Statistical and Dynamical Interpretation of ISIH 
Data from Periodically Stimulated Sensory Neurons 
John K. Douglass and Frank Moss 
Department of Biology and Department of Physics 
University of Missouri at St. Louis 
St. Louis, MO 63121 
Andre Longtin 
Department of Physics 
University of Ottawa 
Ottawa, Ontario, Canada K1N 6N5 
Abstract 
We interpret the time interval data obtained from periodically stimulated 
sensory neurons in terms of two simple dynamical systems driven by noise 
with an embedded weak periodic function called the signal: 1) a bistable 
system defined by two potential wells separated by a barrier, and 2) a Fit- 
zHugh-Nagnmo system. The implementation is by analog simulation: elco- 
tronic circuits which mimic the dynamics. For a given signal frequency, our 
simulators have only two adjustable parameters, the signal and noise intensi- 
ties. We show that experimental data obtained from the periodically stimu- 
lated mechanoreceptor in the crayfish tailfan can be accurately approximated 
by these simulations. Finally, we discuss stochastic resonance in the two 
models. 
1 INTRODUCTION 
It is well known that sensory information is transmitted to the brain using a 
code which must be based on the time intervals between neural firing events or 
the mean firing rate. However, in any collection of such data, and even when 
the sensory system is stimulated with a periodic signal, statistical analyses have 
shown that a significant fraction of the intervals are random, having no coher- 
ent relationship to the stimulus. We call this component the 'boise". It is clear 
993 
994 Douglass, Moss, and Longtin 
that coherent and incoherent subsets of such data must be separated. Moreover, 
the noise intensity depends upon the stimulus intensity in a nonlinear manner 
through, for example, efferent connections in the visual system (Kaplan and 
Barlow, 1980) and is often much larger (sometimes several orders of magnitude 
larger!) than can be accounted for by equilibrium statistical mechanics (Denk 
and Webb, 1992). Evidence that the noise in networks of neurons can dynami- 
cally alter the properties of the membrane potential and time constants has also 
been accumulated (Kaplan and Barlow, 1976; Treutlein and Schulten, 1985; 
Bernander, Koch and Douglas, 1992). Recently, based on comparisons of inter- 
spike interval histograms (ISIH's) obtained from passive analog simulations of 
simple bistable systems, with those from auditory neurons, it was suggested that 
the noise intensity may play a critical role in the ability of the living system to 
sense the stimulus intensity (Longtin, Bulsara and Moss, 1991). In this work, it 
is shown that in the simulations, ISIH's are reproduced provided that noise is 
added to a weak signal, i.e. one that cannot cause firing by itself. All of these 
processes are essentially nonlinear, and they indicate the ultimate futility of 
simply measuring the background spontaneous rate" and later subtracting it 
from spike rates obtained with a stimulus applied. Indeed, they raise serious 
doubts regarding the applicability of any linear transform theory to neural prob- 
lems. 
In this paper, we investigate the possibility that the noise can enhance the ability 
of a sensory neuron to transmit information about periodic stimuli. The present 
study relies on two objects, the ISIH and the power spectrum, both familiar 
measurements in electrophysiology. These are obtained from analog simulations 
of two simple dynamical systems, 1) the overdamped motion of a particle in a 
bistable, quartic potential; and 2) the FitzHugh-Nagumo model. The results of 
these simulations are compared with those from experiments on the mechanore- 
ceptor in the tailfan of the crayfish Procambarus clarkii. 
2 THE ANALOG SIMULATOR 
Previously, we made detailed comparisons of ISIH's obtained from a variety of 
sensory modalities (Longtin, Bulsara and Moss, 1991) with those measured on 
the bistable system, 
/fit) + e sin(cot) (1) 
where e is the stimulus intensity, and  is a quasi white, Gaussian noise, defined 
by ((t)(s)) = (D/')exp(- It-sl/) with D the noise intensity and ' a 
(dimensionless) noise correlation time. Quasi white means that the actual noise 
correlation time is at least one order of magnitude smaller than the integrator 
Statistical & Dynamical Interpretation of ISIH Data from Periodically Stimulated Sensory Neurons 995 
time constant (the "clock" by which the simulator measures time). It was shown 
that the neurophysiological data could be satisfactorily matched by data from 
the simulation by adjusting either the noise intensity or the stimulus intensity 
provided that the other quantity had a value not very different from the height 
of the potential barrier. Moreover, bistable dynamical systems of the type rep- 
resented by Eq. (1) (and many others as well) have been frequently used to 
demonstrate stochastic resonance (SR), an essentially nonlinear process 
whereby the signal-to-noise ratio (SNR) of a weak signal can be enhanced by 
the noise. Below we show that SR can be demonstrated in a typical excitable 
system of the type often used to model sensory neurons. This raises a tantaliz- 
ing question: can SR be discovered as a naturally occurring phenomenon in 
living systems? More information can be found in a recent review and work- 
shop proceedings (Moss, 1993; Chialvo and Apkarian, 1993; Longtin, 1993). 
