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Stabilizing Hopfield neural networks via inhibitory self-connections
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J. Math. Anal. Appl. 292 (2004) 135–147 www.elsevier.com/locate/jmaa
Stabilizing Hopfield neural networks via inhibitory self-connections ✩ Lin Wang 1 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NF, Canada A1C 5S7 Received 4 June 2003 Submitted by H.L. Smith
Abstract The theory of monotone dynamical systems is employed to establish some sufficient conditions for the global attractivity of the Hopfield neural networks with finite distributed delays. The results show that self-inhibitory connections can be used to stabilize a delayed network provided the diagonal delays corresponding to the inhibitory self-connections are small enough. 2003 Elsevier Inc. All rights reserved. Keywords: Attractivity; Distributed delay; Monotone dynamical system; Neural network
1. Introduction In this paper, we study the Hopfield type neural networks with finite distributed delays described by the following system: t n x˙i (t) = −bi xi (t) + aij gj kij (t − s)xj (s) ds + Ji , j =1
i ∈ N(1, n),
t −τij
(1.1)
where n 2 is the number of neurons in the network, xi denotes the state variable associated to the ith neuron, and the n × n connection matrix A = (aij ) tells how the neurons are ✩
Research was partially supported by NSERC of Canada. E-mail addresses: [email protected], [email protected]. 1 Present address: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1. 0022-247X/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2003.11.048
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connected in the network, and the activation function gj shows how the j th neuron reacts to the input, τij 0 for i, j ∈ {1, 2, . . . , n} =: N(1, n) with τ = max{τij , i, j ∈ N(1, n)}, and the kernel functions satisfy τii kij (t) 0,
kii (t) dt = 1,
i, j ∈ N(1, n).
0
When τ = 0, bi (x) = bi x, i ∈ N(1, n), system (1.1) reduces to x˙i (t) = −bi xi (t) +
n
aij gj xj (t) + Ji ,
i ∈ N(1, n),
(1.2)
j =1
which is extensively studied (see, e.g., [3,6,8,11]). Note that system (1.1) also includes the following Hopfield neural networks with fixed discrete delays as a special case: n x˙i (t) = −bi xi (t) + aij gj xj (t − τij ) + Ji ,
i ∈ N(1, n),
(1.3)
j =1
which is also investigated by many authors, for example, [1,2,4,7,12–14,16]. We will employ the ideas and techniques in van den Driessche et al. [12] and van den Driessche and Zou [13] to show how to stabilize the Hopfield neural networks (1.1) with general activation functions and finite distributed delays via the self-inhibitory connections. Without the assumptions on monotonicity and differentiability of the activation function, as pointed out in [13], we assume (S1 ) For each i ∈ {1, 2, . . . , n}, gi : R → R is globally Lipschitz continuous with a Lipschitz constant Li ; (S2 ) For each i ∈ {1, 2, . . . , n}, |gi (x)| Mi , x ∈ R, for some constant Mi > 0. We also assume that for each i ∈ N(1, n), bi (u) − bi (v) mi > 0. bi is continuous with (1.4) u−v Unlike in [12] and [13], we do not require any differentiability for the activation functions and the time delays are not necessary to be fixed constants, indeed they are distributed over a finite interval. The motivation of considering nonsmooth activation functions is that such functions have been widely adopted in network implementations (e.g., piecewise linear functions). We will see, by applying our results to (1.3), that the results in [12] and [13] are reproduced with a better estimate for the smallness of effective delays. The initial conditions associated with (1.2) are set to be xi (0) = φi (0),
i ∈ N(1, n),
(1.5)
and the associated ones for (1.1) are xi (s) = φi (s),
s ∈ [−τ, 0], i ∈ N(1, n).
(1.6)
It is not difficult to establish the following existence result for an equilibrium of system (1.2) by applying the Schauder’s fixed point theorem.
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Theorem 1.1. If (1.4) and (S2 ) hold, then for every input J , there exists an equilibrium for system (1.2).
2. Global stability of (1.2) Let x ∗ be an equilibrium of (1.2). Substituting x(t) = u(t) + x ∗ into (1.2) leads to n ∗ ∗ ∗ ∗ , (2.1) u˙ i (t) = − bi ui (t) + xi − bi xi − aij gj uj (t) + xj − gj xj j =1
which can be denoted by n u˙ i (t) = −βi ui (t) + aij sj uj (t) ,
i ∈ N(1, n),
(2.2)
j =1
where βi (ui (t)) = bi (ui (t) + xi∗ ) − bi (xi∗ ), sj (uj (t)) = gj (uj (t) + xj∗ ) − gj (xj∗ ). If we let u = (u1 , . . . , un )T ∈ R n , A = [aij ]n×n , s(u) = (s1 (u1 ), . . . , sn (un ))T , B(u) = (β1 (u1 ), . . . , βn (un ))T ∈ R n , then system (2.2) can be rewritten as u(t) ˙ = −B u(t) + As u(t) . (2.3) It is obvious that x ∗ is globally asymptotically (exponentially) stable for (1.2) if and only if the trivial solution u = 0 of (2.2) or (2.3) is globally asymptotically (exponentially) stable. Moreover, the uniqueness of the equilibrium of (1.2) follows from its global asymptotic (exponential) stability. From Theorem 2 of [3], we have Theorem 2.1. Suppose (1.4), (S1 ) and (S2 ) are satisfied. Assume also that for each i ∈ N(1, n), usi (u) > 0 when u = 0. Then, for every input J , system (1.2) has a unique equilibrium ymptotically stable if the matrix W ∗ defined by
p1 m1 pn mn 1 ,..., W ∗ = diag − (P T + T T P ) L1 Ln 2
(2.4) x∗
which is globally as-
(2.5)
is positive definite for some P = diag(p1 , . . . , pn ) with pi > 0, i ∈ N(1, n). A direct corollary of this theorem is Corollary 2.1. Suppose (1.4), (S1 ) and (S2 ) and (2.4) hold. If for some pi > 0, mi 1 aii pi + |pi aij + pj aj i | < pi , i ∈ N(1, n), 2 Li j =i
holds, then the equilibrium x ∗ of (1.2) is globally asymptotically stable.
