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Journal of Computational Neuroscience 7, 17–32 (1999)
c
°1999 Kluwer Academic Publishers. Manufactured in The Netherlands.
Low-Dimensional Dynamics in Sensory Biology 2:
Facial Cold Receptors of the Rat
HANS A. BRAUN, MATHIAS DEWALD, KLAUS SCH ¨
AFER AND KARLHEINZ VOIGT
Department of Physiology, University of Marburg, 35037 Marburg, Germany
braun@mailer.uni-marburg.de
XING PEI, KEVIN DOLAN AND FRANK MOSS
Center for Neurodynamics, University of Missouri–St. Louis, St. Louis, MO 63121, USA
mossf@umslvma.umsl.edu
Received April 14, 1998; Revised October 5, 1998; Accepted December 2, 1998
Action Editor: G. Bard Ermentrout
Abstract. We report the results of a search for evidence of unstable periodic orbits in the sensory afferents of the
facial cold receptors of the rat. Cold receptors are unique in that they exhibit a diversity of action potential firing
patterns as well as pronounced transients in firing rate following rapid temperature changes. These characteristics
are the result of an internal oscillator operating at the level of the membrane potential. If such oscillators have three
or more degree of freedom, and at least one of which also exhibits a nonlinearity, they are potentially capable of
complex activity. By detecting the existence of unstable periodic orbits, we demonstrate low-dimensional dynamical
behavior whose characteristics depend on the temperature range, impulse pattern, and temperature transients.
Keywords: unstable periodic orbits, temperature sensitive neurons, internal oscillator, low-dimensional dynamics
1. Introduction
This article is the second in a series on our findings of
unstable periodic orbits (UPOs) in thermally sensitive
neurons. The detection of UPOs in biological prepa-
rations as well as in medical applications has recently
become an important topic for research. One of the
main reasons for this interest arises from the use of
UPOs in the control of both nonchaotic (Christini and
Collins, 1995; Hall et al., 1997) and chaotic (Schiff
et al., 1994) dynamical systems in both biology and
medicine. Since most biological and medical prepa-
rations are contaminated by various levels of high-
dimensional noise, which obscures low-dimensional
dynamical signatures, the very question of detection of
UPOs and of being able to distinguish them from noise
becomes paramount. In this series, we are concentrat-
ing on the detection problem. By finding bifurcations
between states with high and low or zero concentra-
tions of UPOs, we seek to identify external conditions
for which the UPOs appear in various types of ther-
mally sensitive neurons.
We continue the use of our topological method for
finding and counting UPOs in noisy biological dyna-
mics (Pierson and Moss, 1995; Pei and Moss, 1996a,
1996b). In a previous article in this series, we discussed
ambient temperature induced bifurcations between sta-
ble and unstable periodic behavior, as observed experi-
mentally in catfish electroreceptor organs (Braun et al.,
1997). As in the previous work, our data are in the form
of time series of neural action potentials, or spikes. An
example is given in Fig. 1A, where time courses of
the mean firing rate Fare shown before and after step
changes in the temperature as shown below. Figure 1B
shows example spike trains. We collect and analyze the
time intervals ID1,ID2,...,IDn,IDn+1,... between
18 Braun et al.
Figure 1. A: The firing rate of a cold receptor in response to the temperature step changes shown by the arrows below—decrease (left) and
increase (right). B: Example spike trains for the two steps. C (left panel): A scatterplot showing a single encounter indicated by the numbered
sequence of points along the stable (blue) and unstable (red) manifolds. The arrows indicate the directions of the flow. C (right panel): The
complete data set (small open diamonds, black) and all selected encounters, stable manifold (blue squares) and unstable manifold (red diamonds)
taken from the steady-state discharge at 30◦C shown above.D: A sequence of successive interspike intervals, ID plot, extracted from the discharge
before and after the cooling step shown in B, with intervals, identified by enlarged colored symbols, belonging to the single encounter shown in
C (left panel). The midinterval, or common point, that belongs to both approaching and departing sets is shown by the green circle.
Low-Dimensional Dynamics in Sensory Biology 19
spikes. The primary way in which our data are dis-
played is on the familiar two-dimensional scatter plot
of IDn+1versus IDnas shown in the lower left panel
in Fig. 1C. We are concerned not only with the general
shape taken by a collection of such time intervals but
particularly by the temporal sequence of the points.
Specifically, we search data files of time intervals for a
sequence of points that approach the line at 45◦on the
scatter plot (the line of all periodic points) at exponen-
tially decreasing distances followed immediately by a
sequence that departs from it at exponentially increas-
ing distances. Such sequential sets map onto the scatter
plot shown in C, where points approaching along the
stable manifold are shown in red, and those depart-
ing along the unstable manifold are shown in blue. As
explained in detail before (Braun et al., 1997), such
sequences are the signatures of single encounters with
UPOs. Straight lines fit to the approaching and depart-
ing sets identify the directions of the stable and unsta-
ble manifolds.1The unstable periodic point lies on the
45◦line at the intersection of these manifolds. There
may be many such encounters in a complete data set.
We collect and plot all the approaching and departing
sets of points found in the entire data set. The selected
sets form two identifiable populations—those grouped
along the stable and unstable manifolds respectively—
whereas randomized data sets, called surrogates as dis-
cussed below and in the appendix, show no identifiable
groupings.
