendrites of minimal length making contact with only one preceding axon.
3. Long thin dendrites found in the associative memory circuits of the thalamic reticular nucleus.
4. Highly ramified neurites found in all of the signal processing circuits of the feature extraction
engines.
Category 1 is of very little technical interest. The dendrites of category 2 are usually so short that
they can be considered lumped circuits. In the rare case where they are of significant length, they
can be treated mathematically as an axon segment. The long thin neurites of stage 2 neurons
corresponding to category 3 have not been characterized mathematically in the literature. It is likely
that these dendrites, particularly if myelinated, can also be considered mathematically as axon
segments. Attempts to describe the electrical performance of the neurites of stage 2 neurons
corresponding to category 4 have been made by Segev & London108 and by Rall109. Both investigators
ignored the potential inductance of the circuits.
6.3.10.1 The organization aspects of the stage 2 neurites
Segev & London have discussed the top level architecture of the neurites from both architectural and
computational perspectives. Their framework is based on the convention of sodium and potassium
currents and they do not differentiate clearly between pulse and electrotonic circuits. However, many
of their observations are useful. They note; “one should expose the (morphological) ‘face’ and the
(electro-chemical) ‘character’ of neurons hoping that, with persistent experimental effort, an
understanding of the functional consequences of the ‘neuron-ware’ will emerge.” They reference the
earlier work of Rall (1959 & 1964) incorporated in his 1977 paper. Unfortunately, they do not extend
their analyses to include the cytology and electrolytics of the relevant neurons.
Segev & London note the typical stage 2 neurite (a dendrite of a central neuron of the CNS in their
terminology) may synapse on the order of 10,000 times with adjacent axons. However, they do not
recognize that these neurons process electrotonic signals. They suggest each synapse may operate
at bandwidths up to 100 cycles per second They use the notation 100 pulses per second which applies
to stage 3 neural circuits. Such stage 3 circuits seldom involve more than a dozen synapse. Their
subsequent analysis applies primarily to stage 3 (phasic) neurons. They do recognize the utility of
considering the distributed parameters of each branch of a neurite and adopt a compartmental model.
100 Neurons & the Nervous System
110Taylor, W. Mittman, S. & Copenhagen, D. (1996) Passive electrical cable properties and synaptic excitation
of tiger salamander retinal ganglion cells.
They differentiate this model from Rall’s cable model. However, they ignore the inductance of the
conduit involved and continue to rely upon the one-dimensional “heat flow” equation. They develop
seven ‘insights’ based on their passive neurite model. While provocative, they are less than definitive
and each deserves additional study. Their suggestion that ‘dendritic [and poditic] spines are favorable
sites for plastic changes’ is almost certainly relevant to the role of these features in achieving long-
term memory.
Segev and London advance their work with their assertion that “the actual synaptic input is a
conductance change with an associated reversal of potential (battery ) rather than a current input.
Thus, for a strong conductance change, the synaptic potential will saturate as it approaches its
reversal potential.” If their electrolytic model was more defined, they would word this assertion
differently. The synapse exhibits a variable transconductance which reaches a maximum as the
collector potential of the Activa approaches saturation potential. The remainder of their discussion
suffers from a lack of a detailed model of the neuron to the levels of conexus and the Activa. Their
conclusions on page 226-227 reflect the limitations of their models. This work offers answers to most
of their rhetorical questions.
6.3.10.2 The distributed electrical circuit of the neurites
The literature has not provided a large amount of performance data on the neurites as electrical
circuits. In fact, the podites have not been recognized previously in the literature. It appears that
the dendrites of the projection neurons are considerably less important electrically than the dendrites
of the electrotonic neurons. The same thing can be said about the podites. The neurites of
electrotonic neurons are frequently highly arborized. Much of the terminal structure associated with
each arbor are too small to be seen with light microscopy. Rall has addressed this fact and calculated
equivalent areas, volumes and surfaces for arborized dendrites.
From a functional perspective, it is important to differentiate between the dendrites associated with
electrotonic signaling (stage 1 and 2 neurons) and those few associated with pulse signaling (stage
3 neurons).
The neurites of the signal processing neurons are all electrotonic. Their properties have been studied
by Taylor, Mittman & Copenhagen110. The title of their paper appears unrepresentative. They only
discuss the dendrites of ganglion cells and they analyze them in terms of lumped constant
parameters. As an example, they assume the entire soma is equipotential. See Section 2.10.2.1.1.
They also assume the neuron and the input synapse are passive devices. They calculate an
“equivalent resistance” for the dendrite. However, they do not discuss the possibility that what they
measured was actually a time delay, related to conduction velocity in an electrolyte, instead of a
resistance. They do not analyze them from the perspective of an electrical transmission line as
suggested by their title. Taylor, et. al. follow Rall and do not introduce inductance into their model.
Such a circuit is not suitable for signal transmission over distances of more than tens of microns. The
circuit exhibits considerable attenuation and frequency dispersion with distance. This is a Hermann
cable approach.
The compartmental models of Segev & London and the cable models of Rall both highlight the
variation in the diameter and length of segments of neuritic tissue. However, both ignore the
inductance associated with the capacitance of these structures when used in propagation (statistically
a rare occurrence). Their analyses are properly limited to the support of electrotonic signals.
6.4 Pharmacology within the context of the Electrolytic Theory
Figure 6.4.1-1 presents a framework for discussing the effect of pharmacological materials on a
single neuron of the neurological system. The upper gray area of the figure shows the elements
required for the normal operation of the neural portion of the neuron. Not shown are the elements
required for homeostasis.
Conforming to the Electrolytic Theory of the Neuron developed in this work, the only signal carrying
elements (the neurotransmitters) are the electron and its operational construct, the hole. These are
Electrophysiology 6- 101
shown entering the neuron on the left and exiting on the right. The signal carrying
neurotransmitters enter and leave a neuron via a synapse as shown by the black bars at the extremes
of the individual cellular elements of the neuron. The synapse is an active electronic device
electrically biased to operate as a unidirectional electrolytic diode.
