Propagation effects of current and conductance noise in a model neuron with subthreshold oscillations

Abstract

We have examined the effects of current and conductance noise in a single-neuron model which can generate a variety of physiologically important impulse patterns. Current noise enters the membrane equation directly while conductance noise is propagated through the activation variables. Additive Gaussian white noise which is implemented as conductance noise appears in the voltage equations as an additive and a multiplicative term. Moreover, the originally white noise is turned into colored noise. The noise correlation time is a function of the system's control parameters which may explain the different effects of current and conductance noise in different dynamic states. We have found the most significant, qualitative differences between different noise implementations in a pacemaker-like, tonic firing regime at the transition to chaotic burst discharges. This reflects a dynamic state of high physiological relevance.

Full text

Propagation effects of current and conductance noise in a model neuron
with subthreshold oscillations
Christian Finke
a
, Jürgen Vollmer
b
, Svetlana Postnova
a
, Hans Albert Braun
a,*
a
Institute of Normal and Pathological Physiology, University of Marburg, Deutschhausstraße 1-2, D-35037 Marburg, Germany
b
Max Planck Institute for Dynamics and Self-Organization, Bunsenstraße 10, 37073 Göttingen, Germany
article info
Article history:
Received 16 December 2007
Received in revised form 17 March 2008
Accepted 18 March 2008
Available online 1 April 2008
Keywords:
Ornstein–Uhlenbeck
Impulse pattern
Pacemaker
Stochastic dynamics
abstract
We have examined the effects of current and conductance noise in a single-neuron model which can gen-
erate a variety of physiologically important impulse patterns. Current noise enters the membrane equa-
tion directly while conductance noise is propagated through the activation variables. Additive Gaussian
white noise which is implemented as conductance noise appears in the voltage equations as an additive
and a multiplicative term. Moreover, the originally white noise is turned into colored noise. The noise
correlation time is a function of the system’s control parameters which may explain the different effects
of current and conductance noise in different dynamic states. We have found the most significant, qual-
itative differences between different noise implementations in a pacemaker-like, tonic firing regime at
the transition to chaotic burst discharges. This reflects a dynamic state of high physiological relevance.
Ó2008 Elsevier Inc. All rights reserved.
1. Introduction
Biological systems are inherently noisy. All physiological
recording parameters show random fluctuations even under iden-
tical, exactly controlled stimulus conditions. Such fluctuations also
appear when the influence of measurement noise can be excluded
– as it is, for example, in the case of recording neuronal spike trains
from afferent sensory nerve fibers where the individual action
potentials clearly poke out of the recording noise (Fig. 1, lower
trace). In this situation, the spikes are generated at the receptive
endings far away from the recording site. The measurement noise,
which can be seen in the base-line of these recordings, can not
influence the spike-timing at the receptive endings. Nevertheless,
there are always random fluctuations of the interspike-intervals
(Fig. 1, upper trace). These fluctuations can only be caused by the
neurons’ intrinsic randomness and/or stochastic variations in its
environment.
Despite such unavoidable variability in experimental data, neu-
ron models often do not consider the randomness of impulse gen-
eration but focus on the systematic changes of firing rate and
impulse patterns in response to a specific stimulus. Such ‘deter-
ministic’ simulations are physiologically justified. They are in line
with the general goal of biological research which first of all means
to evaluate the principle systems’ dynamics and their responses to
the physiologically relevant control parameters. The influence of
additional, unknown and uncontrollable parameters may be ne-
glected as long as they do not exert systematic effects but only in-
duce some randomness.
When other simulations nevertheless include noise, it often
might be for the simple reason to achieve simulation data which
is looking more realistic. However, most of the ‘noisy’ simulations
are done for the examination of ‘cooperative effects’ between noise
and non-linear systems dynamics (see for example [1]) which
means that a random input, as any input, can significantly change
the system’s output especially around its bifurcations. This is well
known from intensively examined noise-related phenomena like
stochastic resonance [2–4], coherence resonance [5,6] or noise-in-
duced chaos [7].
At this point the question arises how to account for physiolog-
ically appropriate noise realizations in computer models. Indeed,
there already exists an overwhelming literature about the physio-
logical sources of noise and its appropriate realization and numer-
ical implementation (for example [8,9]). These studies address the
questions which type of noise (e.g. Gaussian or Poisson distributed)
and which type of correlations (white or colored noise) should be
applied, and whether this should be done in an additive or multi-
plicative way.
What kind of noise can be implemented, however, also depends
on the type of the neuronal model. The modeling approaches
extend from the experimentally derived Hodgkin–Huxley (HH)
type models [10] via their simplified versions like the FitzHugh–
Nagumo, Morris–Lecar or Izhikevich models [11–13] to the rather
formal integrate-and-fire neurons [14] and even simpler map-
based models [15].
0025-5564/$ - see front matter Ó2008 Elsevier Inc. All rights reserved.
doi:10.1016/j.mbs.2008.03.007
* Corresponding author. Tel.: +49 64 21 28 62305; fax: +49 64 21 28 669 67.
E-mail address: Braun@Staff.Uni-Marburg.De (H.A. Braun).
Mathematical Biosciences 214 (2008) 109–121
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HH-type models mostly do not consider the randomness of sin-
gle ion channels but their compound effects because they usually
also do not consider the single channel currents but the whole cell
currents of specific conductances. Single channel gating with single
ion channel randomness becomes of relevance when the neuron
possesses only a low number of the specific ion channels. The im-
pact of single channel noise therefore has mostly been examined in
studies considering relatively few ion channels (e.g. [16,17] and
references therein).
The conventional HH-type approaches consider the compound
noise effects on whole cell ion currents. Noteworthy, already in
the deterministic simulations the voltage dependent opening and
closing probability of the ion channels is taken into consideration
by exponentially voltage-dependent gating variables which leads
to sigmoid activation curves of voltage-dependent conductances.
This reflects an approximately Gaussian distribution of single
channel activation around a certain membrane voltage which has
experimentally been proven in thousands of voltage/patch-clamp
recordings. What is missing here is the random fluctuation over
time. This is the component which makes a noisy system’s output.
In our study, we use a modified version of the Hodgkin–Huxley
type approach, the so-called Huber–Braun model [18,19]. This
model, on the one hand, is simplified compared to the original
equations. On the other hand, it has functionally relevant exten-
sions. Most importantly, it includes two more currents for sub-
threshold oscillations in addition to the spike-currents. Such a
system can generate a diversity of physiologically relevant impulse
patterns.
The goal of this work is to examine how such a physiologically
realistic system reacts on different noise implementations. In this
type of models one can consider current as well as conductance
noise. Both noise implementations are rooted in the well-known
randomness of single ion channel gating. Random opening and
closing of individual ion channels induces random fluctuations of
the ion conductances and consequently of the ion currents.
So far it is not known what type of noise is more physiologically
realistic. Membrane electrophysiology has proven to be so complex
that even in the well-understood case of Gaussian white noise fluc-
tuations, exact results are extremely hard to obtain. However, it
has been pointed out that the assumption of white noise is not
likely the best way of modeling the stochasticity in biological sys-
tems [20]. Nevertheless, most part of the ongoing research towards
a better understanding of cell and membrane physiology is based
on the assumption and subsequent analysis of a system that is in
some way perturbed by Gaussian white noise, as for example in
[21]. Therefore, for a first approach the use of additive Gaussian
white noise appears justified. Besides being the most widely used
form, it is also the simplest way of noise implementation, and it
can be handled mathematically.
The major question concerns the cooperative effects between
noise and non-linear systems dynamics, i.e. whether and how even
such simple noise terms may be modified when passing through
the set of differential equations and whether this may lead to phys-
iologically relevant alterations of the impulse patterns. In the next
chapter (Section 2) we describe the physiological background of
the model, and then give a description of the model’s structure,
equations and noise implementation. The results section has three
parts. In Section 3.1 we show numerical simulations addressing
relevant noise effects. The mathematical analysis in Section 3.2
identifies some of the essential control parameters for the propaga-
tion of noise. In Section 3.3 the interdependencies between differ-
ent noise implementations and impulse pattern variations are
further examined with non-linear dynamics tools, in order to elu-
cidate the noise sensitivity of the different model variables and
their most sensitive states. Finally, key results are summarized
and discussed in Section 4.
2. Model
2.1. Physiological background
We use a Hodgkin–Huxley (HH) type model which had origi-
nally been developed to simulate the impulse activity of peripheral
cold receptors in response to temperature changes [18,19].In
agreement with the experimental data, this model exhibits an
enormous variety of different impulse patterns. The major activity
patterns are illustrated in Fig. 2.
From low to high temperatures, a cold receptor’s activity
changes from pacemaker-like spike generation (tonic firing,
Fig. 2(A)) to burst discharges (Fig. 2(B)) which presumably goes
through chaotic impulse patterns [22]. With further increasing
temperature again a tonic firing pattern of single spikes appears
(Fig. 2(C)). These spikes, however, are apparently riding on mem-
brane oscillations. At the highest temperatures the oscillations re-
main subhreshold, eventually triggering spikes with the help of
noise (Fig. 2(D), see also [23,24]).
These different patterns can be seen in many other neurons in
the peripheral and central nervous system (PNS and CNS). How-
ever, none of the other neurons, to our knowledge, is able to gen-
erate all of these patterns as cold receptors do in response to
their physiological adequate stimulus. For example, thalamocorti-
cal and hypothalamic neurons exhibit tonic-to-bursting transitions
[25,26] but are missing the tonic firing activity with subthreshold
oscillations. In contrast, transitions from subthreshold oscillations
to tonic firing can be recorded in other CNS structures like the
entorhinal cortex and the amygdala [27] as well as in peripheral
sensory afferents from warm receptors [28] and shark electrore-
ceptors [24]. These neurons, however, do not show the transitions
from pacemaker-like tonic firing to burst discharges.
Altogether, cold receptors appear to be the most flexible single-
neuron pattern generators which is all the more remarkable as
they obviously consist of nothing more than free nerve endings
without any specialized structures and without any synaptic
Fig. 1. An example to illustrate how the interspike-intervals (ISI, upper trace) are
determined from extracellular action potentials recordings (spikes, lowest trace).
Whenever the voltage of the original recording (lowest trace) crosses a certain
threshold value (dashed line), a TTL pulse (mid-trace) is generated. These pulse sets
the spike-times from which the interspike-intervals (ISI, upper trace) are calculated.
Spike-times of individual neurons were only taken when voltage deflections of
similar amplitudes clearly poke out from the recording noise and from eventual
background activity of other neurons (smaller voltage peaks).
110 C. Finke et al. / Mathematical Biosciences 214 (2008) 109–121

