Path Integrals for Stochastic Neurodynamics

Path Integrals for Stochastic Neurodynamics Toru Ohira and Jack D. Cowan TR-94-016 June 30, 1994

Sony Computer Science Laboratory Inc. 3-14-13 Higashi-gotanda, Shinagawa-ku, Tokyo, 141 JAPAN

c 1995 Sony Computer Science Laboratory Inc. Copyright

In Proceedings of the World Congress on Neural Networks, San Diego, June, 1994

Path Integrals for Stochastic Neurodynamics Toru Ohira Sony Computer Science Laboratory 3-14-13 Higashi-gotanda Shinagawa, Tokyo 141, Japan Jack D. Cowan Department of Mathematics, The University of Chicago, Chicago, IL 60637 June 30, 1994

Abstract

We present here a method for the study of stochastic neurodynamics in the framework of the "Neural Network Master Equation" proposed by Cowan. We consider a model neural network composed of two{state neurons subject to simple stochastic kinetics. We introduce a method based on a spin choerent state path integral to compute the moment generating function of such a network. A formal construction of the path integral is presented and the general expression for many neuron networks is obtained. We show explicitly that the method enables us to obtain the exact moment generating function for a single neuron case. Possible directions for the analysis of many neuron networks as well as an alternative path integral formulation are discussed. I. Introduction

The "Neural Network Master Equation" (NNME) was recently introduced by Cowan to describe stochastic neural networks[1]. The NNME is a master equation [2] for neural networks based on the formalism of "second quantization" for classical many{body systems [3][4]. This formulation of the master equation enables us to use techniques developed in quantum eld theory. Two directions of analysis have been investigated in previous works. A hierarchy of moment equations is derived and used to obtain time dependent description of statistical parameters of simple networks[5]. A Feynman diagram representation of the moment generating function (MGF) of a neural network described by the NNME has also been proposed [6]. The purpose of this paper is to present another approach with the NNME. We construct a path integral for the MGF using spin coherent states. The formal expression of such a path integral for a many neuron network is obtained and the exact MGF is recovered for a single neuron with self-excitation. Possible directions for the analysis of networks with many neurons, including results from alternative path integral formulation, are discussed. II. Neural Network Master Equation

We rst review the Neural Network Master Equation. We assume that neurons at each site, say ith site, can be in one of two states, either "active" or "quiescent", denoted by two 1

2 dimensional basis vectors using the Dirac notation j1i > and j0i > respectively. ! ! 0 1 j0i >= 1 ; j1i >= 0 i i

(1)

We de ne the inner product of these states as

< 1i j1i >=< 0i j0i >= 1; < 0i j1i >=< 1i j0i >= 0:

(2)

The transition rate of the ith neuron from one state to the other is given as follows:

j1i >! j0i >;

(Vi ) j0i >! j1i >;

(3)

where  is a uniform "decay rate", Vi is the weighted sum of all inputs from active neurons connected to the ith neuron, and  is the "activation rate" function. In general,  is nonlinear and is assumed to have the standard sigmoid shape[7]. Let the states (or con gurations) of the network be represented by fj >g, the direct product space of each neuron in the network.

j >= jv1 > jv2 > . . . jvN >;

vi = 0 or 1:

(4)

Let P[ , t] be the probability of nding the network in a particular state at time t. We introduce the "neural state vector" for N neurons in a network as X (5) j(t) >= P [ ; t]j >; f g

where the sum is taken over all possible network states. With these de nitions, we can write the NNME for a network with the transition rates given by (3), using the "creation" and "annihilation" operators, i+ and j? in the Pauli spin formalism. ! ! 0 0 0 1 ? + ; j = (6) i = 1 0 j 0 0 i These operators anti-commute at the same site (i = j ) to satisfy the physical assumption of single occupancy, and commute for diering sites (i 6= j ). The NNME then takes the form of an evolution equation: (7) ? @t@ j(t) >= Lj(t) > with the network \Liouvillian" L given by: N N N X X X L =  (i+ ? 1)i? + (i? ? 1)i+( n1 wij j+j?) (8) i=1

i=1

j =1

where n is an average number of connections to each neuron, and wij is the "weight" from the j th to the ith neuron. Thus the weights are normalized with respect to the average number n of connections per neuron. We can derive an equation for the MGF of the network from the NNME using the \spincoherent states"[8]: N N Y X (9) < Z~ j =< 0jjExp[ Zi i?] = (< 0i j + Zi < 1i j); i=1

i=1

3 where the product is taken as a direct product, Zi are complex parameters, and < 0jj =< 01 j < 02 j < 03 j . . . < 0N j:

(10)

Introducing one-point and multiple-point moments as X X vi (t) = vi P [ ; t]; vi vj . . . (t) = (vi vj . . .)P [ ; t]; f g

f g

(11)

