EXPONENTIAL STABILITY AND INSTABILITY OF STOCHASTIC NEURAL NETWORKS 1. X. X. Liao 2 and X. Mao 3

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1 EXPONENTIAL STABILITY AND INSTABILITY OF STOCHASTIC NEURAL NETWORKS X. X. Liao 2 and X. Mao 3 Department of Statistics and Modelling Science University of Strathclyde Glasgow G XH, Scotland, U.K. ABSTRACT In this paper we shall discuss stochastic effects to the stability property of a neural network u(t) = Bu(t) + Ag(u(t)). Suppose the stochastically perturbed neural network is described by an Itô equation dx(t) = [ Bx(t) + Ag(x(t))]dt + σ(x(t))dw(t). The general theory on the almost sure exponential stability and instability of the stochastically perturbed neural network is first established. The theory is then applied to investigate the stochastic stabilization and destabilization of the neural network. Several interesting examples are also given for illustration.. Introduction Much of the current interest in artificial networks stems not only their richness as a theoretical model of collective dynamics but also from the promise they have shown as a practical tool for performing parallel computation (cf. Denker [2]). Theoretical understanding of neural-network dynamics has advanced greatly in the past ten years (cf. [, 4 7, ]). The neural network proposed by Hopfield [4] can be described by an ordinary differential Supported by the Royal Society. 2 Permanent Address: Department of Mathematics, Huazhong Normal University, Wuhan, P.R.China. 3 For any correspondence regarding this paper please address to this author.

2 equation of the form C i u i (t) = R i u i (t) + T ij g j (u j (t)), i n, (.) j= on t 0. The variable u i (t) represents the voltage on the input of the ith neuron. Each neuron is characterized by an input capacitance C i and a transfer function g i (u). The connection matrix element T ij has a value + /R ij when the noninverting output of the jth neuron is connected to the input of the ith neuron through a resistance R ij, and a value /R ij when the inverting output of the jth neuron is connected to the input of the ith neuron through a resistance R ij. The parallel resistance at the input of each neuron is defined R i = ( n j= T ij ). The nonlinear transfer function g i (u) is sigmoidal, saturating at ± with maximum slope at u = 0. In term of mathematics, that is g(u) is nondecreasing, ug i (u) 0 and g i (u) β i u for all < u <, (.2) where β i is the slope of g i (u) at u = 0 and is supposed to be finite. defining b i =, a ij = T ij C i R i C i equation (.) can be re-written as By u i (t) = b i u i (t) + a ij g j (u j (t)), i n, (.3) j= or equivalently where u(t) = Bu(t) + Ag(u(t)), t 0, (.4) u(t) = (u (t),, u n (t)) T, B = diag.(b,, b n ), A = (a ij ) n n, g(u) = (g (u )),, g n (u n )) T. Moreover, we always have b i = a ij, i n. (.5) j= It is clear that whenever given an initial data u(0) = x o R n equation (.4) has a unique global solution on t 0. Especially, the equation admits

