In this paper we shall discuss stochastic effects to the stability property of a neural network Ëu(t) = âBu(t) + Ag(u(t)). Suppose the stochastically perturbed neural ...
EXPONENTIAL STABILITY AND INSTABILITY OF STOCHASTIC NEURAL NETWORKS 1 X. X. Liao2 and X. Mao3 Department of Statistics and Modelling Science University of Strathclyde Glasgow G1 1XH, 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 Itˆ o 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.
1.
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. [1, 4–7, 11]). The neural network proposed by Hopfield [4] can be described by an ordinary differential 1 2
Supported by the Royal Society. Permanent Address: Department of Mathematics, Huazhong Normal University, Wuhan,
P.R.China. 3 For any correspondence regarding this paper please address to this author.
equation of the form n X 1 Ci u˙ i (t) = − ui (t) + Tij gj (uj (t)), Ri j=1
1 ≤ i ≤ n,
(1.1)
on t ≥ 0. The variable ui (t) represents the voltage on the input of the ith neuron. Each neuron is characterized by an input capacitance Ci and a transfer function gi (u). The connection matrix element Tij has a value + 1/Rij when the noninverting output of the jth neuron is connected to the input of the ith neuron through a resistance Rij , and a value − 1/Rij when the inverting output of the jth neuron is connected to the input of the ith neuron through a resistance Pn Rij . The parallel resistance at the input of each neuron is defined Ri = ( j=1 |Tij |)−1 . The nonlinear transfer function gi (u) is sigmoidal, saturating at ±1 with maximum slope at u = 0. In term of mathematics, that is g(u) is nondecreasing, ugi (u) ≥ 0 and |gi (u)| ≤ 1 ∧ βi |u|
for all − ∞ < u < ∞,
(1.2)
where βi is the slope of gi (u) at u = 0 and is supposed to be finite. By defining 1 Tij bi = , aij = Ci Ri Ci equation (1.1) can be re-written as u˙ i (t) = −bi ui (t) +
n X
1 ≤ i ≤ n,
aij gj (uj (t)),
(1.3)
j=1
or equivalently u(t) ˙ = −Bu(t) + Ag(u(t)),
t ≥ 0,
(1.4)
where u(t) = (u1 (t), · · · , un (t))T , B = diag.(b1 , · · · , bn ), A = (aij )n×n , g(u) = (g1 (u1 )), · · · , gn (un ))T . Moreover, we always have bi =
n X
|aij |,
1 ≤ i ≤ n.
(1.5)
j=1
It is clear that whenever given an initial data u(0) = xo ∈ Rn equation (1.4) has a unique global solution on t ≥ 0. Especially, the equation admits
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 [1], Liao [7], Quezz et al. [11]. 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) = xo ∈ Rn ,
(1.6)
where w(t) = (w1 (t), · · · , wm (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. Ft = σ{w(s) : 0 ≤ s ≤ t}), and σ : Rn → Rn×m i.e. σ(x) = (σij (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 (1.6) has a unique global solution on t ≥ 0, which is denoted by x(t; xo ). Moreover, we also assume σ(0) = 0 for the stability purpose of this paper. So equation (1.6) admits an equilibrium solution x(t; 0) ≡ 0. It is also easy to see from the uniqueness that whenever the initial data xo 6= 0, the solution will never be zero with probability one, that is x(t, xo ) 6= 0 for all t ≥ 0 a.s. Now that equation (1.6) is a stochastically perturbed system of equation (1.4), it is interesting to know how the stochastic perturbation effects the stability property of equation (1.4). That is, when equation (1.4) is stable, it is useful to know whether the perturbed equation (1.6) remains stable or becomes unstable; but when equation (1.4) is unstable, it is then useful to know whether the perturbed equation (1.6) 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 (1.6). Theorem 2.1 Assume there exists a symmetric positive definite matrix Q = (qij )n×n and a pair of numbers µ ∈ R and ρ ≥ 0 such that 2xT Q[−Bx + Ag(x)] + trace[σ T (x)Qσ(x)] ≤ µxT Qx,
(2.1)
xT Qσ(x)σ T (x)Qx ≥ ρ(xT Qx)2
(2.2)
for all x ∈ Rn . Then the solution of equation (1.6) satisfies lim sup t→∞
1 µ log(|x(t; xo )|) ≤ −(ρ − ) t 2
a.s.
