By using an integral inequality, we establish some sufficient conditions for the existence and

Since it was proposed by Kosko (see [

But in a real nervous system, it is usually unavoidably affected by external perturbations which are in many cases of great uncertainty and hence may be treated as random. As pointed out by Haykin [

However, the above results are mainly on the stability of considered stochastic neural networks. And it is well known that studies on neural dynamical systems not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillatory behavior. On the other hand, the neural networks are often subject to impulsive effects that in turn affect dynamical behaviors of the systems. Moreover, a leakage delay, which is the time delay in the leakage term of the systems and a considerable factor affecting dynamics for the worse in the systems, is being put to use in the problem of stability for neural networks. However, so far, very little attention has been paid to neural networks with time delay in the leakage (or “forgetting”) term. Such time delays in the leakage term are difficult to handle but have great impact on the dynamical behavior of neural networks. Therefore, it is meaningful to consider neural networks with time delays in the leakage term [

But to the best of our knowledge, there are few papers published on studying the existence of periodic solutions of impulsive stochastic neural networks with time delay in the leakage term. Motivated by the previous discussions, in this paper, we consider the following impulsive stochastic BAM neural networks:

Our main purpose in this paper is using an integral inequality, which is from a lemma in [

Let

For convenience, for an

Throughout this paper, we assume that the following conditions hold:

From

This paper is organized as follows. In Section

In this section, we introduce some definitions and state some preliminary results.

A stochastic process

If

The solution

Under conditions

For any

The periodic solution

Let

Assume that all conditions of Lemma

By Lemmas

Let

Under our assumptions, we consider the following system:

Similar to Lemma 2.1 in paper [

Let

if

if

In this section, we will state and prove the sufficient conditions for the existence and

Let

there exists an integer

Then (

We can rewrite (

Suppose that

In this section, we present an example to illustrate the feasibility of our results obtained in the previous section.

Let

By calculating, we have

By the numerical simulation in Figures

Transient states of stochastic BAM neural networks.

State response of

State response of

This work is supported by the National Natural Science Foundation of People’s Republic of China under Grant 10971183.