3 an equilibrium solution u(t) 0 (i.e. the solution when the initial data u(0) = 0). The stability problem of this equilibrium solution has been studied by many authors e.g. Coben & Crosshery [], Liao [7], Quezz et al. []. The aim of this paper is to investigate the stochastic effects to the stability. Suppose there exists a stochastic perturbation to the neural network and the stochastically perturbed network is described by a stochastic differential equation { dx(t) = [ Bx(t) + Ag(x(t))]dt + σ(x(t))dw(t) on t 0, x(0) = x o R n, (.6) where w(t) = (w (t),, w m (t)) T is an m-dimensional Brownian motion defined on a complete probability space (Ω, F, P ) with a natural filtration {F} t 0 (i.e. F t = σ{w(s) : 0 s t}), and σ : R n R n m i.e. σ(x) = (σ ij (x)) n m. Throughout this paper we always assume that σ(x) is locally Lipschitz continuous and satisfies the linear growth condition as well. So it is known (cf. Friedman [3] or Mao [9]) that equation (.6) has a unique global solution on t 0, which is denoted by x(t; x o ). Moreover, we also assume σ(0) = 0 for the stability purpose of this paper. So equation (.6) admits an equilibrium solution x(t; 0) 0. It is also easy to see from the uniqueness that whenever the initial data x o 0, the solution will never be zero with probability one, that is x(t, x o ) 0 for all t 0 a.s. Now that equation (.6) is a stochastically perturbed system of equation (.4), it is interesting to know how the stochastic perturbation effects the stability property of equation (.4). That is, when equation (.4) is stable, it is useful to know whether the perturbed equation (.6) remains stable or becomes unstable; but when equation (.4) is unstable, it is then useful to know whether the perturbed equation (.6) becomes stable or remains unstable. In following sections we shall discuss these problems in detail. 2. Exponential Stability In this section we shall discuss the exponential stability of the stochastic neural network (.6). Theorem 2. Assume there exists a symmetric positive definite matrix Q = (q ij ) n n and a pair of numbers µ R and ρ 0 such that 2x T Q[ Bx + Ag(x)] + trace[σ T (x)qσ(x)] µx T Qx, (2.) x T Qσ(x)σ T (x)qx ρ(x T Qx) 2 (2.2)

4 for all x R n. Then the solution of equation (.6) satisfies t log( x(t; x o) ) (ρ µ ) a.s. (2.3) 2 whenever x o 0. In particular, if ρ > µ/2 then the stochastic neural network (.6) is almost surely exponentially stable. Proof. Fix any x o 0 arbitrarily and write x(t; x o ) = x(t) simply. Note from the uniqueness of the solution that x(t) 0 for all t 0 a.s. So one can apply the well-known Itô formula to obtain ( ) d log[x T (t)qx(t)] = ( ) x T 2x T (t)q[ Bx(t) + Ag(x(t))] + trace[σ T (x(t))qσ(x(t))] dt (t)qx(t) 2 ( ) [x T (t)qx(t)] 2 x T (t)qσ(x(t))σ T (x(t))qx(t) dt In view of condition (2.) we obtain + 2 x T (t)qx(t) xt (t)qσ(x(t))dw(t). log[x T (t)qx(t)] log[x T o Qx o ] + µt 2 M(t) + 2M(t) a.s. (2.4) for all t 0, where M(t) = t 0 x T (s)qx(s) xt (s)qσ(x(s))dw(s) which is a continuous martingale vanishing at t = 0 and M(t) is its quadratic variation, i.e. M(t) = t o By condition (2.2) it is easy to see that ( ) [x T (s)qx(s)] 2 x T (s)qσ(x(s))σ T (x(s))qx(s) ds. M(t) ρt. (2.5) Now let k =, 2, and let ε (0, ) be arbitrary. Using the well-known exponential martingale inequality (cf Métivier [0]) one can derive that ( P ω : sup [M(t) ε M(t) ] > ) 0 t k 2ε log k k.

5 Hence the Borel-Cantelli lemma yields that for almost all ω Ω there exists a random integer k o (ω) such that for all k k o sup [M(t) ε M(t) ] log k, 0 t k 2ε that is, M(t) ε M(t) + log k, 0 t k. 2ε Substituting this into (2.4) yields log[x T (t)qx(t)] log[x T o Qx o ] + µt (2 ε) M(t) + ε log k for all 0 t k and k k o almost surely. By (2.5) one therefore obtains that log[x T (t)qx(t)] log[x T o Qx o ] [(2 ε)ρ µ]t + ε log k for all 0 t k and k k o almost surely. k t k and k k o then So for almost all ω Ω, if t log[xt (t)qx(t)] [(2 ε)ρ µ] + (log[x T o Qx o ] + ) k ε log k. This implies t log[xt (t)qx(t)] [(2 ε)ρ µ] a.s. Letting ε 0 we obtain One the other hand, note t log[xt (t)qx(t)] (2ρ µ) a.s. (2.6) λ min x 2 x T Qx, x R n since Q is a symmetric positive definite matrix, where λ min > 0 is the smallest eigenvalue of Q. Consequently, it follows from (2.6) that t log( x(t) ) (ρ µ 2 ) a.s. as required. The proof is complete.

