Spatially localized interactive neural populations—I. A mathematical model

Spatially localized interactive neural populations—I. A mathematical model

m7-4985/8l/040427-20 w2.cQ/o PcrpvmonPreysLid (I 19X1Societyfor MathemaliealBiology SPATIALLY LOCALIZED INTERACTIVE NEURAL POPULATIONS-I. A MATHEMA...

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SPATIALLY LOCALIZED INTERACTIVE NEURAL POPULATIONS-I. A MATHEMATICAL MODEL*

~JAMES

R. BRANNAN Department of Mathematical Sciences, Clemson University, Clemson, SC 29631, U.S.A.

n WILLIAM E. BOYCE Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12181, U.S.A.

A probabilistic model of a spatially localized, mutually exitatory (inhibitory) population of neurons is formulated to help explain average evoked potential and post-stimulus time histogram measurements. The model is based on the stochastic activity of single neurons within interactive masses of neurons which exhibit co-operative behavior. Macrostate variables corresponding to the above measurements are related through the model to features of neural operation at the individual and ensemble level. Steady-state solutions are obtained and their physiological implications are discussed.

1. Introduction. In this article we develop a mathematical model to help explain nervous activity as observed through post-stimulus time histogram (PSTH) and average evoked potential (AEP) measurements. The AEP is an average of neural electrical activity in the neighborhood of a recording extracellular electrode over an ensemble of identical input-output trials. The PSTH is an average generated from observations of the pulse train from one or a few neurons over an ensemble of identical input-output trials. While the averaging process helps attenuate noise, it also yields coarsened measurements. However, we take the viewpoint that the measurements still contain information which reflects the basic anatomical and functional constraints upon the system. The purpose of the model is to explicitly relate these constraints to the measurements. *This paper is based on a dissertation submitted by the first author to Rensselaer Polytechnic in partial fulfilment of the requirements for the degree of Doctor of Philosophy.

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428

JAMES R. BRANNAN

AND WILLIAM

E. BOYCE

The heart of the model is a diffusion equation for the probability distribution of the soma transmembrane potentials of the neurons in the system. The diffusion equation is then embedded in a system of feedback equations to account for neural interconnectivity and refractory periods. The model is developed logically from a set of primary assumptions which may serve to help identify the types of neural populations to which the model is applicable. Part of the modeling process is to make reasonable idealizations, possibly at the expense of some biological realism, in order to arrive at a tractable set of equations. This is partly an artistic endeavour with justification provided by experimental validation. In a succeeding paper we show that a very simplified version of the model is capable of predicting results in good agreement with experimental data. Other analytical models of neural sets have been developed by using field theoretic approaches (Beurle, 1956; Griffith, 1963b, 1965; Wilson and Cowan, 1972, 1973; Zetterberg, 1973). The Griffith model is a non-linear parabolic equation for a variable representing the overall “excitation” of the nervous network. The field quantity chosen by Beurle is the proportion of cells becoming active in a small volume per unit time, simply called the activity. The general methodology behind the Beurle model is to write equations to the effect that the probability that a cell at position x at time t will fire is equal to the probability that it is sensitive times the probability that sufficient excitation has arrived to cause the neuron to reach threshold. The Wilson and Cowan and Zetterberg models, descendents of the Beurle model, use similar methodology to write equations for primitive variables representing the proportion of excitatory and inhibitory cells which are firing per unit time. We do not attempt to include the effects of spatial variation in our model at this stage, hence it can only be valid for neurons in close spatial proximity. Beginning with a stochastic differential equation for the transmembrane potential of a neuron in the ensemble, probabilistic arguments are used to derive the macroscopic equations modeling the system. The observables arise naturally in the model. While the model developed in this paper is slightly more complicated than the Wilson and Cowan model, its derivation from more elementary principles helps clarify the relationship between the system and the measurements made on the system. We add that the Brannan and Boyce model also predicts that mutually excitatory populations can have stable, uniform levels of activity other than those of inactivity and maximal activity in agreement with empirical results (Freeman, 1974b). This is in contrast to the Beurle and Griffith models which led Griffith (Griffith, 1963a) to introduce inhibition to rectify this so-called paradox (Ashby et al., 1962).

