Resonance in a Stochastic Neuron Model 
with Delayed Interaction 
Toru Ohira* 
Sony Computer Science Laboratory 
3-14-13 Higashi-gotanda 
Shinagawa, Tokyo 141, Japan 
ohira @csl. sony. co.jp 
Yuzuru Sato 
Institute of Physics, 
Graduate School of Arts and Science, University of Tokyo 
3-8-1 Komaba, Meguro, Tokyo 153 Japan 
ysato @sacral. c. u-tokyo. ac.jp 
Jack D. Cowan 
Department of Mathematics 
University of Chicago 
5734 S. University, Chicago, IL 60637, U.S.A 
cowan@math. uchicago. edu 
Abstract 
We study here a simple stochastic single neuron model with delayed 
self-feedback capable of generating spike trains. Simulations show 
that its spike trains exhibit resonant behavior between "noise" and 
"delay". In order to gain insight into this resonance, we simplify 
the model and study a stochastic binary element whose transition 
probability depends on its state at a fixed interval in the past. 
With this simplified model we can analytically compute interspike 
interval histograms, and show how the resonance between noise and 
delay arises. The resonance is also observed when such elements 
are coupled through delayed interaction. 
I Introduction 
"Noise" and "delay" are two elements which are associated with many natural 
and artificial systems and have been studied in diverse fields. Neural networks 
provide representative examples of information processing systems with noise and 
delay. Though much research has gone into the investigation of these two factors 
in the community, they have mostly been separately studied (see e.g. [1]). Neural 
*Affiliated also with the Laboratory for Information Synthesis, RIKEN Brain Science 
Institute, Wako, Saitama, Japan 
Resonance in a Stochastic Neuron Model with Delayed Interaction 315 
models incorporating both noise and delay are more realistic [2], but their complex 
characteristics have yet to be explored both theoretically and numerically. 
The main theme of this paper is the study of a simple stochastic neural model 
with delayed interaction which can generate spike trains. The most striking feature 
of this model is that it can show a regular spike pattern with suitably "tuned" 
noise and delay [3]. Stochastic resonance in neural information processing has been 
investigated by others (see e.g. [4]). This model, however, introduces a different 
type of such resonance, via delay rather than through an external oscillatory signal. 
It can be classified with models of stochastic resonance without an external signal 
[5]. 
The novelty of this model is the use of delay as the source of its oscillatory dynamics. 
To gain insight into the resonance, we simplify the model and study a stochastic 
binary element whose transition probability depends on its state at a fixed inter- 
val in the past. With this model, we can analytically compute interspike interval 
histograms, and show how the resonance between noise and delay arises. We fur- 
ther show that the resonance also occurs when such stochastic binary elements are 
coupled through delayed interaction. 
2 Single Delayed-feedback Stochastic Neuron Model 
Our model is described by the following equations: 
utu(t) : 
(u(t)) = 
-u(t) + m(u(t - + e(t) 
2 
1 + e-.(v(t) -) 
(1) 
where r/and 0 are constants, and V is the membrane potential of the neuron. The 
noise term L has the following probability distribution. 
1 
: 2 
: 0 (u < -L, > L), (2) 
i.e., z; is a time uncorrelated uniformly distributed noise in the range (-L, L). It 
can be interpreted as a fluctuation that is much faster than the membrane relaxation 
time/. The model can be interpreted as a stochastic neuron model with delayed 
self-feedback of weight W, which is an extension of a model with no delay previously 
studied using the Fokker-Planck equation [6]. 
We numerically study the following discretized version: 
2 
V(t + 1) = 1 + e-,(v(t-') -) - 1 + z; (3) 
We fix r/and 0 so that this map has two basins of attractors of differing size with 
no delay, as shown in Figure i(A). We have simulated the map (3) with various 
noise widths and delays and find regular spiking behavior as shown in Fig i(C) for 
tuned noise width and delay. In case the noise width is too large or too small given 
self-feedback delay, this rhythmic behavior does not emerge, as shown in Figl(B) 
and (D). 
We argue that the delay changes the effective shape of the basin of attractors into 
an oscillatory one, just like that due to an external oscillating force which, as is 
well-known, leads to stochastic resonance with a tuned noise width. The analysis of 
the dynamics given by (1) or (3), however, is a non-trivial task, particularly with 
316 T. Ohira, Y. Sato and J. D. Cowan 
respect to the spike trains. A previous analysis using the Fokker-Planck equation 
cannot capture this emergence of regular spiking behavior. This difficulty motivates 
us to further simplify our model, as described in the next section. 