6 We now employ this theorem to establish a number of useful corollaries. Corollary 2.2 Let (.2) hold. Assume that there exists a positive definite diagonal matrix Q = diag.(q, q 2,, q n ) and two real numbers µ > 0, ρ 0 such that trace[σ T (x)qσ(x)] µx T Qx, x T Qσ(x)σ T (x)qx ρ(x T Qx) 2 for all x R n. Let λ max (H) denote the biggest eigenvalue of the symmetric matrix H = (h ij ) n n defined by h ij = { 2qi [ b i + (0 a ii )β i ] for i = j, q i a ij β j + q j a ji β i for i j. Then the solution of equation (.6) satisfies ( t log( x(t; x o) ) ρ [ µ + λ ]) max(h) 2 min i n q i a.s. (2.7) if λ max (H) 0, or otherwise ( t log( x(t; x o) ) ρ [ µ + λ ]) max(h) 2 max i n q i a.s. (2.8) whenever x o 0. Proof. Compute, by (.2), 2x T QAg(x) = 2 x i q i a ij g j (x j ) i,j= 2 i q i (0 a ii )x i g i (x i ) + 2 i j x i q i a ij β j x j = 2 i q i (0 a ii )β i x 2 i + i j x i (q i a ij β j + q j a ji β i ) x j. Thus, in the case λ max (H) 0, 2x T Q[ Bx + Ag(x)] ( x,, x n ) H ( x,, x n ) T λ max (H) x 2 λ max(h) min i n q i x T Qx,

7 and then conclusion (2.7) follows from Theorem 2. easily. Similarly, in the case λ max (H) < 0, 2x T Q[ Bx + Ag(x)] λ max (H) x 2 λ max(h) max i n q i x T Qx and then conclusion (2.8) follows from Theorem 2. again. complete. The proof is Corollary 2.3 Let both (.2) and (.5) hold. Assume that there exist n positive numbers q, q 2,, q n such that q i [0 sign(a ii )] δ ij a ij q j b j, j n, where β 2 j i= Moreover assume δ ij = { for i = j, 0 for i j. trace[σ T (x)qσ(x)] µx T Qx, x T Qσ(x)σ T (x)qx ρ(x T Qx) 2 for all x R n, where Q = diag.(q, q 2,, q n ) and µ > 0, ρ 0 are both constants. Then the solution of equation (.6) satisfies whenever x o 0. Proof. Compute, by the conditions, 2x T QAg(x) = 2 2 i,j= i,j= i= t log( x(t; x o) ) (ρ µ 2 ) i,j= x i q i a ij g j (x j ) x i q i [0 sign(a ii )] δ ij a ij β j x j q i [0 sign(a ii )] δ ij a ij (x 2 i + β 2 j x 2 j) ( n ) q i a ij x 2 i + j= q i b i x 2 i + i= ( j= β 2 j i= q j b j x 2 j = 2x T QBx. j= a.s. ) q i [0 sign(a ii )] δ ij a ij x 2 j