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Since our theoretical work is based largely on the empirical results and data analysis of Freeman (Freeman, 1974a, 1974b, 1975) we wish to introduce some of his terminology. Freeman categorizes various types of neural mass. The most primitive type is the KO set defined as any set of neurons numbering from lo3 to lo8 having a common source of input and a common sign of output (+, excitatory; -, inhibitory) and having no functional interconnections between neurons in the set. A KI set is any set of neurons having a common source on input, a common sign of output and dense interactions between neurons in the set. There are two types: The KI, set consists of mutually excitatory neurons and the KI, set consists of mutually inhibitory neurons. The KII set is formed by the existence of dense functional interconnections between two KI sets. Although there are more complex levels in the hierarchy of interactive neural masses, we do not mention them since we will not be concerned with them. 2. Assumptions. Consider a system of identical neurons. Let X,,(t)[Xi,(t)J be the stochastic process which represents the electrical potential of the exterior (interior) surface of the soma membrane of a single neuron with respect to a far distant point. Then we represent the state of a typical neuron in the aggregate by the stochastic process

x(t)4xin(t)-XpX(f)

(1)

which is termed the transmembrane potential (TMP). We make the following assumptions about the ensemble of neurons and its constituents. (Al) There are n neurons in the population and n$- 1. (A2) The space occupied by the ensemble is small enough and axon conduction velocities are large enough so that the population is essentially spatially homogeneous with respect to conduction delays. (A3) Postsynaptic potentials (PSPs) arriving at a neuron combine via an operation deviating very little from linear algebraic summation and result in step changes in X(r). (A4) In the absence of deliberate and controlled stimulation there is a background activity due to the random and spontaneous discharging of individual neurons within the ensemble (which may be partially a result of background activity in the afferent pathway). (A5) The neurons within the mass are densely interconnected. (A6) The number of connections between a pre-synaptic cell and a postsynaptic cell is small relative to the total number of synaptic inputs to the post-synaptic cell. (A7) Afferent projections are densely and uniformly distributed across the set.

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(A8) Individual neurons in the mass which are sensitive to input generate an action potential at the instant that X(t) exceeds the threshold potential (THP), 8. (A9) In the absence of input, X(t) decays exponentially to an equilibrium potential c with membrane time constant /I. (AlO) Following the generation of an impulse, a neuron cannot be made to fire again for a period of rmsec, the absolute refractory period (ARP). The TMP is then reset to a new value x0 and the process begins anew. To partially simulate depolarizing after potentials, x0 should be slightly less than c. (All) Excitatory post-synaptic potentials (EPSPs) and inhibitory postsynaptic potentials (IPSPs) are small relative to 8-X0. (Al2) Synaptic and conduction delays are lumped into a single delay. The above list contains the most obvious assumptions about the operations of neural sets which are contained in the model. Note that two important concepts are built into the assumptions. First, there is the principle of divergence whereby the axon(s) of single neurons project to many target neurons. Second, there is the principle of convergence whereby many lines of traffic converge on a single cell. If PC cc in (A9), the single neuron model is called a leaky integrator since the neuron forgets inputs that occurred far in the past. If /~-CD, we obtain a perfect integration model in which all input is remembered until the neuron tires. 3. Stochastic Model of Single Cell Activity. The times of discharge of neurons in interactive masses are individuaily unpredictable. We therefore regard the sequence of epochs corresponding to the occurrence of action potentials in a single neuron as a time series generated by a stochastic point process. Although the spike trains of single neurons are not Poisson processes, a result from queueing theory (Khinchine, 1960) states that the superposition of n stationary, orderly, and mutually independent point processes approaches a Poisson process as n+cz if the intensities of the component streams tend to zero uniformly while the sum of the intensities tends to a finite constant and no large numbers of events occur in short sections of one and the same stream. Since significant changes in the amplitudes of AEPs and PSTHs occur over time intervals ranging upward from 50 msec, over periods of time short compared to 50 msec the point process generated by a single neuron may be regarded as stationary. These considerations along with assumptions (Al), (A5), (A6), (A7), (AlO), and (All) suggest that a reasonable approximation is to assume that in timedependent states of activity, single neurons in the population are bombarded by streams of EPSPs and IPSPs which look like compound