UPOs are present in dissipative chaotic systems
(Cvitanovic, 1991; Badii et al., 1994). They are ar-
ranged in an hierarchical order, with the lowest-order
(period one) orbit for which IDn+1=IDn, followed by
the period two, IDn+2=IDn, and so on comprising a
countable infinity. Biological dynamics are, however,
ubiquitously noisy. Thus after a visit to a particular
UPO, the trajectory may escape from it to wander off
among the higher orders more or less randomly before
returning again to a neighborhood of the original UPO.
Our tasks are to estimate the number of visits in a given
data file and to distinguish their signatures from noise.
Wesearchhereonlyforencounterswith theperiod one
orbits. The first few higher-order orbits (p=1to3)
have been detected experimentally, and a scaling rela-
tion connecting these with the infinite set has been de-
rived (Pei et al., 1998). The method has recently been
used to detect low-dimensional deterministic behav-
ior in records of spontaneous bursting in hippocampal
slice preparations from newborn animals (Menendez
de la Prida et al., 1997), in human epileptic activity
(Le Van Quyen et al., 1997), in human cardiac activity
(Narayanan et al., 1998), and in synaptic discharges
from a central neuron (Faure and Korn, 1997). Al-
ternative methods for finding UPOs in experimental
datafileshaverecentlybeenproposed(Schmelcherand
Diakonos, 1997, 1998; So etal., 1996, 1997; Gluckman
et al., 1997). None of these alternative methods has yet
been extensively applied to biological data, though one
data set from a hippocampal slice preparation has been
used as a demonstration of one of the alternative meth-
ods (So et al., 1997, 1998).
Our experiment was performed using the cold re-
ceptor neurons found in the facial skin surface of the
rat. These neurons respond strongly to transient tem-
perature changes, showing excitation on rapid cooling
and inhibition on rapid warming. In the steady state,
these neurons encode surface temperaure by firing-rate
increases with increasing temperature to a maximum,
followed by subsequent decreases in the firing rate.
The steady-state firing rate is thus ambiguous. The fir-
ing patterns, however, add further information. It has
been shown previously that these patterns are charac-
terized by oscillating processes (Heinz et al., 1990)
and that the signal transduction process is mediated by
noise. For shark multimodal cells with similar temper-
ature dependence, it has been shown that the intrinsic
oscillations and noise are essential to the encoding of
ambient surface temperature (Braun et al., 1994). Thus
we can assume that there exists an endogenous noisy
oscillator (at the level of the membrane potential) gen-
erating more or less regularly spaced action potentials
that mark its threshold crossings (Longtin and Hinzer,
1996).UPOswereobservedandcontrolledwiththeex-
ternal temperature. Normally the temperature was ap-
plied as a steady or linearly ramped slow stimulus. In
studies of the nonstationary responses, however, we ap-
plied both step changes and low-frequency sinusoidal
modulation of an average temperature. The modulation
periods were always much longer than those of the in-
ternal oscillators, but they were either comparable to or
shorter than the adaptation times. Periodically forced
nonlinearoscillatorsaretypicallycapable of bothstable
and chaotic dynamics (Strogatz, 1994).
In the following, we first describe the preparation:
the facial cold receptors of the rat. We then discuss the
recording techniques, show example data, and briefly
describe the analysis method. The method is then ap-
plied to quasistationary files, wherein bifurcations bet-
ween UPOs and more or less purely random behavior
are demonstrated with temperature as the bifurcation
parameter.We then investigatenonstationarysituations
wherein the temperature undergoes either a step or a
20 Braun et al.
ramp change or a low-frequency periodic modulation.
We find that the appearance of UPOs is favored at
certain phases of the modulation cycle and after adap-
tation to step changes. These observations serve also
to demonstrate the suitability of the method for use
with nonstationary data, wherein the time course of
the dynamical behavior is tracked. This is followed
by additional data resulting from slow ramping of the
temperature. We find that UPOs are favored when the
temperature is decreasing from a high value and the fir-
ing patterns change from irregular single spike activity
to bursting but that they disappear when the temper-
ature is ramped up from a lower value and irregular
spike activity is reestablished. These results are mir-
rored by the behavior during sinusoidal temperature
modulation. The UPOs enter as the temperature de-
creases and the first indication of a bifurcation in the
firing pattern occurs. They vanish as further cooling in-
duces more regular bursting. Moreover, the UPOs are
absent during adaptation after steps and fast ramps of
decreasing temperature in the bursting range, but the
reverse is true at the lowest temperature when irregular
spiking is reestablished. Finally, we close with a sum-
mary and discussion.
2. Materials and Methods
Rats weighing between 0.35 and 0.59 kg (mean: 0.48)
were anaesthetized with 50 mg/kg body weight of
sodium pentobarbital. An intravenous insertion was
placed in the left jugular vein in order to administer
additional anaesthetic if needed during the experiment.
Facial hair on the nose and upper lip was removed with
a depilatory cream. The head was fixed with an anchor-
ing device, and a tracheal breathing tube was inserted.
The infraorbital nerve was dissected rostral to the fora-
men infraorbital and prepared as fine filaments. A pool
for mineral oil was formed by sewing the skin flaps
made by the incisions. A small platinum electrode was
used to record single-unit activity. By using cold and
warm brass cylinders with 0.2 mm diameter tips, the
receptive fields of the cold fibers were identified and
delineated.