8 Hence 2x T Q[ Bx + Ag(x)] + trace[σ T (x)qσ(x)] µx T Qx. Then the conclusion follows from Theorem 2.2. The proof is complete. Corollary 2.4 Let both (.2) and (.5) hold. Assume the network is symmetric in the sense Moreover assume a ij = a ji for all i, j n. trace[σ T (x)σ(x)] µ x 2, x T σ(x)σ T (x)x ρ x 4 for all x R n, where both µ > 0 and ρ 0 are both constants. Then the solution of equation (.6) satisfies that if ˇβ, or t log( x(t; x o) ) (ρ + ˆb( ˇβ) µ ) a.s. (2.9) 2 t log( x(t; x o) ) (ρ ˇb( ˇβ ) µ ) a.s. (2.0) 2 if < ˇβ whenever x o 0, where ˇβ = max i n β i, ˇb = max i n b i, ˆb = min i n b i. Proof. Compute 2x T Ag(x) = 2 2 i,j= i= i,j= x i a ij g j (x j ) x i a ij β j x j ˇβ = ˇβ [ n ( n ) a ij x 2 i + j= = ˇβ [ n b i x 2 i + i= i,j= ( n j= i= a ij (x 2 i + x 2 j) ) ] a ji x 2 j ] b j x 2 j = 2 ˇβx T Bx. j=

9 Hence 2x T [ Bx + Ag(x)] 2( ˇβ)x T Bx. Therefore, in the case ˇβ, 2x T [ Bx + Ag(x)] + trace[σ T (x)σ(x)] [ 2ˆb( ˇβ) + µ] x 2, and conclusion (2.9) follows from Theorem 2. with Q = the identity matrix. On the other hand, in the case < ˇβ, 2x T [ Bx + Ag(x)] + trace[σ T (x)σ(x)] [2ˇb( ˇβ ) + µ] x 2, and conclusion (2.0) follows from Theorem 2. again. The Proof is complete. 3. Exponential Instability In this section we shall discuss the exponential instability for the stochastic neural network described by equation (.6). Theorem 3. Assume there exists a symmetric positive definite matrix Q = (q ij ) n n and two real numbers µ R, ρ > 0 such that for all x R n. Then 2x T Q[ Bx + Ag(x)] + trace[σ T (x)qσ(x)] µx T Qx (3.) lim inf x T Qσ(x)σ T (x)qx ρ(x T Qx) 2 (3.2) t log( x(t; x o) ) µ 2 ρ a.s. (3.3) whenever x o 0. In particular, if ρ < µ/2 then the stochastic neural network (.6) is almost surely exponentially unstable. Proof. Fix any x o 0 arbitrarily and again write x(t; x o ) = x(t) simply. By the Itô formula as well as conditions (3.), (3.2) one can derive that log[x T (t)qx(t)] log[x T o Qx o ] + (µ 2ρ)t + 2M(t) a.s. (3.4) for all t 0, where M(t) = t 0 x T (s)qx(s) xt (s)qσ(x(s))dw(s) the same as before. Note from condition (3.2) that M(t) = t o ( ) [x T (s)qx(s)] 2 x T (s)qσ(x(s))σ T (x(s))qx(s) ds ρt.

10 It is known (cf. Liptser & Shiryayev [8]) that M(t)/t 0 almost surely as t. Consequently (3.4) yields But, note lim inf t log[xt (t)qx(t)] µ 2ρ a.s. (3.5) λ max x 2 x T Qx, x R n, where λ max > 0 is the biggest eigenvalue of Q. Hence it follows from (3.5) that lim inf t log( x(t) ) µ 2 ρ a.s. as required. The proof is complete. Corollary 3.2 Let (.2) hold. Assume that there exists a positive definite diagonal matrix Q = diag.(q, q 2,, q n ) and two positive numbers µ, ρ such that trace[σ T (x)qσ(x)] µx T Qx, x T Qσ(x)σ T (x)qx ρ(x T Qx) 2 for all x R n. Let λ min (S) denote the smallest eigenvalue of the symmetric matrix S = (s ij ) n n which is defined by s ij = { 2qi [ b i + (0 a ii )β i ] for i = j, q i a ij β j q j a ji β i for i j. Then the solution of equation (.6) satisfies lim inf whenever x o 0. t log( x(t; x o) ) [ µ + λ ] min(s) ρ a.s. (3.6) 2 min i n q i Proof. In the same way as the proof of Corollary 2.2 one can show that 2x T Q[ Bx + Ag(x)] ( x,, x n ) S ( x,, x n ) T λ min (S) x 2. Note that we must have λ min (S) 0 since all the elements of S are nonpositive. So 2x T Q[ Bx + Ag(x)] λ min(s) x T Qx min i n q i and then conclusion (3.6) follows from Theorem 3. easily. complete. The proof is