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Poisson processes, non-homogeneous in time. Thus, we let N,(t)[Ni(t)] denote the non-homogeneous Poisson process with intensity &(t)[Ai(t)] that is associated with the arrival of EPSPs (IPSPs). Then let E(r)=

2 j=

Wi

1

be the corresponding non-homogeneous compound Poisson process where the Wj(Zj) are independent and identically distributed random variables corresponding to signed step changes in the soma TMP effected by the EPSPs (IPSPs). In accordance with (A3), (A9), and (AlO) we write dX(t)=

+X(t)-c]dt+dE(t)+dr(t)

(2)

X(0)=x().

(3)

with

The solution to equations (2) and (3) can be represented by N.(r)

X(t)=

C exp[-(t-~j)/Bl~j~(t-rj)

(xg- c)exp(-t//?)+c+

j=l

NiW

+ jzI exP C-(t-~j)IPlzju(r-~j)

(4)

are the arrival times of the EPSPs (IPSPs) where r l,*-*,rN,(~) (s I,. . ., Sag) and U( - ) is the unit step function. Denoting the probability density of the random variable ~jlZj) by k,(w)[ki(z)] it follows (Soon& 1973; Feller, 1971) that the equation of evolution for the conditional probability density p(x, t)=p(x, t;xo,O) is

s s cc

+M)

k,(w)p(x-w, t)dw-A(t)p(X, t)

-CC

a

+

ii(t)

ki(z)p(x-Z,t)dZ

-CC

(5)

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E. BOYCE

with

p(x,O)=S(x-x,).

(6)

expectation given X(0) =_yO and Now let Exe denote the conditional j=,/-1 while ~~,[X(t)]=Ex,[{X(t)-E,,[X(t)]}Z]. If d,(t)=& and ~i(t)=~i are constants, not both zero, it can be shown from equations (5) and (6) that the characteristic function

of the normalized process

,(,)=WW,,CW1 62, lx (t )I may be written as

Y(V, 0 =exp(- v2/2)exp WY 0) where

(f W(6M and

M= 3.3!



Note that W and Z are independent of X(0). Therefore, if t zp >O, then y(v, t) converges to exp - v2/2) as fi(n,+IzJ+~~. It follows that the distribution associated with the normalized random variable S(t) converges in distribution to the unit normal distribution (Chung, 1974). A slightly more complicated analysis shows that X(r) will also be approximately normally distributed in the nonhomogeneous case as long as [J.,(t)+ ~i(t)]P~ C% 1 where C is a constant. Then only the mean and variance of X(t) are required to have an estimate of its distribution. We replace equation (2) by a linear Ito stochastic differential equation whose

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solution is in fact a Gaussian process with mean and variance identical to the mean and variance determined by equation (2) (Arnold, 1973): dX(t)=

-i[X(t)-C]+J&)JY[W]+E.,(~)E[Z] P (7)

with

(8)

X(0)=x,.

equation associated with equation (7),

The Fokker-Planck

a

ap(xJ)

-=z

at

i(‘( s

x-c)-i,,(t)E[W]-ii(t)E[Z]

++{Uf)E[W2]+&(t)E[Z'])} is therefore the appropriate

>p(x,t)

1

mx,f) ax2

diffusion approximation

(9)

to equation (5).

The significance of the requirement that the dimensionless quantity [i(t) +&(t)Jfi be large is readily understood from equation (4), where we see that J:-s [J,(s)+;li(s)]d s is the effective number of random variables contributing to the sum X(t). Under the conditions stated,

s I

[iJS)+Ji(S)]dshp

min

{5(S) + J-i(S)}2 C.

SE[1-_8,11

1-B

Thus, we are getting an effect analogous to the central limit theorem. 4. Extension to the Ensemble.