~ t) is given as it can be shown that the moment generating function G(Z; ~ t) =< Z~ j(t) > G(Z;

(12)

from which we can recover the moments as ~ t)jZ~ =1 ; vi (t) = @Z@ G(Z; i

~ t)jZ~ =1 vi vj . . . (t) = @Z@ @Z@ . . . G(Z; i

j

(13)

~ t) by simply projecting the NNME onto the spin-coherent We obtain the equation for G(Z; states: ~ t) =< Z~ jLj(t) > ? @t@ G(Z; (14) III. Construction of a Path Integral

We now describe a path integral representation of the MGF. From Eqs. (7) and (12), we obtain the following expression for the MGF: ~ tf ? ti ) = hZ~ je?L(tf ?ti) j(t = 0)i G(Z;

(15)

~ t) equals It can be shown that with suitable initial conditions and normalization, the MGF G(Z; the "transition element" T given by [9] T = hZ~f jExp[?L(tf ? ti )]jZ~ i i:

(16)

We express the transition element T in path integral form using the resolution of identity: Z ZY N d2Zi ~ ~ d(Z~ )jZ~ ihZ~ j; (17) j Z ih Z j  1 = 2  c c i (1 + Zi Zi )3 where the integration is taken over the entire complex plane of Zi . The construction of a path integral follows closely the derivation in [10]. We rst factorize the operator e?L(tf ?ti) into the product of the short time operators: e?L(tf ?ti) =

M Y e?L

i=1

(18)

with  = (tf ? ti )=M . We now insert the resolution of identity M times between each of the factors to obtain: Z Y M (19) T = ( d(Z~ s))hZ~ f jZ~ M ihZ~ M je?LjZ~ M ?1 i . . . hZ~ s je?L jZ~ s?1i . . . hZ~ 1je?L jZ~ i i: s=1

4 In the limit of M ! 1 and  ! 0, we obtain:

~ s?1

~ s?1

hZ~s je?L jZ~ s?1i = hZ~ s jZ~ s?1i[1 ?  hZ~jsL~jZs?1 i ] = hZ~ s jZ~ s?1iExp[? hZ~ jsL~jZs?1 i ]: hZ jZ i hZ jZ i ~s

~s

Hence, we obtain the expression Z Y M M M X?1 hZ~ s+1jLjZ~ s i Y d(Z~s) hZ~ s+1jZ~ s iExp[? T = lim ~ s+1 ~ s ]; M !1

(20) (21)

hZ jZ i where jZ~ 0 i = jZ~ i i and hZ~ N +1 j = hZ~ f j. We now let jZ~ s+1i = jZ~ s+1 i ? jZ~ s i, then Z Y M M M X?1 hZ~ s+1jLjZ~ s i Y hZ~ s jZ~ s i [ d(Z~ s)hZ~ sjZ~ s i]hZ~ f jZ~ M i[ (1 ? ~ s ~ s )]Exp[? T = lim ~ s+1 ~ s ]: (22) M !1 s=1 hZ jZ i s=0 hZ jZ i s=1 s=0

s=0

s=1

We now assume that the major contribution to the integral comes from those paths for which jZ~s i is of order . Setting dtd jZ~ i = 1 jZ~ i , and using Eq. (22), we obtain the continuous expression: Z Y N 2 2 T= (23) (  (1 +dZZiZ )2 )Exp[S]; i=1

i

i

where the action S is given as Z tf X N 1 )Z  dZi + hZ~ jLjZ~ i ] + ln[hZ~ f jZ~ (t )i]: d [ ( S = ? f ~ ~ 1 + Z  Z i dt ti

i=1

The boundary conditions are:

i i

hZ jZ i

jZ~ (ti)i = jZ~ i i; hZ~ (tf )j = hZ~ f j:

(24) (25)

Eq. (24) can be put into a more symmetric form via integrating by parts to yield the following: Z tf 1 X N  1  dZi ? Zi dZi ) + hZ~ jLjZ~ i ] + 1 ln[hZ~ (ti)jZ~ i ihZ~ f jZ~ (tf )i]: (26) ( Z d [ S = ? i 2 i=1 (1 + Zi Zi ) dt dt ti hZ~ jZ~ i 2 These equations de ne the path integral form of the transition element T or, equivalently, the moment generating function G. IV. Single Neuron Case

We rst consider the case of a single neuron with self-excitation. The Liouville operator is given as follows: + ? ? + L = ( ? 1) + ( ? 1) : (27) The action in this case is: Z tf 1 [ 1 (Z  dZ ? Z dZ  ) + (Z  ? 1)Z + (1 ? Z  )]dt S = ? dt dt ti (1 + Z  Z ) 2 1 + 2 ln[(1 + Z  (ti)Z i )(1 + Z f Z (tf ))]: (28) In this case, we can calculate T exactly using the classical approximation or saddle-point evaluation [11] of the path integral, in a fashion similar to the free particle in quantum mechanics. The classical approximation to the path integral with the action given by (28) yields: @ L~ d @ L~ @ L~ d @ L~ = ; = (29) @Z dt @Z @Z  dt @Z 

5 with

~ ?

