Delay-dependent asymptotic stability for neural networks with distributed delays

https://doi.org/10.1016/j.nonrwa.2005.11.001Get rights and content

Abstract

In this paper, dynamical behavior of a class of neural networks with distributed delays is studied by employing suitable Lyapunov functionals, delay-dependent criteria to ensure local and global asymptotic stability of the equilibrium of the neural networks. Our results are applied to classical Hopfield neural networks with distributed delays and some novel asymptotic stability criteria are also derived. The obtained conditions are shown to be less conservative and restrictive than those reported in the known literature.

Introduction

Networks with connection delays arise naturally in many areas of science, including biology, population dynamics, neuroscience, economics, and so on [5], [2], [6]. In neural networks, for instance, delays result from, i.e., finite axonal transmission speeds [1], [3], [4], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [27], [28], [29], [30]. Such networks lack an intrinsic notion of simultaneity since the present state of the system is inaccessible to the constituent neurons. These networks are nevertheless capable of operating in stability, even when complex units and significantly large delays are involved. In fact, networks with delays can actually stabilize more easily. This stability property is especially relevant for neural networks, circumventing the difficulties in establishing a concept of collective or simultaneous information processing in the presence of delayed information transmission [27], [1].

Recent years have witnessed a growing interest in the dynamics of neural networks with time delays. Particularly, a large number of studies have been devoted to global asymptotical stability or global exponential stability in a variety of neural networks (see [1], [3], [4], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [24], [25], [27], [28], [29], [30] and the references therein), including the Hopfield neural networks introduced by Hopfield, cellular neural networks introduced by Chua and BAM introduced by Kosko. Usually, such systems have been investigated under the assumption of asymmetric connection weight and nonmonotonic activation function. However, monotonicity and differentiability of activation functions come from the experimental results of brain sciences, moreover, they have strong biological background. On the other hand, realistic modeling of many large neural networks with nonlocal interaction inevitably requires connection delays to be taken into account, since they naturally arise as a consequence of finite information transmission and processing speeds among the neurons.

Studies on the importance of time delays in neural networks have so far been confined only to discrete, or constant, delays [4], [13], [21], [12], [7], [8], [9], [16], [17], [18]. In other words, it has been assumed that information reaches from one neuron to another after a fixed time T which is unchanging as the system evolves, and, moreover, the neurons act only on the instantaneous value of the received information and forget any previous values such discrete-delay models often fail to adequately describe biological neural systems by neglecting the possibilities that the neural networks may incorporate “memory” effects by using the past history of the received information [27], [1]. Because of these shortcomings, models based on distributed delays have been proposed as early as the time of Volterra [2], [26], and used in such areas as biology [22], ecology [23], neural networks [1], [3], [4], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [27]. It has especially been pointed out in the biological and brain sciences that distributed delays lead to more realistic models [27], [1].

Generally, the stability criteria for time-delay systems can be classified into two categories, namely delay-independent criteria and delay-dependent criteria, depending on whether they contain the delay argument as a parameter. There have been a number of significant developments in searching the stability criteria for systems with constant delays [4], [13], [21], [12], [7], [8], [9], [16], [17], [18]. Only a few of them are for neural networks with distributed delays; see for instance, [27], [1], [3], [4], [19], [14], [11]. To the best of the authors’ knowledge, the criteria are delay-independent criteria in the case of the distributed delays yet.

In this paper, we will present some new local and global asymptotic stability of the equilibrium of neural networks with distributed delays. Our results essentially show that if the equilibrium of the network remains globally asymptotically stable when the distributed delays are small enough. In order to prove our results, we construct the suitable Lyapunov functionals.

The organization of this paper is as follows. In the following section, we will give the network system to be considered and some needed preliminaries. The proofs of the main results will give in Section 3, and the detailed constructive procedure of the Lyapunov functional V is given in Appendix. Finally, some conclusions are presented in Section 4.

Section snippets

Statement of networks and preliminaries

In this paper, we will consider the following neural networks with distributed delays:u˙i(t)=-di(ui(t))+j=1nwijgj(uj(t))+j=1nwijτ-tKij(t-s)gj(uj(s))ds+Ii,i=1,2,,n,for t0, where wij, wijτ and Ii are real constants. The delay kernels Kij(.),i,j=1,2,,n, satisfy the following conditions (H4). The functions gj are continuously differentiable on R=(-,+) and such that gi(0)=0,j=1,2,,n.

For system (1), we assume that the following conditions are satisfied.

(H1) There exist positive constants D̲i

Stability analysis

In this section, we will consider the stability of the equilibrium (u1*,u2*,,un*) of system (1). We assume that the following conditions (H5).

(H5) There exist positive constants λi,i=1,2,,n, such that the following matrix: R=σ1r12r13r1nr21σ2r23r2nrn1rn2rn3σnis negative-definite, i.e., (-1)iσ1r12r13r1nr21σ2r23r2nrn1rn2rn3σn>0,i=1,2,,n,whereσi=λi-D̲iqi+wii+wiiτ+12j=1nqjD¯jpjλi|wijτ|0+sKij(s)ds+qiD¯ipiλj|wjiτ|0+sKji(s)ds+j=1nqj2λi|wijτ|k=1n(|wjk|+|wjkτ|)0+sKij(s)ds+k=1nλ|w

Conclusions

The importance of asymptotic stability has been noted by many authors in relation to various physical and biological phenomena. Our findings suggest that in certain cases the delay distribution has a stabilizing effect on the interconnected neural systems. This implies that stability is a rather common and robust dynamical behavior for interacting neural systems. Form a practical point, the properties of distributed delays are expected to be helpful in modeling observed phenomena. For instance,

Acknowledgements

The work was supported by Program for New Century Excellent Talents in University, the National Nature Science and Foundation of China under Grant 60573047, the Natural Science Foundation of Chongqing under Grant 8509 and 8986-3.

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