11 Corollary 3.3 Let both (.2) and (.5) hold. Assume the network is symmetric in the sense Moreover assume a ij = a ji for all i, j n. trace[σ T (x)σ(x)] µ x 2, x T σ(x)σ T (x)x ρ x 4 for all x R n, where both µ and ρ are positive numbers. Then the solution of equation (.6) satisfies that lim inf t log( x(t; x o) ) µ 2 ˇb( + ˇβ) ρ a.s. whenever x o 0, where ˇβ = max i n β i and ˇb = max i n b i. Proof. Compute Hence Therefore, 2x T Ag(x) = 2 2 i,j= i= i,j= x i a ij g j (x j ) x i a ij β j x j ˇβ = ˇβ [ n ( n ) a ij x 2 i + j= = ˇβ [ n b i x 2 i + i= i,j= ( n j= i= a ij (x 2 i + x 2 j) ) ] a ji x 2 j ] b j x 2 j = 2 ˇβx T Bx. j= 2x T [ Bx + Ag(x)] 2( + ˇβ)x T Bx 2ˇb( + ˇβ) x 2. 2x T [ Bx + Ag(x)] + trace[σ T (x)σ(x)] [µ 2ˇb( + ˇβ)] x 2,

12 and the conclusion (2.7) follows from Theorem 3. with Q = the identity matrix. The proof is complete. 4. Stabilization by Linear Stochastic Perturbation We know the neural network u(t) = Bu(t) + Ag(u(t)) may not stable sometimes. Perhaps one might imagine that an unstable neural network should behave even worse (more unstable) if the network subjects to stochastic perturbation. However, this is not always true. In fact, as every thing has two sides, stochastic perturbation may make the given unstable network nicer (stable). In this section we shall show that any neural network of form (.4) can be stabilized by stochastic perturbation. From the practical point of view we restrict ourselves to linear stochastic perturbation only. In other words we only consider the stochastic perturbation of the form σ(x(t))dw(t) = B k x(t)dw k (t), i.e. σ(x) = (B x, B 2 x,, B m x), where B k, k m are all n n matrices. In this case, the stochastically perturbed network (.6) becomes dx(t) = [ Bx(t) + Ag(x(t))]dt + B k x(t)dw k (t) on t 0, (4.) x(0) = x o R n. Note that and trace[σ T (x)qσ(x)] = x T Bk T QB k x x T Qσ(x)σ T (x)qx = trace[σ T (x)qxx T Qσ(x)] = x T Bk T Qxx T QB k x = (x T QB k x) 2. We immediately obtain the following useful result from Theorem 2.. Theorem 4. Assume there exists a symmetric positive definite matrix Q = (q ij ) n n and a pair of numbers µ R and ρ 0 such that 2x T Q[ Bx + Ag(x)] + x T Bk T QB k x µx T Qx

13 and (x T QB k x) 2 ρ(x T Qx) 2 for all x R n. Then the solution of equation (4.) satisfies t log( x(t; x o) ) (ρ µ 2 ) a.s. whenever x o 0. In particular, if ρ > µ/2 then the stochastic neural network (4.) is almost surely exponentially stable. Let us now explain through examples how one can apply this theorem to stabilize a given neural network. Example 4. Let B k = θ k I for k m, where I is the identity matrix and θ k, k m are all real numbers. Then equation (4.) becomes dx(t) = [ Bx(t) + Ag(x(t))]dt + θ k x(t)dw k (t) (4.2) (the initial data is omitted here). One can see that the numbers θ k, k m represent the intensity of the stochastic perturbation. Choose Q to be the identity matrix. Note in this case that and x T Bk T QB k x = (x T QB k x) 2 = Moreover, in view of (.2) we have B k x 2 = (x T θ k x) 2 = θk x 2 2 (4.3) θk x 2 4. (4.4) 2x T QAg(x) 2 x A g(x) 2 ˇβ A x 2, where ˇβ = max k n β k and denotes the operator norm of a matrix, i.e. A = sup{ Ax : x R n, x = }. Hence 2x T Q[ Bx + Ag(x)] 2( ˇβ ˆb) x 2, (4.5)