(2.3)
whenever xo 6= 0. In particular, if ρ > µ/2 then the stochastic neural network (1.6) is almost surely exponentially stable. Proof. Fix any xo 6= 0 arbitrarily and write x(t; xo ) = x(t) simply. Note from the uniqueness of the solution that x(t) 6= 0 for all t ≥ 0 a.s. So one can apply the well-known Itˆ o formula to obtain d log[xT (t)Qx(t)] =
1 T T 2x (t)Q[−Bx(t) + Ag(x(t))] + trace[σ (x(t))Qσ(x(t))] dt xT (t)Qx(t) 2 T T − T x (t)Qσ(x(t))σ (x(t))Qx(t) dt [x (t)Qx(t)]2 +
2 xT (t)Qx(t)
xT (t)Qσ(x(t))dw(t).
In view of condition (2.1) we obtain log[xT (t)Qx(t)] ≤ log[xTo Qxo ] + µt − 2hM (t)i + 2M (t)
a.s.
(2.4)
for all t ≥ 0, where Z M (t) = 0
t
1 xT (s)Qσ(x(s))dw(s) T x (s)Qx(s)
which is a continuous martingale vanishing at t = 0 and hM (t)i is its quadratic variation, i.e. Z t 1 T T hM (t)i = x (s)Qσ(x(s))σ (x(s))Qx(s) ds. T 2 o [x (s)Qx(s)] By condition (2.2) it is easy to see that hM (t)i ≥ ρt.
(2.5)
Now let k = 1, 2, · · · and let ε ∈ (0, 1) be arbitrary. Using the well-known exponential martingale inequality (cf M´etivier [10]) one can derive that 1 1 P ω : sup [M (t) − εhM (t)i] > log k ≤ . 2ε k 0≤t≤k
Hence the Borel-Cantelli lemma yields that for almost all ω ∈ Ω there exists a random integer ko (ω) such that for all k ≥ ko sup [M (t) − εhM (t)i] ≤ 0≤t≤k
1 log k, 2ε
that is, M (t) ≤ εhM (t)i +
1 log k, 2ε
0 ≤ t ≤ k.
Substituting this into (2.4) yields log[xT (t)Qx(t)] ≤ log[xTo Qxo ] + µt − (2 − ε)hM (t)i +
1 log k ε
for all 0 ≤ t ≤ k and k ≥ ko almost surely. By (2.5) one therefore obtains that 1 log[xT (t)Qx(t)] ≤ log[xTo Qxo ] − [(2 − ε)ρ − µ]t + log k ε for all 0 ≤ t ≤ k and k ≥ ko almost surely. So for almost all ω ∈ Ω, if k − 1 ≤ t ≤ k and k ≥ ko then 1 1 1 T T log[x (t)Qx(t)] ≤ −[(2 − ε)ρ − µ] + log[xo Qxo ] + log k . t k−1 ε This implies lim sup t→∞
1 log[xT (t)Qx(t)] ≤ −[(2 − ε)ρ − µ] t
a.s.
Letting ε → 0 we obtain lim sup t→∞
1 log[xT (t)Qx(t)] ≤ −(2ρ − µ) t
a.s.
(2.6)
One the other hand, note λmin |x|2 ≤ xT Qx,
x ∈ Rn
since Q is a symmetric positive definite matrix, where λmin > 0 is the smallest eigenvalue of Q. Consequently, it follows from (2.6) that lim sup t→∞
1 µ log(|x(t)|) ≤ −(ρ − ) t 2
as required. The proof is complete.
a.s.