(2.6)
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If the condition (2.6) is strengthened a little bit, we can actually obtain the global exponential stability, as stated in the following theorem. Theorem 2.2. Suppose (1.4), (S1 ) and (S2 ) and (2.4) hold. If mi pi aii + |pj aj i | < pi , i ∈ N(1, n), Li
(2.7)
j =i
holds for some positive numbers p1 , p2 , . . . , pn , then, for every input J , system (1.2) has a unique equilibrium which is globally exponentially stable in the sense that n xi (t) − x ∗ C1 e−σ1 t ,
(2.8)
i
i=1
where C1 , σ1 will be specified later. Proof. Let δ > 0 and σ1 > 0 be given as
pj |aj i | , i ∈ N(1, n) , δ := min mi pi − max 0, Li pi aii +
(2.9)
j =i
σ1 := max σ > 0: δ − pi σ > 0, i ∈ N(1, n) .
(2.10)
Define V (t) = V (u(t)) by V (t) =
n
pi eσ1 t |ui |.
(2.11)
i=1
Then the upper-right derivative of V can be estimated as n n + σ1 t D V (t) = pi e aij sj uj (t) σ1 ui (t) − sign ui (t) βi ui (t) − (1.2)
i=1 n
j =1
eσ1 t pi σ1 ui (t) − pi mi ui (t) + pi aii si ui (t)
i=1
+ =
j =i n σ1 t
e
i=1
+ =
eσ1 t pi |aij | sj uj (t)
pi σ1 ui (t) − pi mi ui (t) + pi aii si ui (t)
eσ1 t pj |aj i | si ui (t)
j =i n σ1 t
e
pi σ1 ui (t) − pi mi ui (t) + pi aii + pj |aj i | si (ui ) j =i
i=1
−(δ − pi σ1 )eσ1 t
n i=1
ui (t) .
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This shows that V (t) is a Liapunov function and hence V (t) V (0) =
n
pi φi (0) − xi∗ =: C0 < ∞
(2.12)
i=1
and thus we have n n ui (t) = xi (t) − x ∗ C1 e−σ1 t i
i=1
(2.13)
i=1
with C1 := C0 /min{pi , i ∈ N(1, n)}.
2
Remark 2.1. Corollary 2.1 and Theorem 2.2 show that the self-inhibitory connections do play an important role in stabilizing network (1.2) when ignoring delay.
3. Global attractivity of HNNs with finite distributed delays Note that the delays do not change the values of equilibria, but the stability may be lost when the delays increase. It is natural to expect that the stable equilibrium of the system without delays remains stable for the delayed system when the delays are sufficiently small. In order to give an estimation for the smallness, we will use the powerful theory of monotone dynamic systems. To be more precise, we shall use the related theory about the nonstandard ordering–exponential ordering. To this end and for reader’s convenience, we first introduce the theorem that we will employ below. In the following, as in [9], the partial order in Rn will be the usual componentwise ordering. The partial ordering φ ψ in C := C([−τ, 0], Rn ) will mean φ(θ ) ψ(θ ) for each θ ∈ [−τ, 0]. The inequality x < y (x y) between two vectors in Rn will mean x y and xi < yi for some (all) i ∈ N(1, n). The inequality φ < ψ in C will mean that φ ψ and φ = ψ, and φ ψ will mean that φ(θ ) ψ(θ ) for all θ ∈ [−τ, 0]. Let D be an n × n essentially nonnegative matrix, that is, D + λI is entry-wise nonnegative for all sufficiently large λ. Define KD = ψ ∈ C: ψ 0 and ψ(t) eD(t −s)ψ(s), −τ s t 0 . It can be seen that KD is a normal cone in C and thus it induces a partial ordering in C, denoted by D , in the usual way, that is, φ D ψ if and only if ψ − φ ∈ KD and φ
(3.1)
where f : C → Rn is globally Lipschitz continuous. By the fundamental theory of FDES [5], the system (3.1) generates a semi-flow Φ on C by Φ(t, φ) = Φt (φ) = xt (φ),
t 0, φ ∈ C,
for those t for which xt (φ) is defined. The following theorem is from Smith and Thieme [10].
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Theorem 3.1. Assume the following conditions are satisfied: (ID ) If φ, ψ ∈ C satisfy φ D ψ and K is a proper subset of N(1, n) such that φk ψk , k ∈ K, and φk (0) = ψk (0) for k ∈ N(1, n) − K, then for some p ∈ N(1, n) − K, fp (ψ) > fp (φ). (SMD ) If φ, ψ ∈ C satisfy φ D ψ and φ ψ, then f (ψ) − f (φ) D ψ(0) − φ(0) . Then Φ is strongly order preserving (SOP) on C. Using the above theorem, we will show that the semiflow Φ generated by the solution of (1.1) is SOP under the exponential ordering, if the diagonal delays corresponding to negative self-connections are sufficiently small. Theorem 3.2. Assume that bi (u) is Lipschitz continuous with Lip(bi ) = γi for i ∈ N(1, n), aij 0 for i = j , A is irreducible and gi satisfies gi (u) − gi (v) Li , i ∈ N(1, n). (3.2) u−v If the diagonal delays τii corresponding to negative aii are sufficiently small satisfying 0
τii
1 , ri∗
(3.3)
where ri∗ will be given below, then the semi-flow Φ generated by the solution of (1.1) is SOP under D . Proof. Take D = diag(d1 , . . . , dn ) with di = −γi − ri ,
i ∈ N(1, n),
where ri > 0, i ∈ N(1, n), are constants to be specified later. Then D is essentially nonnegative. For system (1.1), we have 0 n fi (φ) = −bi φi (0) + aij gj kij (−s)φj (s) ds + Ji , i ∈ N(1, n). (3.4) j =1
−τij
Let φ, ψ ∈ C satisfy φ D ψ and φ ψ. Then 0 n fi (ψ) − fi (φ) = −bi ψi (0) + aij gj kij (−s)ψj (s) ds j =1
+ bi φi (0) −
n j =1
−τij
0 aij gj
kij (−s)φj (s) ds −τij
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= − bi ψi (0) − bi φi (0) 0 n + aij gj kij (−s)ψj (s) ds j =1
−τij
0 − gj
kij (−s)φj (s) ds −τij
−γi ψi (0) − φi (0) 0 + aii gi
kii (−s)ψi (s) ds − gi −τii
0 + aij gj kij (−s)ψj (s) ds j =i
0 kii (−s)φi (s) ds −τii
−τij
0 − gj
kij (−s)φj (s) ds
.