There is, however, a significant difference between the dynamics represented by 
Eq. (1) and the more usual neuron models which are excitable systems. A 
simple example of the latter is the FitzHugh-Nagumo (FN) model, the ISIH's of 
which have recently been studied (Longtin, 1993). The FN model is an excit- 
able system controlled by a bifurcation parameter. When the voltage variable is 
perturbed past a certain boundary, a large excursion, identified with a neural 
firing event, occurs. Thus a deterministic refractory period is built into the 
model as the time required for the execution of a single firing event. By con- 
trast, in the bistable system, a firing event is represented by the transition from 
well A to well B. Before another firing can occur, the system must be reset by 
a reverse transition from B to A, which is essentially stochastic. The bistable 
system thus exhibits a statistical distribution of refractory periods. The FN 
system is not bistable, but, depending on the value of the bifurcation parameter, 
it can be either periodically firing (oscillating) or residing on a fixed point. The 
FN model used here is defined by (Longtin, 1993), 
v = v(v- 0.5)(1 - v) - w + 
(2) 
v = v- w - [b + e sin(o01, 
(3) 
where v is the fast variable (action potential) to which the noise//is added, w is 
the recovery variable to which the signal is added, and b is the bifurcation par- 
ameter. The range of behaviors is given by: b >0.65, fixed point and b _> 0.65, 
oscillating. We operate far into the fixed point regime at b = 0.9, so that 
bursts of sustained oscillations do not occur. Thus single spikes at more-or- 
less random times but with some coherence with the signal are generated. A 
schematic diagram of the analog simulator is shown in Fig. 1. The simulator is 
constructed of standard electronic chips: voltage multipliers (X) and operational 
996 Douglass, Moss, and Longtin 
Flu-tpui:P(T), P(w) 
 PC !1 Action potential: fast 
0.5 
variable 
L--_" N oi s e 
L/I gen. 
+0.5) 
.=I0-5s 
Recovery: slow variable 
tb = v-w-b- E sintut 
 b=0.1 to 1.0 
v-w-b 
.=IO-s 
Signal I 
gen. ,,.] 
 sinwt 
Fig. 1 Analog simulator of FitzHugh-Nagumo model. The characteris- 
tic response times are determined by the integrator time constants as 
shown. The noise correlation time was 'n = 10'5 s. 
amplifier summers (_+). The fast variable, v(0, was digitized and analyzed for 
the ISIH and the power spectrum by the PC shown. Note that the noise correla- 
tion time, 10 '5 s, is equal to the fast variable integrator time constant and is 
much larger than the slow variable time constant. This noise is, therefore, col- 
ored. Analog simulator designs, nonlinear experiments and colored noise have 
recently been reviewed (Moss and McClintock, 1989). Below we compare data 
from this simulator with electrophysiological data from the crayfish. 
Statistical & Dynamical Interpretation of ISIH Data from Periodically Stimulated Sensory Neurons 997 
3 EXPERIMENTS WITH CRAYFISH MECHANORECEPTOR CELLS 
Single hair mechanoreceptor cells of the crayfish tailfan represent a simple and 
robust system lacking known efferents. A simple system is necessary, since we 
are searching for a specific dynamical behavior which might be masked in a 
more complex physiology. In this system, small motions of the hairs (as small 
as a few tens of nanometers) are transduced into spike trains which travel up 
the sensory neuron to the caudal ganglion. These neurons show a range of 
spontaneous firing rates (internal noise). In this experiment, a neuron with a 
relatively high internal noise was chosen. Other experiments and more details 
are described elsewhere (Bulsara, Douglass and Moss, 1993). The preparation 
consisted of a piece of the tailfan from which the sensory nerve bundle and 
ganglion were exposed surgically. This appendage was sinusoidally moved 
through the saline solution by an electromagnetic transducer. Extracellular 
recordings from an identified hair cell were made using standard methods. The 
preparations typically persisted in good physiological condition for 8 to 12 
hours. An example ISIH is shown in the upper panel of Fig. 2. The stimulus 
period was, T O = 14 ms. Note the peak sequence at the integer multiples of T O 
(Longtin, et al, 1991). This ISIH was measured in about 15 minutes for which 
about 8K spikes were obtained. An ISIH obtained from the FN simulator in the 
same time and including about the same number of spikes is shown in the lower 
panel. The similarity demonstrates that neurophysiological ISIH's can easily be 
mimicked with FN models as well as with bistable models. Our model is also 
able to reproduce non renewal effects (data not shown) which occur at high fre- 
quency and/or low stimulus or noise intensity, and for which the first peak in 
the ISIH is not the one of maximum amplitude. 