In semiconductor physics, the saltatory motion of an electron through the lattice of a crystalline (or
liquid crystalline) material exhibiting at least one void in its lattice, can be described by a conceptual
particle, the hole, having the opposite polarity and moving through the lattice in the opposite
direction. A hole typically exhibits a lower transport velocity, moving through the valence band
(defined by the lattice structure of the material), than a free electron moving through the same
material in its conduction band.
102 Neurons & the Nervous System
Figure 6.4.1-1 A framework for discussing the pharmacology of the neuron. The
individual neuron operates effectively with only the elements in the upper gray
area. The materials in the lower gray area can impact that operation in a variety
of ways. These materials may attack the neuron through pores or in areas of type
2 membrane. Each element of the neuron offers potential attack points.
Electrophysiology 6- 103
111Von Bohlen, O. & Dermietzel, R. (2006) Neurotransmitters and Neuromodulators, 2nd Ed. Weinheim,
Germany: Wiley-VCH Verlag Page 4
To support the operation of the neuron, electrical energy is required. This energy is obtained through
the decarboxylation of glutamic acid (aspartic acid is a backup) at an electrostenolytic site associated
with a section of type 2 lemma discussed in Section 2.1.4.2.1. The residue of this decarboxylation is
GABA (alanine is the alternate associated with the backup–aspartic acid). It should be noted that
glutamic acid and aspartic acid are the only acidic amino acids that can provide a negative potential
to the inside of the neuron. This is a unique feature of the neural system.
To support the decarboxylation mechanism and generate a suitable electrical potential within the
neuron, two conditions must be met.
1. A physical receptor site attractive to glutamic acid must exist on the surface of the neuron.
2. The site must be in electrical contact with the interior of the lemma.
These two conditions are uniquely satisfied by a region of lemma formed of two lipids that are
asymmetric at the molecular level (Section 2.2.1). Such a region is defined as a type 2 lemma in this
work.
It should also be noted that the above decarboxylation is performed without the presence of an
enzyme, as frequently described in the literature,
A major feature of the neuron not previously recognized in the literature, is that every neuron
contains a three-terminal active device at the junction of the dendritic arm, the poditic arm and the
axon (as indicated by the vertical black bar in the center of the neuron). As noted earlier, this device
is called an Activa. This configuration provides a feature not previously documented in the
neurological literature. Every neuron is fundamentally a three-terminal device. It cannot be properly
described as a two-terminal device.
The lower portion of the figure structures the neuromodulators reported to modify the operation of
neurons within the CNS. This tabulation is similar to one in von Bohlen & Dermietzel111. Their text
is more appropriately titled Neuromodulators of the CNS since they do not address the many
neuromodulators associated with the PNS and enteric systems and do not address the actual signal
carrying neurotransmitters of the neural system.
Figure 1.1 of Katz can be modified easily to reflect the actual situation, Figure 6.4.1-2.
104 Neurons & the Nervous System
Electrons enter the neuron through the synapses (shown as black vertical bars on the left). This
synapse is an active electrolytic device formed by an Activa and electrically biased to form a
unidirectional diode for electrical signals (electrons or holes). To be quite precise, these devices are
formed of PNP semiconductor material and their dominant form of charge propagation is by
(relatively slow moving) holes. A current entering the dendrite arm of the neuron through a synapse
results in a current of the same polarity at the synapse associated with the pedicle of the axon of the
neuron shown by the black vertical bar on the right. The dendritic signal path is said to be non-
inverting. A current of the same polarity entering the podite arm of the neuron through a synapse
Figure 6.4.1-2 Hallmarks of neurotransmission. Lower figure; the probes are
presumed to be contacting the axoplasm of each of the neurons. The signals at
the pedicles of both A & B are identical except shifted in time. However, they are
applied to the differential inputs of the three-terminal neuron at C. Neuron C is a
stage 3 phasic neuron shown operating below threshold in all cases except for
the one action potential generated at the end of stimulation by neuron A. Neurons
A & B appear to be stage 3 phasic neurons. The undershoots in the waveforms
from neurons A and B can be explained by an inadequate test protocol. Inset,
electrolytic equivalent circuit of the lower figure with both A and B driving C.
Modified from Katz, 1999.
Electrophysiology 6- 105
112Katz, P. (1999) Beyond Neurotransmission. Oxford: Oxford University Press preface and pg 1.
results in a current of the opposite polarity at the synapse associated with the pedicel of the axon of
the neuron. The poditic signal path is said to be inverting.
No other chemicals are required for the routine operation of the neural function of the neuron, not
even oxygen. However, a wide range of chemicals are required to support the homeostasis of the
neuron.
The definition of neurotransmission in the past has been difficult. Katz112 has provided recent
definitions of neurotransmission and neuromodulation attempting to be as precise as possible.
However, his assumptions are inappropriate; he relies on the two terminal neuron of the chemical
theory of the neuron. The internal three-terminal character of the neuron results in a fundamental
change in the functional definition of a neurotransmitter. The fact that only electrical charges are
signal carrying entities within the neural system results in two additional fundamental changes.
Most of the chemicals associated pharmacologically with the neuron result in modulation of the
operation of the overall neuron but are not involved in the primary signal-carrying function. Thus
a modified form of Katz’s definition is as follows:
Neurotransmission is carried out fundamentally by electrical means. It consists of the end-to-end
propagation of information. Such communications between neurons via a synapse is fast
(microsecond range), point-to-point (neuron-to-neuron) and simple (no change in polarity of the
information at a synapse). Within a specific neuron, the polarity and amplitude of the information
may be adjusted arbitrarily.
The wide use of the term neurotransmitter in pharmacology introduces a significant
problem. The biological materials defined as neurotransmitters (sometimes labeled
biogenics) in pharmacology have no direct relationship with the information
(encoded as neural signals) being transmitted within the neural system. These
biological materials have been identified primarily by their frequent presence near
neurons, not their function.
However, it would take a major revolution to upset the use of the term
neurotransmitter to describe these non-neuro-signal-transmitting materials. It may
be necessary to adopt a synonym to avoid incessant confusion. Neuro-signaling-
agent could be such a term.