connections [29]. Hence, the manifold of the dynamics obviously
arises from the intrinsic membrane properties of the receptive
endings.
These particular conditions have also been considered in the
cold receptor model which, therefore, has become a frequently
used tool, partly with slight modifications, to examine general as-
pects of impulse pattern variability and to evaluate the underlying
dynamics [30–33] or their implications for neuronal synchroniza-
tion [6,34–36]. The most detailed description of the model and
its physiological background can be found in [18]. A summary is gi-
ven in the remainder of this section.
2.2. Model structure and equations
To account for the manifold of activity patterns the original
Hodgkin–Huxley approach, on the one hand, had to be extended
while it, on the other hand could be simplified. In the end we have
arrived, despite of the extensions, at a four-dimensional model
which also is the case for the original HH model.
2.2.1. Model structure
The original HH model examines the time-course of the action
potentials while the cold receptor model focuses on the mecha-
nism of impulse generation which obviously arises from sub-
threshold membrane potential oscillations [23]. Therefore, in
addition to the spike generating sodium and potassium currents
(I
Na
and I
K
) two more currents (I
Nap
and I
KðCaÞ
) were included which
activate below the threshold of spike generation (hence, subthresh-
old currents) and have significantly slower activation kinetics than
the spike-currents.
I
Nap
stands for a persistent, i.e. non-inactivating Na-current and
I
KðCaÞ
for a calcium-dependent potassium current with voltage
dependent activation of the calcium channels. These subthreshold
currents generate slow potential oscillations. Slow depolarization
occurs due to the persistent Na-current which, in turn, activates
a slow repolarizing current with a certain time delay. The slow
repolarizing current is modelled as a simplified version of a Ca-
dependent K-current with voltage dependent activation of the
Ca-currents (for details see [18,19]).
Simplifications compared to the conventional HH-approach
have been made in several parts. First of all we do not separately
calculate the ion channel activation and inactivation probabilities
in the steady-state but relate the voltage dependent ion currents
directly to the membrane potential in form of a sigmoid function
which reflects the approximately Gaussian probability distribution
of ion channel activation (see Section 1).
Moreover, we have modeled the fast Na-current I
Na
, which carries
the upstroke of the spike, as instantaneously activating because it
activates much faster (in the sub-millisecond range) than any other
current. We also neglect the inactivation of I
Na
and we do not con-
sider the voltage dependencies of the activation time constants. Both
factors would change the shape of the action potential a little bit
which – as said above – is of minor interest for our study.
We are examining the mechanisms of impulse pattern genera-
tion and their alterations. The relevant currents in this respect
are the subthreshold currents although their interactions with
the spike generating currents may play an important role in spe-
cific situations, e.g. at the transition to chaos and in the pace-
maker-like tonic firing regime [31]. However, this is certainly not
related to their shape.
Our model had originally been developed with regard to cold
receptor transduction, i.e. it referred to specific types of ion chan-
nels. Nevertheless, it can also account for similar activity patterns
in many other neurons. The relevant dynamics do not depend on
specific types of ion channels but arise from their interactions
which is reflected in the model’s structure.
From a more general point of view, the model consists of two
potentially oscillating subsystems, the spike-generator and a slow
oscillator, which are operating at different time scales and are acti-
vated at different voltage levels. The resonances between these
two subsystems are the essential determinants for the system’s
dynamics.
This general aspect shall be considered by the use of more gen-
eral terms for the activation variables and parameters of the di-
verse ion currents:
d: refers to the fast depolarizing spike-current I
Na
,
r: refers to the fast repolarizing spike-current IK,
sd: stands for the slow depolarizing current INap,
sr: stands for the slow repolarizing current IKðCaÞ.
2.2.2. Model equations
These assumptions lead to the following set of model equations.
The time evolution of the membrane potential is given by
C
M
dV
dt¼X
i
I
i
¼I
l
I
d
I
r
I
sd
I
sr
;ð1Þ
where C
M
is the membrane capacitance, Vis the membrane voltage
and I
i
are the ion currents.
Fig. 2. Examples of extracellular recorded impulse pattern from peripheral cold
receptors as they typically occur around specific temperature ranges as indicated in
the figures. The diagrams show impulse sequences (spikes with normalized height)
together with the distribution of interspike-intervals which were taken from longer
time series.
C. Finke et al. / Mathematical Biosciences 214 (2008) 109–121 111

The leak current
I
l
¼g
l
ðVV
l
Þ;ð2Þ
can be calculated based on the leak conductance g
l
and the equilib-
rium potential V
l
.
The other currents I
d
;I
r
;I
sd
and I
sr
are calculated through the
equation:
I
i
¼qg
i
a
i
ðVðtÞV
i
Þfor i¼d;r;sd;sr;ð3Þ
with V
i
being the equilibrium potentials, g
i
the maximum conduc-
tances and a
i
the activation parameters. The parameter qis used
for the temperature scaling of the ion currents.
Asymptotically the voltage dependences of I
d
;I
r
, and I
sd
are gi-
ven by sigmoid activation parameters:
a
i1
¼1
1þexpðs
i
ðVV
0i
ÞÞ for i¼d;r;sd;ð4Þ
where the V
0i
are the half-activation potentials, and s
i
the slopes of
the steady state activation curves.
However, the system only slowly approaches these values, such
that the time-dependent activation parameters a
r
and a
sd
evolve
like
da
i
dt¼/
s
i
ða
i1
a
i
Þfor i¼r;sd:ð5aÞ
In contrast there is instantaneous activation for the fast Na-current
a
d
¼a
d1
:ð5bÞ
Activation of the slow repolarizing current is modeled as a simpli-
fied version of a Ca-dependent K-current which comprises the volt-
age dependent Ca-currents and the changes of Ca-concentrations
with subsequent alterations of the K-current in one equation which
directly couples the slow repolarizing current to the slow depolar-
izing current:
da
sr
dt¼/
s
sr
ðgI
sd
ka
sr
Þ;ð5cÞ
with gas a coupling constant and kas a relaxation factor.
Irrespective of the details, this model is principally made up by
two oscillating subsystems, the spike-generator and a slow oscilla-
tor, which are operating at different time scales and are activated
at different voltage levels. The resonances between these two sub-
systems are the essential determinants for the system’s dynamics
and impulse pattern.
Alteration of any control parameter can lead to significant
changes which will be demonstrated here with temperature scal-
ing. This mainly means scaling of the time constants of current
activation which, according to experimental data, have a Q
10
of
3.0. Minor temperature dependencies of the maximum conduc-
tances with a Q
10
of only 1.3 are also considered.
Accordingly, the temperature dependencies are given by
/¼3:0
ðTT
0
Þ=10