We now employ this theorem to establish a number of useful corollaries. Corollary 2.2 Let (1.2) hold. Assume that there exists a positive definite diagonal matrix Q = diag.(q1 , q2 , · · · , qn ) and two real numbers µ > 0, ρ ≥ 0 such that trace[σ T (x)Qσ(x)] ≤ µxT Qx, xT Qσ(x)σ T (x)Qx ≥ ρ(xT Qx)2 for all x ∈ Rn . Let λmax (H) denote the biggest eigenvalue of the symmetric matrix H = (hij )n×n defined by hij =
2qi [−bi + (0 ∨ aii )βi ] qi |aij | βj + qj |aji | βi
for i = j, for i = 6 j.
Then the solution of equation (1.6) satisfies 1 1h λmax (H) i lim sup log(|x(t; xo )|) ≤ − ρ − µ + 2 min1≤i≤n qi t→∞ t
a.s.
(2.7)
a.s.
(2.8)
if λmax (H) ≥ 0, or otherwise 1 1h λmax (H) i lim sup log(|x(t; xo )|) ≤ − ρ − µ + 2 max1≤i≤n qi t→∞ t whenever xo 6= 0. Proof. Compute, by (1.2), T
2x QAg(x) = 2
n X
xi qi aij gj (xj )
i,j=1
≤2
X
=2
X
qi (0 ∨ aii )xi gi (xi ) + 2
i
X
|xi | qi |aij | βj |xj |
i6=j
qi (0 ∨ aii )βi x2i +
i
X
|xi |(qi |aij | βj + qj |aji | βi )|xj |.
i6=j
Thus, in the case λmax (H) ≥ 0, 2xT Q[−Bx + Ag(x)] ≤ (|x1 |, · · · , |xn |) H (|x1 |, · · · , |xn |)T ≤ λmax (H)|x|2 ≤
λmax (H) T x Qx, min1≤i≤n qi
and then conclusion (2.7) follows from Theorem 2.1 easily. Similarly, in the case λmax (H) < 0, 2xT Q[−Bx + Ag(x)] ≤ λmax (H)|x|2 ≤
λmax (H) T x Qx max1≤i≤n qi
and then conclusion (2.8) follows from Theorem 2.1 again. The proof is complete. Corollary 2.3 Let both (1.2) and (1.5) hold. Assume that there exist n positive numbers q1 , q2 , · · · , qn such that βj2
n X
qi [0 ∨ sign(aii )]δij |aij | ≤ qj bj ,
1 ≤ j ≤ n,
i=1
where
δij =
1 0
for i = j, for i = 6 j.
Moreover assume trace[σ T (x)Qσ(x)] ≤ µxT Qx, xT Qσ(x)σ T (x)Qx ≥ ρ(xT Qx)2 for all x ∈ Rn , where Q = diag.(q1 , q2 , · · · , qn ) and µ > 0, ρ ≥ 0 are both constants. Then the solution of equation (1.6) satisfies lim sup t→∞
µ 1 log(|x(t; xo )|) ≤ −(ρ − ) t 2
a.s.
whenever xo 6= 0. Proof. Compute, by the conditions, 2xT QAg(x) = 2
n X
xi qi aij gj (xj )
i,j=1
≤2
n X
|xi | qi [0 ∨ sign(aii )]δij |aij | βj |xj |
i,j=1
≤
n X
qi [0 ∨ sign(aii )]δij |aij |(x2i + βj2 x2j )
i,j=1 n n n n X X X X 2 2 ≤ qi |aij | xi + βj qi [0 ∨ sign(aii )]δij |aij | x2j i=1
≤
n X i=1
j=1
qi bi x2i +
j=1 n X j=1
i=1
qj bj x2j = 2xT QBx.
Hence 2xT Q[−Bx + Ag(x)] + trace[σ T (x)Qσ(x)] ≤ µxT Qx. Then the conclusion follows from Theorem 2.2. The proof is complete. Corollary 2.4 Let both (1.2) and (1.5) hold. Assume the network is symmetric in the sense |aij | = |aji |
for all 1 ≤ i, j ≤ n.