We turn now to the question of whether SR, based on the power spectrum, can 
be demonstrated in such excitable systems. The power spectrum typically 
shows a sharp peak due to the signal at frequency Oo, riding on a broad noise 
background. An example, measured on the FN simulator, is shown in the left 
panel in Fig. 3. This spectrum was obtained for a constant signal intensity set 
just above threshold and for the stated external noise intensity. The SNR, in 
decibels, is defined as the ratio of the strength S(o) of the signal feature to the 
noise amplitude, N(o), measured at the base of the signal feature: SNR = 10 
1OgoS/N. The panel on the right of Fig. 3 shows the SNR's obtained from a 
large number of such power spectra, each measured for a different noise inten- 
sity. Clearly there is an optimal noise intensity which maximizes the SNR. 
This is, to our knowledge, the first demonstration of SR based on the power 
spectra in an excitable system. Just as for the bistable systems (Moss, 1993), 
when the external noise intensity is too low, the signal is not "sampled" fre- 
quently enough and the SNR is low. By contrast, when the noise intensity is too 
998 Douglass, Moss, and Longtin 
 27.0 
 24.0 
.,-, 21.0 
18.0 
t5.0 
2.0 
9.00 
6.00 
3.00 
I 
t6.0:32.0 48.0 64.0 80.0 96.0 tt2. t28. t44.xE60. 
Time [ms) 
'270 
24 0 
2t 0 
18 o 
t5o 
Dt20 
9.00 
6.00 
3.00 
tB.O 32.0 48.0 64.0 OO.O gB.O tt2. t28. t44.xE)BO. 
Time (ms) 
Fig. 2. ISIH's obtained from the crayfish stimulated at 68.6 Hz 
(upper); and the FN simulator driven at the same frequency with b = 
0.9, Vnois e = 0.022 Vrm s, and Vsig -' 0.53 grm s (lower). 
high, the signal becomes randomized. The occurrence of a maximum in the 
SNR is thus motivated. SR has also been studied using well residence time pro- 
bability densities, which are analogous to the physiological ISIH's (Longtin, el 
al, 1991), and was further studied in the FN system (Longtin, 1993). In these 
cases, it is observed that the individual peak heights pass through maxima as the 
noise intensity is varied, thus demonstrating SR, similar to that shown in Fig. 3, 
based on the ISIH (or residence time probability density). 
4 DISCUSSION 
We have shown that physiological measurements such as the familiar ISIH pat- 
terns obtained from periodically stimulated sensory neurons can be easily mim- 
icked by analog simulations of simple noisy systems, in particular bistable sys- 
Statistical & Dynamical Interpretation of ISIH Data from Periodically Stimulated Sensory Neurons 999 
29.3 
22.0 
t4.7 
7.33 
1.47 
.733 
4.00 
2.00 
- - - 000  
.0 .100 .1 . .50 .300 .350 .4 .45E0 ooo o.oo ooo ooo 4o.oo 5o.oo 
frequency IMIz} os vm.tac 
Fig. 3. A power spectrum from the FN simulator stimulated by a 20 
Hz signal for b = 0.9, e = 0.25 V and V,,ois = 0.021 Vm s (left). 
The SNR's versus noise voltage measured in the FN system showing 
SR at V,,oi = 10 mV,,,, (right). Similar SR results based on the ISIH 
have been obtained by Longtin (1993) and by Chialvo and Apkarian 
(1993). 
terns for which the refractory period is strictly stochastic and excitable systems 
for which the refractory period is deterministic. Further, we have shown that 
SR, based on SNR's obtained from the power spectrum, can be demonstrated for 
the FitzHugh-Nagumo model. It is worth emphasizing that these results are 
possible only because the systems are inherently nonlinear. The signal alone is 
too weak to cause firing events in either the bistable or the excitable models. 
Thus these results suggest that biological systems may be able to detect weak 
stimuli in the presence of background noise which they could not otherwise 
detect. Careful behavioral studies will be necessary to decide this question, 
however, a recent and interesting psychophysics experiment using human in- 
terpretations of ambiguous figures, presented in sequences with both coherent 
and random components points directly to this possibility (Chialvo and 
Apkarian, 1993). 
Acknowledgements 
This work was supported by the Office of Naval Research grant N00014-92-J- 
1235 and by NSERC (Canada). 
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