SINiLE NEURON MODEL: RESPONSE TO WEAK 
MODUI, ATION IN THE PRESENCE OF NOISE 
A. R. Bulsara and E. W. Jacobs 
Naval Ocea Systems Center, Materials Research Brach, Sa Diego, CA 92129 
F. Moss 
Physics Dept., Univ. of Missouri, St. Louis, MO 63121 
ABSTRAOT 
We consider a noisy bistable single neuron model driven by a periodic 
external modulation. The modulation introduces a correlated switching 
between states driven by the noise. The information flow through the sys- 
tem from the modulation to the output switching events, leads to a succes- 
sion of strong peks in the power spectrum. The signal-to-noise ratio (SNR) 
obtained from this power spectrum is a measure of the information content 
in the neuron response. With increasing noise intensity, the SNR passes 
through a maximum, an effect which has been called stochastic resonance. 
We treat the problem within the framework of a recently developed approx- 
imate theory, valid in the limits of wek noise intensity, wek periodic forc- 
ing and low forcing frequency. A comparison of the results of this theory 
with those obt6ned from a linear system FFT is also presented. 
INTRODUCTION 
Recently, there has been an upsurge of interest in single or few-neuron nonlinear 
dynamics (see e.g. Li and Hopfield, 1989; Tuckwell, 1988; Paulus, Gass and Mandell, 1990; 
Aihara, Takake and Toyoda, 1990). However, the precise relationship between the many- 
neuron connected model and a single effective neuron dynamics has not been examined in 
detail. Schieve, Bulsara and Davis (1991) have considered a network of N symmetrically 
interconnected neurons embodied, for example in the 'connectlonist' models of Hopfield 
(1982, 1984) or Shamma (1989) (the latter corresponding to a mammalian auditory net- 
work). Through an adiabatic elimination procedure, they have obtained, in closed form, the 
dynamics of a single neuron from the system of coupled differential equations describing the 
N-neuron problem. The problem has been treated both deterministically and stochastically 
(through the inclusion of additive and multiplicative noise terms). It is important to point 
out that the work of Schieve, Bulsara, and Davis does not include a priori a self-coupling 
term, although the inclusion of such a term can be readily implemented in their theory; this 
has been done by Bulsara and Schieve (1991). Rather, their theory results in an explicit 
form of the self-coupling term, in terms of the parameters of the rem6ning neurons in the 
network. This term, in effect, renormalizes the self-coupling term in the Shamma and Hop- 
field models. The reduced or 'effective' neuron model is expected to reproduce some of the 
gross features of biological neurons. The fact that simple single neuron models, such as the 
model to be considered in this work, can indeed reproduce several features observed in bio- 
logical experiments has been strikingly demonstrated by Longtin, Bulsara and Moss (1991) 
through their construction of the inter-spike-interval histograms (ISIHs) using a Schmidt 
trigger to model the neuron. The results of their simple model agree remarkably well with 
data obtained in two different experiments (on the auditory nerve fiber of squirrel monkey 
(Rose, Brugge, Andersen and Hind, 1967) ad on the cat visual cortex (Siegal, 1990)). 
In this work, we consider such a 'reduced' neural element subject to a weak periodic 
external modulation. The modulation introduces a correlated switching between the 
67 
68 Bulsara, Jacobs, and Moss 
bistable states, driven by the noise with the signal-to-noise ratio (SNR) obtained from the 
power spectrum, being taken as a measure of the information content in the neuron 
response. As the aclditive noise variance increases, the SNR passes through a maximum. 
This effect has been called 'stochastic resonance' and describes a phenomenon in which the 
noise actually enhances the information content, i.e., the observability of the signal. Sto- 
chastic resonance has been observed in a modulated ring laser experiment (McNamara, 
Wiesenfeld and Roy, 1988; Vemuri and Roy, 1989) as well as in electron parsmagnetic reso- 
nance experiments (Gammaitoni, Martinelli, Pardi and Santucci, 1991) and in a modulated 
magnetoselastic ribbon (Spano and Ditto, 1991). The introduction of multiplicative noise 
(in the coefficient of the sigmoid transfer function) tends to degracle this effect. 