The receptive fields of individual identified cold re-
ceptors were stimulated using a metal thermode with a
3 mm tip diameter. The temperature of the thermode
was regulated by thermostat-controlled circulating
water. Seven thermostats, connected to the water by
multiposition valves, were used to provide both steady
temperatures and steps between 10 and 40◦C in steps
of 5◦C. Successive temperature steps could be applied
with the time period between steps being 3 to 5 min.
In a second series of experiments, ramp-shaped and
sinusoidal temperature changes were applied using an
electronically controlled thermode. The temperature of
the preparation was measured with a fine thermocou-
ple attached to the stimulating surface. Data in the form
of time series of action potentials were collected and
stored on magnetic tape for later analysis. A more de-
tailed description of these methods can be found in
Heinz et al. (1990).
3. Data Analysis
The method has been previously discussed (Braun
et al., 1997); hence we outline it only briefly here. We
deal with the time intervals Tibetween action poten-
tials. In Fig. 1C, the small open diamonds show the
familiar two-dimensional scatter plot of Tn+1versus Tn
(the period one orbit) for an example 1,000 time inter-
vals recorded from the cold receptor. From these, all
encounters are extracted, as shown by the cloud of blue
triangles (approach along the stable manifold) and red
triangles (departure along the unstable manifold).
We define an encounter to be a sequence of time
intervals that satisfy the following criterion: (1) three
points that approach the 45◦line with successively de-
creasing (perpendicular) distances from it, followed
immediately by three that depart with successively in-
creasing distances; (2) straight lines that are matched
(linear least-squares fit) to these two sets and that
have slopes that fall in the ranges 0≥ms>−1 and
−1≥mus >−∞, where the subscripts represent the
stable and unstable manifolds, respectively; and finally,
(3) an intersection of these two lines that lies within
a distance εof the 45◦line (ε=half the mean per-
pendicular distance of the five intervals from the 45◦
lines). All encounters found in the file using the selec-
tion criterion, together with the two lines that identify
the manifolds, are shown in Fig. 1C.
There will be Nencounters in a given data file that
depend on the selection criterion used. Moreover, us-
ing exactly the same selection criterion, one can also
find a certain number of “encounters” in a file of purely
random numbers. It is thus necessary to establish the
statistical significance of N. This can be done by com-
paring the actual data file to a surrogate file that is a
randomized version of the original file (Theiler et al.,
1992). In our case, the dynamical information is con-
tained in the sets of time interval sequences as defined
Low-Dimensional Dynamics in Sensory Biology 21
above. Thus we can destroy the dynamical information
by randomizing the sequence of all data points in the
original file. The surrogate file is then searched using
the same selection criterion, with the result that Nsen-
counters are found. In order to obtain a good estimate
of Ns, we make and search 100 surrogates of the origi-
nalfileusingdifferentrealizationsof the randomization
process and obtain hNsiand the standard deviation σ.
A statisticalmeasure of thesignificanceof Nis givenby
K=N−hNsi
σ.(1)
For Gaussian statistics K≥2 represents the detection
ofunstable orbits with a greaterthan95% levelofconfi-
dence (Bevington, 1969), whereas K≈0 signifies the
Figure 2. A time course showing the response of a cold receptor to step changes in the temperature. The lowest panel shows the firing rate
and the temperature steps. At each step decrease in temperature, the firing rate shows a spike-like increase followed by an adaptive decay to a
new steady value. The record of all interspike intervals ID is shown above. Selected encounters are shown by enlarged colored symbols (blue,
stable; red, unstable manifolds). The single interval common to both stable and unstable sets (labeled “3, 4” in Fig. 1C, left) is shown in green.
The K-values measured over the lengths of file segments indicated by the solid bars are shown. A histogram of the percentage of encounter
points found in 200 point sequential file segments is shown in the top panel.
absence of low-dimensional dynamical behavior with
statistical significance.
4. Experimental Results
4.1. Step Changes in Temperature
The receptor is a cold-sensing modality. This means
that on a decrease of temperature, the firing rate in-
creases. Figure 2 shows the results of a sequence of
temperature decreases commencing at 35◦C and ter-
minating at 10◦C. The actual steps, shown by the fir-
ing rates Fin the lowest panel, are marked by sharp
increases followed by decays, or adaptations, whose
characteristic time is temperature dependent. This
22 Braun et al.
decay time generally becomes longer at the lower tem-
peratures. The decay consists of a fast component fol-
lowed by a slower component that sometimes nearly
linearly approaches the steady state. The time course
of the interspike intervals, ID, is shown just above the
firing rates. These data have been analyzed for UPOs.
Timeintervalsequences found by themethoddescribed
above are identified by color: blue and red triangles in-
dicating the stable and unstable manifolds respectively
on the return maps. We have colored the selected inter-
vals accordingly on the interval plot, wherein the inter-
val common to points 3 and 4 is identified by a green
point. The segments of the data set that were analyzed
for UPOs are identified by solid lines above which the
K-values are shown. The segments for which K≥2
show statisticallysignificant indicationsof UPOs; how-
ever, only the segments at 25 and 20◦C strongly show
the presence of UPOs. Note also from the colored sym-
bols that UPOs are entirely absent during the adapta-
tion periods just following a step temperature change.
However, this behavior is reversed for the last adapta-
tion after the 15 to 10◦C step.