(A) I (E) l.q 
_ _  i(b)v q P 
(B) 
v(t) x(t) 
(c) 
v(t) 
(D) 
v(t) 
(G) 
x(t) 
x(t) 
Figure 1: (A) The shape of the sigmoid function qb (b) for r/= 4 and 0 = 0.1. The 
straight line (a) is qb = V and the crossings of the two lines indicate the stationary 
point of the dynamics. Also, the typical dynamics of V(t) from the map model are 
shown as we change noise width L. The values of L are (B) L = 0.2, (C) L = 0.4, 
(D) L = 0.7. The data is taken with r = 20, r/ = 4.0, 0 = 0.1 and the initial 
condition V(t) = 0.0 for t C [-r, 0]. The plots are shown between t = 0 to 1000. 
(E) Schematic view of the single binary model. Some typical dynamics from the 
binary model are also shown. The values of parameters are r = 10, q = 0.5, and 
(F) p = 0.005, (G) p = 0.05, and (H) p = 0.2. 
3 Delayed Stochastic Binary Neuron Model 
The model we now discuss is an approximation of the dynamics that retains the 
asymmetric stochastic transition and delay. The state X(t) of the system at time 
step t is either -1 or 1. With the same noise , the model is described as follows: 
f(.) 
= o[f(x(t- r)) + 
= + b) + - b)), 
= 1 (0_<n), -1 (0>n), 
(4) 
where a and b are parameters such that ]a[ <_ L and lb[ _< L, and r is the delay. 
This model is an approximate discretization of the space of map (3) into two states 
Resonance in a Stochastic Neuron Model with Delayed Interaction 317 
with a and b controlling the bias of transition depending on the state of X r steps 
earlier. When a  b, the transition between the two states is asymmetric, reflecting 
the two differing sized basins of attractors. 
We can describe this model more concisely in probability space (Figure i(E)). The 
formal definition is given as follows: 
P(1, t+l) = p, X(t-r)=-l, 
: i-q, X(t-r)=l, 
P(-1, t+l) = q, X(t-r)=l, 
= i-p, X(t-r)--1, 
p = (1+), 
q = (1-), (5) 
where P(s, t) is the probability that X(t) = s. Hence, the transition probability of 
the model depends on its state r steps in the past, and is a special case of a delayed 
random walk [7]. 
We randomly generate X(t) for the interval t = (-r, 0). Simulations are performed 
in which parameters are varied and X(t) is recorded for up to 10 6 steps. They 
appear to be qualitatively similar to those generated by the map dynamics (Figure 
i(F),(G),(H)). From the trajectory X(t), we construct a residence time histogram 
h(u) for the system to be in the state -1 for u consecutive steps. Some examples 
of the histograms are shown in Figure 2 (q = I - q = 0.5, r - 10). We note that 
with p << 0.5, as in Figure 2(A), the model has a tendency to switch or spike to 
the X -- 1 state after the time step interval of r. But the spike trains do not last 
long and result in a small peak in the histogram. For the case of Figure 2(C) where 
p is closer to 0.5, we observe less regular transitions and the peak height is again 
small. With appropriate p as in Figure 2(B), spikes tend to appear at the interval r 
more frequently, resulting in higher peaks in the histogram. This is what we mean 
by stochastic resonance (Figure 2(D)). Choosing an appropriate p is equivalent to 
"tuning" the noise width L, with other parameters appropriately fixed. In this 
sense, our model exhibits stochastic resonance. 
This model can be treated analytically. The first observation to make with the 
model is that given r, it consists of statistically independent r + 1 Markov chains. 
Each Markov chain has its state appearing at every r+l interval. With this property 
of the model, we label time step t by the two integers s and k as follows 
t=s(r+l)+k, (0< s,0<k <r) (6) 
Let P+(t) -- P+(s, k) be the probability for the state to be in the 4-1 state at time 
t or (s, k). Then, it can be shown that 
: 
= 
p 
p+q 
- q, 
p+q 
' = 1-(p+q). (7) 
In the steady state, P+(s -, o, k) -- P+ = a and P_(s -, o, k) =_ P_ = . The 
steady state residence time histogram can be obtained by computing the following 
318 T. Ohira, Y. Sato and J. D. Cowan 
quantity, h(u) _= P(+;-,u; +), which is the probability that the system takes 
consecutive -1 states u times between two +1 states. With the definition of the 
model and statistical independence between Markov chains in the sequence, the 
following expression can be derived: 
p(+;_,.; +) = p+(p_).p+ = (1 _<. < 7.) (s) 
-- P+(P_)'(1 - q) = (/)'(a)(1 - q) (u = 7') (O) 
-- P+(P_)(q)(1 _p)U-(p)_ (/)U(p)2 (u > 7.) (10) 
With appropriate normalization, this expression reflects the shape of the histogram 
obtained by numerical simulations, as shown in Figure 2. Also, by differentiating 
equation (9) with respect to p, we derive the resonant condition for the peak to 
reach maximum height as 
q--pt (11) 
or, equivalently, 
L - a = (L + b)7.. (12) 
In Figure 2(D), we see that maximum peak amplitude is reached by choosing pa- 
rameters according to equation (11). We note that this analysis for the histogram is 
exact in the stationary limit, which makes this model unique among those showing 
stochastic resonance. 