THE MODEL; STOCHASTIC RESONANCE 
The reduced neuron model consists of a single Hopfield-type computational element, 
which may be modeled as a R-C circuit with nonlinear feedback provided by an operational 
amplifier having a sigmoid transfer function. The equation (which may be rigorously 
derived from a fully connected network model as outlined in the preceding section) may be 
cast in the form, 
+ a x - b taxhx: s0 + F(t), (1) 
where F(t) is Gaussian delta-correlated noise with zero mean and variance 2D, z0 being a 
dc input (which we set equal to zero for the remainder of this work). An analysis of (1), 
including multiplicative noise effects, has been given by Bulsara, Boss and Jacobs (1989). 
For the purposes of the current work, we note that the neuron may be treated as a particle 
in a one-dimensional potential given by, 
u(x)= 2 b (2) 
x being the one-dimensional .tate variable representing the membrane potential. In gen- 
eral, the coefficients a and b depend on the details of the interaction of our reference neu- 
ron to the remaining neurons in the network (Schieve, Bulsara and Davis, 1990). The 
potential described by (2) is bimodal for n) 1 with the extrema occurring at (we set 
throughout the remainder of this work), 
e=0, + [1 1-tanhb lbtanhb (3) 
1 - b sech2b ' 
the approximation holding for large b. Note that the N-shaped characteristic inherent in 
the firing dynamics derived from the Hodgkin-Huxley equations (Rinzel and Ermentrout, 
1990) is markedly similar to the plot of dU/dz vs. z for the simple bistable system (1). 
For a stationary potential, and for D << U0 where U0 is the depth of the deterministic 
potential, the probability that a switching event will occur in unit time, i.e. the switching 
rate, is given by the Kramers frequency (Kramers, 1940), 
0 
r0-- zf_,dyexp(V(y)/Z) f_'dzexp(-U(z)/V) , (44 
which, for small noise, may be cast in the form (the local equilibrium assumption of Kra- 
mers), 
where Ulal(z} = daU/dz a 
We now include a periodic modulation term esinwt on the right-hand-side of {1) (note 
that for e<2(b-1)/(3b) one does not observe switching in the noise-free system). This 
leads to a modulation (i.e. rocking) of the potential (2) with time: an aclditional term 
-zesinwt is now present on the right-hand-side of (2). In this case, the Kramers rate (4) 
becomes time-dependent: 
which is accurate only for  << U0 and co << {Ulal()}/a. The latter condition is referred to 
as the adiabatic approzimation. It ensures that the probability density corresponding to 
Single Neuron Model: Response to Weak Modulation in the Presence of Noise 69 
the time-modulated potential is approximately stationary (the modulation is slow enough 
that the instantaneous probability density can 'adiabatically' relax to a succession of 
quasi-stationary states). 
We now follow the work of McNamara and Wiesenfeld (1989), developing a two-state 
model by introducing a probability of finding the system in the left or right well of the 
potential. A rate equation is constructed based on the Kramers rate r(t) given by (5). 
Within the framework of the adiabatic approximation, this rate equation may be integrated 
to yield the time-dependent conditional probability density function for finding the system 
in a given well of the potential. This leads directly to the autocorrelation function 
< z(t) z(t + r) > and finally, via the Wiener-Khinchine theorem, to the power spectral den- 
sity P(fl). The details are given by Bulsara, Jacobs, Zhou, Moss and Kiss (1991): 
P(n)= vs(4r0 +as) 4r0 +a s + v2(r +as) 
where the first term on the right-hand-side represents the noise background, the second 
term being the signal strength. Taking into account the finite bandwidth of the measuring 
system, we replace (for the purpose of comparison with experimental results) the delta- 
function in (6) by the quantity (Aco)- where Aco is the width of a frequency bin in the 
(experimental) Fourier transformation. We introduce signal-to-noise ratio SNR = 10log R in 
decibels, where R is given by 
4c%0s - . (7) 
R  1 + DS(4ro  +w2) D s(4ro  +w) 8cSr  
In writing down the above expressions, the approximate Kramers rate (4b) has been used. 
However, in what follows, we discuss the effects of replacing it by the exact expression (4a). 
The location of the maximum of the SNR is found by differentiating the above equation; it 
depends on the amplitude  and the frequency co of the modulation, as well as the additive 
noise variance D and the parameters a and b in the potential. 