Katz was unable to give a definition of neuromodulation using positive terms. He defined it as not
fast, not point-to-point and not simple. Neuromodulation, other than that carried out
electrolytically within a neuron, is defined here as caused by the release of a neuromodulator (a
chemical agent) within the interneural matrix. A Neuromodulator is a chemical that is temporarily
able to impact the operation of one or more neurons within a nerves or local area slowly (action
arising after milliseconds to seconds). The interneural matrix (INM) is frequently described as the
pool in the pharmacological literature.
Such neuromodulation typically involves a neuroreceptor site that supports an enzymatic action on
the lemma of the neuron. The site is generally considered modulator-specific. The neuroreceptor
site may support an electrostenolytic process in conjunction with a region of type 2 lemma or it may
support the transfer of molecular material through the lemma at a type 3 lemma site.
6.4.1 Interneuron signal communications
The exchange of signaling information between neurons is carried out within the synapses. The
synapses are formed by an Activa electrolytically biased as a unidirectional diode. The underlying
bandwidth of the Activa and its associated biasing exhibits a bandwidth in the multi-megacycle range.
Thus, it is capable of passing signals along the neural system at sub-microsecond intervals. In
practice, the impedances associated with the post-synaptic neurites limit the bandwidth of the signals
transmitted to a few kilohertz. As a result, the measured transit time of a synapse (between the
axoplasm of the preceding neuron and the associated orthodromic neuroplasm) is usually described
as in the sub-millisecond range.
Contrary to common descriptions of the synaptic mechanism (based on a two-terminal neuron), the
106 Neurons & the Nervous System
synapse itself does not change the character of the message it transmits. There is no such thing as
separate excitatory (EPSP) and inhibitory (IPSP) postsynaptic pulses or signals. The manipulation
of the character of the signal is controlled by which terminal of the three-terminal Activa, within the
orthodromic neuron, is stimulated by the signal.
6.4.2 Neuromodulators of the CNS
As shown in the figure, neuromodulators can be grouped into several major families, the biogenic
amines, the biogenic peptides, and the biogenic fatty acids. There are also a few simple molecules,
nitric oxide and carbon monoxide in particular, that have been considered neuromodulators.
The family of biogenic amines is quite large. Acetylcholine is a tertiary amine alcohol with a simpler
structure than the other biogenic amines.
The biogenic peptides is also a large family that included many hormone-releasing hormones. The
released hormones are also generally peptides and are considered hormones of the endocrine system.
The opioids are a large group of peptides all as extensions of the structure of Leu-enkephalin (Bohlen,
pg 199). The resulting peptides can also be considered as distinct sections extracted from a larger
peptide labeled prodynorphin.
The peptides of the cholecystokinin (CCK) family are formed by degradation of the cholecystokinin
molecule (Bohlen, pg 184).
A number of biogenic fatty acids are derived from the 20-carbon arachidonic acid.
6.4.3 Neuromodulators of the PNS & Enteric Systems
The Neuromodulators of the PNS & Enteric Systems are beyond the scope of this work.
6.4.4 Neuroreceptors
The physical character and method of operation of neuroreceptors supporting the activity of
neuromodulators are poorly defined in the literature. Conceptually, neuroreceptors can be integral
parts of the lipid molecules forming the lemma of a neuron (as in the case of the glutamate receptor
site of electrostenolysis), distinctly separate molecules in stereo-bond with a portion of the lemma of
a neuron, or a physical aperture created within the lemma of a neuron by the neuroreceptor. Most
discussions of neuroreceptors have not defined specifically what the action is that is of the
neuroreceptor-neuromodulator combination. Is the action to inject electrical charge into the plasma
within the lemma? Is the action to insert a biogenic molecule into the plasma within the lemma?
What is the action of the combination?
Bohlen & Dermietzel discuss potential solutions to this problem at the conceptual level (page 8). In
many discussions, a “second messenger” is introduced to account for the action of the neuromodulator-
neuroreceptor combination. These second messengers remain inadequately defined.
Figure 6.4.1-1 also introduces another major problem related to neuroreceptors. Which of the three
plasmas of the typical neuron does the specific neuroreceptor associate. Is its goal to affect the
electrical parameters associated with the axoplasm? Alternately, is its goal to affect the parameters
of the dendroplasm or the podaplasm? While the chemical theory of the neuron has previously
considered the neuron a two-terminal device with only one significant plasma, this is demonstrably
not the case.
6.4.5 Neuromodulation blockers
A variety of neuromodulator blockers are recognized. Their action has not been clearly defined in the
literature. In some cases, the blockers may occupy neuroreceptor sites and prevent the targeted
neuromodulator from occupying that site. In other cases, it is conceivable, the blocker interacts with
the target neuromodulator in the INM surrounding the neurons of interest. Resolution of these
Electrophysiology 6- 107
113Storm, J. (1989) An after-hyperpolarization of medium duration in rat hippocalmpal pyramidal cells J Physiol
vol 409, pp 171-190 fig 4
114Washburn, M. & Moises, H. (1992) Muscarinic responses of rat basolateral amygdaloid neurons recorded
in vitro. J Physiol vol 449, pp 121-154
115Womble, M. & Moises, H. (1992) Muscarinic inhibition of M-current and a potassium leak conductance in
neuronse of the rat basolateral amygdala J Physiol vol 457, pp 93-114
116Moises, H. & Womble, M. (1995) Acetylcholine-operated ionic conductances in central neurons In Stone,
T. ed. CNS Neurotransmitters and Neuromodulators: Acetylcholine. Boca Raton, FL: CRC Press pp 129-148
alternatives remains a subject of study.
6.4.6 Neuron operation with neuromodulators present
A key understanding can be based on the work of many. A common experimental protocol is to
establish a control experiment (exhibiting a specific waveform), introduce a neuromodulator by
perfusion, and then demonstrate a return to the baseline waveform following a “wash.” With the
wash containing no neuromodulator, it is clear the basic waveform is not dependent on the presence
of any soluble neuromodulator. The protocols seldom define the perfusion and wash durations.