Moreover assume trace[σ T (x)σ(x)] ≤ µ|x|2 , xT σ(x)σ T (x)x ≥ ρ|x|4 for all x ∈ Rn , where both µ > 0 and ρ ≥ 0 are both constants. Then the solution of equation (1.6) satisfies that lim sup t→∞
1 ˇ − µ) log(|x(t; xo )|) ≤ −(ρ + ˆb(1 − β) t 2
a.s.
(2.9)
ˇ or if 1 ≥ β, lim sup t→∞
1 µ log(|x(t; xo )|) ≤ −(ρ − ˇb(βˇ − 1) − ) t 2
a.s.
if 1 < βˇ whenever xo 6= 0, where βˇ = max βi , 1≤i≤n
ˇb = max bi , 1≤i≤n
ˆb = min bi . 1≤i≤n
Proof. Compute T
2x Ag(x) = 2
n X
xi aij gj (xj )
i,j=1
≤2
n X
|xi | |aij | βj |xj | ≤ βˇ
i,j=1
n X
|aij |(x2i + x2j )
i,j=1
n X n n X n hX i X 2 ˇ =β |aij | xi + |aji | x2j i=1
j=1
j=1
i=1
n n hX i X ˇ T Bx. = βˇ bi x2i + bj x2j = 2βx i=1
j=1
(2.10)
Hence ˇ T Bx. 2xT [−Bx + Ag(x)] ≤ −2(1 − β)x ˇ Therefore, in the case 1 ≥ β, ˇ + µ]|x|2 , 2xT [−Bx + Ag(x)] + trace[σ T (x)σ(x)] ≤ [−2ˆb(1 − β) and conclusion (2.9) follows from Theorem 2.1 with Q = the identity matrix. ˇ On the other hand, in the case 1 < β, 2xT [−Bx + Ag(x)] + trace[σ T (x)σ(x)] ≤ [2ˇb(βˇ − 1) + µ]|x|2 , and conclusion (2.10) follows from Theorem 2.1 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 (1.6). Theorem 3.1 Assume there exists a symmetric positive definite matrix Q = (qij )n×n and two real numbers µ ∈ R, ρ > 0 such that 2xT Q[−Bx + Ag(x)] + trace[σ T (x)Qσ(x)] ≥ µxT Qx
(3.1)
xT Qσ(x)σ T (x)Qx ≤ ρ(xT Qx)2
(3.2)
for all x ∈ Rn . Then µ 1 log(|x(t; xo )|) ≥ − ρ t 2
lim inf t→∞
a.s.
(3.3)
whenever xo 6= 0. In particular, if ρ < µ/2 then the stochastic neural network (1.6) is almost surely exponentially unstable. Proof. Fix any xo 6= 0 arbitrarily and again write x(t; xo ) = x(t) simply. By the Itˆ o formula as well as conditions (3.1), (3.2) one can derive that log[xT (t)Qx(t)] ≥ log[xTo Qxo ] + (µ − 2ρ)t + 2M (t)
a.s.
(3.4)
for all t ≥ 0, where Z M (t) = 0
t
1 xT (s)Qσ(x(s))dw(s) xT (s)Qx(s)
the same as before. Note from condition (3.2) that Z t 1 T T hM (t)i = x (s)Qσ(x(s))σ (x(s))Qx(s) ds ≤ ρt. T 2 o [x (s)Qx(s)]
It is known (cf. Liptser & Shiryayev [8]) that M (t)/t → 0 almost surely as t → ∞. Consequently (3.4) yields lim inf t→∞
1 log[xT (t)Qx(t)] ≥ µ − 2ρ t
a.s.
(3.5)
But, note λmax |x|2 ≥ xT Qx,
x ∈ Rn ,
where λmax > 0 is the biggest eigenvalue of Q. Hence it follows from (3.5) that 1 µ lim inf log(|x(t)|) ≥ − ρ a.s. t→∞ t 2 as required. The proof is complete. Corollary 3.2 Let (1.2) hold. Assume that there exists a positive definite diagonal matrix Q = diag.(q1 , q2 , · · · , qn ) and two positive numbers µ, ρ such that trace[σ T (x)Qσ(x)] ≥ µxT Qx, xT Qσ(x)σ T (x)Qx ≤ ρ(xT Qx)2 for all x ∈ Rn . Let λmin (S) denote the smallest eigenvalue of the symmetric matrix S = (sij )n×n which is defined by sij =
2qi [−bi + (0 ∧ aii )βi ] −qi |aij | βj − qj |aji | βi
for i = j, for i = 6 j.