The SNR computed via the above expression increases as the modulation frequency is 
lowered relative to the Kramers frequency. Lowering the modulation frequency also shar- 
pens the resonance peak, and shifts it to lower noise values, an effect that has been demon- 
strated, for example, by Bulsara, Jacobs, Zhou, Moss and Kiss (i991). The above may be 
readily explained. The effect of the weak modulating signal is to alternately raise and lower 
the potential well with respect to the barrier height U0. In the absence of noise and for 
 << U0, the system cannot switch states, i.e. no information is transferred to the output. In 
the presence of noise, however, the system can switch states through stochastic activation 
over the barrier. Although the switching process is statistical, the transition probability is 
periodically modulated by the external signal. Hence, the output will be correlated, to some 
degree, with the input signal (the modulation 'clocks' the escape events and the whole pro- 
cess will be optimized if the noise by itself produces, on average, two escapes within one 
modulation cycle). 
Figure 1 shows the SNR as a function of the noise variance 2D. The potential barrier 
height U0 = 2.4 for the b = 2.5 case considered. Curves corresponding to the adiabatic expres- 
sion (7), as well as the SNR obtained through an exact (numerical) calculation of the Kra- 
mers rate, using (4a) are shown, along with the data points obtained via direct numerical 
simulation of (1). The Kramers rate at the maximum (2D u U0) of the SNR curve is 0.72. 
This is much greater than the driving frequency co=0.0393 used in this plot. The curve 
computed using the exact expression (4a) fits the numerically obtained data points better 
than the adiabatic curve at high noise strengths. This is to be expected in light of the 
approximations used in deriving (4b) from (4a). Also, the expression (6) has been derived 
from a two-state theory (taking no account of the potential). At low noise, we expect the 
two-state theory to agree with the actual system more closely. This is reflected in the reso- 
nance curves of figure 1 with the adiabatic curve differing (at the maximum) from the data 
points by approximately ldb. We reiterate that the SNR, as well as the agreement between 
the data points and the theoretical curves improves as the modulation frequency is lowered 
relative to the Kramers rate (for a fixed frequency this can be achieved by changing the 
potential barrier height via the parameters a and b in (2)). On the same plot, we show the 
SNR obtained by computing directly the Fourier transform of the signal and noise. At very 
70 Bulsara, Jacobs, and Moss 
low noise, the ~ideal linear filter ~ yields results that are considerably better than stochastic 
resonance. However, at moderate-to-high noise, the stochastic resonance, which may be 
looked upon as a ~nonlinear filter ~, offers at least a 2.5db improvement for the parameters 
of the figure. As indicated above, the improvement in performance achieved by stochastic 
resonance over the ~ideal linear filter' may be enhanced by raising the Kramers frequency 
of the nonlinear filter relative to the modulation frequency w. In fact, as long as the basic 
conditions of stochastic resonance axe realized, the nonlinear filter will outperform the best 
linear filter except at very low noise. 
ZZ.5. 
20.0 
17.5 
15.0 
1Z.5 
0.00 
\ o 
 o o 
\o o o 
o o 
o o o 
1.Z5 Z.50 3.75 5.00 
Noise Variance 2D 
Fig 1. SNR using adiabatic theory, 
eqn. (7), with (b ,w,e)= 
(2.5,0.0393,0.3) and r0 given 
by (4b) (solid curve) and (4a) 
(dotted curve). Data points 
correspond to SNR obtained 
via direct simulation of (1) 
(frequency resolution =6.1x 10 - Hz). 
Dashed curve corresponds to best 
possible linear filter (see text). 
Multiplicative Noise Effects 
We now consider the case when the neuron is exposed to both additive and multipli- 
cative noise. In this case, we set b(t) = bo+ (t) where 
=o, < > = 2D. -,). (s) 
In a reai system such fluctuations might arise through the interaction of the neuron with 
other neurons in the network or with external fluctuations. In fact, Schieve, Bulsara and 
Davis (1991) have shown that when one derives the 'reduced' neuron dynamics in the 
form (1) from a fully connected N-neuron network with fluctuating synaptic couplings, then 
the resulting dynamics contain multiplicative noise terms of the kind being discussed here. 