However, Storm113 mentions indirectly intervals of 3 to 10 minutes when using 5 mM-
tetraethylammonium (TEA).
Section 13.1.2.2 provides information on the voltage clamp procedure as well as information on other
probe procedures.
Section 6.4 has discussed the operation of a simple neuron as typified in Figure 6.4.1-1. The critical
requirement for signal propagation from the neurites (dendrite and/or podite) to the axon is that the
Activa be properly/adequately biased electrolytically. In the configuration shown, the dendrite
terminal (the emitter) must be positive with respect to the podite terminal (the base). This insures
the input diode of the Activa is forward biases. At the same time, the axon (the collector) must be
negative with respect to the podite terminal. This insures the output diode of the Activa is reverse
biased. Under these conditions, any signal applied between the dendrite and podite terminals will
result in a change in the potential of the axoplasm representing that signal. By arranging the
impedances associated with the individual terminals, the output signal representation can be made
an analog representation or a phasic pulse keyed to the applied signal.
The above electrolytic biasing is easily achieved with only two sources of electricity, a large amplitude
source capable of making the collector more negative than the base and a smaller source capable of
making the emitter (dendrite) more positive than the base. It is possible to satisfy both of these
requirement using only negative power sources by connecting a large one to the collector (axon), a
small one to the base (podite) and connecting the emitter (dendrite) through an impedance to the
circuit ground point. This is most easily achieved by establishing a glutamate receptor site on the
type 2 lemma of the axon and a second glutamate receptor site on the type 2 lemma of the podite. The
neuron will operate as either an analog or phasic neural circuit under these conditions without the
presence of any other biochemical agents (including any neuromodulators).
Many investigators have achieved neural operation in this manner in-vitro and without the presence
of any known neuromodulators. The Moises team has collected considerable data on the operation
of similar neurons using slices of CNS tissue. An analysis of their data can provide considerable
information about the effect of neuromodulators on their neurons114,115,116. The last paper is a
comprehensive review that contains new material but calls on the earlier two papers. They employed
two forms of pathological intervention. They changed the resting potential of the (presumed)
axoplasm of the neurons using patch-clamp procedures and they introduced a variety of
neuromodulators individually into the INM (the pool surrounding the neuron under observation).
The first point to note is that their “control” situations involve the operation of their neurons in the
absence of any known neuromodulator. The second point to note is that the application of a
neuromodulator using their bulk or ionophoretic methods had minimal impact on the detailed shape
of the signals appearing at their point of experimental observation (presumably the potential of the
axoplasm of the neuron). The third point is that varying the potential of the axoplasm by the patch-
clamp procedure is a pathological situation. It can be best described as introducing a parasitic
108 Neurons & the Nervous System
stimulus into the system in order to measure various parameters of the neural circuit. By varying
the resting potential of the axoplasm in the absence of any applied neuromodulator, the neuron
continued to reproduce analog signals or to generate action potentials as expected. Failure to operate
in this manner was cause for rejection of the data set associated with that neuron. The patch-clamp
technique is discussed in greater detail in Section 13.1.2 & Section 13.1.3.
Figure 6.4.6-1 reproduces frames B and C of figure 1 in Washburn & Moises. It shows the measured
performance of two typical small pyramid neurons from slices from the basolateral amygdala of
Sprague-Dawley rats. Their morphological sketch in A has been replaced by a circuit diagram of the
neuron and a potential output characteristic.
The performance is easily correlated with that presented in Section 6.4 from other neural
experiments. They assert the operation of these small pyramid neurons were typical of those in other
parts of the brain. They also noted the effect of using a perfusion of 100 M of cadmium can be
reproduced using an equivalent perfusion of caesium at 2 M or of the calcium-chelating agent EGTA
(not known as a biogenic material).
Electrophysiology 6- 109
6.4.7 The roles of the Stage 6 Neuro-effectors
Section 6.4.2 discussed the role of materials modulating neuron output activity but did not discuss
the neurosecretory outputs of various stage 6 “end-effector neurons.” A framework needs to be
developed that describes these neurosecretory outputs and shows that some of them are the source
of the neuromodulators defined above. This framework requires a definition of the neural-glandular
system interface.
While the output of neurons within the neural system is typically an electrolytic current, the output
Figure 6.4.6-1 Operation of small pyramid neurons from the amygdala of rat under
voltage-clamp conditions. A; Electrolytic circuit of the pyramid neuron and its
output characteristic. See text. Ba; hyperpolarizing steps of 0.1 nA over the range
from -0.8 to +0.3 nA. Bb; action potentials generated in response to short current
pulses. Trace 1 was generated by a single depolarizing pulse of 0.7 nA for 10 ms.
Trace 2 was generated by a single depolarizing pulse of 0.8 nA for 10 ms. The
second trace was rising toward the threshold for a second action potential but did
not reach that point. The two pulse waveform on the right was the result of a 0.9
nA depolarizing pulse for 10 ms. Bc; three-pulse spike train generated in
response to a prolonged depolarizing pulse of 0.5 nA for 450 ms. C; operation of
a different small pyramid neuron when biased to –60 mV from a normal resting
potential of –69 mV. Ca; intracellular injection of a depolarizing 1.0 nA for 150 ms
elicited a burst of action potentials followed by a significant overshoot
polarization followed by a decay back to the quiescent potential. Cb; action
potential pattern showing “accommodation” following application of a
depolarizing 0.5 nA for 150 ms. Cc & Cd; patterns equivalent to Ca & Cb obtained
following superfusion by a cadmium containing medium. For purposes of
comparison, the trace of Ca is reproduced as trace 2 in Cc. Frames B & C from
Washburn & Moises, 1992.
110 Neurons & the Nervous System
of the stage 6 neuro-effector neurons can involve a variety of chemicals. These chemicals can affect:
Specific muscular tissue, particularly smooth muscle tissue.
Other neural tissue within the brain (and typically related to the encephalocrine system and
described as neuromodulators).
Other neural tissue outside of the brain (a poorly defined mechanism).