14 where ˆb = min k n b k. Combining (4.3) (4.5) and applying Theorem 4. we see that the solution of equation (4.2) satisfies t log( x(t; x o) ) ( 2 θk 2 ( ˇβ ˆb) ) a.s. whenever x o 0. In particular, if choose θ k s large enough such that θk 2 > 2( ˇβ ˆb) then the stochastic neural network (4.2) is almost surely exponentially stable. Now if we choose θ k = 0 for 2 k m, then equation (4.2) becomes an even simpler one dx(t) = [ Bx(t) + Ag(x(t))]dt + θ x(t)dw (t). (4.6) That is we only use a scalar Brownian motion as the source of stochastic perturbation. This stochastic network is almost surely exponentially stable provided θ 2 > 2( ˇβ ˆb). From this simple example we see that if a strong enough stochastic perturbation is added onto a neural network u(t) = Bu(t) + Ag(u(t)) in a certain way then the network can be stabilized. In other words we have already obtained the following theorem. Theorem 4.2 Any neural network of the form u(t) = Bu(t) + Ag(u(t)) can be stabilized by Brownian motion provided (.2) is satisfied. Moreover, one can even use only a scalar Brownian motion to do so. Theorem 4. ensures that there are many choices for the matrices B k in order to stabilize a given network. Of course the choices in Example 4. are just the simplest ones. For illustration one more example is given here. Example 4.2 that For each k, choose a positive definite n n matrix D k such x T D k x 3 2 D k x 2,

15 Obviously, there are lots of such matrices. Let θ be a real number and define B k = θd k. Then equation (4.) becomes dx(t) = [ Bx(t) + Ag(x(t))]dt + θ Again let Q = identity matrix. Note and x T Bk T QB k x = D k x(t)dw k (t). (4.7) θd k x 2 θ 2 m (x T QB k x) 2 = θ 2 (x T D k x) 2 3θ2 4 m D k 2 x 2 D k 2 x 4. Combining these together with (4.5) and then applying Theorem 4. we obtain that the solution of equation (4.7) satisfies ( θ 2 t log( x(t; x o) ) 4 D k 2 ( ˇβ ˆb) ) a.s. whenever x o 0. So if θ 2 > 4( ˇβ ˆb) ( m D k 2) then the stochastic network (4.7) is almost surely exponentially stable. From the above examples one can see that in order to stabilize an unstable network the linear stochastic perturbation should be strong enough. This is not surprising since if the stochastic perturbation is too weak it may not be able to change the instability property of the network. 5. Destabilization by Linear Stochastic Perturbation In the previous section we have discussed the stochastic stabilization problem. Let us now turn to consider the opposite problem stochastic destabilization. That is, we shall add stochastic perturbation onto a given stable network in the hope that the perturbed network becomes unstable. Obviously the stochastic perturbation should be strong enough otherwise the stability property will not be destroyed. However, the strength of the perturbation is not the only effect. As a matter of fact, the way how the stochastic perturbation is added onto the network is more important. As seen in the previous