C
;ð6aÞ
q¼1:3
ðTT
0
Þ=10

C
:ð6bÞ
The model is implemented for a unit membrane area of 1 cm
2
. The
numerical parameter values are:
(1) equilibrium potentials: V
sd
¼V
d
¼50 mV;V
sr
¼V
r
¼
90 mV;V
l
¼60mV;
(2) ionic conductances: g
l
¼0:1;g
d
¼1:5;g
r
¼2:0;g
sd
¼0:25;
g
sr
¼0:4 (in
mS
cm
2
);
(3) membrane capacitance: C
M
¼1
lF
cm
2
; gives a passive time con-
stant M¼C
M
=g
l
¼10 ms;
(4) activation time constants: s
r
¼2ms;s
sd
¼10 ms;s
sr
¼20 ms;
(5) slope of steady state activation: s
d
¼s
r
¼0:25;s
sd
¼0:09;
(6) half activation potentials: V
0d
¼V
0r
¼25 mV;V
0sd
¼
40 mV;
(7) coupling and relaxation constants for I
sr
:g¼0:012;
k¼0:17;
(8) reference temperature: T
0
¼25

C.
These numerical values are important for the system’s dynam-
ics which essentially depend on the resonances between the spike-
generator and the subthreshold oscillator, These two subsystems
operate at different voltage levels (given by the voltage-dependent
steady-state activation curves) and time scales (determined by the
activation time constants). Alterations of the numerical values
which in this study result from temperature scaling can dramati-
cally change the system’s behavior and, expectedly, also the noise
effects.
2.2.3. Noise implementation: current versus conductance noise
The general equation for noise implementation is
dx
dt¼fðxÞþwgðxÞþn;ð7Þ
wherein wand nare the noise terms which may simply be added to
the equation (as additive noise n) or scaled with the system variable
(as multiplicative noise wgðxÞ), see [37] for more details.
This work is confined to the analysis of additive noise:
dx
dt¼fðxÞþn;
and only considers Gaussian white noise.
The question is whether and how additive white noise can in-
duce significant changes in the system’s dynamics. Significant
noise effects can first of all be expected at the system’s bifurcation
points. Moreover, it has to be considered that physiological sys-
tems generally cannot be modeled with a single equation as given
above – or only with enormous simplification. All of the more real-
istic physiological models, especially those of HH-type, consist of a
set of equations. However, in a set of equations, when noise is
added to one variable, it also appears to a certain extent and
expectedly in modified form in the other variables. Due to the
closed feedback loops it even comes back into the variable where
it originally has been added. There it will be superimposed with
the actual noise input. Accordingly, even when the noise is uncor-
related, the system carries a memory of previous noise inputs.
Therefore, the question arises: of what type is the effective noise
for the system’s output. In this paper, we compare the effects of
current and conductance noise.
Current noise. Current noise goes directly into the membrane
equation through which the membrane voltage is calculated. The
membrane voltage is the relevant variable which determines the
spiking patterns, i.e. our system’s output. In contrast, conductance
noise is added to the activation variables of the ionic currents
which is farther away from the system’s output and therefore
may undergo additional modifications.
Moreover, there is only one membrane voltage while we have
five different conductances (including the leak conductance). Of
course, there are also five current terms. However summing up dif-
ferent noise terms at this point would not make a qualitative dif-
ference but simply attenuate the noise amplitude. Therefore, the
noise from the different currents are generally comprised in a sin-
gle noise current I(compare (1))
C
M
dV
dt¼X
i
I
i
¼I
l
I
d
I
r
I
sd
I
sr
þI:ð8Þ
Conductance noise. The ionic conductances are calculated indi-
vidually which leads to several differential equations to describe
the time-delays of ion channel activation. However, in this study,
we will not examine the situation when noise is added to all acti-
vation variables. It anyhow would not be possible to implement
112 C. Finke et al. / Mathematical Biosciences 214 (2008) 109–121

compound noise to all conductances because the leak current and
the fast depolarizing current do not have time dependent activa-
tion variables, i.e. are missing the relevant differential equations
ðg
l
¼const;a
d
¼a
d1
Þ.
We also do not specifically consider noise which is going to the
fast repolarizing conductance because numerical simulations have
demonstrated that these effects are immediately overwhelmed by
much lower noise from the slowly activating conductances. Apart
from that, the mathematical statements which will be given for
the slow repolarizing current also apply for the fast repolarizing
current because their equations are of the same form. Therefore,
we will focus on noise implementation in the differential equations
for the slow depolarizing and the slow repolarizing activation vari-
ables a
sd
and a
sr
which is in line with the assumption that the sub-
threshold system, anyhow, constitutes the more relevant part for
spike-generation and impulse pattern alteration.
Noise is added to the activation variable of the slow depolariz-
ing current in the following form:
da
sd
dt¼/
s
sd
ða
sd1
a
sd
Þþn:ð9Þ
The slow repolarizing current is not directly voltage dependent but
only indirectly connected to the membrane voltage via the activa-
tion of the slow depolarizing current to which it is coupled by the
factor g. This requires a second term for its relaxation which is
scaled with the factor k. Again, conductance noise goes into the dif-
ferential equation as a single additive term
da
sr
dt¼/
s
sr
ðgI
sd
ka
sr
Þþn:ð10Þ
2.2.4. Noise implementation in numerical simulations
In all numerical simulations, the noise terms are calculated and
implemented according to the Box–Müller algorithm as described
in [9]:
n¼4D
DtlnðaÞ

1=2
cosð2pbÞ;ð11Þ
where Dtis the time step of the integrator, and a;b2½0;1are uni-
formly distributed random numbers. The noise intensity is adjusted
by the parameter D.
We have run the simulations with Euler as well as fourth order
Runge–Kutta integration with time-steps varying between 0.01
and 0.1 ms without any recognizable differences.
3. Results
3.1. Examples from numerical simulations
Temperature scaling of the model neuron leads to typical vari-
ations of the impulse patterns. When the interspike-intervals are
plotted versus temperature a characteristic bifurcation structure
appears which has been repeatedly described and thoroughly
examined by our group and others. Most of the studies consider
the deterministic situation (Fig. 3, upper trace) for an evaluation
of the underlying dynamics [19,30–32].
The deterministic simulation starts, at low temperatures, with
pacemaker-like tonic activity. Noticeably, this is not accelerated
but slightly slowed with increasing temperature before it goes
through a period doubling scenario into chaotic discharges until
a broad range of burst activity appears. The bursts are triggered
by clearly visible oscillations. The oscillations are continuously
accelerated with increasing temperature. Due to the shortening
of the oscillation cycles the number of spikes per burst is succes-
sively reduced until each oscillation cycle can only trigger a single
spike. A tonic firing pattern remains, but it arises from different
dynamics than the tonic firing activity at low temperatures. These
spikes are riding on subthreshold oscillations while single spike
activity at low temperatures is generated by pacemaker-like
depolarisations.
Noisy simulations of this model, to our knowledge, so far have
only been run with the implementation of current noise (Fig. 3,
second trace). Distinct noise effects, of course, shall be expected
around the bifurcations (e.g. [33]). The most pronounced and
immediately obvious noise effect can be seen when the determin-
istic model operates in the range where the subthreshold oscilla-
tions are close to the threshold of spike generation. In this
situation, the deterministic model either triggers periodic spike se-
quences or completely fails to trigger a spike (below and above
34 °C, respectively). Addition of noise leads to a typical impulse
pattern where the interspike-intervals are scattered around integer
multiples of the oscillation period (from around 29 °Cupto37°C).
The oscillation period determines the timing of the spikes while
noise determines whether a spike is triggered or not (‘skipping’).
The probability of spike generation, of course, depends on how
close the deterministic oscillations are approaching the threshold
of spike generation, i.e. on the oscillation amplitude and base-line.
These exceptional noise effects have been thoroughly examined
mostly with a slightly different model [38] which has been de-
signed to account for the activity pattern of shark electroreceptor
because these are exclusively operating in the range of noisy sub-
threshold oscillations under all stimulus conditions [24]. Similar
pattern have also been seen in divers brain areas at the transitions
form a steady state to tonic firing (e.g. [27]).
The modeling study by Huber and Braun [38] has compared the
effects of current and conductance noise. Conductance noise was
Fig. 3. Bifurcation diagrams of interspike-intervals (ISI, ordinates) from determin-
istic and noisy computer simulations with temperature scaling (abscissea). The
noise intensities Dare shown in the figures. The upper diagram is from the deter-
ministic simulation (D = 0). In the noisy simulations below, the type of noise nis
indicated in parentheses: I, current noise; sd, conductance noise added to the slow
depolarizing variable; sr, conductance noise added to the slow depolarizing
variable.
C. Finke et al. / Mathematical Biosciences 214 (2008) 109–121 113

