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The stability issue is investigated for a class of stochastic neural networks with time delays in the leakage terms. Different from the previous literature, we are concerned with the almost sure stability. By using the LaSalle invariant principle of stochastic delay differential equations, Itô’s formula, and stochastic analysis theory, some novel sufficient conditions are derived to guarantee the almost sure stability of the equilibrium point. In particular, the weak infinitesimal operator of Lyapunov functions in this paper is not required to be negative, which is necessary in the study of the traditional moment stability. Finally, two numerical examples and their simulations are provided to show the effectiveness of the theoretical results and demonstrate that time delays in the leakage terms do contribute to the stability of stochastic neural networks.

During the past decades, a great deal of attention has been paid to investigate the dynamics behaviors such as stability, periodic oscillatory behavior, almost periodic oscillatory behavior, and chaos and bifurcation of neural networks. Particularly, the stability of neural networks is one of the best topics since many important applications depend heavily on the stability of the equilibrium point. Therefore, there have appeared a large number of works on the stability of the equilibrium point of various neural networks such as Hopfield neural networks, cellular neural networks, recurrent neural networks, Cohen-Grossberg neural networks, and bidirectional associative memory (BAM) neural networks [

As is well known, time delay is one of the most significant phenomena that occur in many different fields such as biology, chemistry, economy, and communication networks. Moreover, it is inevitably encountered in both neural processing and signal transmission due to the limited bandwidth of neurons and amplifiers. However, the existence of time delays may cause oscillation, divergence, chaos, instability, or other poor performance in neural networks, which are usually harmful to the applications of neural networks. Therefore, the stability analysis for neural networks with time delays has attracted many researchers’ much attention in the literature. The existing works on the stability of neural networks with time delays can be simply classified into four categories: constant delays, time-varying delays, distributed delays, and mixed time delays.

It should be mentioned that a new class of delays, called leakage delays (also named time delays in the “forgetting” or leakage terms), was initially introduced by Gopalsamy [

On the other hand, noise disturbance is a major source of instability and poor performances in neural networks. Usually, many real nervous systems are affected by external perturbations which in many cases are of great uncertainty and hence may be treated as random. Just as Haykin pointed out, the synaptic transmission can be regarded as a noisy process introduced by random fluctuations from the release of neurotransmitters and other probabilistic causes. Therefore, we should consider the effect of noise disturbances when studying the stability of neural networks. Generally speaking, neural networks with noise disturbances are called stochastic neural networks. Recently, there have appeared a large number of results on the stability of stochastic neural networks (see, e.g., [

Motivated by the above discussion, in this paper we study the stability problem for a class of stochastic neural networks with time delays in the leakage terms. Different from the previous literature, we aim to remove the restriction of

The remainder of this paper is organized as follows. In Section

The notations used in this paper are quite standard.

In this paper, we consider a class of neural networks with mixed time delays, which is described by the following integrodifferential equations:

Throughout this paper, the following assumptions are assumed to hold.

Let

Now we give the concept of almost sure stability for system (

The equilibrium point of (

The following lemma is needed to prove our main results.

For any positive definite matrix

In this section, the almost sure stability of the equilibrium point for system (

Under Assumptions H1–H3, the equilibrium point of (

Fixing

If we ignore the effect of time delays in the leakage terms, then system (

Correspondingly, we revise Assumptions H2 and H3 as follows.

Under Assumptions H1, H′2, and H′3, we have the following result.

Under Assumptions H1, H′2, and H′3, the equilibrium point of (

Consider the following Lyapunov-Krasovskii functional:

Theorems

It is worth pointing out that

In this section, two numerical examples are given to illustrate the effectiveness of the obtained results.

Consider a two-dimensional stochastic neural network with time delays in the leakage terms:

Other parameters of network (

By using the Euler-Maruyama numerical scheme, simulation results are as follows:

The state response of network (

Consider a two-dimensional stochastic neural network without time delays in the leakage terms:

All other parameters of network (

By using the Matlab LMI toolbox, we can obtain the following feasible solution for LMIs (

By using the Euler-Maruyama numerical scheme, simulation results are as follows:

The state response of network (

Examples

In this paper, we have investigated the almost sure stability analysis problem for a class of stochastic neural networks with time delays in the leakage terms. Some novel delay-dependent conditions are obtained to ensure that the suggested system is almost surely stable, which is quite different from the moment stability. Our method is mainly based on the LaSalle invariant principle of stochastic delay differential equations, Itô’s formula, and stochastic analysis theory. Moreover, the stability criteria given in this paper are expressed in terms of LMIs, which can be solved easily by recently developed algorithms. In addition, we use two examples to show that time delays in the leakage terms do contribute to the stability of stochastic neural networks. Finally, we point out that it is possible to generalize our results to some more complex stochastic neural networks with time delays in the leakage terms (e.g., consider the effect of fractional-order factor [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was jointly supported by the Alexander von Humboldt Foundation of Germany (Fellowship CHN/1163390), the National Natural Science Foundation of China (61374080), Qing Lan Project of Jiangsu, the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Key University Science Research Project of Anhui Province (KJ2016A705), the Key Projects of Anhui Province University Outstanding Youth Talent Support Program (gxyqZD2016317), the Natural Science Foundation of Jiangsu Province (BK20140089), and the Play of Nature Science Fundamental Research in Nanjing Xiao Zhuang University (2012NXY12).