16 section, sometimes, the stronger the stochastic perturbation is added the more stable the network becomes. From the practical point of view, we again restrict ourselves to linear stochastic perturbation only. In other words we still assume the stochastically perturbed network is described by equation (4.). Applying Theorem 3. to equation (4.) we immediately obtain the following useful result. Theorem 5. Assume there exists a symmetric positive definite matrix Q = (q ij ) n n and a pair of numbers µ R and ρ > 0 such that and 2x T Q[ Bx + Ag(x)] + x T Bk T QB k x µx T Qx (x T QB k x) 2 ρ(x T Qx) 2 for all x R n. Then the solution of equation (4.) satisfies lim inf t log( x(t; x o) ) µ 2 ρ whenever x o 0. In particular, if ρ < µ/2 then the stochastic neural network (4.) is almost surely exponentially unstable. Let us now apply this theorem to show how one can use stochastic perturbation to destabilize a given network. Example 5. First of all, let the dimension of the network n 3. Let m = n, i.e. choose an n-dimensional Brownian motion (w (t), w 2 (t),, w n (t)) T. Let θ be a real number. For each k =, 2,, n, define B k = (b kij ) n n by b kij = θ if i = k and j = k + or otherwise b kij = 0; and moreover define B n = (b nij ) n n by b nij = θ if i = n and j = or otherwise b kij = 0. Then the stochastic network (4.) becomes a.s. x 2 (t)dw (t) dx(t) = [ Bx(t) + Ag(x(t))]dt + θ. x n (t)dw n (t). (5.) x (t)dw n (t) Let Q = the identity matrix. Note x T Bk T QB k x = B k x 2 = θx k 2 = θ 2 x 2. (5.2)

17 Also, setting x n+ = x, 2θ2 3 n (x T QB k x) 2 = θ 2 x 2 kx 2 k+ x 2 kx 2 k+ + θ2 6 (x 4 k + x 4 k+) θ2 3 x 4. (5.3) Moreover, by (.2), 2x T Q[ Bx + Ag(x)] 2(ˇb + ˇβ A ) x 2, (5.4) where ˇb = max k n b k and ˇβ = max k n β k. Combining (5.2) (5.4) and then applying Theorem 5. we see that the solution of equation (5.) satisfies lim inf t log( x(t; x o) ) θ2 2 (ˇb + ˇβ A ) θ2 3 = θ2 6 (ˇb + ˇβ A ) a.s. whenever x o 0. So the stochastic neural network (5.) is almost surely exponentially unstable if θ 2 > 6(ˇb + ˇβ A ). Example 5.2 Secondly, let us consider the case when the dimension of the network n is an even number, say n = 2p (p ). Let m =, that is choose a scalar Brownian motion w (t). Let θ be a real number. Define 0 θ θ 0 B = 0 Then equation (4.) becomes... dx(t) = [ Bx(t) + Ag(x(t))]dt + θ Let Q = identity matrix again. Note 0 0 θ θ 0. x 2 (t) x (t). x 2p (t) x 2p (t) dw (t). (5.5) x T B T QB x = θ 2 x 2 and (x T QB x) 2 = 0. (5.6)

18 Combining (5.6) with (5.4) and then applying Theorem 5. we see that the solution of equation (5.5) satisfies lim inf t log( x(t; x o) ) θ2 2 (ˇb + ˇβ A ) a.s. whenever x o 0. So the stochastic neural network (5.5) is almost surely exponentially unstable if θ 2 > 2(ˇb + ˇβ A ). Summarizing the above two examples we obtain the following conclusion. Theorem 5.2 Any neural network of the form ẋ(t) = Bx(t) + Ag(x(t)) can be destabilized by Brownian motion provided the dimension n 2 and (.2) is satisfied. Naturally, one would ask what happens when the dimension n =. Although from the practical point of view one-dimensional networks are rare, the question needs to be answered for the completeness of theory. So let us consider a one-dimensional network u(t) = bu(t) + ag(u(t)), (5.7) where b > 0 and a = b or b, and g(u) is a sigmoidal real-valued function such that ug(u) 0 and g(u) β u for all < u <. Assume β <. Then it is easy to verify that the solution u(t; x o ) of equation (5.7) with initial data u(0) = x o 0 satisfies t log( u(t; x o) ) b [ β ( 0 sign(a) )] < 0. In other words, network (5.7) is exponentially stable. Now perturb this network stochastically and assume the perturbed network is described by dx(t) = [ bx(t) + ag(x(t))]dt + θ k x(t)dw k (t), (5.8)