added to the slow repolarizing variable which, in this model is
voltage-dependent. Apart from the different noise intensities
which are needed to produce comparable effects in the spike pat-
terns the results suggest that there is no qualitative difference be-
tween the different noise implementations. On a first view, this
may be expected because both current and conductance noise re-
flect, although in different ways, the stochastic opening and closing
of ion channels.
In the corresponding range of subthreshold oscillations with
skippings also the actual, more complex model does not exhibit
significant differences in comparison of current and conductance
noise. Similar to the electroreceptor simulations it is only the re-
quired noise intensity which can be 10
3
to 10
4
times smaller for
conductance than for current noise. Apart from these differences,
also in the bursting range (which does not occur in the electrore-
ceptor model), the effects of different noise implementations ap-
pear to be qualitatively the same. In contrast, there are clearly
visible differences with different noise implementations when
the model neuron approaches the range of pacemaker-like tonic
firing at low temperatures.
Remarkably strong noise effects have previously been described
for this type of tonic firing regime in comparison with others [39].
Particularly strong fluctuations of the interspike-intervals can be
seen close to the bifurcation to chaos. This may be due to noise in-
duced transitions across the bifurcation point (at around 6.7 °C)
which is additionally suggested by the serial interdependencies
as indicated by the shape of the return maps ([39],Fig. 6). How-
ever, the fluctuations are becoming weaker and the particular
structure of the return maps disappears the farther the neurons
are tuned away from the bifurcation point. Below 5 °C, noise essen-
tially leads to random fluctuations of approximately Gaussian dis-
tribution of the interspike-intervals around the deterministic
value. These studies were done with current noise.
Addition of conductance noise, in the same temperature range,
leads to completely different interval distributions (Fig. 3, lower
diagrams). These distributions are characterized by an accumula-
tion of short intervals, mostly below the deterministic value, in
parallel with the occurrence of a significant number of long inter-
vals which spread far above the deterministic value. Especially
when noise is added to the slow repolarizing variable (a
sr
,Fig. 3
lowest diagram), the burst pattern from the mid-temperature
range seems to be preserved, continuing across the range of deter-
ministic chaos down to the deterministically tonic firing regime.
Surprisingly, the further the model is tuned into this regime, the
more the interval distributions are separated in short and long
intervals. This is just the opposite effect as observed with current
noise where the distribution is narrowed.
We have additionally illustrated this particular situation in
Fig. 4(A) where the voltage traces from 4 °C are plotted. It illus-
trates the periodic spike generation of the deterministic simulation
(D¼0, upper trace) and the randomness which is introduced by
current noise (n
I
, second trace from top) as well as the extraordi-
narily irregular activity which occurs with conductance noise
when it is added to the slow depolarising variable (a
sd
, second trace
from bottom) and to the slow repolarising variable (a
sr
, lowest
trace).
With conductance noise, the activity pattern, indeed, shows a
burst-like grouping of impulses. Nevertheless, the pattern is com-
pletely different from the periodic burst discharge at mid-temper-
atures where the periodicities of impulse generation can still be
seen even under very noisy conditions. In this situation the transi-
tions between high frequent spikes and periods of silence seem to
be related to a stochastic force which randomly drives the mem-
brane voltage above and below the threshold of spike generation
for longer periods of time. When these voltage shifts are signifi-
cantly longer than the interspike-intervals in the high frequency
discharge a series of impulses will be triggered which then appears
as a burst discharge. Longer downward deflections of the mem-
brane voltage generate the burst pauses.
For further insight into the noise effects on the spike-triggering
processes, in Fig. 4(B) on the left hand side, we have plotted cut-
outs of the noisy sequences from Fig. 4(A) with higher time resolu-
tion. For the graphs on the right hand side we have specifically se-
lected even shorter sequences without spike-generation and also
have enhanced the voltage scale for a better comparison of the
noisy fluctuation in the membrane voltage.
It is immediately evident that noise is differently manifested in
the different curves. Current noise (n
I
, upper traces in Fig. 4(B)) ap-
pears as comparably fast, random fluctuations in the voltage traces.
It is mainly the plot on the right which elucidates that the fast
noise deflections are superimposed by slower and stronger fluctu-
ations. With conductance noise in the slow depolarising variable
(a
sd
, mid traces in Fig. 4(B)), the voltage fluctuations are of similar
amplitude but the tendency towards slow fluctuations with a
reduction of the high frequency components is considerably
strengthened.
This is even more the case when conductance noise is added to
the slow repolarizing variable (a
sr
, lower traces in Fig. 4(B)). The
voltage traces appear very smooth and only tiny fluctuations seem
to remain which are only recognizable with higher voltage resolu-
tion (please note the different scale of the lower diagram on the
right hand side of Fig. 4(B)). These voltage deflections hardly ex-
ceed 1 mV which is much less compared to approximately 6 mV
deflections in the other diagrams.
This outcome is particularly remarkable because just this type
of noise which is implemented via the slow repolarizing variable
a
sr
leads to the strongest fluctuations in the interspike-intervals
Fig. 4. Voltage traces from numerical simulations at different noise implementa-
tions of different intensities as indicated in the figure. (A) From top to bottom:
deterministic simulation, current noise, conductance noise in the slow depolarizing
variable, conductance noise in the slow repolarizing variable. (B) Cut-outs from the
noisy voltage traces in (A) with enhanced time-scale (left) and additionally enha-
nced voltage scale (right, voltage traces without spikes).
114 C. Finke et al. / Mathematical Biosciences 214 (2008) 109–121

(see Fig. 3). These effects can only be understood on the basis of the
smooth and long lasting deflections which can be seen in the long-
er simulations in Fig. 4(A). The high frequent components of the
originally Gaussian white noise seem to be almost eliminated
and obviously do not play a significant role for the generation of
the burst-like pattern. These random transitions between spiking
periods and periods of silence is obviously the result of slow, long
lasting deflections which can reach much higher amplitudes
(clearly more than 10 mV) than all the faster voltage deflections
in the other simulations.
Altogether, different noise implementations can have a similar
effect on the membrane voltage and the spiking pattern but can
also be manifested quite differently. This especially is the case in
the pacemaker-like tonic firing regime at low temperatures which
here has been examined in more detail with numerical simula-
tions. The modification of an originally Gaussian white noise cur-
rent may be understood from the low-pass filtering due to the
membrane capacitance C
M
(compare Eq. (1)). The phenomena
which are seen with addition of conductance noise are not so easy
to explain. This especially concerns the question why conductance
noise has a considerably smoothed, less noisy appearance in the
voltage traces, especially when it is added to the slow repolarizing
variable a
sr
, while it induces much stronger fluctuations in the
interspike-intervals with much smaller noise intensities. These
questions shall be addressed in the next chapters with an analyti-
cal approach.
3.2. Propagation of noise
In the past 30 years, much work has been done in the develop-
ment of tools for the analysis of stochastic effects in dynamical sys-
tems and their applications to biological and biophysical systems
(see for example [40]). Rather sophisticated methods and results
from mathematics and theoretical physics like the use of the Fok-
ker–Planck equation and the theory of Itô and Stratonovich integral
calculus have been integrated successfully into Mathematical Biol-
ogy, as featured for example in the very general context of [41] on
the theory of Ornstein–Uhlenbeck processes in neuronal modeling.
The general strategy of this work is dedicated to the help of build-
ing a better understanding of the propagation of noise terms in the
Huber–Braun system of coupled differential equations and their
large-scaleeffect on the system’s dynamics ratherthan to obtain gen-
eral equations for statistical properties of solutions to this system.
All equations in this work are to be understood in the context of
Itô calculus. Let nðtÞdenote Gaussian white noise with the statistics
hnðtÞi ¼ 0;hnðtÞnðt
0
Þi ¼ 2Ddðtt
0
Þ;
for two time points t;t
0
where all frequencies of its power spectrum
S
n
ðxÞ¼R
1
1
hnðtÞnð0Þi e
ixt
dt2Dare present with equal weight.
We now consider a stochastic differential equation (SDE) for the
rate of change of the activation variable a
sd
ðtÞ. Let BðtÞdenote the
Brownian motion process from which the white noise process is
derived. The SDE of our interest then reads
da
sd
ðtÞ¼ /
s
sd
1
1þexpðs
sd
ðVðtÞV
0sd
ÞÞ a
sd
ðtÞ

dtþDdBðtÞ:
ð12Þ
To solve (12) for a
sd
ðtÞ, we apply a method described in [42]. Here,
we content ourselves with a short description of the necessary steps
towards a solution. The interested reader is invited to a more thor-
ough and general mathematical derivation in Appendix A.
The first step is to compute the (stochastic) exponential UðtÞ,
which in our case means solving the ODE
dUðtÞ¼ /
s
sd
UðtÞdt;
yielding UðtÞ¼e