−τij
Clearly if the matrix A is a nonnegative matrix, then (SMD ) holds. In the following we therefore may assume that aii < 0 for some i ∈ N(1, n). Then for such i, fi (ψ) − fi (φ) − di φi (0) − ψi (0) −γi ψi (0) − φi (0) 0 0 kii (−s)ψi (s) ds − gi
+ aii gi −τii
+ (γi + ri ) ψi (0) − φi (0) = ri ψi (0) − φi (0) 0 + aii gi
−τii
0
kii (−s)ψi (s) ds − gi −τii
kii (−s)φi (s) ds
kii (−s)φi (s) ds −τii
0
ri ψi (0) − φi (0) + aii Li
kii (−s) ψi (s) − φi (s) ds.
−τii
On the other hand φ D ψ implies that ψ − φ ∈ KD and hence ψ(0) − φ(0) e−Ds ψ(s) − φ(s) for s ∈ [−τ, 0], that is
ψi (s) − φi (s) e−(ri +γi )s ψi (0) − φi (0) for i ∈ N(1, n).
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This indicates that 0
0
kii (−s) ψi (s) − φi (s) ds −τii
kii (−s)e−(ri +γi )s ψi (0) − φi (0) ds
−τii
ψi (0) − φi (0) e(ri +γi )τii .
Therefore we have fi (ψ) − fi (φ) − di ψi (0) − φi (0) (ri + aii Li eτii (ri +γi ) ) ψi (0) − φi (0) > 0, provided that ri > |aii |Li eτii (ri +γi ) ,
i ∈ N(1, n).
(3.5)
Inequality (3.5) is satisfied if and only if τii <
ln |aiiri|Li γi + ri
,
i ∈ N(1, n).
(3.6)
Now let τi (s) :=
ln |aiis|Li γi + s
,
i ∈ N(1, n).
(3.7)
A simple calculation shows that τ¨i (s) < 0 for s > |aii |Li and τ˙i (|aii |Li ) > 0 and τi (|aii |Li ) = 0. Therefore τi (s) attains its maximal value 1/ri∗ at ri∗ , where ri∗ is the unique positive root of equation s γi h(s) := 1 + − ln =0 s |aii |Li for s > |aii |Li . Let ri = ri∗ for i ∈ N(1, n). This shows that (SMD ) holds if the diagonal delays τii corresponding to the negative aii satisfy τii <
1 , ri∗
i ∈ N(1, n).
(3.8)
(ID ) can be easily verified under the assumption that the connection matrix T is irreducible. Thus the proof is complete. 2 For system (1.1), the phase space is X = C = C([−τ, 0], Rn ). Therefore, every nonequilibrium point in X can be approximated from below and from above under the exponential ordering. (Indeed, for any x ∈ X, the sequences {xn }, {yn } defined by xn (s) := x(s) − eDs /n, yn (s) := x(s) + eDs /n, s ∈ [−τ, 0] serve this purpose.) Note that from the boundedness of the activation functions, it is easy to show that every bounded set B ⊂ X has a bounded orbit and thus the relatively weak compactness requirement (C) in [9] is met. Notice also that Corollary 2.1 and Theorem 2.2 imply the uniqueness of the equilibrium x ∗ of system (1.2), which shows that under the same assumptions, the equilibrium x ∗ of system (1.1) is unique. Thus Theorem 3.2, together with Theorem 3.1 of Smith [9], immediately gives
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Theorem 3.3. Assume that (3.2) holds and bi satisfies mi
bi (u) − bi (v) γi , u−v
i ∈ N(1, n),
A is irreducible and aij 0 for i = j . If either (2.6) or (2.7) holds, then the system (1.1) has a unique equilibrium which is globally attractive provided the diagonal delays τii corresponding to negative aii are sufficiently small satisfying (3.3). Letting s = e|aii |Li in (3.7), we have Corollary 3.1. Under the same assumptions as the above theorem except that (3.3) is replaced by τii
1 , γi + e|aii |Li
i ∈ N(1, n).
(3.9)
Then we have the same conclusion. Note that in Theorem 3.3 and Corollary 3.1, the connection matrix A is supposed to be irreducible and the off-diagonal terms aij 0, j = i. Motivated by [12], using the embedding technique, which was early used in [15] by Wu and Zhao, we will remove those restrictions. Indeed, we have Theorem 3.4. If either (2.7) or pi aii +
n pi |aij | + pj |aj i | j =i
2
< pi
mi , Li
i ∈ N(1, n),
(3.10)
for some pi > 0, i ∈ N(1, n), then system (1.1) has a unique equilibrium which is globally attractive provided the diagonal delays τii corresponding to negative aii are sufficiently small such that (3.3) holds. Proof. Let a + = max(a, 0) and a − = max(−a, 0). Then a + and a − are nonnegative and a = a + − a − , |a| = a + + a − . Define n × n matrices A∗ = (Aij ) and B = (Bij ) by aii 0 for j = i, for j = i, Aij = = B (3.11) ij + − aij + s for j = i, aij + s for j = i, where s > 0 will be specified later. Now system (1.1) can be rewritten as n Aij gj x˙i (t) = −bi xi (t) + j =1
−
n j =1
t
t kij (t − s)xj (s) ds t −τij
kij (t − s)xj (s) ds + Ji .