19 where θ k s are all real unmbers. It is not difficult to show by Theorem 4. that the solution x(t; x o ) of equation (5.8) with initial data x(0) = x o 0 satisfies t log( x(t; x o) ) b [ β ( 0 sign(a) )] 2 θ 2 k < 0 So the stochastic neural network (5.8) becomes even more stable. We therefore see that a one-dimensional stable network may not be destabilized by Brownian motions if the stochastic perturbation is restricted to be linear. 6. Open Problems It has been showed that for any given unstable neural network of the form u(t) = Bu(t) + Ag(u(t)) (6.) satisfying (.2), one can always choose suitable matrices B, B 2,, B m such that the stochastically perturbed network dx(t) = [ Bx(t) + Ag(x(t))]dt + B k x(t)dw k (t) (6.2) is almost surely exponentially stable, and moreover the choices for such B k s are plenty. One the other hand, stabilization is expensive and the cost is generally proportional to m trace(b kbk T ). In practice, it is important to find the best B k s which minimize the cost. Let us now describe such problem in a strictly mathematical way. For each λ > 0 and m, denote by S λ,m the family of matrices (B, B 2,, B m ) such that the top Lyapunov exponent of the solution of equation (6.2) is not greater than λ. Obviously S λ,m is not empty. Define r λ,m = inf (B,,B m ) S λ,m trace(b k Bk T ). The first open problem is: Is there an optimal ( B,, B m ) S λ,m in the sense r λ,m = trace( B k BT k )? Now let S λ = m= S λ,m and r λ = inf{r λ,m : m < }. The second open problem is: Is there an optimal ( B,, B m ) S λ in the sense r λ = trace( B k BT k )? a.s.

20 Furthermore, define r = inf{r λ : λ > 0}. In the case when network (6.) is exponentially unstable, it is not very difficult to show r > 0. The mean of r is that if matrices (B, B 2,, B m ) for some m are such that trace(b k Bk T ) < r, then the stochastic network (6.2) is definitely not almost surely exponentially stable. Should m trace(b kbk T ) be called the intensity of the stochastic perturbation, then the intensity must not be less than r in order to stabilize the given network. So we can call r the minimum intensity of stochastic perturbation for stabilization. The question is: What is the value of r? Acknowledgement The authors would like to thank the Royal Society for the financial support so that X. Mao is able to invite X.X. Liao to visit the University of Strathclyde to carry out this joint research. REFERENCES [] Coben, M.A. and Crosshery S., Absolute stability and global pattern formation and patrolled memory storage by competitive neural networks, IEEE Trans. on Systems, Man and Cybernetics 3 (983), [2] Denker, J.S.(Editor), Neural Networks for Computing (Snowbird, UT, 986), Proceedings of the Conference on Neural Networks for Computing, AIP, New York, 986. [3] Friedman, A. Stochastic Differential Equations and Applications, Academic Press, Vol., 975. [4] Hopfield, J.J., Neural networks and physical systems with emergent collect computational abilities, Proc. Natl. Acad. Sci. USA, 79(982), [5] Hopfield, J.J., Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci. USA, 8(984), [6] Hopfield, J.J. and Tank, D.W., Computing with neural circuits, Model Science, 233(986), [7] Liao, X.X., Stability of a class of nonlinear continuous neural networks, Proceedings of the First World Conference on Nonlinear Analysis, WC33, 992. [8] Liptser, R.Sh. and Shiryayev, A.N., Theory of Martingales, Kluwer Academic Publishers, 986.

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