/t
ssd
. We now make the ansatz
a
sd
ðtÞ¼UðtÞWðtÞ;ð13Þ
where dWðtÞ¼aðtÞdtþbðtÞdBðtÞand dUðtÞas above. The coeffi-
cient functions aðtÞand bðtÞcan be obtained from their relation to
UðtÞ, resulting
aðtÞ¼ /
s
sd
expð/t=s
sd
Þ
1þexpðs
sd
ðVðtÞV
0sd
ÞÞ and bðtÞ¼expð/t=s
sd
Þ:
This gives the solution for WðtÞ, and by inserting into (13) we arrive
at the expression:
a
sd
ðtÞ¼að0Þexpð/t=s
sd
ÞþZ
t
0
expð/ðstÞ=s
sd
Þ
1þexpðs
sd
ðVðsÞV
0sd
ÞÞ ds
þDZ
t
0
expð/ðstÞ=s
sd
ÞdBðsÞ:ð14Þ
Now let
Xðt;sÞDexpð/ðstÞ=s
sd
Þand Y
sd
ðtÞZ
t
0
Xðt;sÞdBðsÞ:
It is physiologically reasonable to set the initial value a
sd
ð0Þ¼0; Eq.
(14) thus becomes
a
sd
ðtÞ¼Z
t
0
expð/ðstÞ=s
sd
Þ
1þexpðs
sd
ðVðsÞV
0sd
ÞÞ dsþY
sd
ðtÞa

sd
ðtÞþY
sd
ðtÞ;
ð15Þ
which gives a deterministic integral a

sd
ðtÞdriven by the stochastic
process Y
sd
ðtÞ. Although we cannot solve the integral of the process
Y
sd
ðtÞanalytically, it is possible to establish the following statistical
properties:
Proposition 1. Let Y
sd
ðtÞ¼R
t
0
Dexpð/ðstÞ=s
sd
ÞdBðsÞ. Then
(1) Y
sd
ðtÞis a Gaussian process with continuous paths.
(2) hY
sd
ðtÞi ¼ 0.
(3) hY
sd
ðtÞY
sd
ðt
0
Þi ¼
D
2
s
sd
2/
e
/jtt
0
j=s
sd
ð1e
2/t=s
sd
Þ.
These statements are a direct consequence of Theorem 4.12 in
[42], which is cited in full in Appendix A, and can be verified imme-
diately. We note that
hY
sd
ðtÞY
sd
ðt
0
Þi ! D
2
s
sd
2/e
/jtt
0
j=s
sd
for ts
sd
;
and thus for reasonably long observations, Proposition 1 shows
that although the rate of change of the activation variable is sub-
ject to Gaussian white noise, the time evolution of the variable
itself is perturbed by colored noise that obeys statistics of an
Ornstein–Uhlenbeck process. This is due to the fact that even
though the Ornstein–Uhlenbeck correlation function is multiplied
by a factor 1 e
2/t=s
sd
(which can be viewed as the memory
term of the autocorrelation since it does not depend on the abso-
lute correlation time jtt
0
j), this factor vanishes very quickly as
time evolves.
We calculate the time constants appearing here at a tempera-
ture of 4 °C because as stressed in Section 2, this is the range where
the dynamics are the most interesting in the sense that the most
significant differences could be seen between different noise
implementations. The time constant of the correlation at 4 °Cis
s
sd
/
100 ms and can be read off statement (3) in Proposition 1. This
time constant is already prominent in Eq. (12) which is the defin-
ing equation for the activation variable a
sd
ðtÞ.
The activation variable of the slow repolarizing current a
sr
ðtÞis
coupled to the slow depolarizing current I
sd
ðtÞvia
da
sr
ðtÞ
dt¼/
s
sr
ðgI
sd
ðtÞka
sr
ðtÞÞ;
C. Finke et al. / Mathematical Biosciences 214 (2008) 109–121 115

and writing I

sd
ðtÞqg
sd
a

sd
ðtÞðVðtÞV
sd
Þ, the corresponding SDE
reads
da
sr
ðtÞ¼ /
s
sr
ðgðI

sd
ðtÞþqg
sd
VðtÞY
sd
ðtÞqg
sd
V
sd
Y
sd
ðtÞÞka
sr
ðtÞÞdt:
ð16Þ
We thus find that the equation for I
sr
ðtÞis modified in a non-trivial
way by Y
sd
ðtÞ. To take a mental note of this fact we denote the
resulting slow repolarizing current by I
noise
sr
ðtÞ.
Entering (15) into the equation for the slow depolarizing ionic
current, the rate of change of the membrane potential reads
C
m
dVðtÞ¼ g
l
ðtÞðVðtÞV
l
Þdt
qg
d
1
1þexpðs
d
ðVðtÞV
0d
ÞÞ ðVðtÞV
d
Þdt
qg
sd
Z
t
0
expð/ðstÞ=s
sd
Þ
1þexpðs
sd
ðVðsÞV
0sd
ÞÞ ds

þDZ
t
0
expð/ðstÞ=s
sd
ÞdBðsÞðVðtÞV
sd
Þdt
qg
r
a
r
ðVðtÞV
r
ÞdtI
noise
sr
ðtÞdt:ð17Þ
Employing our previous notation Y
sd
ðtÞ, expanding and reordering
the terms gives
C
m
dVðtÞ¼g
l
ðtÞðVðtÞV
l
ÞdtI
noise
sr
ðtÞdt
qg
d
ðVðtÞV
d
Þ1
1þexpðs
d
ðVðtÞV
0d
ÞÞ dt
qg
r
a
r
ðVðtÞV
r
Þdt
qg
sd
ðVðtÞV
sd
ÞdtZ
t
0
expð/ðstÞ=sÞ
1þexpðs
sd
ðVðsÞV
0sd
ÞÞ ds
qg
sd
VðtÞY
sd
ðtÞdtqg
sd
V
sd
Y
sd
ðtÞdt
¼ðI
l
ðtÞI
d
ðtÞI
r
ðtÞI

sd
ðtÞÞdt
I
noise
sr
ðtÞdtqg
sd
VðtÞY
sd
ðtÞdtqg
sd
V
sd
Y
sd
ðtÞdt:ð18Þ
This is a stochastic differential equation in VðtÞdriven directly by a
multiplicative noise term qg
sd
VðtÞY
sd
ðtÞand an additive noise
term qg
sd
V
sd
Y
sd
ðtÞas well as an indirect noise source in the cur-
rent I
noise
sr
ðtÞ.
With the numerical values given in Section 2for q;g
sd
and V
sd
,
we can calculate the scaling factor of the noise intensity Dto be
qg
sd
V
sd
¼7:2 in the additive term and qg
sd
¼0:14 in the multipli-
cative term.
While Eq. (18) may not be treatable in a convenient way analyt-
ically, we see that the effect of even a simple perturbation in the
time evolution of a single activation variable – turning it into a
SDE by adding Gaussian white noise – results in a stochastic differ-
ential equation for the membrane potential which is driven by col-
ored noise in a highly non-trivial way. Comparing this directly with
‘current noise’, i.e. Gaussian white noise added to the rate of
change of the membrane potential:
C
m
dVðtÞ¼ðI
l
ðtÞI
d
ðtÞI
r
ðtÞI
sr
ðtÞI
sd
ÞðtÞdtþDdBðtÞ;ð19Þ
we see that conductance noise even in only one dynamic variable
gives rise to an apparently much more complicated, richer structure
of the equation that determines the observables of the system.
For a similar approach to the slow repolarizing activation vari-
able a
sr
ðtÞ, we again add Gaussian white noise nðtÞwith
nðtÞdt¼dBðtÞto the differential equation that determines a
sr
ðtÞ.
In order to derive an analytical solution, by averaging we may as-
sume I
sd
to be constant [43]. The SDE then reads
da
sr
ðtÞ¼ /
s
sr
ðgI
sd
ka
sr
ðtÞÞdtþDdBðtÞ:ð20Þ
Employing the same strategy as in the case of dealing with the slow
depolarizing variable a
sd
ðtÞearlier in this chapter, we find the exact
solution to Eq. (20) to be
a
sr
ðtÞ¼gI
sd
k1exp /kt
s
sr