If we interpret (Al), (A5), (A6) and (All) to mean that at time t the TMPs of the neural set are essentially independent, identically distributed random variables, then the Law of Large Numbers (Chung, 1974) suggests interpreting p(x, t)Ax as approximately the proportion of neurons with TMPs in (x,x + Ax] at time t. Equation (9) can be written in the form of a balance equation

ap(x,f)

-=

at

-&WI

(10)

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AND WILLIAM

E. BOYCE

where

(11) is the amount of probability crossing x in the positive direction per unit time at time t, i.e., the probability flux at (x, t). An ensemble of neurons may be divided into two disjoint sets-those that are receiving excitation and/or inhibition, and those that have fired and are in an absolute refractory state. According to assumption (A8), the process terminates at the moment X(t)=f3. The appropriate boundary condition at the absorbing barrier x=6’ is ~(0, t)=O (Feller, 1971). Therefore the probability flux J(fl, t)= -~{A,(t)E[W2] +I, (t)E[Z2])

T

(12)

is the proportion of neurons discharging per unit time at time t. Now let v(t) be the proportion of neurons in the absolute refractory state. Direct insertion of this feature into the model yields an explicit delayed time argument in the equations. To keep the model reasonably simple we idealize the times ‘_r spent in the absolute refractory state to be independent random variables distributed according to P( ?; > t) = e-t’r. Using the Law of Large Numbers it is not difficult to show that the equation for u(t) is

W) -=

-;u(t)+J(O,t)

dt

(13)

where (l/r)u(t) is the rate at which neurons leave the absolute refractory state. According to (Al) we add the source term (l/r)u(t)&x-x0) to the right hand side of equation (10) to obtain aP(x, -=

at

Conservation

t)

-;[J(x,t)]+;u(t)6(x-x0).

(14)

of probability requires that

s e

u(t)+

p(x,t)dx=l,

-30

tz0.

(15)

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Now assume that excitatory input to each neuron is the superposition of two Poisson processes, one due to excitatory afferent activity with intensity i,(t) and the other originating from within the set with intensity p(t). If two non-homogeneous compound Poisson processes are superposed, the resulting process is a non-homogeneous compound Poisson process with an intensity equal to the sum of the intensities of the component processes. Let K be the random variable representing the magnitude of the effect on the TMP at the soma of the endogenically generated PSPs. It follows that the expression for J(x, t ) in equation (10) is now

ah, t) +A{(t)E[Z2])~*

(16)

A neuron will be said to be in the state S if its TMP has exceeded its THP but its neurotransmitter has not yet induced any PSPs in its target cells. Roughly speaking, following the generation of an action potential, a neuron is in the state S for a length of time corresponding to the combined conduction and synaptic delay. By reasoning similar to that in the derivation of equation 113) it can be shown that the equation for the proportion of neurons Q(t) in state S is

dQW -= dt

-jQ(t)[email protected],r),

where d is the expected waiting time in state S. It then follows that

where q equals the average number of neurons within the set to which a typical neuron is postsynaptic, and a is the average number of distinct postsynaptic contacts between pre-synaptic and post-s~aptic members of the set.

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JAMES R. BRANNAN AND WILLIAM E. BOYCE

Using equation (16) and rearranging terms in equation (17) yields

I

p(t)

(18) The equations proposed as a model for K 1 sets are equations (13), (14), and (18) with J(x, t) given by equation ( 16). It is required that EC WJ 20 and E[Zj 50. If E[K] > 0 the equations model a KI, set. If E[K] ~0, the equations are a model for a KIi set. For more realistic models it is possible to modify equations (13), (14), and (18) to include any or ail of the following features of neural operations: (i) non-random absolute refractory periods; (ii) relative retractory periods; (iii) PSPs that depend on the level of the TMP; (iv) multiple-time constant membrane decays; (v) a random variable reset value; (vi) intrinsic fluctuations in the THP. In addition, models of KII sets may be constructed from models of KI sets. 5. lnrerpretation of the Variables. Recall that J(#,t) is the proportion of neurons becoming active per unit time at time t. This is the primitive variable chosen by E3eurle (Bet&e, 1956) and subsequently by Wilson and Cowan (Wilson and Cowan, 1972, 1933) to characterize the activity of a population of neurons. The variable J(8, t) can also be related to the PSTH. The PSTH gives the following measure: given that a stimulus was initiated at time t=O, what is the probability of firing in a single unit as a function of time? A peak on a PSTH shows a preferred time of firing relative to the stimulus. The amplitude of a PSTH is in fact a measure of the average firing frequency of a single unit. If a stimulus was applied to the ensemble of n identical neurons at time t=O, then nJ(B, r)Ar~number of neurons which fire during (t, t-t-At]. This would also be the number of times a single neuron fired during (t, t + At] measured over n independent and identical trials. It follows that J(B,t) can be interpreted as the average pulse frequency of a single neuron in the mass. It can therefore be related directly to Freeman’s experiments. In what follows we refer to the flux term J(t-i,t) as the activation rate and denote it by R. Although R depends on several parameters, for convenience the dependence will not be explicitly denoted.