 ....... A ......... 
p 
Figure 2: Residence time histogram and dynamics of X(t) as we change p. The 
values of p are (A) p = 0.005, (B) p = 0.05, (C) p = 0.2. The solid lie in the 
histogram is from the analytical expression given in equatiOns(8-10 ). Also, in (D) 
we show a plot of peak height by varying p. The solid line from equation (9). 
The parameters are 7. = 10, q = 0.5. 
4 Delay Coupled Two Neuron Case 
We now consider a circuit comprising two such stochastic binary neurons coupled 
with delayed interaction. We observe again that resonance between noise and delay 
Resonance in a Stochastic Neuron Model with Delayed Interaction 319 
takes place. The coupled two neuron model is a simple extension of the model in 
the previous section. The transition probability of each neuron is dependent on the 
other neuron's state at a fixed interval in the past. Formally, it can be described in 
probability space as follows. 
Pl(1, t + 1) 
t+ 1) 
P2(1, t + 1) 
t + 1) 
= Pl, 
= i -- ql, 
---- ql, 
-- 1 - 
= P2, 
-- 1 -- q2, 
= q2, 
= 1 -- P2, 
X2(t- r2) = -l, 
X2(t - r2) = 1, 
X2(t - r2) = 1, 
X2(t- r2)= -1, 
Xl(t - rl) = -1, 
xl(t- rl)= 1, 
xl(t- rl) = 1, 
X(t- rl) = -1 
(13) 
Pi(s, t) is the probability that the state of the neuron i is Xi(t) = s. We have 
performed simulation experiments on the model and have again found resonance 
between noise and delay. Though more intricate than the single neuron model, we 
can perform a similar theoretical analysis of the histograms and have obtained ap- 
proximate results for some cases. For example, we obtain the following approximate 
analytical results for the peak height of the interspike histogram of X for the case 
r = r2 = r. ( The peak occurs at r + r2 + 1.) 
H(pl, P2, ql, q2) 
P (P, P2, ql, q2) 
//2 (Pl, P2, ql, q2) 
Pa (Pl, P2, q, q2) 
P4 (Pl, P2, ql, q2) 
fl (Pl, P2, ql, q2 ) 
f 2 (Pl , P2 , q , q2) 
S(pl, P2, ql, q2) 
: {P3(Pl,P2,ql,q2)ql + P4(Pl,P2,ql,q2)(1 -pl)}  (14) 
{Pl(Pl,P2, ql,q2)(qlq2Pl + q1(1 - q2)(1 - ql)) 
+P2(Pl,P2,ql,q2)((1 - P)q2P + (1 - pl)(1 - q2)(1 - 
fl (Pl, P2 , ql , q2 ) f 2 (Pl, P2 , ql , q2 ) 
= (15) 
s(p , P2 , q , q2) 
fl (Pl, P2, ql, q2) 
= S(pl,P2,ql,q2) (16) 
f2 (Pl, P2, ql, q2) 
= (17) 
s (Pl, P2, ql, q2) 
1 
= s(pl,P2,ql,q2) (18) 
pl(1 --p2) +P2(1 -- ql) 
= (19) 
ql(1 --q2)+ q2(1 --q) 
P2 q- Pl (1 - P2 -- q2 ) 
= q2+q(1-p2-q2) (20) 
= fl (Pl, P2, ql, q2) f2 (Pl, P2, q, q2) 
+ fl (Pl, P2, ql, q2 ) + f2 (P, P2, ql, q2 ) + i (21) 
These analytical results are compared with the simulation experiments, examples 
of which are shown in Figure 3. A detailed analysis, particularly for the case of 
rl  r2, is quite intricate and is left for the future. 
5 Discussion 
There are two points to be noted. Firstly, although there are examples which may 
indicate that stochastic resonance is utilized in biological information processing, 
it is yet to be explored if the resonance between noise and delay has some role in 
320 T. Ohira, Y. Sato andJ. D. Cowan 
h('c) 
o.oo 
o.oo 
(A) .... 
  ..... h() ..... 
0.006 
0.004 
0.002 
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 
B) o.o1 
*C 0.000 
h() o.0o 
0.004 
0.002 
(c) 
o.1 0.2 0.3 0.4 o.s o.x 0.2 0.3 0.4, o.5 
Figure 3: A plot of peak height by varying pa. The solid line is from equation (14- 
20). The parameters are r = r2 = 10, q = q2 = 0.5, (A)p = pa, (B) p = 0.005, 
(C) p - 0.025. 
neural information processing. Secondly, there are many investigations of spiking 
neural models and their applications (see e.g., [8]). Our model can be considered 
as a new mechanism for generating controlled stochastic spike trains. One can 
predict its application to weak signal transmission analogous to recent research 
using stochastic resonance with a larger number of units in series [9]. Investigations 
of the network model with delayed interactions are currently underway. 
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