L

1

1  dZ (1 + Z  Z ) [ 2 f(Z dt



? dZdt Z )g + (Z  ? 1)Z + (1 ? Z  )]:

(30)

This classical equation of motion leads to equations for Z and Z  : dZ = (?Z + )(Z + 1); dt

dZ  = ( ? Z  )(Z  ? 1): dt

(31)

with the boundary conditions: Z (ti ) = Z i ; Z  (tf ) = Z f  . Substitution of (31) into (28) leads to the classical action in the simple form: Z tf 1 1 dtf? + [Z (t) + Z  (t)]g + + ln[(1 + Z  (ti)Z i )(1 + Z f  Z (tf ))] (32) SCL = 2 2 ti We can nally show, via a tedious but straightforward calculation of explicitly solving the classical equation of motion and of the classical action, the transition element given by T = Exp[SCL] with the following identi cation Zi f (1 + Z i ) ! P+; Z ! Z:

(33)

(P+ is the probability of being in the active state at time ti .) gives the exact moment generating function: G(Z; tf ? ti ) = (1 ? P (tf ? ti )) + ZP (tf ? ti ); (34) where   P (t) = ( ) + (( ) ? P+)e?(+)t : (35) + +

We note here that the end point factors appearing in the path integral play a role in obtaining this solution. V. Many Neuron Case

We now consider the many neuron case with L given by Eq. (8). The path integral representation is given by (26) with N (Z  ? 1)Z N N 1 hZ~ j(X hZ~ jLjZ~ i = X i i ? ? 1)+( 1 X wij +?)jZ~ i: [  (  ] + i j j n j =1 hZ~ jZ~ i i=1 (1 + Zi Zi ) hZ~ jZ~ i i=1 i

(36)

N N (Z  ? 1)Z 1 X (1 ? Zi ) (Zi Zi ) ]: hZ~ jLjZ~ i = X i i ] + [ w [  ij hZ~ jZ~ i i=1 (1 + Zi Zi ) n i=1;j =1 (1 + Zi Zi ) (1 + Zi Zi )

(37)

The second term requires a Taylor expansion for , and cannot be written in a concise form. Here, we proceed with the case (x) = x (the linear case without self-activation). Then

Eqs. (26) and (37) formally de ne the path integral for the linear case. We note that this path integral is de ned on a curved surface with non-linear factors 1=(1 + Zi Zi ). Because of this, the algebraic manipulations required to deal with the many neuron case are rather complicated [12].

References

6 VI. Discussion

We now wish to discuss what can be done with these formal derivations of the spin coherent state path integral for the NNME. In principle, even for the many neuron case, we can proceed with saddle point approximations and other approximation schemes developed for path integrals. The diculty associated with a curved surface complicates such calculations. It turns out that we can alternatively formulate an equivalent path integral using boson coherent states, which circumvents this diculty at the expense of using twice as many variables. In such formulation, we have shown that we can obtain an equivalent of the mean eld approximation via a saddle point approximation to such a path integral. The corresponding derivation with the spin coherent state path integral presented here as well as higher order approximation to both formulations of path integrals are currently being investigated.

References [1] J.D.Cowan, in Advances in Neural Information Processing Systems 3, edited by R. P. Lippman, J. E. Moody, and D. S. Touretzky (Morgan Kaufmann Publishers, San Mateo, 1991), p.62. [2] N.van Kampen, Stochastic Processes in Physics and Chemistry(North Holland, Amsterdam, 1981) [3] M.Doi, J.Phys.A 9,1465 (1976) [4] P. Grassberger and M .Scheunert, Fortachritte der Physik 28, 547, (1980) [5] T.Ohira and J.D.Cowan, Phys.Rev.E 48, 2259, (1993) [6] T.Ohira and J.D.Cowan, to be published in Proceedings of Fifth Australian Conference of Neural Networks, 1994 [7] J. D. Cowan, in Neural Networks, edited by E. R .Caianiello(Springer, New York 1968), p. 181. [8] A. Perelomov, Generalized Coherent States and Their Applications (Springer, New York, 1986) [9] M.Doi, J.Phys.A 9,1479 (1976) [10] J.Blaizot and G.Ripka, Quantum Theory of Finite Systems(The MIT Press, Cambridge, 1986) [11] M. Swanson, Path integrals and quantum processes, Academic Press, San Diego,, 1992. [12] H. Kuratsuji, Path integrals in the SU(2) coherent state representation and related topics, Path Integrals and Coherent States of SU(2) and SU(1,1) (Singapore) (A. Inomata, H. Kuratsuji, and C. C. Gerry, eds.), World Scienti c, Singapore, 1992.