Other non-neural tissue outside of the brain (typically related to the endocrine system).
Exuding of chemicals into the external environment (typically related to the exocrine system).
Using this framework, most of the many neuromodulators (including the neurotransmitters of
pharmacology) described in Section 6.4.2 can be described as fundamentally neuro-effectors released
by the neuro-secretory neurons of the neural system. It includes the neuron/glandular intrface
(Section 16.3 and Chapter 23).
In the context of this work, a gland is a mass of epithelium supporting a large group of neuro-effector
neurons that are releasing one or more chemicals into the brain cavity, or the blood and lymph
channels. If connected to the blood and/or lymph channels, the material is considered part of the
endocrine system.
Glands have been described (Chapter 23) as structures capable of creating a specialized substance
(a neuro-effector) and excreting the substance onto a surface (exocrine type) or into the blood or
lymph streams (endocrine type). This listing must be expanded to describe excretion into the brain
itself (encephalocrine) where the materials act as neuromodulators on an extended area basis.
Further categorized as to whether the material is passed through the cell wall (merocrine type),
breaks through the wall or separates along with part of the wall (apocrine type) or holocrine type
where the cell is destroyed in the process of freeing the specialized substance.
Electrophysiology 6- 111
TABLE OF CONTENTS 12/15/22
6. Electrophysiology of Action Potential Propagation ...........................................2
6.3 Reformulating the concept of signal transmission in neuroscience . . . . . . . . . . . . . . . . . . . . . . . . . . 2
6.3.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
6.3.1.1 Limit of mathematics used in chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
6.3.1.2 Background from the literature–Hermann vs Maxwell . . . . . . . . . . . . . . . . . . 7
6.3.1.2.1 The contribution of Maxwell–The General Wave Equation . . . . . 9
6.3.1.2.2 Rationalizing the Hermann & Maxwell approaches--The concept of
the local circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
6.3.1.2.3 Problems in the recent literature relying upon the old Hermann
concept .............................................11
6.3.1.3 Background from the literature–Rushton & others, 1950's . . . . . . . . . . . . . . 13
6.3.2 Technical background related to the action potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.3.2.1 The electrical characteristics associated with the action potential (as generated)
..........................................................16
6.3.2.1.1 The graphical description of an action potential . . . . . . . . . . . . . 16
6.3.2.1.2 Potential monopulse oscillators for generating action potentials
...................................................17
6.3.2.1.3 The equations for a switching mode monopulse oscillator . . . . . . 19
6.3.2.2 The action potential versus temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.3.2.3 Overshoot, undershoot and polarity reversal . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.3.2.3.1 Precise definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.3.2.3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.3.2.4 Summary characteristics of the action potential (as generated) . . . . . . . . . . . 24
6.3.2.4.1 Difference between action potential vs its derivative . . . . . . . . . . 25
6.3.2.4.2 Difference between action potential vs generator potential . . . . . 27
6.3.2.4.3 Difference between action potential vs pseudo-action potentials
...................................................27
6.3.3 Mathematics of signal conduction and propagation in neural circuits . . . . . . . . . . . . . . 28
6.3.3.1 Importation of the morphological model into electrophysiology . . . . . . . . . . 29
6.3.3.1.1 Packaging of functional neurons based on morphology . . . . . . . . 30
6.3.3.1.2 Neural signals are orthodromic (unidirectional) . . . . . . . . . . . . . . 31
6.3.3.1.3 Electrophysiological/histological model of a neuron . . . . . . . . . . 33
6.3.3.1.4 Myelin as a critical element in signal propagation . . . . . . . . . . . . 34
6.3.3.1.5 The transversal filter and time dispersion in the neural system . . 35
6.3.3.2 Introduction to Modern Cable Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.3.3.2.1 Efficient transmission over long distances–propagation . . . . . . . 37
6.3.3.2.2 Adequate transmission over short distances–conduction . . . . . . . 37
6.3.3.3 Propagation over the ideal coaxial cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.3.3.4 Propagation over a lossy coaxial cable–real axon . . . . . . . . . . . . . . . . . . . . . 43
6.3.3.4.1 The theoretical intrinsic propagation velocity and attenuation on a
lossy line ............................................44
6.3.3.4.2 The theoretical impedance of a lossy cable . . . . . . . . . . . . . . . . . 44
6.3.3.4.3 Intrinsic pulse dispersion along a lossy line . . . . . . . . . . . . . . . . . 45
6.3.3.4.4 The mobility coefficients of typical electrolytes & semiconductor
materials ............................................45
6.3.4 Transmission by conduction of electrotonic signals within Stage 1,2 & 4 neurons . . . . 47
6.3.4.1 Conduction of signals within axons of stage 1,2 & 4 neurons . . . . . . . . . . . . 47
6.3.4.1.1 Calculation of the signal conducted along an axon . . . . . . . . . . . 49
6.3.4.2 Delay associated with conduction of signals within stage 1,2 & 4 neurons . . 50
6.3.5 Transmission by propagation of phasic signals within Stage 3 neurons . . . . . . . . . . . . . 51
6.3.5.1 Background–the electrical circuit of the axon as a transmission line . . . . . . 51
6.3.5.1.1 The physiological model for a signal projection neuron . . . . . . . 52
6.3.5.1.2 The electrical transmission line model of a signal projection neuron
EDIT ............................................... 53
6.3.5.1.3 The cylindrical transmission line . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.3.5.2 The interface between a distributed line and an Activa . . . . . . . . . . . . . . . . . 56
6.3.5.2.1 Importance of impedance matching at junctions . . . . . . . . . . . . 56
6.3.5.2.2 Half-section filter design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.3.5.2.3 The impedance of the terminal portions of an axon segment . . . . 57
6.3.5.3 The electrolytics/electrostatics of the axon segment . . . . . . . . . . . . . . . . . . . 57
112 Neurons & the Nervous System
6.3.5.3.1 The geometry and electrostatics of the axon segment . . . . . . . . . 58
6.3.5.3.2 The electrodynamics of the axon segment . . . . . . . . . . . . . . . . . . 60
6.3.5.3.3 The impact of the electrolytic skin effect on the axon segment . . 60
6.3.5.3.4 The electrodynamics of the axon associated with propagation . . 61
6.3.5.4 The impedances of the distributed part of the axon . . . . . . . . . . . . . . . . . . . . 61
6.3.5.4.1 The axolemma-myelin junction . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.3.5.4.2 Capacitance of the distributed part of an axon . . . . . . . . . . . . . . . 62
6.3.5.4.3 The capacitance of an unmyelinated axon . . . . . . . . . . . . . . . . . . 63
6.3.5.4.4 The inductance of the distributed part of an axon . . . . . . . . . . . . 63
6.3.5.4.5 The effective resistance of the distributed part of an axon . . . . . . 64
6.3.5.4 The conduction of signals in a demyelinated, or stripped, axon . . . . . . . . . . 65
6.3.6 Importance of compensated probes to avoid artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.3.6.1 Background regarding Probes EMPTY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.3.6.1.1 Single wire Probes, extraneural & Intraneural EMPTY . . . . . . . . 65
6.3.6.1.2 Multi wire Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.3.6.2 Probes used in patch-clamp investigations . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.3.6.2.1 Background relating to the patch-clamp techniques . . . . . . . . . . . 70
6.3.6.2.2 Patch-clamp technique used with a stage 3A neuron . . . . . . . . . . 70
6.3.6.2.3 Patch-clamp technique used with a NON stage 3A neuron EMPTY
...................................................70
6.3.7 Computational anatomy as a flexible tool in the neural system . . . . . . . . . . . . . . . . . . . 71