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A second variable we wish to relate to Freeman’s work is

s @

xp(x,t)dx+Ou(t)

-CC

(191

where as usual ~$1 denotes the number of neurons in the population When a neural set is stimulated by pulses arriving along an afferent tract, PSPs are extracellular manifestations of loop currents with electromotive force in the dendritic membranes. The passage of the current across the extracellular resistance generates an extracellular potential field. Under certain conditions depending on the geometry of the nerve cells and their relative positions and orientations, the fields of potential sum. Then the amplitude of the potential difference between a selected point extracellularly located in the set and a far distant point is proportional to the average over the ensemble of TMPs at the somas of the neurons in the set. Over a sequence of identical and independent input-output trials an AEP is statistically generated. We therefore identify (rb(t) defined by equation (19) to be proportional to the AEP in appropriate situations. 6. Viff~i~it~)of the disunion Model. Roughly speaking, the diffusion model should be applicable to KI sets in which a neuron must receive input from many pre-synaptic neurons before it can generate an impulse. In other words, there should be a very weak dependence between the state of one neuron and the state of its neighbor. This is not always the case since there is evidence of neurons which may fire upon receiving a single excitatory pulse (Sabah and Murphy, 1971). On the other hand, numbers of synapses on individual neurons in the cortex are usually measured in hundreds or thousands, the highest recorded being around 80,~. If neurons were highly sensitive to single EPSPs, it would suggest that we have a very delicately balanced system, and stability would be a major problem, particularly in the presence of spontaneous background activity. Freeman’s work (Freeman, 1975) does yield evidence of weak dependence in some neural sets. It may require several trials before a volley of action potentials induced in an afferent tract by an electrical stimulus induces an action potential in a neuron in the target population. A better understanding of what is occurring in an interactive set of neurons can be obtained by closer inspection of the operations of dendrites. The function performed by a dendrite (or dendritic tree) is to transmit the effects of PSPs to the soma or axon hillock where an action potential may be generated. In doing this, the dendrite sums over space and time and smooths out the resultant. The PSPs are degraded by the time they reach

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E. BOYCE

the soma due to the cable properties of the passive membrane. As a result, the input is converted to a continuously varying graded wave of potentlar arriving at the soma. Hence, the receptor field of a neuron performs pulse to wave conversion. Since X(t) is the TMP at the soma, continuous variation of X(t) due to afferent activity suggests that the diffusion model may be an appropriate approximation in some situations.

7. Non-dimensionalized Equations. Equations (13), (14) and (18) are nondimensional&d by introducing the new independent and dependent dimensionless variables

and

Dimensionless coefficients and parameters are defined by

A

ECW21

6=2(e-xo)2 A

ECz21

p=2(e-xo)z * EIXI K=B_X,

di=aq

$=

ECK'I

2(8-x0)2

SPATIALLY In

the

LOCALIZED

resulting non-dim~nsionalized

s

@(x,.t) -=--St

c?Xi[

INTERACTJVE

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equations the hats are omitted:

1

P ‘-x)-~(r)~--;i,(f)~+~i(t)O d2p(x,t )

p(X,t) 1

I

1

+C~u(r)II/+n,(t)a+;li(f)Pl +~v(f)6(x-1) dX2

W) _ dt

439

(20)

(21 (221

where 0 O. ~(0, r)=O and p(o0, t) < x. must now decrease to 0 to The definitions of the summarized below.