Bij gj t −τij
(3.12)
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Let yi = −xi , i ∈ N(1, n). Then (3.12) can be embedded into the following system with dimension 2n: t n x˙i (t) = −bi (xi (t)) + j =1 Aij gj ( t −τij kij (t − s)xj (s) ds) t + nj=1 Bij fj ( t −τij kij (t − s)yj (s) ds) + Ji , t (3.13) y˙i (t) = −qi (yi (t)) + nj=1 Bij gj ( t −τij kij (t − s)xj (s) ds) t + nj=1 Aij fj ( t −τij kij (t − s)yj (s) ds) − Ji , i ∈ N(1, n), where qi (x) and fi (x) are defined by qi (x) = −bi (−x) and fi (x) = −gi (−x), respectively, for i ∈ N(1, n) and x ∈ R. Clearly qi has the same property as bi does and fi has the same property as gi does. Define ui (t), i ∈ N(1, 2n), by ui (t) = xi (t),
un+i (t) = yi (t),
i ∈ N(1, n),
bi∗ (u), i ∈ N(1, 2n), by bi∗ (u) = bi (u),
∗ bn+i (u) = qi (u),
i ∈ N(1, n),
and hi (u), i ∈ N(1, 2n), by hi (u) = gi (u),
hn+i (u) = fi (u),
i ∈ N(1, n).
Then (3.13) can be rewritten as u˙ i (t) = −bi∗
ui (t) +
2n
t
Kij (t − s)uj (s) ds + Ii ,
wij hj
j =1
t −δij
i ∈ N(1, 2n),
(3.14)
where Ii = Ji and In+i = −Ji for i ∈ N(1, n), the 2n × 2n matrix W = (wij ) is given by ∗
A B W= B A∗ and δij , Kij for i, j ∈ N(1, 2n) are given by δij = δn+i,j = δi,n+j = δn+i,n+j = τij ,
i, j ∈ N(1, n),
and Kij = Kn+i,j = Ki,n+j = Kn+i,n+j = kij ,
i, j ∈ N(1, n).
It is obvious now that Aij > 0 and Bij > 0 for i, j ∈ N(1, n) and j = i and thus wij > 0 for i, j ∈ N(1, 2n) and W is irreducible. So if condition (3.10) holds for some pi > 0, i ∈ N(1, n), then we are able to choose sufficient small s > 0 such that aii pi +
n pi |aij | + pj |aj i | j =i
2
s mi (pi + pj ) < pi , 2 Li n
+
j =i
i ∈ N(1, n),
(3.15)
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145
∗ which implies that for pi∗ > 0, i ∈ N(1, 2n), with pi∗ = pi and pn+i = pi for i ∈ N(1, n), we have
pi∗ wii +
2n p∗ |w | + p∗ |w | ij ji i j j =i
2
< pi∗
mi , Li
i ∈ N(1, 2n).
(3.16)
If (2.7) is true for some pi > 0, i ∈ N(1, n), then similarly we can find sufficient small s > 0 such that n n mi aii pi + pj |aj i | + s pj < pi , i ∈ N(1, n), (3.17) Li j =i
j =i
∗ which implies that for pi∗ > 0, i ∈ N(1, 2n), with pi∗ = pi and pn+i = pi for i ∈ N(1, n), we have
pi∗ wii +
2n j =i
pi∗ |wj i | < pi∗
mi , Li
i ∈ N(1, 2n).
(3.18)
Applying Theorem 3.3 to system (3.14) immediately completes the proof. 2
4. Examples Example 4.1. Consider
tanh √2 x1 (t)
x1 (t) 1/2 −1/2 J1 x˙1 (t) 3 =− + . 2 + 1/2 1/2 J2 x˙2 (t) x2 (t) tanh √ x2 (t)
(4.1)
3
√ 1/2 −1/2 Here, A = 1/2 1/2 , bi (xi (t)) = xi (t), si (xi ) = tanh(2xi / 3) with mi = 1, Li = √ 2/ 3 for i = 1, 2. Clearly, choosing p1 = p2 = 1, all conditions of Corollary 2.1 are satisfied, therefore, system (4.1) has an equilibrium, which is globally asymptotically stable. We point out the Corollary 2 of [7] cannot be applied to system (4.1). Note that the stabilization role of inhibitory self-connection given in (2.6) and (2.7) are conditional for the delayed network (1.1), in the sense that the corresponding delays τii must be small. This is demonstrated in the following example. Example 4.2. Consider a neural network with two neurons x˙1 (t) = −x1 (t) − 0.5f (x1 (t − τ11 )) + 1.7f (x2 (t − τ12 )), x˙2 (t) = −x2 (t) + 1.25f (x1 (t − τ21 )) − 0.6f (x2 (t − τ22 )),
(4.2)
where f (s) = (1/2)(|s + 1| − |s − 1|). Applying Theorem 3.3 to this example, we obtain that (x1 , x2 ) = (0, 0) is the unique equilibrium of system (4.2) which is globally asymptotically stable whenever the diagonal delays are small enough with τ11 0.463, τ22 0.408. This conclusion is shown by Fig. 1. However, if (3.3) is not satisfied, then it is possible for system (4.2) to have a solution which
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L. Wang / J. Math. Anal. Appl. 292 (2004) 135–147
Fig. 1. Numeric solution for (4.2) with τ11 = 0.45 and τ22 = 0.4.
Fig. 2. Numeric solution for (4.2) with τ11 = 1.2 and τ22 = 1.0.
is not asymptotically stable. Indeed, if we let τ11 = 1.2, τ12 = 0.5, τ21 = 2.0, τ22 = 1.0, the numerical simulation (Fig. 2) shows that system (4.2) has a periodic solution, while Corollary 3.1 gives a smaller estimation of the diagonal delays as τ11 0.424, τ22 0.38.