6.3.7.1 Delay–based computational anatomy between the retina and the visual cortex
..........................................................72
6.3.7.2 The special case of a syncytium formed by the ganglia of the leech . . . . . . . 75
6.3.7.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.3.7.2.2 An explicit configuration used in swimming . . . . . . . . . . . . . . . . 77
6.3.7.2.3 Summary of the neural system of Hirudo medicinalis . . . . . . . . . 78
6.3.7.3 The special case of the giant axons of the 3rd stellate ganglia in the squid . . 80
6.3.7.3.1 Measured and calculated action potentials of the squid . . . . . . . . 80
6.3.7.3.2 A reinterpretation of the Hodgkin & Huxley waveforms . . . . . . . 80
6.3.8 Correlation of the transmission theory of axons with the measured data . . . . . . . . . . . . 81
6.3.8.1 Rigor required when transitioning from exploratory to exploitive research . 82
6.3.8.1.1 The special case of the giant axons of the 3rd stellate ganglia in the
squid ............................................... 83
6.3.8.1.2 The equivalent circuit of the giant axon-Cole . . . . . . . . . . . . . . . 84
6.3.8.2 Node of Ranvier test circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.3.8.3 Redefinition of axon dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3.8.3.1 Specialized glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3.8.3.2 Transmission of the Action Potential . . . . . . . . . . . . . . . . . . . . . . 91
6.3.8.3.3 Derivation of the average transmission velocity . . . . . . . . . . . . . . 92
6.3.8.3.4 The intrinsic propagation velocity on a lossy line . . . . . . . . . . . . 95
6.3.8.3.5 Performance of the pulse signal repeater . . . . . . . . . . . . . . . . . . . 96
6.3.8.3.6 Performance of the axon segment . . . . . . . . . . . . . . . . . . . . . . . . 96
6.3.8.4 Effect of pharmacological agents on the Action Potential . . . . . . . . . . . . . . . 96
6.3.8.5 The pathological condition of transmission over a demyelinated axon . . . . . 97
6.3.8.5.1 Chemical demyelination & regrowth of frog sciatic nerves . . . . . 97
6.3.8.5.2 Matching a Node of Ranvier to a cable . . . . . . . . . . . . . . . . . . . . 98
6.3.9 The time delays relevant to the neural system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3.9.1 Sources of signal delay in the external sensory neurons . . . . . . . . . . . . . . . . 99
6.3.9.2 Effects of temperature on each source of signal delay . . . . . . . . . . . . . . . . . . 99
6.3.9.3 Primary uses and effects of signal delay within the visual system . . . . . . . . 100
6.3.10 The complex electrical structure of the ramified neurites . . . . . . . . . . . . . . . . . . . . . . 100
6.3.10.1 The organization aspects of the stage 2 neurites . . . . . . . . . . . . . . . . . . . . 105
6.3.10.2 The distributed electrical circuit of the neurites . . . . . . . . . . . . . . . . . . . . 106
6.4 Pharmacology within the context of the Electrolytic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4.1 Interneuron signal communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4.2 Neuromodulators of the CNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4.3 Neuromodulators of the PNS & Enteric Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4.4 Neuroreceptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.4.5 Neuromodulation blockers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4.6 Neuron operation with neuromodulators present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4.7 The roles of the Stage 6 Neuro-effectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Electrophysiology 6- 113
Chapter 6 List of Figures 12/15/22
Figure 6.3.1-1 Hermann’s conception of the electrophysiology of a nerve . . . . . . . . . . . . . . . . . . . 8
Figure 6.3.1-2 The electrical and magnetic fields associated with a coaxial transmission line . . . 9
Figure 6.3.2-1 Comparison of recorded and calculated action potentials. . . . . . . . . . . . . . . . . . . 18
Figure 6.3.2-2 Measured action potentials as a function of temperature . . . . . . . . . . . . . . . . . . 20
Figure 6.3.2-3 Interpretation of experimental stage 3 waveforms generated by the tactile sensors of
the leech, Hirudo medicinalis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Figure 6.3.3-1 The basic electronic forms of a fundamental neural path . . . . . . . . . . . . . . . . . . . 32
Figure 6.3.3-2 Fully implemented electrophysiological/histological Neuron . . . . . . . . . . . . . . . . 34
Figure 6.3.3-3 General plan of a transversal filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Figure 6.3.3-4 Ideal cable Equations based on inner/outer radii of dielectric . . . . . . . . . . . . . . . 38
Figure 6.3.3-5 The capacitance and inductance of an ideal axon segment . . . . . . . . . . . . . . . . . . 39
Figure 6.3.3-6 The characteristic impedance and phase velocity along an axon . . . . . . . . . . . . . 40
Figure 6.3.3-7 Electric and magnetic fields within an axon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Figure 6.3.3-8 The electrical circuits used to describe an axon (a coaxial cable) . . . . . . . . . . . . . 42
Figure 6.3.3-9 Measured impedance (inductive & capacitive) of a real axon . . . . . . . . . . . . . . . . 43
Figure 6.3.4-1 Models of signal conduction within unmyelinated stage 1, 2 & 4 neurons . . . . . . 48
Figure 6.3.4-2 Net voltage versus distance along a conduit with time as a parameter . . . . . . . . 50
Figure 6.3.5-1 Generic neurological transmission line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Figure 6.3.5-2 Importance of impedance matching at a boundary . . . . . . . . . . . . . . . . . . . . . . . . 56
Figure 6.3.5-3 The geometry and electrostatics of the initial axon segment of a neuron . . . . . . 59
Figure 6.3.6-1 Assortment of multi wire probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Figure 6.3.6-2 “Spontaneous extracellular unit activity recorded from a microelectrode . . . . . . 67
Figure 6.3.6-3 Effect of probe compensation on the recorded oscilloscope . . . . . . . . . . . . . . . . . . 68
Figure 6.3.6-4 Current driven circuit, to generate a parasitic action potential . . . . . . . . . . . . . . 74
Figure 6.3.7-1 Cartoon of the syncytium in Leech based on the Electrolytic Theory of the Neuron
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Figure 6.3.7-2 The nature of the giant axon of Loligo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Figure 6.3.7-3 Records of the action potentials in squid, Loligo forbesi . . . . . . . . . . . . . . . . . . . . 78
Figure 6.3.7-4 The calculated voltage and current profiles of the same action potentials using the
empirical equations of Hodgkin & Huxley. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Figure 6.3.7-5 Tracings from the test configuration of Hodgkin & Huxley purported to be action
potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Figure 6.3.8-1 The RLC character of a giant axon of squid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Figure 6.3.8-2 The currents measured in the return paths (the INM) related to the signals
propagating along the axon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Figure 6.3.8-3 Saltatory conduction in a normal rat ventral root fiber . . . . . . . . . . . . . . . . . . . . 102
Figure 6.4.1-1 A framework for discussing the pharmacology of the neuron . . . . . . . . . . . . . . . 104
Figure 6.4.1-2 Hallmarks of neurotransmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Figure 6.4.6-1 Operation of small pyramid neurons from the amygdala of rat under voltage-clamp
conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
114 Neurons & the Nervous System
(Active) SUBJECT INDEX (using advanced indexing option)
2nd lateral processing matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
50% ................................................................................50, 81
60% ................................................................................... 92
95% ...........................................................................1, 28, 57, 65
98% ................................................................................... 42
99% ................................................................................... 30
acetylcholine........................................................................106, 107
action potential . . . . . 1, 2, 12-21, 23-28, 45, 46, 50, 52, 60, 62, 67, 72-75, 77-81, 84-86, 88-93, 95, 96, 104, 109