Then the solution of equation (1.6) satisfies 1 1h λmin (S) i −ρ lim inf log(|x(t; xo )|) ≥ µ + t→∞ t 2 min1≤i≤n qi
a.s.
(3.6)
whenever xo 6= 0. Proof. In the same way as the proof of Corollary 2.2 one can show that 2xT Q[−Bx + Ag(x)] ≥ (|x1 |, · · · , |xn |) S (|x1 |, · · · , |xn |)T ≥ λmin (S)|x|2 . Note that we must have λmin (S) ≤ 0 since all the elements of S are nonpositive. So λmin (S) T 2xT Q[−Bx + Ag(x)] ≥ x Qx min1≤i≤n qi and then conclusion (3.6) follows from Theorem 3.1 easily. The proof is complete.
Corollary 3.3 Let both (1.2) and (1.5) hold. Assume the network is symmetric in the sense |aij | = |aji |
for all 1 ≤ i, j ≤ n.
Moreover assume trace[σ T (x)σ(x)] ≥ µ|x|2 , xT σ(x)σ T (x)x ≤ ρ|x|4 for all x ∈ Rn , where both µ and ρ are positive numbers. Then the solution of equation (1.6) satisfies that lim inf t→∞
µ 1 ˇ −ρ log(|x(t; xo )|) ≥ − ˇb(1 + β) t 2
a.s.
whenever xo 6= 0, where βˇ = max βi 1≤i≤n
and ˇb = max bi . 1≤i≤n
Proof. Compute T
2x Ag(x) = 2
n X
xi aij gj (xj )
i,j=1
≥ −2
n X
|xi | |aij | βj |xj | ≥ −βˇ
i,j=1
n X
|aij |(x2i + x2j )
i,j=1
n X n n X n hX i X 2 ˇ = −β |aij | xi + |aji | x2j i=1
j=1
j=1
i=1
n n hX i X ˇ T Bx. = −βˇ bi x2i + bj x2j = −2βx i=1
j=1
Hence ˇ T Bx ≥ −2ˇb(1 + β)|x| ˇ 2. 2xT [−Bx + Ag(x)] ≥ −2(1 + β)x Therefore, 2 ˇ 2xT [−Bx + Ag(x)] + trace[σ T (x)σ(x)] ≥ [µ − 2ˇb(1 + β)]|x| ,
and the conclusion (2.7) follows from Theorem 3.1 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 (1.4) 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) =
m X
Bk x(t)dwk (t),
k=1
i.e. σ(x) = (B1 x, B2 x, · · · , Bm x), where Bk , 1 ≤ k ≤ m are all n×n matrices. In this case, the stochastically perturbed network (1.6) becomes m X dx(t) = [−Bx(t) + Ag(x(t))]dt + Bk x(t)dwk (t) on t ≥ 0, (4.1) k=1 x(0) = xo ∈ Rn . Note that trace[σ T (x)Qσ(x)] =
m X
xT BkT QBk x
k=1
and xT Qσ(x)σ T (x)Qx = trace[σ T (x)QxxT Qσ(x)] m m X X T T T = x Bk Qxx QBk x = (xT QBk x)2 . k=1
k=1
We immediately obtain the following useful result from Theorem 2.1. Theorem 4.1 Assume there exists a symmetric positive definite matrix Q = (qij )n×n and a pair of numbers µ ∈ R and ρ ≥ 0 such that T
2x Q[−Bx + Ag(x)] +
m X k=1
xT BkT QBk x ≤ µxT Qx
and
m X
(xT QBk x)2 ≥ ρ(xT Qx)2
k=1
for all x ∈ Rn . Then the solution of equation (4.1) satisfies lim sup t→∞
1 µ log(|x(t; xo )|) ≤ −(ρ − ) t 2
a.s.