It is easier to follow the changes in the density of
UPOs from the histograms shown in black in the top
panel. These are simply the raw number of points com-
prising the encounters found in 200 interval segments
of the file expressed as a percentage of the total num-
ber of points in the segment. No surrogates were sub-
tracted. However, it has previously been shown that
the number of (false) encounters found in files of ran-
dom numbers is always around 3.7%, a number that is
also insensitive to the distribution, whether Gaussian
or uniform (Dolan, 1997). Thus vertical bars in this
histogram that are greater than about 4 to 5% can be
regarded as statistically significant indicators of UPOs.
4.2. Ramp Changes in Temperature
We have also applied slow linear changes in the tem-
perature (ramps) but to a different fiber. Figure 3 shows
theresults.Thetimecourse of the temperature is shown
in the bottom panel. Above is the mean firing rate F,
shown as a black jagged curve. All interspike time
intervals are recorded as before by small black dots.
Sequences of points belonging to encounters with the
period one UPOs are shown by the enlarged open sym-
bols: stable, squares with crosses; unstable, diamonds,
and common point, circle. The horizontal, solid black
bars show the portions of the file analyzed for UPOs
with the resulting K-values.
Reading the time course from the left, the temper-
ature was held constant at 34◦C for long enough for
all adaptive behaviors to be completed. At this tem-
perature no statistically significant UPOs are present
(K=0.33). Then the temperature was ramped down
to 24◦C during 200 sec. We notice that an excess firing
rate occurs. The firing rate curve lies above the linear
connection, shown by thedotted line, between the equi-
librium values of Fatthe beginning andending temper-
atures. This is because the rate of temperature change
is faster than the characteristic time of adaptation in
this region. We note also from the ID diagram above
that UPOs enter (K=7.08) about halfway through this
ramp after some initial adaptation where bursting ap-
pears. The temperature is then held constant at 24◦C.
After adaptation, about 40 sec, there is a strong indi-
cation of UPOs (K=10.32). The temperature is then
ramped up. At the very beginning of thisramp-up, stati-
stically significant UPOs vanish (K=−0.03) and re-
main absent for the remainder of the temperature-time
course. This includes a adaptive time at the end of the
ramp-up. Note that the firing rates during the ramp-up
lie below the linear connection (dotted line) between
the two equilibrium values.
A histogram of the encounter rates in 200 interval
sequential file segments is shown in the top panel. The
histogram easily tracks the appearance and disappear-
ance of the UPOs during the ramps and steady temper-
ature regions.
In Fig. 4A are shown the results of a further test
with a faster temperature ramp and two trials with slow
sinusoidal temperature modulations. Again the fiber is
different from those of the previous figures. The re-
sults of a fast ramp down from 30 to 25◦C are shown in
Fig. 4A, where, in addition, we have shown the interval
histograms and the scatterplots for the two tempera-
ture regions on the right. As before, the enlarged open
squareand diamond symbols indicate thestable and un-
stable manifolds respectively, and the common point is
shown as an open circle. Note the adaptation time of
about 40 sec. A few UPOs (but not in statistically signi-
ficant numbers) are present before the ramp-down. As
before, they vanish during adaptation and reappear in
greater concentration (K=6.63) at the lower temper-
ature after adaptation. The spiking behavior bifurcates
from irregular to bursting after the ramp-down. This
is also shown by the transition from monomodal to bi-
modalshape of theinterval histograms.The scatterplots
show two distinct populations of encounter points for
both temperature regions. Note the histograms above
Low-Dimensional Dynamics in Sensory Biology 23
Figure 3. The response of a different cold receptor to moderately slow ramp changes in the temperature. The time course of the temperature
T, the firing rate F, and the ID record are shown in the lower panels with the encounter points shown by the enlarged open symbols, with
K-values marked above. The top panel shows the histogram of encounters in 200 point file segments. The response to lowered temperature after
adaptation is the appearance of a large concentration of UPOs that largely vanish as the temperature is ramped up.
which again show the UPOs entering and departing in
clumps after adaptation in the low temperature region.
4.3. Periodic Changes in Temperature
In Fig. 4B and C, we demonstrate the utility of the
detection method for nonstationary data. Here we are
modulating the temperature with a slow sinusoidal
wave of period T=100 sec with an average tempera-
tureof 30◦C anda peak-to-peak modulationof 2◦C. The
ID diagram and the histogram clearly show the UPOs
appearing during the decreasing temperature epochs
andwhere the firing rates were increasing. Theconcen-
trationofUPOs is thus phase locked to the temperature.
Theydisappear almost completelynear the temperature
maxima (firing-rate minima). The interval histogram is
monomodal and the scatterplot shows two well-defined
populations of encounter points. Note that the average
temperature is the same as the initial constant tem-
perature shown in Fig. 4A, but the K-value is much
higher (8.04 compared to the insignificant 1.35). This
suggests that the UPOs are essentially a nonstationary
24 Braun et al.
Figure 4. A: The response of a different receptor to a more rapid ramp down in temperature. Note the rapid increase in Ffollowed by the
adaptive decay and the total absence of UPOs that reenter after equilibrium. The histogram of UPOs is above. The interval histograms, which
show a bifurcation to bimodality after the temperature down ramp, and the scatterplots are shown on the right. B: The response to a slow periodic
modulation of the temperature, showing phase locking of the UPO concentration (see histogram above) with the decreasing temperature phase.
The interval histogram and scatterplot (right) show that the discharge is monomodal prior to the temperature down ramp; however, the ID
diagram indicated a tendency to bifurcate toward irregular bursting, or bimodality, after the ramp. C: The response to a fast periodic modulation
(period comparable to the adaptive decay time) showing the near absence of UPOs.