Activa . . 1, 4, 15, 17-19, 21, 23-25, 27-30, 49, 51-53, 56, 57, 59, 62, 63, 77, 78, 81, 83, 92, 95-98, 100, 103-107
adaptation.........................................................................51, 76, 98
adaptation amplifier .......................................................................51
amercine ................................................................................ 98
amplification.............................................................................25
amygdala .......................................................................... 107-109
anatomical computation ....................................................................80
archaic .................................................................................81
Arrhenius ...............................................................................98
associative memory .......................................................................99
attention ................................................................................ 28
average velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 13, 14, 21, 37, 41, 44, 52, 60, 61, 84, 86, 88-92, 96-98
axon segment . . . . . . . . . . . . . . . 5, 14, 21, 27-31, 35, 37, 39-41, 44, 46, 52, 54-60, 73, 77, 81, 85, 86, 88-97, 99
axoplasm . . . . . . . . . . . . . . . 15-21, 23, 24, 27, 35, 38, 40, 41, 48, 49, 51, 52, 57-64, 76, 77, 81, 82, 96, 104-108
baseball.................................................................................35
bilayers ................................................................................. 30
biogenic ...........................................................................106, 108
bipolar ..............................................................................30, 47
boundary layer ............................................................................4
broadband............................................................................... 93
cable theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3, 12, 36, 63
Calcium ............................................................................77, 108
calibration............................................................................79, 87
Central Nervous System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70, 71, 94
clock ...................................................................................15
coaxial cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6, 9, 10, 29, 37, 38, 40, 42-44, 58, 90, 93, 96
cochlear nucleus ..........................................................................36
compensation .........................................................................65, 68
computation .....................................................................3, 13, 14, 80
computational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70, 75, 76, 99
computational anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70, 75, 76
conduction velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 14, 37, 84, 86, 88-90, 100
conexus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4, 15, 23, 24, 28-30, 40, 45, 48, 49, 51, 56, 57, 73-75, 77, 100
cross section .......................................................................60, 63, 93
CRUCIAL .....................................................................12, 14, 42, 51
diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 25, 30, 31, 33, 49, 57, 63, 80, 93, 95, 101, 104, 105, 107
dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12, 25, 35, 44, 45, 54, 56, 70, 81, 84, 88, 89, 91, 92, 100
double layer .......................................................................... 58-60
dynamic range ...........................................................................17
E/D ..................................................................................1, 97
EGTA .................................................................................108
Electrolytic Theory of the Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47, 51, 71, 74, 81, 83, 96, 100
electromagnetic propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
electrostenolytic process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15, 19, 27, 28, 59, 95, 105
endocrine ..........................................................................106, 110
endothermic animals........................................................................ 1
entorhinal ...............................................................................67
exocrine ...............................................................................110
exothermic animals........................................................................51
expanded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6, 17, 28, 40, 59, 83, 110
Electrophysiology 6- 115
external feedback .........................................................................18
feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15, 17, 18, 24, 25, 27, 72, 77
fibrillation...............................................................................97
free running ...........................................................................1, 95
freeze-fracture ...........................................................................94
GABA ................................................................................103
ganglion neuron ..........................................................................25
General Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2, 3, 5, 6, 8-10, 12, 36, 42, 60
glance ..................................................................................49
glutamate ..........................................................................106, 107
group velocity............................................................................84
GWE .............................................................................2, 51, 60
H&H................................................................................... 12
half-amplitude ...........................................................................50
half-section.....................................................................42, 56, 57, 80
Hermann cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 12, 54, 55, 100
hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2, 46, 91, 95, 100, 101
hole transport .........................................................................91, 95
hormone ...............................................................................106
hydronium ........................................................................30, 33, 45
hyperpolarization ........................................................................107
ice .....................................................................................46
impedance measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43, 81
in vitro ................................................................................107
inhomogeneous...........................................................................77
instantaneous velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84, 90
internal feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15, 18, 72
interneuron ..........................................................................85, 105
internode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14, 62, 84, 88, 89, 93, 94
inverting ........................................................................30, 104, 105
in-vitro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20, 37, 82, 83, 95, 107
in-vivo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17, 20, 49, 56, 72, 85, 95
IRIG ....................................................................................1
lactate ..................................................................................35
latency ...........................................................................50, 51, 97
lateral geniculate ......................................................................30, 70
Limulus................................................................................. 71
liquid-crystalline...........................................................................4
local circuit........................................................................10, 11, 13
locomotion ..............................................................................75