The boundary conditions for equation (20) are Note that the resting TMP is y > 0 and TMPs fire. dimensionless variables and parameters are

p(x, t)Ax ~proportion of neurons in the KI set with TMPs in (x,x +AxJ. v(t) =proportion of neurons in the KI set in the ARS. p(t)=intensity (pulses per unit time) of pulses arriving at the dendritic tree and soma of a typical neuron in the population due to firings of other neurons in the population. i,,(t)=intensity (pulses per unit time) of afferent excitatory activity arriving at the dendritic tree and soma of a typical neuron in the population (controlled input). ;i(t) =intensity (pulses per unit time) of afferent inhibitory activity arriving at the dendritic tree and soma of a typical neuron in the population (controlled input). 4 = average value of the effect of EPSPs on the TMP. ct)= average value of the effect of IPSPs on the TMP. ~=average value of the effect of endogenous PSPs on the TMP. a=second moment about zero of the effect of EPSPs on the TMP. p = second moment about zero of the effect of IPSPs on the TMP. tj =second moment about zero of the effect of endogenous PSPs on the TMP. fi = passive membrane decay time constant.

BMB

C

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E. BOYCE

that the TMP decays to in the absence of incoming activity. v = absolute refractory period (ARP). c~=average number of post-synaptic contacts that a typical neuron in the population makes with all other members in the population (strength of interaction). y

=value

8. Steady-State Solutions. Equilibrium stable states which last for time periods ranging from 0.1 set to several seconds are often observed in the mammalian nervous system (Freeman, 1975). The steady states for the model described in this paper are effected by maintaining constant levels of excitatory and/or inhibitory bias. Only KI, sets are treated here, so K>O. In addition, we assume that the statistics of the effects on the TMP of the EPSPs generated from within the population are identical to those of afferent EPSPs, so K=V and I,$=g. In this case the solution(s) to the steady-state equations obtained from equations (20) to (22) is (are) implicitly represented by

l’[(11+L, )a + &PI p(x)=

s x

v

H(x)

H-‘(s)ds,

05x2

1

0

(23)

1

V H(X)

vC(cC + I, )a + &PI

s0

H-‘(s)ds,

l~xcco

I, + [email protected](O) 1 [email protected](O)

/4+&=

(24)

V

r-m=vC(P+Aeb+AiP1

(25)

s co

v+ WllCl-C

H(x)=exp and

-X2

p(x)dx=l

(26)

0

+

2[y + &Ofi - (p +

;l,)tjp]X

2BC(fl + 42b + &PI

1

(27)

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Two special cases of p(x) and u that were obtained numerically are shown in Figures 1 and 2. In addition, the output R (activation rate) is a sigmoid

V

0

X I

2

Figure I. Steady-state probability of a Kl, set with excitatory bias.

3

density function of transmembrane

potentials

Dtmens~onlesstrommembmne potental

Figure 2. Steady-state probability of a KI, set with inhibitory bias.

density function of transmembrane

potentials

function of the input (Figure 3). The shape of this curve is consistent with both theoretical and experimental results (Freeman, 1975; Wilson and Cowan, 1972, 1973). Notice that increasing q the strength of interaction, increases the slope of the response curve at the inflexion. Thus, one would expect that highly interconnected KI, sets would respond more quickly to incoming excitatory activity than would weakly interconnected KI, sets. By increasing M sufficiently, the sigmoid response curve can be made to “break” and we observe that there are two steady states for some range of input (Figure 4). One of the steady states is characterized by extremely high activity and the other by extremely low activity. It is not known

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JAMES R. BRANNAN AND WILLIAM E. BOYCE

5 Afferent

Figure 3. Input-output

I

-5

I

15

bias,

c

mV/msec

curves for a KI, set.

d’ 0 Affetwt

Figure 4. Input-output

excitatory

I

IO

excitatwy

I

I

5

IO

b!as,

*

mV/msec

curve for a KI, set with excessive excitatory feedback.

whether there is any physiological significance to the occurrence of the multiple steady states due to dense functional interconnectivity. By varying some of the other parameters, such as fi or a, the multiple steady states can be made to disappear. Until the model has been determined to be

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sufficiently valid, and more accurate estimates of the parameter ranges are specified, it seems inappropriate to draw any conclusions. about the appearance of the multiple steady states. Several of the qualitative features of the solution(s) determined by equations (23) to (27) are revealed by asymptotic representations. In the following cases we set ii (t ) = 0 and y = 1. Case 1. ie $1. The asymptotic solutions in this case of intense exogeneous activity are

v - 1 - l/vy&

(29)

and R-l/v

(31)

as i,,--+r;. Thus, under large excitatory bias the neurons of a KI, set spend most _of their time in the absolute refractory state. Decreasing v, the amount of time spent in the absolute refractory state tends to increase p(x) and decrease u, as it should. The maximum average firing rate is l/v as expected in this ideahzed model. Finally, note that the limitation on firing rate is controlled essentially by the absolute refractory state, while the effects of excitatory feedback are negligible since neither p nor 01appear in the first two terms of the expressions for p(x), t’, and R. If asymptotic expansions are sought of the form Case 2. p&i.

