Stochastic Differential Equation Models for Spatially Distributed Neurons and Propagation of Chaos for Interacting Systems GOPINATH KALLIANPUR Department of Statistics, University of North Carolina, Chapel Hill, North Carolina 27599-3260 Received 2 January 1992; reuised 9 June 1992

ABSTRACT Distribution or nuclear space-valued stochastic differential equations (SDEs) (diffusions as well as discontinuous equations) are discussed as stochastic models for the behavior of voltage potentials of spatially distributed neurons. A propagation of chaos result is obtained for an interacting system of Hilbert space-valued SDEs.

1.

INTRODUCTION

There is a considerable literature devoted to stochastic methods in neurophysiology, in particular to the study of the fluctuation of voltage potentials. Until recently, almost all of the work ignored the spatial geometry of the neuron, regarding it merely as a point. In reality, the neuron cell together with the dendrites and axons that connect it with other neurons form a complicated system. The papers of Wan [1.5] and Wan and Tuckwell [161 were among the earliest to take into account the spatial extent of the neuron, followed by the work of Walsh [13,14] and Kallianpur and Wolpert [7]. The aims of this article are (1) to present a brief discussion of the stochastic partial differential equations (SPDES) and the infinite-dimensional stochastic differential equations @DES) that model the stochastic behavior of .the voltage potentials of spatially distributed neurons and (2) to study the asymptotic behavior of large assemblages of interacting neurons. For reasons of space, only a small part of the theory developed for the latter is presented here (in Section 4) and references to other work on propagation of chaos results are provided. An SPDE of a simplified model of two neurons in parallel fiber interaction is considered in Section 4.

MATHEMATICAL

BIOSCIENCES

112~207-224 (1992)

OElsevier Science Publishing Co., Inc., 1992 65.5 Avenue of the Americas, New York, NY 10010

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GOPINATH

208

2.

STOCHASTIC

CABLE EQUATION

KALLIANPUR

MODELS

We begin with the well-known example treated in [7] and [13] where the neuron is regarded as a thin linear segment and the deterministic model upon which the stochastic behavior is imposed is the cable equation dli -=_._

L?t

*ll+pfi 2X2 ’

OO

(2Sb)

and u(x,O) = uo(x),

(2Sc)

The segment [O,bl represents the membrane of the neuron, and condition (2.lb) implies that there is no leakage at the end points. The simplest stochastic description of the fluctuation of the voltage potential is obtained by adding the effect of the random impulses to (2.la). The latter might originate as synaptic inputs or be carried through axons from other neuron cells. It is natural to regard the generalized Poisson process as an appropriate model for the incident stream of random disturbances to the quiescent neuron. These Poisson impulses are small enough in amplitude and frequent enough to suggest that a Gaussian process will be a good approximation, a fact to which we shall return at the end of Section 3. Accordingly, to get our stochastic model, we replace Eq. (2.la) by (2Sa’) where &t,x> is space-time Gaussian white noise, that is, it is the fictitious derivative of the space-time Wiener process or Brownian sheet W(t,x>, t 2 0, 0 G x G b. The Brownian sheet is a Gaussian stochastic process depending on t and II spatial parameters x,, . . . , x, (in the present case, fz = 1) such that mv(t,x

I,...,

x,)=0

and

(2-2)

SPATIALLY

DISTRIBUTED

NEURONS

209

The symbol E in (2.2) stands for the mean or expected value of the space random variable w(t, x1,. . . , xn; o) defined on some probability (a, lp), and w E 0 is the probability parameter. The Green’s function of the Neumann problem (2.1) is t > 0,

where A,=a+p(nn/b)*

A,,= ff >

for n>l,

and p,,(x)

= b-l’“,

C&(X) = 2”2b-“2cos(n~x/b),

II a 1.

For f E H = L2[0, bl, define =jobC(t;x,y)f(y)dy

(Tf)(x)

for t > 0; T, = I.

Then T, is a contraction semigroup on H whose generator - L CL a 0) has dense domain and agrees with - al + /? d2/dx2 on smooth functions. Equation (2.la’) can be rigorously written as an It6 stochastic partial differential equation (SPDE) dx(x,t,

co) = - LX(x,t;

X(O,X,W)

3.

the second

DISTRIBUTION STOCHASTIC

(2.3)

=2.+(x).