long-term memory .......................................................................100
Lord Kelvin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2, 8, 10, 11, 13, 45, 49
marker ..................................................................................1
Maxwell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2, 5, 7, 9-11, 42, 60
mesotopic ...............................................................................27
midbrain ................................................................................70
Mind ...................................................................................84
mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2, 45-47, 60, 64, 82
modulation .............................................................................105
monopolar ..............................................................................17
monopulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 14, 15, 17-20, 23, 25, 27, 74, 98
monopulse oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14, 15, 17-20, 23, 25, 74, 98
MT .................................................................................... 67
multi-probe ...........................................................................35, 65
muscarinic .............................................................................107
myelin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12, 14, 15, 34, 35, 58-63, 75, 76, 83, 92-94, 96
Myelination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4, 28, 38, 51, 53, 56, 57, 61, 62, 72, 75, 78, 89, 94, 96
neurite..............................................................................99, 100
neurites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 31, 99, 100, 105, 107
neuromodulator ..................................................................... 105-108
neurotransmitter .........................................................................105
neuro-effector...........................................................................110
nitric oxide .............................................................................106
Node of Ranvier . . . . . . . . . . . . . . . . . . 10, 13-15, 21, 25, 27, 31, 33, 41, 45, 52, 57, 62, 82-86, 89, 91, 92, 94-99
116 Neurons & the Nervous System
noise ..........................................................................24, 26, 65, 83
non-inverting ........................................................................30, 104
opioids ................................................................................106
optic nerve ........................................................................15, 70, 90
P/D equation.............................................................................98
parametric............................................................................65, 80
patch-clamp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 67, 107, 108
pedicel ................................................................................105
pedicle ..........................................................................27, 51, 104
phase velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40, 41, 44-46, 52, 60, 61, 63, 84-86, 88-94, 98, 99
pnp ................................................................................... 104
poda ...................................................................................95
podites .......................................................................... 31, 99, 100
poditic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16, 17, 21, 24, 31, 78, 103, 105
probe compensation .......................................................................68
propagation velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12, 40, 41, 44, 46, 54-56, 60, 61, 80, 90
protocol ........................................................................ 80, 104, 107
pseudo-action potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-27, 73, 77
pulvinar ................................................................................99
Pulvinar pathway .........................................................................99
pyramid neuron .........................................................................109
pyruvate ................................................................................35
quantum-mechanical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4, 28, 29, 45, 97, 98
quiescent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-21, 23-25, 49, 50, 77, 95, 109
reading .................................................................................99
recall...................................................................................47
rectifier ................................................................................. 33
refractory period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25, 62, 89, 92, 96
residue ................................................................................103
saliency map.............................................................................35
saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-19, 23, 24, 100
Schwann cell ......................................................................62, 85, 92
second messenger ........................................................................106
smooth muscle ..........................................................................110
spines .................................................................................100
squid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12, 20, 43, 75-78, 80-82
stage 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 5, 28, 30, 35, 47, 48, 50, 51, 54, 71, 75, 80, 98, 100
stage 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 40, 71, 74, 75, 99
stage 3 . . . . . . . . 1, 4, 5, 14, 17, 18, 23-30, 35-38, 40, 42, 50, 51, 55, 57, 58, 61, 70-72, 74, 75, 80, 98-100, 104
stage 3A ..............................................................................1, 27
stage 3B .................................................................................1
stage 3C .................................................................................1
stage 4 .................................................................................30
stage 5 .................................................................................35
stage 6 ............................................................................109, 110
stellate ...........................................................................75, 76, 80
stress...................................................................................58
synapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15, 28, 29, 31, 49-52, 56, 72, 99-101, 104-106
syncytium ......................................................................70, 72, 74, 75
thalamic reticular nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
thalamus ................................................................................15
three-terminal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 28, 103-106
threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-17, 21, 25, 27, 37, 45, 75, 77, 86, 92, 104, 109
topography ...........................................................................31, 88
topology ..........................................................................71, 72, 75
transduction .............................................................................97
transistor action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15, 24, 25, 30, 33, 49
transversal filter .......................................................................35, 36
type 1 ......................................................................29, 35, 58, 61, 63
type 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 58, 61, 63, 96, 102, 103, 105, 107
type 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 35, 58, 61, 63, 105
V2.....................................................................................49
visual cortex ..........................................................................70, 99
Electrophysiology 6- 117
voltage clamp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 21, 83, 107
Wikipedia ............................................................................... 58
xxx ..................................................................................... 3