whenever xo 6= 0. In particular, if ρ > µ/2 then the stochastic neural network (4.1) is almost surely exponentially stable. Let us now explain through examples how one can apply this theorem to stabilize a given neural network. Example 4.1 Let for 1 ≤ k ≤ m,
Bk = θk I
where I is the identity matrix and θk , 1 ≤ k ≤ m are all real numbers. Then equation (4.1) becomes dx(t) = [−Bx(t) + Ag(x(t))]dt +
m X
θk x(t)dwk (t)
(4.2)
k=1
(the initial data is omitted here). One can see that the numbers θk , 1 ≤ k ≤ m represent the intensity of the stochastic perturbation. Choose Q to be the identity matrix. Note in this case that m X
T
x
BkT QBk x
=
k=1
and
m X k=1
m X
2
|Bk x| =
k=1
T
2
(x QBk x) =
m X k=1
m X
θk2 |x|2
(4.3)
θk2 |x|4 .
(4.4)
k=1
T
2
(x θk x) =
m X k=1
Moreover, in view of (1.2) we have ˇ 2xT QAg(x) ≤ 2|x| ||A|| ||g(x)|| ≤ 2β||A|| |x|2 , where βˇ = max1≤k≤n βk and || · || denotes the operator norm of a matrix, i.e. ||A|| = sup{|Ax| : x ∈ Rn , |x| = 1}. Hence 2xT Q[−Bx + Ag(x)] ≤ 2(βˇ − ˆb)|x|2 ,
(4.5)
where ˆb = min1≤k≤n bk . Combining (4.3)–(4.5) and applying Theorem 4.1 we see that the solution of equation (4.2) satisfies m 1 X 1 2 ˇ ˆ θk − (β − b) lim sup log(|x(t; xo )|) ≤ − 2 t→∞ t
a.s.
k=1
whenever xo 6= 0. In particular, if choose θk ’s large enough such that m X
θk2 > 2(βˇ − ˆb)
k=1
then the stochastic neural network (4.2) is almost surely exponentially stable. Now if we choose θk = 0 for 2 ≤ k ≤ m, then equation (4.2) becomes an even simpler one dx(t) = [−Bx(t) + Ag(x(t))]dt + θ1 x(t)dw1 (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 θ12 > 2(βˇ − 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 (1.2) is satisfied. Moreover, one can even use only a scalar Brownian motion to do so. Theorem 4.1 ensures that there are many choices for the matrices Bk in order to stabilize a given network. Of course the choices in Example 4.1 are just the simplest ones. For illustration one more example is given here. Example 4.2 For each k, choose a positive definite n × n matrix Dk such that √ 3 T x Dk x ≥ ||Dk || |x|2 , 2
Obviously, there are lots of such matrices. Let θ be a real number and define Bk = θDk . Then equation (4.1) becomes dx(t) = [−Bx(t) + Ag(x(t))]dt + θ
m X
Dk x(t)dwk (t).
(4.7)
k=1
Again let Q = identity matrix. Note m X
T
x
BkT QBk x
=
k=1
and
m X
2
|θDk x| ≤ θ
k=1
(xT QBk x)2 = θ2
k=1
m X
m X
2
m X
||Dk ||2 |x|2
k=1
(xT Dk x)2 ≥
m 3θ2 X ||Dk ||2 |x|4 . 4 k=1
k=1
Combining these together with (4.5) and then applying Theorem 4.1 we obtain that the solution of equation (4.7) satisfies m θ2 X 1 2 ˇ ˆ ||Dk || − (β − b) lim sup log(|x(t; xo )|) ≤ − 4 t→∞ t
a.s.