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[3] M. Forti, On global asymptotic stability of a class of nonlinear systems arising in neural network theory, J. Differential Equations 113 (1994) 246–264. [4] K. Gopalsamy, X. He, Stability in asymmetric Hopfield nets with transmission delays, Phys. D 76 (1994) 344–358. [5] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. [6] J. Hopfield, Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. Nat. Acad. Sci. USA 81 (1984) 3088–3092. [7] H. Lu, On stability of nonlinear continuous-time neural networks with delays, Neural Networks 13 (2000) 1135–1143. [8] K. Matsuoka, Stability conditions for nonlinear continuous neural networks with asymmetric connection weights, Neural Networks 5 (1992) 495–500. [9] H.L. Smith, Monotone Dynamical Systems, an Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995. [10] H.L. Smith, R. Thieme, Strongly order preserving semiflows generated by functional differential equations, J. Differential Equations 93 (1991) 332–363. [11] D. Tank, J. Hopfield, Simple neural optimization networks: An A/D converter, signal decision circuit and a linear programming circuit, IEEE Trans. Circuits Systems 33 (1986) 533–541. [12] P. van den Driessche, J. Wu, X. Zou, Stabilization role of inhibitory self-connections in a delayed neural network, Phys. D 150 (2001) 84–90. [13] P. van den Driessche, X. Zou, Global attractivity in delayed Hopfield neural network models, SIAM J. Appl. Math. 58 (1998) 1878–1890. [14] L. Wang, X. Zou, Harmless delays in Cohen–Grossberg neural networks, Phys. D 170 (2002) 162–173. [15] J. Wu, X. Zhao, Permanence and convergence in multi-species competition systems with delay, Proc. Amer. Math. Soc. 126 (1998) 1709–1714. [16] H. Ye, A.N. Michel, K. Wang, Global stability and local stability of Hopfield neural networks with delays, Phys. Rev. E 50 (1994) 4206–4213.
J. Math. Anal. Appl. 292 (2004) 135–147 www.elsevier.com/locate/jmaa
Stabilizing Hopfield neural networks via inhibitory self-connections ✩ Lin Wang 1 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NF, Canada A1C 5S7 Received 4 June 2003 Submitted by H.L. Smith
Abstract The theory of monotone dynamical systems is employed to establish some sufficient conditions for the global attractivity of the Hopfield neural networks with finite distributed delays. The results show that self-inhibitory connections can be used to stabilize a delayed network provided the diagonal delays corresponding to the inhibitory self-connections are small enough. 2003 Elsevier Inc. All rights reserved. Keywords: Attractivity; Distributed delay; Monotone dynamical system; Neural network
1. Introduction In this paper, we study the Hopfield type neural networks with finite distributed delays described by the following system: t n x˙i (t) = −bi xi (t) + aij gj kij (t − s)xj (s) ds + Ji , j =1
i ∈ N(1, n),
t −τij
(1.1)
where n 2 is the number of neurons in the network, xi denotes the state variable associated to the ith neuron, and the n × n connection matrix A = (aij ) tells how the neurons are ✩
Research was partially supported by NSERC of Canada. E-mail addresses: [email protected], [email protected]. 1 Present address: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1. 0022-247X/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2003.11.048
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connected in the network, and the activation function gj shows how the j th neuron reacts to the input, τij 0 for i, j ∈ {1, 2, . . . , n} =: N(1, n) with τ = max{τij , i, j ∈ N(1, n)}, and the kernel functions satisfy τii kij (t) 0,
kii (t) dt = 1,
i, j ∈ N(1, n).
0
When τ = 0, bi (x) = bi x, i ∈ N(1, n), system (1.1) reduces to x˙i (t) = −bi xi (t) +
n
aij gj xj (t) + Ji ,
i ∈ N(1, n),
(1.2)
j =1
which is extensively studied (see, e.g., [3,6,8,11]). Note that system (1.1) also includes the following Hopfield neural networks with fixed discrete delays as a special case: n x˙i (t) = −bi xi (t) + aij gj xj (t − τij ) + Ji ,
i ∈ N(1, n),
(1.3)
j =1
which is also investigated by many authors, for example, [1,2,4,7,12–14,16]. We will employ the ideas and techniques in van den Driessche et al. [12] and van den Driessche and Zou [13] to show how to stabilize the Hopfield neural networks (1.1) with general activation functions and finite distributed delays via the self-inhibitory connections. Without the assumptions on monotonicity and differentiability of the activation function, as pointed out in [13], we assume (S1 ) For each i ∈ {1, 2, . . . , n}, gi : R → R is globally Lipschitz continuous with a Lipschitz constant Li ; (S2 ) For each i ∈ {1, 2, . . . , n}, |gi (x)| Mi , x ∈ R, for some constant Mi > 0. We also assume that for each i ∈ N(1, n), bi (u) − bi (v) mi > 0. bi is continuous with (1.4) u−v Unlike in [12] and [13], we do not require any differentiability for the activation functions and the time delays are not necessary to be fixed constants, indeed they are distributed over a finite interval. The motivation of considering nonsmooth activation functions is that such functions have been widely adopted in network implementations (e.g., piecewise linear functions). We will see, by applying our results to (1.3), that the results in [12] and [13] are reproduced with a better estimate for the smallness of effective delays. The initial conditions associated with (1.2) are set to be xi (0) = φi (0),
i ∈ N(1, n),
(1.5)
and the associated ones for (1.1) are xi (s) = φi (s),
s ∈ [−τ, 0], i ∈ N(1, n).
(1.6)
It is not difficult to establish the following existence result for an equilibrium of system (1.2) by applying the Schauder’s fixed point theorem.
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Theorem 1.1. If (1.4) and (S2 ) hold, then for every input J , there exists an equilibrium for system (1.2).
2. Global stability of (1.2) Let x ∗ be an equilibrium of (1.2). Substituting x(t) = u(t) + x ∗ into (1.2) leads to n ∗ ∗ ∗ ∗ , (2.1) u˙ i (t) = − bi ui (t) + xi − bi xi − aij gj uj (t) + xj − gj xj j =1
which can be denoted by n u˙ i (t) = −βi ui (t) + aij sj uj (t) ,
i ∈ N(1, n),
(2.2)
j =1
where βi (ui (t)) = bi (ui (t) + xi∗ ) − bi (xi∗ ), sj (uj (t)) = gj (uj (t) + xj∗ ) − gj (xj∗ ). If we let u = (u1 , . . . , un )T ∈ R n , A = [aij ]n×n , s(u) = (s1 (u1 ), . . . , sn (un ))T , B(u) = (β1 (u1 ), . . . , βn (un ))T ∈ R n , then system (2.2) can be rewritten as u(t) ˙ = −B u(t) + As u(t) . (2.3) It is obvious that x ∗ is globally asymptotically (exponentially) stable for (1.2) if and only if the trivial solution u = 0 of (2.2) or (2.3) is globally asymptotically (exponentially) stable. Moreover, the uniqueness of the equilibrium of (1.2) follows from its global asymptotic (exponential) stability. From Theorem 2 of [3], we have Theorem 2.1. Suppose (1.4), (S1 ) and (S2 ) are satisfied. Assume also that for each i ∈ N(1, n), usi (u) > 0 when u = 0. Then, for every input J , system (1.2) has a unique equilibrium ymptotically stable if the matrix W ∗ defined by
p1 m1 pn mn 1 ,..., W ∗ = diag − (P T + T T P ) L1 Ln 2
(2.4) x∗
which is globally as-
(2.5)
is positive definite for some P = diag(p1 , . . . , pn ) with pi > 0, i ∈ N(1, n). A direct corollary of this theorem is Corollary 2.1. Suppose (1.4), (S1 ) and (S2 ) and (2.4) hold. If for some pi > 0, mi 1 aii pi + |pi aij + pj aj i | < pi , i ∈ N(1, n), 2 Li j =i
holds, then the equilibrium x ∗ of (1.2) is globally asymptotically stable.