Even Langevin noise by itself can introduce a pitchfork bifurcation into the long-time 
dynamics of such a reduced neuron model under the appropriate conditions (Bulsaxa and 
Schieve, 1991). In an earlier publication (Bulsara, Boss and Jacobs, 1989), it was shown 
that these fluctuations can qualitatively alter the behavior of the stationary probability 
density function that describes the stochastic response of the neuron. In particular, the 
multijlicative noise may induce additional peaks or erase peaks already present in the den- 
sity (see for example Horsthemke and Lefever 1984). In this work we maintain D= suffi- 
ciently small that such effects axe absent. 
In the absence of modulation, one can write down a Fokker Planck equation for the 
probability density function p (z,t) describing the neuron response: 
or a  
at =- a'" '[(z)r l+ t az [/(z)r ], (9) 
where 
c(x) =- x + botanhz + D= tanhz sech2z, 
/(z) =_ 2(D + Dr. tanh%:), (10) 
D being the additive noise intensity. In the steady state, (9) may be solved to yield a 
'macroscopic potential' function analogous to the function U(z) defined in (2): 
u(x)-- - 2 f' dx + 
Single Neuron Model: Response to Weak Modulation in the Presence of Noise 71 
From (11), one obtains the turning points of the potential through the solution of the tran- 
scendental equation 
z - b0tahz + D= tanhz sech2z =0. (12) 
The modified Kramers rate, r=, for this z-dependent diffusion process has been derived by 
Englund, Snapp and Schieve (1984): 
_/(o) [ u(2)(x,) i u(2)(o) i i/ exp[ U(z,)- u(o)] 03) 
tom -- 2' ' 
where the maximum of the potential occurs at z=0 and the left minimum occurs at 
If we now assume that a weak sinusoidal modulation esinut is present, we may once 
again introduce this term into the potential as in the preceding case, again making the adi- 
abatic approximation. We easily obtain for the modified time-dependent Kramers rate, 
r+(e)---fi(O) [U(2)(z,)[U{2)(O)ll'/2exp U(z,)-U(O)2fo'eSimv--t dz . (14) 
4, ( z ) 
Following the same procedure as we used in the additive noise case, we can obtain the ratio 
R -- 1 + $/Au N, for the case of both noises being present. The result is, 
200 (15) 
where, 
0 - I ull() i ull(o) i/ exp[ u()- u(o) l, 
(16a) 
and 
 , dz  [ ,12 rn,l: ] (16b) 
rl---efo fi(z) -2(D+D=) xi+rn tan-'( tanhs,) , 
with rn -- D=/D. 
0.3 0.6 0.9 t .2 1.5 
D 
Fig 2. Effect of multiplicative 
noise, eqn. (15). (b ,c,e)= 
(2,0.31,0.4) and D,, =0 (top 
curve), o.1 (middle curve) and 
0.2 (bottom curve). 
In figure 2 we show the effects of both additive and multiplicative noise by plotting 
the SNR for a fixed external frequency cv=0.31 with (b0,e)--(2,0.4) as a function of the 
additive noise intensity D. The curves correspond to different values of D=, with the upper- 
most curve corresponding to D==0, i.e., for the case of additive noise only. We note that 
increasing D= leads to a decrease in the SNR as well as a shift in its maximum to lower 
values of D. These effects are easily explained using the results of Bulsara, Boss and Jacobs 
72 
Bulsara, Jacobs, and Moss 
(1989), wherein it was shown that the effect of multiplicative noise is to decrease, on aver- 
age, the potential barrier height and to shift the locations of the stable steady states. This 
leads to a degradation of the stochastic resonance effect at large D, while shifting the loca- 
tion of the maximum toward lower D. 
THE POWER SPECTRUM 
We turn now to the power spectrum obtained via direct numerical simulation of the 
dynamics (1). It is evident that a time series obtained by numerical simulation of (1) would 
display switching events between the stable states of the potential, the residence time in 
each state being a random variable. The intrawell motion consists of a random component 
superimposed on a harmonic component, the latter increasing as the amplitude e of the 
modulation increases. In the low noise limit, the deterministic motion dominates. However, 
the adiabatic theory used in deriving the expressions (6) and (7) is a two-state theory that 
simply follows the switching events between the states but takes no account of this 
intrawell motion. Accordingly, in what follows, we draw the distinction between the full 
dynamics obtained via direct simulation of (1) and the 'equivalent two-state dynamics' 
obtained by passing the output through a two-state filter. Such a filter is realized digitally 
by replacing the time series obtained from a simulation of (1) with a time series wherein 
the z variable takes on the values z = + c, depending on which state the system is in. Fig- 
ure 3 shows the power spectral density obtained from this equivalent two-state system. The 
top curve represents the signal-free case and the bottom curve shows the effects of turning 
on the signal. Two features are readily apparent: 
5O o 
35 75 
Zl.5O 
-7 oo 
oo 
0.04 0.08 0  0.,s 
frequency (Hz) 
Fig 3. Power spectral density via 
direct simulation of (1). 