k=1
whenever xo 6= 0. So if m −1 X θ2 > 4(βˇ − ˆb) ||Dk ||2 k=1
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
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 (4.1). Applying Theorem 3.1 to equation (4.1) we immediately obtain the following useful result. Theorem 5.1 Assume there exists a symmetric positive definite matrix Q = (qij )n×n and a pair of numbers µ ∈ R and ρ > 0 such that 2xT Q[−Bx + Ag(x)] +
m X
xT BkT QBk x ≥ µxT Qx
k=1
and
m X
(xT QBk x)2 ≤ ρ(xT Qx)2
k=1
for all x ∈ Rn . Then the solution of equation (4.1) satisfies lim inf t→∞
1 µ log(|x(t; xo )|) ≥ − ρ t 2
a.s.
whenever xo 6= 0. In particular, if ρ < µ/2 then the stochastic neural network (4.1) 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 5.1 First of all, let the dimension of the network n ≥ 3. Let m = n, i.e. choose an n-dimensional Brownian motion (w1 (t), w2 (t), · · · , wn (t))T . Let θ be a real number. For each k = 1, 2, · · · , n − 1, define Bk = (bkij )n×n by bkij = θ if i = k and j = k + 1 or otherwise bkij = 0; and moreover define Bn = (bnij )n×n by bnij = θ if i = n and j = 1 or otherwise bkij = 0. Then the stochastic network (4.1) becomes x (t)dw (t) 2 1 .. . . dx(t) = [−Bx(t) + Ag(x(t))]dt + θ xn (t)dwn−1 (t) x1 (t)dwn (t)
(5.1)
Let Q = the identity matrix. Note n X k=1
T
x
BkT QBk x
=
n X k=1
2
|Bk x| =
n X k=1
|θxk |2 = θ2 |x|2 .
(5.2)
Also, setting xn+1 = x1 , n X
≤
2θ 3
T
2
(x QBk x) = θ
k=1 n 2 X
2
n X
x2k x2k+1
k=1
x2k x2k+1 +
k=1
n 2 X
θ 6
(x4k + x4k+1 ) ≤
k=1
θ2 4 |x| . 3
(5.3)
Moreover, by (1.2), 2 ˇ 2xT Q[−Bx + Ag(x)] ≥ −2(ˇb + β||A||)|x| ,
(5.4)
where ˇb = max1≤k≤n bk and βˇ = max1≤k≤n βk . Combining (5.2)–(5.4) and then applying Theorem 5.1 we see that the solution of equation (5.1) satisfies lim inf t→∞
1 θ2 θ2 θ2 ˇ ˇ log(|x(t; xo )|) ≥ − (ˇb + β||A||) − = − (ˇb + β||A||) t 2 3 6
a.s.
whenever xo 6= 0. So the stochastic neural network (5.1) is almost surely ˇ exponentially unstable if θ2 > 6(ˇb + β||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 ≥ 1). Let m = 1, that is choose a scalar Brownian motion w1 (t). Let θ be a real number. Define
0 θ −θ 0 B1 = 0
0 ..
. 0 −θ
. θ 0
Then equation (4.1) becomes dx(t) = [−Bx(t) + Ag(x(t))]dt + θ
x2 (t) −x1 (t) .. . x2p (t) −x2p−1 (t)
dw1 (t).
(5.5)
Let Q = identity matrix again. Note xT B1T QB1 x = θ2 |x|2
and
(xT QB1 x)2 = 0.
(5.6)
Combining (5.6) with (5.4) and then applying Theorem 5.1 we see that the solution of equation (5.5) satisfies lim inf t→∞
θ2 1 ˇ log(|x(t; xo )|) ≥ − (ˇb + β||A||) t 2
a.s.
whenever xo 6= 0. So the stochastic neural network (5.5) is almost surely ˇ exponentially unstable if θ2 > 2(ˇb + β||A||). Summarizing the above two examples we obtain the following conclusion. Theorem 5.2 Any neural network of the form x(t) ˙ = −Bx(t) + Ag(x(t)) can be destabilized by Brownian motion provided the dimension n ≥ 2 and (1.2) is satisfied. Naturally, one would ask what happens when the dimension n = 1. 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)| ≤ 1 ∧ β|u|
for all −∞ < u < ∞.