It can be shown that the solution

where

w) dt + dU/;,( w),

to (2.3) is given by

term on the right-hand

side is a Wiener

integral.

OR NUCLEAR SPACE-VALUED NEURONAL MODELS

The SDE discussed in the previous section needs to be generalized in two directions. The partial differential equation obtained by Hodgkin and Huxley in their celebrated paper is nonlinear [4]. If a stochastic

210

GOPINATH

KALLIANPUR

model of neuronal activity is to be a reasonable description of reality, it has to incorporate at least some of the nonlinearities inherent in the complex behavior of voltage potentials such as equilibrium or reversal potentials. A second derivation in which generalization is almost forced upon us stems from the fact that if the spatial parameter is ndimensional where II > 2 (e.g., if the neuron is thought of as a twodimensional patch or part of a manifold), then the SDE corresponding to (2.3) has no solution. This was shown by Walsh [14] for the extension of (2.la’) to two spatial parameters

(3-l) with initial

and boundary

$+PY)

conditions

=

g( t;?i-,y)

=fqt;x,7i)=O JY

~(0; x, y) = 0,

=$(t;x,O) for t > O,x,y.

(3.2)

The contraction semigroup T, on L’([O, ~1’) has the generator - L, which equals A - I on the smooth functions on [O, rrl* (A = Laplacian). L has the eigenfunctions $~~~(x,y) = +j(_r)+k(y), where +ji(x) is the same as in the example of Section 1 (with b = n-> and eigenvalues hjk = 1+ j* + k*. The method for obtaining a random field solution to the SPDE corresponding to (3.1)-(3.2) fails because X(1/A,,) = a. One way to accommodate examples of this kind is to formulate and seek a random distributional solution in the sense of Schwarz. This is a situation not unfamiliar (though the context is different) in the (deterministic) theory of partial differential equations. Technically speaking, what this means is that the state space of the stochastic process has to be enlarged to some suitable space of distributions. Combining the two requirements described above leads to (1) the specification of a space of distributions (denoted here by a’) and (2) the study of two kinds of SDEs whose solutions for each time instant t take values in @ ‘. The space @’ is the topological dual of @, the space of test functions, taken to be a countably Hilbertian nuclear space (CHNS). The latter is a linear vector space whose topology is given by an increasing sequence of compatible Hilbertian norms or seminorms II*llr

SPATIALLY

DISTRIBUTED

NEURONS

211

such that if Qr denotes the Il*ll,-completion of a, then

and

@c.*.

c@,c@o=@;,cQ,_,c**.

c@‘,

where a_, is identified with the Hilbert space dual @; and Qr. For r G s, Qs c Qr. Furthermore, the important property of nuclearity says that for every r there exists s > r such that the injection map Qs + Qr is Hilbert-Schmidt. We impose the further mild condition that there exists a complete orthonormal system (CONS) (hi) in Q0 that is complete and orthogonal in Qr for every r. For definitions of the relevant concepts and other details we refer the reader to [5] and [7]. A well-known example of @ is the space ~lRd) of rapidly decreasing functions on Rd. The dual @’ in this case consists of the Schwartz distributions. A useful and wide enough class of spaces Q, and a’ for our purpose is suggested by the cable equation model considered at the outset. Let 2 represent the cell membrane and L2(2? the Hilbert space of functions defined on 2 and square integrable with respect to some natural measure m on 2. When 2 is a finite segment [O,bl or a d-dimensional cube, m is usually a Lebesgue measure. The deterministic part of voltage potential activity is described by a continuous contraction semigroup T, whose generator - L (L a 0) has the property that (I + L)-‘1 is a Hilbert-Schmidt operator for some rl > 0. The space Q, natural to the problem is defined as @=

4EL2(2): i

5 (1+Aj)2r(~,~j)2

Stochastic differential equation models for spatially distributed neurons and propagation of chaos for interacting systems.

Distribution or nuclear space-valued stochastic differential equations (SDEs) (diffusions as well as discontinuous equations) are discussed as stochas...
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