(2.6)
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If the condition (2.6) is strengthened a little bit, we can actually obtain the global exponential stability, as stated in the following theorem. Theorem 2.2. Suppose (1.4), (S1 ) and (S2 ) and (2.4) hold. If mi pi aii + |pj aj i | < pi , i ∈ N(1, n), Li
(2.7)
j =i
holds for some positive numbers p1 , p2 , . . . , pn , then, for every input J , system (1.2) has a unique equilibrium which is globally exponentially stable in the sense that n xi (t) − x ∗ C1 e−σ1 t ,
(2.8)
i
i=1
where C1 , σ1 will be specified later. Proof. Let δ > 0 and σ1 > 0 be given as
pj |aj i | , i ∈ N(1, n) , δ := min mi pi − max 0, Li pi aii +
(2.9)
j =i
σ1 := max σ > 0: δ − pi σ > 0, i ∈ N(1, n) .
(2.10)
Define V (t) = V (u(t)) by V (t) =
n
pi eσ1 t |ui |.
(2.11)
i=1
Then the upper-right derivative of V can be estimated as n n + σ1 t D V (t) = pi e aij sj uj (t) σ1 ui (t) − sign ui (t) βi ui (t) − (1.2)
i=1 n
j =1
eσ1 t pi σ1 ui (t) − pi mi ui (t) + pi aii si ui (t)
i=1
+ =
j =i n σ1 t
e
i=1
+ =
eσ1 t pi |aij | sj uj (t)
pi σ1 ui (t) − pi mi ui (t) + pi aii si ui (t)
eσ1 t pj |aj i | si ui (t)
j =i n σ1 t
e
pi σ1 ui (t) − pi mi ui (t) + pi aii + pj |aj i | si (ui ) j =i
i=1
−(δ − pi σ1 )eσ1 t
n i=1
ui (t) .
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139
This shows that V (t) is a Liapunov function and hence V (t) V (0) =
n
pi φi (0) − xi∗ =: C0 < ∞
(2.12)
i=1
and thus we have n n ui (t) = xi (t) − x ∗ C1 e−σ1 t i
i=1
(2.13)
i=1
with C1 := C0 /min{pi , i ∈ N(1, n)}.
2
Remark 2.1. Corollary 2.1 and Theorem 2.2 show that the self-inhibitory connections do play an important role in stabilizing network (1.2) when ignoring delay.
3. Global attractivity of HNNs with finite distributed delays Note that the delays do not change the values of equilibria, but the stability may be lost when the delays increase. It is natural to expect that the stable equilibrium of the system without delays remains stable for the delayed system when the delays are sufficiently small. In order to give an estimation for the smallness, we will use the powerful theory of monotone dynamic systems. To be more precise, we shall use the related theory about the nonstandard ordering–exponential ordering. To this end and for reader’s convenience, we first introduce the theorem that we will employ below. In the following, as in [9], the partial order in Rn will be the usual componentwise ordering. The partial ordering φ ψ in C := C([−τ, 0], Rn ) will mean φ(θ ) ψ(θ ) for each θ ∈ [−τ, 0]. The inequality x < y (x y) between two vectors in Rn will mean x y and xi < yi for some (all) i ∈ N(1, n). The inequality φ < ψ in C will mean that φ ψ and φ = ψ, and φ ψ will mean that φ(θ ) ψ(θ ) for all θ ∈ [−τ, 0]. Let D be an n × n essentially nonnegative matrix, that is, D + λI is entry-wise nonnegative for all sufficiently large λ. Define KD = ψ ∈ C: ψ 0 and ψ(t) eD(t −s)ψ(s), −τ s t 0 . It can be seen that KD is a normal cone in C and thus it induces a partial ordering in C, denoted by D , in the usual way, that is, φ D ψ if and only if ψ − φ ∈ KD and φ
(3.1)
where f : C → Rn is globally Lipschitz continuous. By the fundamental theory of FDES [5], the system (3.1) generates a semi-flow Φ on C by Φ(t, φ) = Φt (φ) = xt (φ),
t 0, φ ∈ C,
for those t for which xt (φ) is defined. The following theorem is from Smith and Thieme [10].