þZ
t
0
Dexp /kðstÞ
s
sr

dBðsÞ:
ð21Þ
For the noise term
Z
t
0
Dexp /kðstÞ
s
sr

dBðsÞ;
we can establish the following properties:
Proposition 1. Let Y
sd
ðtÞ¼R
t
0
Dexpð/ðstÞ=s
sd
ÞdBðsÞ. Then
(1) Y
sr
ðtÞis a Gaussian process with continuous paths.
(2) hY
sr
ðtÞi ¼ 0.
(3) hY
sr
ðtÞY
sr
ðt
0
Þi ¼
D
2
s
sr
2/k
e
/kjtt
0
j=s
sr
ð1e
2/kt=s
sr
Þ.
The proof is again a fairly straightforward consequence of
Theorem 4.12 in [42]. In particular, we are again dealing with an
Ornstein–Uhlenbeck correlation, damped by a memory term
1e
2/kt=s
sr
which vanishes rather quickly as argued earlier, to
the extent that
hY
sr
ðtÞY
sr
ðt
0
Þi ! D
2
s
sr
2/ke
/kjtt
0
j=s
sr
for ts
sr
:
The observations stated in Propositions 1 and 2 can be interpreted
in such a way that both Eqs. (12) and (20) that determine the
dynamics of the slow oscillator in the Huber–Braun model act as
a low-pass filter on Gaussian white noise.
The time constant of the autocorrelation of Y
sr
ðtÞcan be read off
statement (3) of Proposition 2 and is found to be
s
sr
/k
1200 ms at
4°C.
It is worthwhile to note that the differential Eq. (20) for the acti-
vation variable a
sr
ðtÞfeatures two time constants: the time con-
stant of activation via I
sd
ðtÞis given by
s
sr
/g
17 000 ms while the
time constant of relaxation
s
sr
/k
1200 ms is also the time constant
of noise correlation. Both time constants were calculated for
T=4°C.
Entering (21) into the differential equation for the membrane
voltage, we get
C
m
dVðtÞ¼ðI
l
ðtÞI
d
ðtÞI
r
ðtÞI
sd
ðtÞÞdt
qg
sr
gI
sd
k1exp /kt
s
sr

ðVðtÞV
sr
Þdt
qg
sr
Y
sr
ðtÞVðtÞdtqg
sr
V
sr
Y
sr
ðtÞdt;ð22Þ
which may be viewed as the analogon to Eq. (18) for noise in the
slow repolarizing variable.
Inserting numerical values as before, we find that the intensity
of the multiplicative noise term is scaled by a factor 0.23 while the
additive noise term is scaled by a factor 21.
3.3. Insights from non-linear dynamics
To gain some more insight into the origin of the differences and
relations between the different prescriptions to introduce noise in
the Huber–Braun model, we take a closer look at the structure of
the attracting limit cycle of the system. Again our focus is laid on
the dynamics at 4 °C to which we also refer in the previous chap-
ters because this is the range where the most significant differ-
ences could be seen between different noise implementations.
We first concentrate on the slow degrees of freedom a
sd
and a
sr
which govern the dynamics in between spikes. The graphs in the
respective columns of Fig. 5 show three projections of the limit
cycle on these variables while the rows correspond to the different
implementations of noise. In order to more clearly discern the
effects of noise, its strength Dis smaller by a factor of 50 with
respect to the values considered in the previous sections.
116 C. Finke et al. / Mathematical Biosciences 214 (2008) 109–121

In between spikes the voltage across the membrane is almost
constant – it takes values of about 50 mV and drifts only slowly
towards larger values. Initially, both a
sd
and a
sr
decay. Due to the
voltage drift the asymptotic value a
sd1
ðVÞincreases, however,
and eventually becomes larger than a
sd
. Subsequently, a
sd
increases
again, leads to a larger (negative) current I
sd
, and as a consequence
also to an increase of a
sr
. A bit later Vapproaches 30 mV where
the asymptotic values a
d1
ðVÞand a
r1
ðVÞtake on non-trivial values
and a spike triggers.
Apparently, in all implementations of noise there are only min-
or effects on the traces of V;a
r
and a
sd
, while a
sr
is strongly affected.
Moreover, in spite of the fact of the very different values of Din the
different implementations of noise, the effects on the fluctuations
around the limit cycle are of the same order of magnitude. The
main qualitative difference seems to be that noise added to _
V
and to _
a
sd
leads to smooth changes of a
sr
, while a
sr
shows
pronounced high-frequency oscillations when white noise is added
directly to _
a
sr
. This is a clear consequence of the result of Section
3.2 concerning the correlations building up in the noise. When it
is added to _
Vand to _
a
sd
the effective noise in _
a
sr
is of Ornstein–
Uhlenbeck type with a long memory.
Fig. 6 demonstrates that for all noise strengths up to the phys-
iologically relevant ones the structure of the limit cycle remains
the same. Changes in the spike intervals are therefore not due to
the proximity of a bifurcation.
Since the changes in the spike pattern, and in particular the
occasional appearance of very large time intervals in between
spikes is not related to a bifurcation (i.e. to a topological change
of the flow), we have to resolve to closer inspection of metric prop-
erties of the limit cycle. Fig. 7 shows that the effect of the fast vari-
ables on VðtÞcan safely be neglected in the time intervals in
between spikes, and that the net current due to the slow variables
only amounts to about 1lA/cm
2
. Noticeable effects of the inters-
pike-intervals appear when the current noise is of the same or a
slightly larger order of magnitude (like it is the case for the phys-
iological noise). In between the spikes, in a large part of the inters-
pike interval Vchanges only very little which implies that we are
dealing with a situation close to an unstable fixed point. Within
the accuracy of the noise level, we are hence dealing with the
problem of estimating the dwell time of a trajectory close to an
unstable fixed point in the presence of noise. In the present situa-
tion we are slightly off this point such that full stabilization can
never be achieved, but it is well-known that the dwell times near
an unstable fixed point can substantially increase due to noise.
1
This means that for large noise amplitudes, the system can reside
close to the fixed point for very long times.
4. Discussion
We have examined the effects of current and conductance noise
in a HH-type model with subthreshold oscillations. This model has
been developed to generate different types of physiologically rele-
vant impulse patterns. It includes, in addition to the spike-generat-
ing currents, two subthreshold currents. These subthreshold
currents are modeled according to physiologically important acti-
vation characteristics. There is a slowly depolarizing current which
is directly activated by the membrane voltage, and there is a slowly
repolarizing current which is indirectly coupled to the voltage via
the slow depolarizing current.
Due to the different activation time delays, these currents de-
velop slow voltage oscillations which, in combination with the fast
spike-currents, can introduce rather complex dynamics and can go
through a fascinating range of impulse patterns. This is illustrated
through temperature scaling of the model which mainly means
tuning the activation time constants.
The impulse pattern is essentially determined by the slowly
activating currents. Therefore we have focused on the examination
of conductance noise which is added to one of the slow activation
variables. We have compared these effects with the effects of cur-
rent noise which is added to the equation for the membrane
voltage.
The immediately obvious difference between the various noise
implementations concerns the different noise intensities which are
required to obtain comparable spike pattern fluctuations which
-100
-50
0
50
0.3 0.4 0.5 0.6
V [mV]
asd
-100
-50
0
50
0.352 0.356 0.36 0.364
V [mV]
asr
0.3
0.4
0.5
0.6
0.352 0.356 0.36 0.364
asd
asr
-100
-50
0
50
0.3 0.4 0.5 0.6
V [mV]
asd
-100
-50
0
50
0.352 0.356 0.36 0.364
V [mV]
asr
0.3
0.4
0.5
0.6
0.352 0.356 0.36 0.364
asd
asr
-100
-50
0
50
0.3 0.4 0.5 0.6
V [mV]
asd
-100
-50
0
50
0.352 0.356 0.36 0.364
V [mV]
asr
0.3
0.4
0.5
0.6
0.352 0.356 0.36 0.364
asd
asr
Fig. 5. Three views of limit cycle for T=4°C: noise D¼10
3
added to _
V(top); noise D¼510
8
added to _
a
sd
(centre); noise D¼510
9
added to _
a
sr
(bottom).
1
This is used by artists stabilizing a unicycle by rapid back and forth motion, which
corresponds to the statilization of an inverted pendulum in the (otherwise unstable)
upright position by exerting high frequency periodic [44,45] or noisy [46–48]
perturbations.
C. Finke et al. / Mathematical Biosciences 214 (2008) 109–121 117