Low-Dimensional Dynamics in Sensory Biology 25
phenomenon, appearing preferentially during temper-
ature decreases but after adaptation. However, these
effects are quite dependent on the average temperature
and the firing pattern as the following trial shows.
In Fig. 4C, we show another test with periodic mod-
ulation at a lower average temperature of 26◦C. The
modulation period has been decreased to 50 sec. Com-
paring to the low temperature region in Fig. 4A, it
is evident that the modulation now has quenched the
UPOs. The cooling phase of the cycle has pushed out
the UPOs, and they do not enter because the modu-
lation period is now shorter than the adaptation time.
The transient in Fig. 4A shows that the UPOs do not
appear after a temperature decrease until after an adap-
tation time of about 40 sec. But the period in Fig. 4C
is only 50 sec, and indeed a few encounters do appear
about 40 sec after the temperature decreases (near the
temperature maxima of the following cycle) and near
the minima of the firing rate. The peaks in the bimodal
interval histogram are sharper and more separated than
the ones in Fig. 4A, indicating a more regular bursting
spike pattern.
4.4. Periodic Temperature Modulation Plus
a Slow Transient
In Fig. 5 we show (for yet a different fiber) the effect of
precedingthe periodic modulation with aslowtransient
temperature decrease from an average of about 35 to
26◦C with a superimposed 2◦C sinusoidal modulation
of period 200 sec. The interval histograms are shown
in the top panel. The four scatterplots, shown in the
second panel from the top, correspond to the four seg-
ments of the file marked by K-values. These show well-
defined populations of squares (stable) and diamonds
(unstable) points for the segments with large K-values.
The first scatterplot (corresponding to the single low
K-value) shows less well-defined populations, as ex-
pected. The histogram, interval record, firing rate, and
temperaturetime courses are shownin the lowerpanels.
The general behavior shown in Fig. 5 is consistent with
all the previous observations but with several effects
superimposed. The initial slow temperature ramp in-
duces irregular bursting, and, consistent with Fig. 3, a
large number of encounters occur. This corresponds to
the steady-state discharges in the upper bursting range
in Fig. 1 (25◦C, 20◦C) and to the irregular pattern oc-
curring toward the end of the recording in Fig. 4A
(27◦C, steady discharge). After this ramp, 4 cycles of
slow sinusoidal modulation are added together with an
additional down shift of the average temperature, and
at t=300 sec large concentrations of UPOs enter. The
firing-rate graph and the firing patterns shown by the
ID curve show the continuous effects of further adap-
tation. On the ID plot the two bands of intervals are
well separated during the initial cooling phase, indi-
cating more regular bursting. But as adaptation pro-
ceeds, the bands tend to merge indicating more irreg-
ularity in the bursts. Likewise, the histograms, after
about t=300 sec, are initially in phase with the cool-
ing cycles. But as adaptation proceeds, the sinusoidal
modulation is no longer able to eliminate the UPOs on
cooling. Thus the histogram becomes dephased while
still indicating large concentrations of UPOs. In con-
trast to the results shown in Fig. 4C (sinusoidal mod-
ulation at the same average and peak-to-peak temper-
ature), UPOs are here strongly present after the initial
transient because the period of the modulation is now
considerably longer than the 40 sec adaptation period.
5. Discussion
In an earlier article we demonstrated the presence of
UPOsover certain ranges of thetemperature in the mul-
timodal, temperature-dependent electroreceptors of the
catfish (Braun et al., 1997). In the present article we
continue our studies of the occurrences of these low-
dimensional dynamical objects in sensory biology to
include the facial cold receptors of the rat.
We have exploited the capability of our topologi-
cal method to detect small numbers of encounters with
UPOs in nonstationary files in order to study transient
appearances and disappearances of these objects in re-
sponse to time-varying stimuli. The cold receptor re-
sponds to a decrease in temperature with a sharp and
substantial increase in firing rate followed by a slow,
temperature dependent, decay, during adaptation, to a
final,higher, equilibrium firingrate (Fig. 2). Inequilib-
rium, we have found that UPOs can exist at all temper-
atures studied but are more likely to be found when the
discharge is bimodal—that is, in the presence of burst-
ing (Figs. 3 and 4A), an observation similar to that of
the group in Madrid(Menendez de la Pridaet al., 1997).
This is shown more clearly in Fig. 6, which is the same
data as shown in Fig. 2 except that we have eliminated
the temperature transients and subsequent adaptive de-
cay times in order to focus on the steady-state behav-
ior. The appearance or absence of UPOs is associated
with both the firing pattern and the temperature. At
35◦C, there is a very irregular firing pattern and low
26 Braun et al.
Figure 5. The response of a different receptor to a slow down ramp with subsequent slow periodic modulation of the temperature. After the
down ramp, the system is strongly chaotic, as shown by the K-values, UPO histogram, and scatterplots, which show nicely divided stable and
unstable populations for the larger K-values. The period of the modulation is here much longer than the adaptation time. Nevertheless, effects
of adaptation are discernible throughout the record as the initially well-separated burst intervals become increasingly irregular. The histogram
above shows initial phase locking with the cooling phase after 300 sec, which becomes gradually dephased as adaptation continues.