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Theorem 3.1. Assume the following conditions are satisfied: (ID ) If φ, ψ ∈ C satisfy φ D ψ and K is a proper subset of N(1, n) such that φk ψk , k ∈ K, and φk (0) = ψk (0) for k ∈ N(1, n) − K, then for some p ∈ N(1, n) − K, fp (ψ) > fp (φ). (SMD ) If φ, ψ ∈ C satisfy φ D ψ and φ ψ, then f (ψ) − f (φ) D ψ(0) − φ(0) . Then Φ is strongly order preserving (SOP) on C. Using the above theorem, we will show that the semiflow Φ generated by the solution of (1.1) is SOP under the exponential ordering, if the diagonal delays corresponding to negative self-connections are sufficiently small. Theorem 3.2. Assume that bi (u) is Lipschitz continuous with Lip(bi ) = γi for i ∈ N(1, n), aij 0 for i = j , A is irreducible and gi satisfies gi (u) − gi (v) Li , i ∈ N(1, n). (3.2) u−v If the diagonal delays τii corresponding to negative aii are sufficiently small satisfying 0
τii
1 , ri∗
(3.3)
where ri∗ will be given below, then the semi-flow Φ generated by the solution of (1.1) is SOP under D . Proof. Take D = diag(d1 , . . . , dn ) with di = −γi − ri ,
i ∈ N(1, n),
where ri > 0, i ∈ N(1, n), are constants to be specified later. Then D is essentially nonnegative. For system (1.1), we have 0 n fi (φ) = −bi φi (0) + aij gj kij (−s)φj (s) ds + Ji , i ∈ N(1, n). (3.4) j =1
−τij
Let φ, ψ ∈ C satisfy φ D ψ and φ ψ. Then 0 n fi (ψ) − fi (φ) = −bi ψi (0) + aij gj kij (−s)ψj (s) ds j =1
+ bi φi (0) −
n j =1
−τij
0 aij gj
kij (−s)φj (s) ds −τij
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141
= − bi ψi (0) − bi φi (0) 0 n + aij gj kij (−s)ψj (s) ds j =1
−τij
0 − gj
kij (−s)φj (s) ds −τij
−γi ψi (0) − φi (0) 0 + aii gi
kii (−s)ψi (s) ds − gi −τii
0 + aij gj kij (−s)ψj (s) ds j =i
0 kii (−s)φi (s) ds −τii
−τij
0 − gj
kij (−s)φj (s) ds
.
−τij
Clearly if the matrix A is a nonnegative matrix, then (SMD ) holds. In the following we therefore may assume that aii < 0 for some i ∈ N(1, n). Then for such i, fi (ψ) − fi (φ) − di φi (0) − ψi (0) −γi ψi (0) − φi (0) 0 0 kii (−s)ψi (s) ds − gi
+ aii gi −τii
+ (γi + ri ) ψi (0) − φi (0) = ri ψi (0) − φi (0) 0 + aii gi
−τii
0
kii (−s)ψi (s) ds − gi −τii
kii (−s)φi (s) ds
kii (−s)φi (s) ds −τii
0
ri ψi (0) − φi (0) + aii Li
kii (−s) ψi (s) − φi (s) ds.
−τii
On the other hand φ D ψ implies that ψ − φ ∈ KD and hence ψ(0) − φ(0) e−Ds ψ(s) − φ(s) for s ∈ [−τ, 0], that is
ψi (s) − φi (s) e−(ri +γi )s ψi (0) − φi (0) for i ∈ N(1, n).
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This indicates that 0
0
kii (−s) ψi (s) − φi (s) ds −τii
kii (−s)e−(ri +γi )s ψi (0) − φi (0) ds
−τii
ψi (0) − φi (0) e(ri +γi )τii .
Therefore we have fi (ψ) − fi (φ) − di ψi (0) − φi (0) (ri + aii Li eτii (ri +γi ) ) ψi (0) − φi (0) > 0, provided that ri > |aii |Li eτii (ri +γi ) ,
i ∈ N(1, n).
(3.5)
Inequality (3.5) is satisfied if and only if τii <
ln |aiiri|Li γi + ri
,
i ∈ N(1, n).
(3.6)
Now let τi (s) :=
ln |aiis|Li γi + s
,
i ∈ N(1, n).
(3.7)
A simple calculation shows that τ¨i (s) < 0 for s > |aii |Li and τ˙i (|aii |Li ) > 0 and τi (|aii |Li ) = 0. Therefore τi (s) attains its maximal value 1/ri∗ at ri∗ , where ri∗ is the unique positive root of equation s γi h(s) := 1 + − ln =0 s |aii |Li for s > |aii |Li . Let ri = ri∗ for i ∈ N(1, n). This shows that (SMD ) holds if the diagonal delays τii corresponding to the negative aii satisfy τii <
1 , ri∗
i ∈ N(1, n).
(3.8)
(ID ) can be easily verified under the assumption that the connection matrix T is irreducible. Thus the proof is complete. 2 For system (1.1), the phase space is X = C = C([−τ, 0], Rn ). Therefore, every nonequilibrium point in X can be approximated from below and from above under the exponential ordering. (Indeed, for any x ∈ X, the sequences {xn }, {yn } defined by xn (s) := x(s) − eDs /n, yn (s) := x(s) + eDs /n, s ∈ [−τ, 0] serve this purpose.) Note that from the boundedness of the activation functions, it is easy to show that every bounded set B ⊂ X has a bounded orbit and thus the relatively weak compactness requirement (C) in [9] is met. Notice also that Corollary 2.1 and Theorem 2.2 imply the uniqueness of the equilibrium x ∗ of system (1.2), which shows that under the same assumptions, the equilibrium x ∗ of system (1.1) is unique. Thus Theorem 3.2, together with Theorem 3.1 of Smith [9], immediately gives
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Theorem 3.3. Assume that (3.2) holds and bi satisfies mi
bi (u) − bi (v) γi , u−v
i ∈ N(1, n),
A is irreducible and aij 0 for i = j . If either (2.6) or (2.7) holds, then the system (1.1) has a unique equilibrium which is globally attractive provided the diagonal delays τii corresponding to negative aii are sufficiently small satisfying (3.3). Letting s = e|aii |Li in (3.7), we have Corollary 3.1. Under the same assumptions as the above theorem except that (3.3) is replaced by τii
1 , γi + e|aii |Li
i ∈ N(1, n).
(3.9)
Then we have the same conclusion. Note that in Theorem 3.3 and Corollary 3.1, the connection matrix A is supposed to be irreducible and the off-diagonal terms aij 0, j = i. Motivated by [12], using the embedding technique, which was early used in [15] by Wu and Zhao, we will remove those restrictions. Indeed, we have Theorem 3.4. If either (2.7) or pi aii +
n pi |aij | + pj |aj i | j =i
2
< pi
mi , Li
i ∈ N(1, n),
(3.10)
for some pi > 0, i ∈ N(1, n), then system (1.1) has a unique equilibrium which is globally attractive provided the diagonal delays τii corresponding to negative aii are sufficiently small such that (3.3) holds. Proof. Let a + = max(a, 0) and a − = max(−a, 0). Then a + and a − are nonnegative and a = a + − a − , |a| = a + + a − . Define n × n matrices A∗ = (Aij ) and B = (Bij ) by aii 0 for j = i, for j = i, Aij = = B (3.11) ij + − aij + s for j = i, aij + s for j = i, where s > 0 will be specified later. Now system (1.1) can be rewritten as n Aij gj x˙i (t) = −bi xi (t) + j =1
−
n j =1
t
t kij (t − s)xj (s) ds t −τij
kij (t − s)xj (s) ds + Ji .