somehow resemble the experimentally observed patterns. Current
noise has to be 10
3
to 10
4
times higher than conductance noise
which is added to the slow depolarizing and the slow repolarizing
variable, respectively.
Nevertheless, the resulting noise effects are qualitatively the
same over a broad activity range. It reaches from high tempera-
tures with subthreshold oscillations and skippings via the oscilla-
tory tonic firing regime down to the range of burst activity. It is
all the more remarkable that the noise induced patterns exhibit
significant differences when the pacemaker-like tonic firing regime
at low temperatures is approached.
We specifically have addressed the question why these differ-
ences appear and therefore have focused our numerical simula-
tions, including non-linear dynamics analysis, on the pacemaker-
like tonic firing regime. We have chosen a temperature of 4 °C
which is in the neighbourhood but not too close to the period-dou-
bling bifurcation in the interval plot to check whether the effects
may arise from noise induced transitions to chaos. It can be said
that this is not the case because there is no change in the trajecto-
ries’ topology even with very high noise intensities (Fig. 6, the sec-
ond last).
The essentialhints towards a better understandingof the different
noise effects came from the combination of the numerical and bio-
physical approaches with a mathematical analysis which examines
how noise propagates from the point where it is added to the differ-
ential equations for the slow activation variables to its appearance in
the voltage equation which finally determines the spike pattern.
The combination of these different approaches provided new
insights into the modfications of noise and its manifestation in
the impulse patterns. It even allows us to make some assumptions
about the different sensitivity of the system to current and conduc-
tance noise and also to speculate why the qualitatively different
noise effects appear in the range of pacemaker-like tonic firing
while the effects are principally the same when the spikes are trig-
gered by voltage oscillations.
4.1. Noise modification
To summarize the major outcomes of the mathematical analy-
sis, it can be shown that conductance noise arrives in the voltage
equations significantly modified:
(1) Even simple additive noise appears in the voltage equation
with both an additive and multiplicative term (Eqs. (18),
(22)).
-100
-50
0
50
0.352 0.356 0.36 0.364
V [mV]
asr
-100
-50
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6
V [mV]
ar
-100
-50
0
50
0.352 0.356 0.36 0.364
V [mV]
asr
-100
-50
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6
V [mV]
ar
-100
-50
0
50
0.336 0.344 0.352 0.36 0.368
V [mV]
asr
-100
-50
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6
V [mV]
ar
Fig. 6. Two views of the limit cycle with noise of different amplitudes added to a
sd
: deterministic case D¼0 (top); D¼510
8
also displayed in Fig. 5 (centre); physiological
noise D¼210
5
(bottom).
-5
0
5
10
9000 9200 9400 9600
I [μA / cm2]
t [ms]
sr
sd
slow
fast
Fig. 7. Evolution of the quantities appearing on the right hand side of the system of
differential Eqs. (1) and (5c) for a period of the oscillation in a system with no noise
added. The left hand side shows the currents I
sr
and I
sd
together with their sum, and
the sum of the currents of the fast variables. The right hand side shows the time
derivatives of the three non-trivial activation variables a
sr
;a
sd
and a
r
.
118 C. Finke et al. / Mathematical Biosciences 214 (2008) 109–121

(2) The multiplicative part is downscaled but the additive part is
upscaled.
(3) The white noise is converted into colored noise of Ornstein–
Uhlenbeck correlation with a memory term.
It is essentially the last point which we consider to be of partic-
ular relevance for the different noise effects. The reasons are:
(1) We can find clear relations between the mathematical out-
comes and the phenomena which we can observe in numer-
ical simulations.
(2) The mathematically derived scaling parameters have their
counterparts in functionally most relevant model parame-
ters, including the determinant factor for temperature
scaling.
Concerning the first point, the relation of the different correla-
tion times to the outcomes of the numerical simulations, it can
immediately be seen, as already discussed in Section 3.1, that con-
ductance noise, when it appears in the voltage traces, is far away
from being white noise but is highly correlated. As we now are
even able to estimate the numerical values of the correlation times
(from the mathematical analysis) and can see that these are signif-
icantly longer than the interspike-intervals, of course, it can be ex-
pected that random burst-like discharges are triggered. These
‘bursts’ are not generated by slow oscillations from the system’s
dynamics (this would lead to periodic bursts) but by the slowly
varying highly correlated noise.
Regarding the second point, the connection is given by the time
constants of noise correlation which are the same for the activation
parameters of the respective conductance, including the factor for
temperature scaling. Indeed, this should not be too much of a sur-
prise considering that the differential equations for current activa-
tion constitute a low pass filter, which is also reflected
mathematically in the results stated in Propositions 1 and 2.
4.2. Noise propagation
For a comparison of the different noise effects on the basis of
these findings, let’s have a closer look on the different pathways
of noise propagation for the different noise implementations.
Current noise enters the membrane equations directly. Its
appearance in the voltage traces indicates some correlations of
the originally white noise. Although we cannot offer an analytical
solution for this case, these serial interdependencies may be ex-
plained by the membrane properties which constitute a low pass
filter with a maximum, passive membrane time-constant of
M= 10 ms, compare Section 2.
Conductance noise has to pass additional low pass filters before
it arrives in the voltage equation. The pathway which we were able
to describe analytically is going from noise in the slow depolarizing
variable directly to the membrane voltage. When Gaussian white
noise is added to the differential equation of the slow depolarizing
variable, it appears in the voltage equation as colored noise with a
correlation time constant which corresponds to the time constant
of the activation variable of the slow depolarizing conductance.
At 4 °C, the temperatures at which the numerical simulations were
run, this time constant is approximately 100 ms, i.e. about one or-
der of magnitude higher than the membrane time constant.
However, there is another pathway through which conductance
noise in the slow depolarizing variable can enter the membrane
equation. This is via the slow repolarizing current. The slow repo-
larizing current is not directly voltage-dependent but is activated
by the slow depolarizing current and a separate relaxation term.
The activation time for a
sr
is again two orders of magnitude higher
than that for a
sd
, going up to about 17 000 ms at 4 °C.
We cannot calculate how this additional low-pass filter changes
the correlation time of the noise but it may be expected that it is
also related to this filter’s time-constant. This relation, at least,
could again be shown for the effects of the relaxation part in the
differential equation of a
sr
for which the correlation time, although
through the use of some approximations, could be determined. It is
identical with the relaxation time constant and lies in the range of
1000 ms at 4 °C.
Its not yet clear how the two time constants for activation and
relaxation in the differential equations of the slow repolarizing
variable a
sr
interfere with each other and which of them might
be the determinant factor for the correlation time. However, both
time constants are at least one order of magnitude higher than
the time constants for a
sd
. Accordingly, a significantly higher corre-
lation time can be expected, at least in the range of 1000 ms. Such
slow voltage deflections cannot be recognized easily in the voltage
traces. They are, on a much shorter time scale, repeatedly inter-
rupted by the occurrence of spikes. What can be seen occasionally
are slow downward deflections indicating that the deterministi-
cally depolarizing pacemaker current is overwhelmed by a long-
lasting, noise induced hyperpolarization.
4.3. Noise intensity
The previous discussion relates the different manifestations of
different noise implementations essentially to the alterations of
the correlation times resulting from noise propagation. This may
also be an important factor to explain why such different noise
intensities for current and conductance noise are required. Uncor-
related noise is immediately smoothed out while correlated noise
can sum up over longer times – estimated by the correlation time
– and thereby can reach significantly higher values.
However, the system’s parameters may play an additional role.
The variables to which noise is applied vary, in absolute numbers,
over significantly different ranges. In our examples, at 4 °C (see tra-
jectories in Fig. 6), the voltage passes a range of around 100 mV
which is significantly higher than the range which is covered by
the activation variables. These are in the range of 0.2 for the slow
depolarizing and 0.02 for the slow repolarizing activation variable.
To have the relative noise effects in a comparable range, the
absolute values of the noise-induced fluctuation should correspond
to the absolute values of the range. In comparison of the two types
of conductance noise this means a factor of 10. In comparison of
conductance noise with current noise it is a factor of 10
3
to 10
4
.
This is approximately the relation between the different noise
intensities which we have applied.
4.4. Cooperative effects
The impact of the model’s structure on the correlation time of
noise due to the model’s low-pass components have been dis-
cussed above as the determinant factor which leads to qualitative
alterations of the impulse patterns with conductance noise. This is
the situation at 4 °C, i.e. in the pacemaker-like tonic firing regime.
In contrast, no qualitative differences can be seen over the much
broader range of oscillatory spike generation, it may lead to burst
discharges or single spike activity with skippings.
To get an idea about the reasons for these differences across the
activity range we have to have a closer look on the model’s dynam-
ics. Again, we focus especially on the pacemaker-like range where
the most pronounced differences appear. We use the term ‘pace-
maker-like’ because these dynamics are for sure more complicated
than those of a conventional pacemaker where the depolarization
is driven by a specific pacemaker current. In this model, the pace-
maker-current comprises both types of subthreshold currents with
complicated interactions including significant disturbances from
C. Finke et al. / Mathematical Biosciences 214 (2008) 109–121 119