Low-Dimensional Dynamics in Sensory Biology 27
Figure 6. In order to focus on the steady-state behavior, we have
eliminatedthe transients andadaptive decay timesfrom Fig. 2 andre-
plotted the same data. Encounter points are marked by enlarged open
symbols on the ID record with K-values and temperatures shown
below. Note that the concentration of UPOs, as indicated by the
K-values, depends both on the temperature and the firing pattern,
with irregular bursting patterns favoring high K-values except at the
lower temperature, and the absence of UPOs favored by irregular or
regular, nonbursting firing patterns.
Figure 7. An enlarged view of the temperature step at 20◦C and part of the preceding step at 25◦C showing the adaptive decay in firing rate F,
after the temperature down step and the subsequent equilibrium. Encounter points are marked by the enlarged open symbols on the ID record
with K-values and file segment lengths shown above. The K-values and encounter points indicate the total absence of UPOs during the adaptive
decay in firing rate.
concentration of UPOs. At 30◦C, the firing pattern has
bifurcated to a considerably more regular one, but its
interval histogram is monomodal. UPOs are absent at
30◦C. As the temperature is decreased, large K-values
begin to appear during irregular bursting (25◦C), and
their concentration increases as the bursting becomes
more irregular and the modes of the interval histogram
become more separated (20◦C). At 15◦C, the firing pat-
tern indicates even more irregular bursting with more
widely spaced modes in the interval histogram, but the
concentration of UPOs is becoming small. At the low-
est temperature (10◦C), the pattern has become again
very irregular with a monomodal interval histogram,
and UPOs are completely absent.
UPOs are very sensitive to transients. They are likely
to appear during decreases in the temperature (in-
creases in the firing rate) but only after an adaptive
period during which they are absolutely quenched. In
order to better show the entrance of the UPOs at the
end of the adaptive period, on Fig. 7 we show an ex-
panded view of the 25 and 20◦C temperature regions
in Fig. 2. In the 25◦C region, there is strong UPO
activity indicated by K=6.30. These UPOs are sud-
denly quenched by the downward temperature step to
20◦C that is marked simultaneously by the sharp pos-
itive spike in the firing rate. The UPOs are entirely
absent during the subsequent adaptive decay, as shown
28 Braun et al.
by the complete absence of colored symbols overly-
ing the ID data lying between the dotted vertical lines.
(K=−0.62 <0 indicates that a few false encounters
were found in the surrogate files.) They abruptly enter
again at the end of the adaptive decay and are strongly
present (K=12.66) throughout the remainder of the
20◦C region. A correlation between frequency adapta-
tion and occurrence of UPOs is also seen at the lowest
temperature (10◦C), but it is inverted compared to the
bursting range (Fig. 2). The UPOs enter during adap-
tation and are absent in the steady state at 10◦C. This
inversion is also correlated with a considerable alter-
ation of the firing patterns, the bursting having been
replaced by broadly scattered intervals.
Note also in both Figs. 6 and 7 that in both regions
where Kis large (25 and 20◦C), the UPOs appear and
disappear in clumps. This nonstationary behavior has
been noted before (Pei and Moss, 1996a). Figure 7 is
an expanded view of this temperature range taken from
Fig. 2. It shows the complete disappearance of UPOs
during the accommodation time after a step change in
temperature bounded by two regions with large UPO
concentrations.
The generic behaviors are indicated by Figs. 2, 3,
and 4A. These show the relative absence of UPOs at
high temperatures where the interval distributions are
monomodal; their strong appearance at lower temper-
atures where bursting occurs, as indicated by bimodal
interval distributions; and their absence during adapta-
tion after or during rapid temperature decreases except
at the lowest temperature. These observations are fur-
ther confirmed by the results shown in Fig. 4B and
C and Fig. 5, where the effects of periodic tempera-
ture modulations are shown. For low-frequency modu-
lations such that the modulation period is much longer
than the adaptation time, we observe the appearance of
UPOs during and after temperature decreasing (firing-
rate increasing) phases and their disappearance during
temperature increases. Their concentrations, as shown
by the K-values and the histograms are phase locked
with the temperature modulation (Fig. 4B). If the mod-
ulation frequency is high enough that the period be-
comes comparable to or shorter than the characteristic
time of adaptation, the UPOs are quenched (Fig. 4C).
Finally, slow transients of decreasingtemperature favor
the strong appearance of UPOs (Fig. 5).
Our main object here has been to simply char-
acterize the UPOs in relation to the more familiar
measures—the firing rates and the modalities of the in-
terval histograms. It is difficult to speculate regarding
their function; however, in our earlier work (Braun
et al., 1997), we did note that large concentrations of
UPOswere evident for temperatureswhere the electric-
field sensitivities of the catfish electroreceptors were
maximum. In cold receptors, the majority of UPOs
again was found in the temperature range of maximum
steady-state firing rate that is associated with the max-
imum sensitivity to fast temperature changes (Sch¨afer
et al., 1990). During strong transients the UPOs dis-
appear, while the existence of UPOs in the steady state
might contribute to heightened altertness or readiness
for change.