Bij gj t −τij
(3.12)
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Let yi = −xi , i ∈ N(1, n). Then (3.12) can be embedded into the following system with dimension 2n: t n x˙i (t) = −bi (xi (t)) + j =1 Aij gj ( t −τij kij (t − s)xj (s) ds) t + nj=1 Bij fj ( t −τij kij (t − s)yj (s) ds) + Ji , t (3.13) y˙i (t) = −qi (yi (t)) + nj=1 Bij gj ( t −τij kij (t − s)xj (s) ds) t + nj=1 Aij fj ( t −τij kij (t − s)yj (s) ds) − Ji , i ∈ N(1, n), where qi (x) and fi (x) are defined by qi (x) = −bi (−x) and fi (x) = −gi (−x), respectively, for i ∈ N(1, n) and x ∈ R. Clearly qi has the same property as bi does and fi has the same property as gi does. Define ui (t), i ∈ N(1, 2n), by ui (t) = xi (t),
un+i (t) = yi (t),
i ∈ N(1, n),
bi∗ (u), i ∈ N(1, 2n), by bi∗ (u) = bi (u),
∗ bn+i (u) = qi (u),
i ∈ N(1, n),
and hi (u), i ∈ N(1, 2n), by hi (u) = gi (u),
hn+i (u) = fi (u),
i ∈ N(1, n).
Then (3.13) can be rewritten as u˙ i (t) = −bi∗
ui (t) +
2n
t
Kij (t − s)uj (s) ds + Ii ,
wij hj
j =1
t −δij
i ∈ N(1, 2n),
(3.14)
where Ii = Ji and In+i = −Ji for i ∈ N(1, n), the 2n × 2n matrix W = (wij ) is given by ∗
A B W= B A∗ and δij , Kij for i, j ∈ N(1, 2n) are given by δij = δn+i,j = δi,n+j = δn+i,n+j = τij ,
i, j ∈ N(1, n),
and Kij = Kn+i,j = Ki,n+j = Kn+i,n+j = kij ,
i, j ∈ N(1, n).
It is obvious now that Aij > 0 and Bij > 0 for i, j ∈ N(1, n) and j = i and thus wij > 0 for i, j ∈ N(1, 2n) and W is irreducible. So if condition (3.10) holds for some pi > 0, i ∈ N(1, n), then we are able to choose sufficient small s > 0 such that aii pi +
n pi |aij | + pj |aj i | j =i
2
s mi (pi + pj ) < pi , 2 Li n
+
j =i
i ∈ N(1, n),
(3.15)
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145
∗ which implies that for pi∗ > 0, i ∈ N(1, 2n), with pi∗ = pi and pn+i = pi for i ∈ N(1, n), we have
pi∗ wii +
2n p∗ |w | + p∗ |w | ij ji i j j =i
2
< pi∗
mi , Li
i ∈ N(1, 2n).
(3.16)
If (2.7) is true for some pi > 0, i ∈ N(1, n), then similarly we can find sufficient small s > 0 such that n n mi aii pi + pj |aj i | + s pj < pi , i ∈ N(1, n), (3.17) Li j =i
j =i
∗ which implies that for pi∗ > 0, i ∈ N(1, 2n), with pi∗ = pi and pn+i = pi for i ∈ N(1, n), we have
pi∗ wii +
2n j =i
pi∗ |wj i | < pi∗
mi , Li
i ∈ N(1, 2n).
(3.18)
Applying Theorem 3.3 to system (3.14) immediately completes the proof. 2
4. Examples Example 4.1. Consider
tanh √2 x1 (t)
x1 (t) 1/2 −1/2 J1 x˙1 (t) 3 =− + . 2 + 1/2 1/2 J2 x˙2 (t) x2 (t) tanh √ x2 (t)
(4.1)
3
√ 1/2 −1/2 Here, A = 1/2 1/2 , bi (xi (t)) = xi (t), si (xi ) = tanh(2xi / 3) with mi = 1, Li = √ 2/ 3 for i = 1, 2. Clearly, choosing p1 = p2 = 1, all conditions of Corollary 2.1 are satisfied, therefore, system (4.1) has an equilibrium, which is globally asymptotically stable. We point out the Corollary 2 of [7] cannot be applied to system (4.1). Note that the stabilization role of inhibitory self-connection given in (2.6) and (2.7) are conditional for the delayed network (1.1), in the sense that the corresponding delays τii must be small. This is demonstrated in the following example. Example 4.2. Consider a neural network with two neurons x˙1 (t) = −x1 (t) − 0.5f (x1 (t − τ11 )) + 1.7f (x2 (t − τ12 )), x˙2 (t) = −x2 (t) + 1.25f (x1 (t − τ21 )) − 0.6f (x2 (t − τ22 )),
(4.2)
where f (s) = (1/2)(|s + 1| − |s − 1|). Applying Theorem 3.3 to this example, we obtain that (x1 , x2 ) = (0, 0) is the unique equilibrium of system (4.2) which is globally asymptotically stable whenever the diagonal delays are small enough with τ11 0.463, τ22 0.408. This conclusion is shown by Fig. 1. However, if (3.3) is not satisfied, then it is possible for system (4.2) to have a solution which
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L. Wang / J. Math. Anal. Appl. 292 (2004) 135–147
Fig. 1. Numeric solution for (4.2) with τ11 = 0.45 and τ22 = 0.4.
Fig. 2. Numeric solution for (4.2) with τ11 = 1.2 and τ22 = 1.0.
is not asymptotically stable. Indeed, if we let τ11 = 1.2, τ12 = 0.5, τ21 = 2.0, τ22 = 1.0, the numerical simulation (Fig. 2) shows that system (4.2) has a periodic solution, while Corollary 3.1 gives a smaller estimation of the diagonal delays as τ11 0.424, τ22 0.38.
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