the spike-current. As shown in Fig. 7 and discussed there, both cur-
rents, and still more their compound value, are coming very close
to zero, indicating the neighbourhood of an unstable fixed point.
Such a situation can additionally be stabilized by noise.
Moreover, the noise induced fluctuations are higher than the
remaining pacemaker currents. This means the noise effects can con-
siderably enhance the pacemaker current and accelerate spike-gen-
eration (shorter interspike-intervals than in the deterministic case,
see Fig. 3). It even can turn the depolarizing (negative) pacemaker
current into a positive, i.e. hyperpolarizing current.
Especially correlated noise, as discussed earlier, can induce
strong and long-lasting voltage changes which occasionally can
be seen as downward deflections in the hyperpolarizing direction
(Fig. 3). Upward deflection of this type cannot be observed because
they are immediately interrupted by spikes. The long-term effects
in this direction is indicated by the occurrence of longer, burst-like
impulse sequences of varying time. These alterations lead to the
typical distributions with short and long interspike-intervals
which can be seen with conductance noise and which makes the
qualitative difference in comparison with current noise.
Hence it is a combination of
(1) this specific system’s dynamics with ion currents close to
zero (close to an unstable fixed point) and
(2) effects of highly correlated, colored noise as the result of the
system’s time-delays.
Both factors are considerably changing when the model is tuned
to higher temperatures. The relevant parameters or temperature
scaling are the activation time-constants which also determine
the correlation time. Over a temperature range of 30 °C it goes
down by a factor of 27.
The more important effect, however, may come from the sys-
tem’s dynamics. The appearance of slow oscillations indicate that
the dynamics are going away from pacemaker currents which are
close to zero towards pronounced limit cycles with stable altera-
tions between de- and repolarizing currents. These comparably
stable dynamics cannot easily be changed by noise. Noise effects
can be seen in the spike-timing with some stronger effects at the
period adding bifurcations where noise can decide whether an
additional spike is triggered or not.
This, of course, makes a big difference in deciding whether the
oscillation triggers a spike or remains below the threshold of
spike-generation (skipping). This leads to the pronounced noise
effects with multimodal interval distributions in the upper tem-
perature range. Nevertheless, these distributions are qualitatively
the same with all types of noise implementations. The cooperative
effects in this situation are brought in by the specific situation in
which the system operates but are obviously not significantly
influenced by modifications of the noise.
4.5. Conclusions and outlook
The different effects of current and conductance noise can be
ascribed to their modifications by model parameters which essen-
tially determine also the systems dynamic’s. Distinct effects of con-
ductance noise with qualitative changes of the spiking pattern can
be related to the transformation of Gaussian white noise into col-
ored noise (of Ornstein–Uhlenbeck type) which appears when
noise is propagating from the activation variables to the voltage
equation by low pass filtering.
As low pass filtering, i.e. time-delayed activation, is a general prop-
erty of HH-type neurons and many other models it may be assumed
that similar effects of noise implementation play a significant role in
other systems as well. This, however, needs further evaluation. More-
over, a particular difference between current and conductance noise
can only be seen in specific situations which clearly emphasizes the
relevance of cooperative effectsbetween noiseand non-lineardynam-
ics. In many situations the difference may be neglected.
However, the situation which we have specifically emphasized
(the pacemaker-like tonic firing regime) is of interest not only be-
cause of its particular dynamics. It is of high relevance also from a
physiological point of view. The transitions from this type of tonic
firing activity to bursting activity reflects a situation which plays a
major role in many physiological and pathophysiological functions.
Correct noise implementation therefore can play a major role to
achieve the physiologically relevant data. Hence, to evaluate which
is the appropriate type of noise, further studies shall also be done
to examine which of the resulting impulse pattern of the model fits
best to the experimental data.
Acknowledgment
This work was supported by the EU through the Network of
Excellence BioSim, No. LSHB-CT-2004-005137.
Appendix A. Solving a linear stochastic differential equation
The main technical tool in solving a linear stochastic differential
equation is the stochastic exponential. We do not assign a physio-
logical meaning to this mathematical object but merely use it for
the sake of calculation.
Assume the process XðtÞto have a stochastic differential. Then,
any function UðtÞsatisfying
dUðtÞ¼UðtÞdXðtÞand Uð0Þ¼1;or UðtÞ¼1þZ
t
0
UðsÞdXðsÞ;
ð23Þ
is called the stochastic exponential of X.
For a process XðtÞof finite variation, the solution to (23) is given
by UðtÞ¼expðXðtÞÞ.IfXðtÞis an Itô process, it is a non-trivial result
that the only solution is given by
UðtÞ¼exp XðtÞXð0Þ1
2½X;XðtÞ

;ð24Þ
where ½X;XðtÞdenotes the quadratic variation of X. For a proof, see
for example [42].
Now consider a stochastic differential equation of the form
dXðtÞ¼ðaðtÞþbðtÞXðtÞÞdtþðcðtÞþdðtÞXðtÞÞdBðtÞ;ð25Þ
where a;b;cand dare continuous functions of t.
Now in order to find a solution in the general setting of (25),it
will be helpful to look for a solution of the form
XðtÞ¼UðtÞWðtÞ;ð26Þ
where
dUðtÞ¼bðtÞUðtÞdtþdðtÞUðtÞdBðtÞ;ð27Þ
and
dWðtÞ¼aðtÞdtþbðtÞdBðtÞ:ð28Þ
Set Uð0Þ¼1 and Wð0Þ¼Xð0Þ. We observe that (27) is the stochas-
tic exponential of some new process ZðtÞby virtue of
dUðtÞ¼UðtÞdZðtÞand dZðtÞ¼bðtÞdtþdðtÞdBðtÞ:ð29Þ
We can thus apply (24) and find that the stochastic exponential of Z
is given by
UðtÞ¼exp ZðtÞZð0Þ1
2½Z;ZðtÞ

¼exp Z
t
0
bðsÞdsþZ
t
0
dðsÞdBðsÞ1
2Z
t
0
d
2
ðsÞds

¼exp Z
t
0
bðsÞ1
2d
2
ðsÞ

dsþZ
t
0
dðsÞdBðsÞ

:
120 C. Finke et al. / Mathematical Biosciences 214 (2008) 109–121

It is now our task to find coefficients aðtÞand bðtÞin such a way that
the relation XðtÞ¼UðtÞWðtÞholds. In order to fulfill it, we take the
differential of the product. Note that up to this point we have made
no limiting assumptions about the processes UðtÞand WðtÞwhich
means the stochastic differential of the product is formally given by
dðUðtÞWðtÞÞ ¼ UðtÞdWðtÞþdUðtÞWðtÞþr
U
ðtÞr
W
ðtÞdt;
where the standard deviations are r
U
ðtÞ¼cðtÞand r
W
ðtÞ¼dðtÞ. The
coefficients can then be found to satisfy relations
aðtÞUðtÞ¼aðtÞcðtÞdðtÞand bðtÞUðtÞ¼cðtÞ:
Taking into account that we already have an expression for UðtÞ,we
can obtain WðtÞfrom (28) and thus find XðtÞto be
XðtÞ¼UðtÞXð0ÞþZ
t
0
aðsÞcðsÞdðsÞ
UðsÞdsþZ
t
0
cðsÞ
UðsÞdBðsÞ

:ð30Þ
We now cite Theorem 4.12 in [42] which yields Propositions 1 and 2:
Theorem. Xðt;sÞbe a regular non-random function with
R
t
0
X
2
ðt;sÞds <1. Then the process YðtÞ¼R
t
0
Xðt;sÞdBðsÞis a Gaussian
process with mean zero and covariance function
CovðYðtÞ;YðtþuÞÞ ¼ Z
t
0
Xðt;sÞXðtþu;sÞdsðt;uP0Þ:
For the rather technical proof of this result, we refer the reader
to [42].
The following concluding remarks will briefly describe how the
method described in this section relates to Section 3.2. Note that
Eq. (12) (as well as Eq. (20)) is a special case of (25). It can be
obtained by setting XðtÞ¼a
sd
ðtÞand
aðtÞ¼ /
s
sd
1
1þexpðs
sd
ðVðtÞV
0sd
ÞÞ ;bðtÞ¼ /
s
sd
;
cðtÞ¼D;dðtÞ¼0:
By inserting a;b;cand dinto (30) and carrying out the integra-
tions, we arrive at Eq. (15) (and Eq. (21), respectively) in Section
3.2.
We are furthermore presented with noise terms
Y
sd
ðtÞ¼Z
t
0
Dexp /ðstÞ
s
sd

dBðsÞand
Y
sr
ðtÞ¼Z
t
0
Dexp /kðstÞ
s
sr

dBðsÞ;
that stem from the stochastic part R
t
0
cðsÞ
UðsÞ
dBðsÞof (30). In order to
gain some insight into the behaviour of these stochastic processes,
we make use of Theorem 4.12 of [42] as cited above to derive Prop-
ositions 1 and 2.
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