Appendix
Testing the Method and Surrogates Against
Exponentially Correlated and Harmonic Noise
The following question is often raised: Is the topologi-
calrecurrence method used here to detect UPOs or their
absence, confused by exponentially correlated noise
(so-called colored noise)? This is implausible, since
we observe bifurcations between states with noise only
and states that contain noise plus significant concen-
trations of UPOs mediated by temperature changes as
smallas2◦C—for example, as shown in Fig. 4. Itseems
veryunlikely thatthe always-present noisewould bifur-
cate from short correlation times (for which we would
see no encounters) to longer correlation times (which
would presumably confuse the search algorithm into
reporting false encounters). In this appendix, we show
that the method is not confused by colored noise or by
the existence in the data of stable periodic orbits con-
taminated by noise, or the so-called harmonic noise
(Neiman and Schimansky-Geier, 1994). Moreover, we
further discuss the suitability of surrogates obtained by
simple shuffling with those that retain the correlation
function. In the latter regard it is necessary to define
the event or encounter to be sought in the data files,
from which follows the proper null hypothesis. The
construction of an appropriate surrogate (which tests
the null hypothesis) follows also from the definition of
an event.
The event sought in the data files is the signature
in the phase plane of an encounter with a periodic un-
stable fixed point. Such a signature is indicated by the
example in Fig. 1C. Thus the definition of a single en-
counter is any set of three points that approach the fixed
point with sequentially decreasing distances,followed
Low-Dimensional Dynamics in Sensory Biology 29
by a set of three points that depart from it with sequen-
tially increasing distances. One point, labeled 3, 4 in
Fig. 1C, is common to the two sets. Thus an encounter
consists of a sequence of six time intervals that satisfy
the foregoing definition. In the body of this article, and
in our previous work (Braun et al., 1997) additional
restrictions were applied in the search algorithm; how-
ever, in this appendix we wish to adopt the simplest
conceivable definition of an encounter event.
As demonstrated in our earlier publications (Pierson
and Moss, 1995; Pei and Moss, 1996a, 1996b; Braun
et al., 1997), the approach is to send the search algo-
rithm through a data file in order to count the number N
of encounters, that satisfy the definition. This number
is compared to the average number hNsiand standard
deviation σfound in suitably randomized versions of
the original data file, called surrogates (Theiler et al.,
1992; Schreiber, 1998). The result is the statistic K,
given by Eq. (1). The determination of the presence of
UPOs is made with greater than 99% confidence when
K≥3. (That is, the probability ofa false determination
is p≤0.01.) This statistic depends for its accuracy on
the hypothesis that the surrogates destroy events that
were actually the result of low-dimensional dynamics,
leaving, in the mean, only those occurring by chance,
or the so-called false positive events. It is important
to realize that the encounter event defined above is a
specific sequence of time intervals resulting from low-
dimensional dynamics. It is properly destroyed only by
shuffling the order of the time intervals making up the
sequence. Thus surrogates constructed simply by ran-
domly shuffling the order of the time intervals as they
appear in the file are appropriate.
We now demonstrate that this search algorithm with
simple shuffled surrogates is not confused by colored
noise. The exponentially correlated noise is gener-
ated by the following linear filter, or two-dimensional
Ornstein-Uhlenbeck process (Uhlenbeck and Ornstein,
1930; Jung, 1994):
dy
dt =−
1
τ[y−x],(A1)
dx
dt =−
1
τ[x−W],(A2)
where τis the correlation time and Wis a Gaussian,
white (delta correlated) noise with zero mean and unit
standard deviation. This equation is solved numerically
for four correlation times spanning the biologically rel-
evant range from 25 to 100 ms. The solutions y(t)are
Table 1. Test using second-order linear filter (two-dimensional
Ornstein-Uhlenbeck process).
τ=25 msec 50 msec 75 msec 100 msec
A. Noise alone
N180 187 185 190
hNsi174 174 175 174
σ15.6 15.3 17.5 15.2
K0.39 0.85 0.57 1.05
K(AAFT) 0.22 1.01 0.55 0.96
B. Noise plus added encounters (E)
N+E(E)275 (95) 287 (100) 292 (107) 290 (100)
hNsi172 173 177 175
σ13.9 18.8 14.8 16.5
K7.38 6.07 7.75 6.99
K(AAFT) 6.44 12.35 9.72 12.32
then “thresholded” (Sauer, 1994, 1995)—that is, the
sequencesof time intervalsbetween positivegoingzero
crossings of y(t)are obtained. The two-dimensional
process was necessary because of the thresholding
(Jung, 1994). This point will be discussed further in
a longer version of this work (Dolan, Witte, Neiman,
and Moss, to be published). These files are truncated
to a standard length of 3000 time intervals. The results
of analyzing single noise files to obtain Nand 100 sur-
rogates of the same file to obtain hNsiand the standard
deviation σ, are shown in Table 1A. The values of K,
shown in the bottom row, hover around zero, correctly
indicating that this file contains no UPOs—that is, it is
only noise, albeit colored noise.
In order to demonstrate that the method correctly de-
tects UPOs, we inserted a certain number Eof defined
encounter events into the four noise files generated
above. The defined event was an actual encounter de-
tected in the cold receptor neuron and shown in Fig. 1C
(with point number 3 omitted and point number 4 taken
as the common point 3, 4). The specific sequence of
time intervals in this encounter is
[IDn,IDn+5]≡[50,108,72,87,73,121] ms.(A3)
Before the insertion, these numbers were scaled so that
their mean equaled that of the noise file into which
they were inserted. The events were inserted midway
between the “false positive” events already occurring
in the file. In order to avoid overlaps, these insertions
weremade only when the distancebetween a sequential
pair of false positive events was 10 intervals or greater.
30 Braun et al.
Table 2. Test using harmonic noise.