(b ,ca ,e ,20) = (1.6056,0.03,0.65,0.25). 
Bottom curve: e=O case. 
1. The power spectrum displays odd harmonics of the modulation; this is a hallmark of sto- 
chastic resonance (Zhou and Moss, 1990). If one destroys the symmetry of the potential (1) 
(through the introduction of a small dc driving term, for example), the even harmonics of 
the modulation appear. 
2. The noise floor is lowered when the signal is turned on. This effect is particularly striking 
m the two-state dynamics. It stems from the fact that the total area under the spectral 
density curves in figure 3 (i.e. the total power) must be conserved (a consequence of 
Parseval's theorem). The power in the signal spikes therefore grows at the expense of the 
background noise power. This is a unique feature of weakly modulated bistable noisy sys- 
tems of the type under consideration in this work, and graphically illustrates the ability of 
noise to assist information flow to the output (the signal). The effect may be quantified on 
exaznining equation (6) above ?he noise power spectral density (represented by the first 
 ' '  2 2 2 2 2 -1 
term on the right-hand-side) decreases as the term 2re c {D (4r 0 +lq )} approaches 
unity. This reduction in the noise floor is most pronounced when the signal is of low fre- 
quency (compared to the Kramers rate and large amplitude A similar, effect be 
) . . . may 
observed in the spectral density corresponding to the full system dynamxcs. In this case, the 
total power is only approximately conserved (in a finite bandwidth) and the effect is not so 
Single Neuron Model: Response to Weak Modulation in the Presence of Noise 73 
pronounced. 
DISCUSSION 
In this paper we have presented the details of a cooperative stochastic process that 
occurs in nonlinear systems subject to weak deterministic modulating signals embedded in 
a white noise background. The so-called ~stochastic resonance" phenomenon may actually 
be interpreted as a noise-assisted flow of information to the output. The fact that such sim- 
ple nonlinear dynamic systems (e.g. an electronic Schmidt trigger) are readily realizeable in 
hardware, points to the possible utility of this technique (fax beyond the application to sig- 
nal processing in simple neural networks) as a nonlinear filter. We have demonstrated that, 
by suitably adjusting the system parameters (in effect changing the Kramers rate), we can 
optimize the response to a given modulation frequency and background noise. In a practical 
system, one can move the location and height of the bell-shaped response curve of figure 1 
by changing the potential parameters and, possibly, infusing noise into the system. The 
noise-enhancement of the SNR improves with decreasing frequency. This is a hallmark of 
stochastic resonance and provides one with a possible filtering technique at low frequency. 
It is important to point out that all the effects reported in this work have been reproduced 
via analog simulations (Bulsara, Jacobs, Zhou, Moss and Kiss, 1991: Zhou and Moss, 1990). 
Recently a new approach to the processing of information in noisy nonlinear dynamic sys- 
tems, based on the probability density of residence times in one of the stable states of the 
potential, has been developed by Zhou, Moss and .lung (1990). This technique, which offers 
an alternative to the FFT, was applied by Longtin, Moss and Bulsara (1991) in their con- 
struction of the inter-spike-interval histograms that describe neuronal spike trains in the 
central nervous system. Their work points to the important role played by noise in the 
procesing of information by the central nervous system. The beneficial role of noise has 
already been recognized by Buhmann and Schulten (1986, 87). They found that noise, deli- 
berately added to the deterministic equations governing individual neurons in a network 
significantly enhanced the network's performance and concluded that ...the noise...is an 
essential feature of the information processing abilities of the neural network and not a 
mere source of disturbance better suppressed..." 
Acknowledgements 
This work was carried out under funding from the Office of Naval Research grant nos. 
N00014-90-AF-00001 and N000014-90-J-1327. 
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