1 U-Uv,+U,-+... P

R-R,+R,$+...

444 we

JAMES

R. BRANNAN

AND WILLIAM

E. BOYCE

find that

P*(x)=

1

OSxSl

Cl-exp(-ulxl~)l~,/v~(~l~+~,),

exp ( - WwCexp

w+-

f]~,/~vy(p*

+ E,,),

~~,a3(01

po =1 - aa& (0)

15 x < ‘CC

(32

(34)

and (35) The quantity Ei,(O) is determined

to be

1+a~+v~,~-J[(1+~q+v~,~)Z-4a~l 2cio

EidO)=

(36)

By considering the case of c1+0 we find that

(37) and

&r2

&YI

Ro- l+V~~q+(l+V~,?)3C(+,...

(38)

Therefore, for large p and small a, Y and R increase if the strength of interaction 01 increases. Expressions for pl(x), ui, pi, and R, are quite complicated and are not inctuded. Case 3. O
p(x ) - k t expC- (x - 1)2/2A,@1

v-o

(40)

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445

24-O

(41)

R-O

(42)

k; l -J(2xA,[email protected])

(43)

and

where

as 830. It is clear that the passive membrane decay property tends to attract the population TMPs toward the common resting potential. This is also obvious from the stochastic differential equation (2). I I. Discussion. The model developed in this paper is intended to provide a quantitative and conceptual framework for exploring the possible types of information contained in PSTH and AEP measurements. While the equations are only an approximate description, our preliminary results indicate that the model implicitly contains the (approximate) functional relationship between the following features of operation observed in spatially localized, interactive neural populations: passive membrane decay, refractory periods, synaptic delays, feedback, and randomness. If the model can be shown to possess a sufficient degree of validity, then it can be used to study the influence or contribution of these features of neural operation on PSTHs, AEPs, and possibly EEGs. Further elucidation of the nature of the information contained in these measurements could then be obtained by determining the dependence of these features of operation on biochemical changes at cellular and subcellular levels. In a succeeding paper we study the dynamical characteristics of the model and deal with the problem of stability of steady-state solutions. We further show that the model yields a very good explanation of data obtained from experiments on a KI, set, the periglomerular neurons in the olfactory bulb of the cat. This work was supported in part by a Fellowship sponsored by the IBM Corporation and in part by the National Science Foundation under Grant MCS75-08328. LITERATURE Arnold, I_.. 1974. Stochastic ~i~~~~l7riftl Equations. New York: John Wiley. Ashby, W. R., H. von Foerster and C. C. Walker. 1962. “Instabiiity of Pulse Activity in a Net with Threshold.” Narure 196, 561-62.

446

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AND WILLIAM

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Methods in the Theory of Queueing. Griffrn’s Statistical Monographs & Courses. New York: Hafner. Sabah, N. H. and J. T. Murphy. 1971. “A Superposition Model of the Spontaneous Activity of Cerebellar Purkinje Cells.” Biophys. J. 11,415-427. Soong, T. T. 1973. Random Differential Equations in Science and Engineering. New York: Academic Press. Wilson, H. R. and J. D. Cowan. 1972. “Excitatory and Inhibitory Interactions in Localized Populations of Model Neurons.” Biophys. J. 12, l-24. 1973. “A Mathematical Theory of the Functional Dynamics of Cortical and -. and Thalamic Nervous Tissue.” Kybernetic 13,55-80. Zetterberg, L. H. 1973. “Stochastic Activity in a Population of Neurons-A Systems Analysis Approach.” Rep. Inst. Med. Phys. TNO, Utrecht 1, 153..

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