ω=0.251 K rad/s 0.126 K rad/s 0.084 K rad/s 0.063 K rad/s
A. Noise alone
N163 164 196 171
hNsi161 179 182 184
σ20.1 20.1 20.5 21.0
K0.10 −0.75 0.68 −0.62
K(AAFT) −0.55 1.21 −0.50 0.46
B. Noise plus added encounters (E)
N+E(E)234 (71) 226 (62) 253 (57) 226 (55)
hNsi167 179 183 185
σ16.8 19.0 20.3 12.2
K3.98 2.48 3.44 3.36
K(AAFT) 3.32 3.33 3.64 3.12
In order to preserve the 3000 interval file length, the six
existing time intervals, located at the midpoint, were
overwritten by the sequence (A3) in the course of an
insertion. The actual numbers of encounter events in-
serted into the noise files are shown in parentheses in
Tables 1B and 2B. Thus data files contaminated with
colorednoise but containing knownnumbers of defined
encounter events are created.
At this point we can state the null hypothesis: Ran-
dom files contain the defined encounter events in num-
bers statistically indistinguishable from those to be
found in data files containing the signatures of UPOs.
The results of our search of colored noise files contain-
ing known numbers E, of encounter events (A3) are
shown in Table 1B. We note that the K-values, ranging
from 3 to 5, correctly detect the presence of encoun-
ters at confidence levels greater than 99%. The null
hypothesis can therefore be rejected.
It is often asserted that surrogates that retain tempo-
ral correlations are necessary in order to avoid false in-
dicationsof low-dimensionaldynamical activity infiles
containing correlated noise. Such surrogates are neces-
sary when the indication of low-dimensional behavior
is based exclusively on the existence of correlations—
as, for example, techniques for determining the corre-
lation dimension (Grassberger and Procaccia, 1983).
In contrast, the encounter events defined above, an ex-
ample of which is given by (A3), though correlated,
contain more information than simple exponential cor-
relations. A proper surrogate is one that destroys this
detailedinformation. Since theinformation is conveyed
by the specific order of the time intervals, it is this
order that must be randomized in the surrogate. Simple
shuffling is therefore sufficient and unambiguous, as
we have shown in Table 1. By contrast, the use of sur-
rogates that retain short time correlations is incorrect
since valid encounter events (which are indeed cor-
related) can be retained in the surrogate. Surrogates
that falsely retain defined events do not correctly test
the null hypothesis and may underestimate the statis-
tical confidence levels of the findings. An example is
shownin thebottom rowofTable1B, forwhich we used
the same search algorithm but with amplitude-adjusted
fourier transform (AAFT) surrogates, which do retain
short time correlations (Theiler et al., 1992). As before,
the average hNsiand standard deviation σ, from 100
surrogates were used for each determination of K.We
note that the results are similar. Thus for data sets of
this size, both surrogates seem to work equally well.
However, in this and our preceding article we are
dealing with receptor organs that contain not only
colored noise but oscillators (or pacemakers) as well
(Braun et al., 1994). Thus, in order to further test our
topological method, we generate data files containing
stable periodic orbits (SPOs) as well as UPOs. Stable
periodic orbits show a different topology in the phase
plane: they are stable from all directions. It is this dif-
ference that allows our algorithm to distinguish UPOs
from SPOs. For this test, we make use of harmonic
noise (Neiman and Schimansky-Geier, 1994) gener-
ated by a linear harmonic oscillator driven by noise,
d2x
dt2=−ω2x−αdx
dt +W,(A4)
Low-Dimensional Dynamics in Sensory Biology 31
where ωis the natural frequency, αis the damping, and
Wis again a Gaussian, white noise with zero mean and
unit standard deviation. We have generated data files
from this equation for four natural frequencies with
periods equal to the four correlation times used with
Eqs. (A1) and (A2)—that is, frequencies ranging from
10 to 40 Hz (63 to 251 rad/sec). In order to maintain
the same width of the maximum in the power spectrum
for the different natural frequencies, we set the damp-
ing α=ω/2. Using the same thresholding procedure
for x(t)as described above, we generated files of 3000
time intervals. Our algorithm found no statistically sig-
nificant UPOs in these files as shown by Table 2A.
Following the same procedure to insert Eencounter
events (A3) into these files, we again obtain K-values
ranging from about 3 to nearly 4, again indicating
detection of the UPOs at high confidence levels. The
use of AAFT surrogates again produces about the same
results as the simple shuffled surrogates.
We therefore conclude that the simple search algo-
rithm with shuffled surrogats used here and in our pre-
viouswork efficientlydetects UPOs and isnotconfused
byeither correlated noise or stable periodic orbits. This
demonstration is important to research wherein UPOs
are used in control or for online determinations of their
presence in biological preparations. The reason is ob-
vious: in such applications, the search algorithms must
run rapidly. Algorithms and surrogates that are not nu-
merically intensive, such as the ones used here, thus
offer distinct advantages.
The colored and harmonic noise files used to gener-
ate the results in the two tables have been posted on our
web site and can be downloaded from http://neurodym.
umsl.edu/colorednoise.html
Acknowledgments
We are grateful to L. Menendez de la Prida for inform-
ing us of her interesting results prior to publication.
This work was supported by the European Science
Foundation and the U.S. Office of Naval Research
Physics Division.
Note
1. As in our previous work, we confine this discussion to three-
dimensionalsystems for which themanifolds are one-dimensional
when projected on a two-dimensional plane, such as the scatter
plot or first return map. The manifolds of complex systems of
higher order project onto higher-dimensional spaces.
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