Assume β < 1. Then it is easy to verify that the solution u(t; xo ) of equation (5.7) with initial data u(0) = xo 6= 0 satisfies lim sup t→∞
1 log(|u(t; xo )|) ≤ −b 1 − β 0 ∨ sign(a) < 0. t
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 +
m X k=1
θk x(t)dwk (t),
(5.8)
where θk ’s are all real unmbers. It is not difficult to show by Theorem 4.1 that the solution x(t; xo ) of equation (5.8) with initial data x(0) = xo 6= 0 satisfies m 1 X 1 lim sup log(|x(t; xo )|) ≤ −b 1 − β 0 ∨ sign(a) − θk2 < 0 2 t→∞ t
a.s.
k=1
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))
(6.1)
satisfying (1.2), one can always choose suitable matrices B1 , B2 , · · · , Bm such that the stochastically perturbed network dx(t) = [−Bx(t) + Ag(x(t))]dt +
m X
Bk x(t)dwk (t)
(6.2)
k=1
is almost surely exponentially stable, and moreover the choices for such Bk ’s are plenty. One the otherPhand, stabilization is expensive and the cost is m generally proportional to k=1 trace(Bk BkT ). In practice, it is important to find the best Bk ’s which minimize the cost. Let us now describe such problem in a strictly mathematical way. For each λ > 0 and m ≥ 1, denote by Sλ,m the family of matrices (B1 , B2 , · · · , Bm ) such that the top Lyapunov exponent of the solution of equation (6.2) is not greater than −λ. Obviously Sλ,m is not empty. Define rλ,m =
m X
inf (B1 ,···,Bm )∈Sλ,m
trace(Bk BkT ).
k=1
˜1 , · · · , B ˜m ) ∈ Sλ,m in the The first open problem is: Is there an optimal (B sense m X ˜k B ˜kT )? rλ,m = trace(B k=1
S∞
Now let Sλ = m=1 Sλ,m and rλ = inf{rλ,m : 1 ≤ m < ∞}. The second ˜1 , · · · , B ˜m open problem is: Is there an optimal (B ˜ ) ∈ Sλ in the sense rλ =
m ˜ X k=1
˜k B ˜ T )? trace(B k
Furthermore, define r = inf{rλ : λ > 0}. In the case when network (6.1) is exponentially unstable, it is not very difficult to show r > 0. The mean of r is that if matrices (B1 , B2 , · · · , Bm ) for some m ≥ 1 are such that m X
trace(Bk BkT ) < r,
k=1
then the stochastic Pmnetwork (6.2) is definitely not almost surely exponentially stable. Should k=1 trace(Bk BkT ) 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 [1] 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 13 (1983), 815–826. [2] Denker, J.S.(Editor), Neural Networks for Computing (Snowbird, UT, 1986), Proceedings of the Conference on Neural Networks for Computing, AIP, New York, 1986. [3] Friedman, A. Stochastic Differential Equations and Applications, Academic Press, Vol.1, 1975. [4] Hopfield, J.J., Neural networks and physical systems with emergent collect computational abilities, Proc. Natl. Acad. Sci. USA, 79(1982), 2554–2558. [5] Hopfield, J.J., Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci. USA, 81(1984), 3088–3092. [6] Hopfield, J.J. and Tank, D.W., Computing with neural circuits, Model Science, 233(1986), 3088–3092. [7] Liao, X.X., Stability of a class of nonlinear continuous neural networks, Proceedings of the First World Conference on Nonlinear Analysis, WC313, 1992. [8] Liptser, R.Sh. and Shiryayev, A.N., Theory of Martingales, Kluwer Academic Publishers, 1986.
[9] Mao, X., Exponential Stability of Stochastic Differential Equations, Marcel Dekker Inc., 1994. [10] M´etivier, M., Semimartingales, Walter de Gruyter, 1982. [11] Quezz, A., Protoposecu V. and Barben, J., On the stability storage capacity and design of nonlinear continuous neural networks, IEEE Trans. on Systems, Man and Cybernetics 18 (1983), 80–87.
