By using the concept of differential equations with piecewise constant argument of generalized type, a model of stochastic cellular neural networks with piecewise constant argument is developed. Sufficient conditions are obtained for the existence and uniqueness of the equilibrium point for the addressed neural networks.

Since Chua and Yang introduced cellular neural networks (CNNs) [

In real nervous systems, there are many stochastic perturbations that affect the stability of neural networks. The results in [

The theory of differential equations with piecewise constant argument was initiated in Cooke and Wiener [

To the best of our knowledge, the equations with piecewise constant arguments were not considered as models of neural networks, except possibly in [

Motivated by the discussion above, our paper attempts to fill the gap by considering stochastic cellular neural networks with piecewise constant arguments. In this paper, criteria on the

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

In this paper, let

We study stochastic cellular neural networks with piecewise constant arguments described by the differential equations:

Throughout this paper, the following standard hypotheses are needed.

Functions

for all

There exists a positive number

Assume that

There exist nonnegative constants

for all

In the following, for further study, we give the following definitions and lemmas.

The equilibrium point

In such a case,

When

Let

In particular, one may take

if

if

if

If

A particular form of (

Assuming that there exists constant

In this section, we will study the existence and uniqueness of the equilibrium point of neural networks (

Assume that (H1)–(H4) are fulfilled. Then, for every

For each

The Gronwall inequality implies

Now we claim that, for all

Firstly, we compute

Next, assume (

That is, (

One can see from (

It remains to show that

From the proof of Theorem

Assume that (H1)–(H3) are fulfilled. Then, for every

When the system (

In this section, we will establish some sufficient conditions ensuring the

Let

Let

It is clear that the stability of the zero solution of (

For simplicity of notation, we denote

In order to obtain our results, the following assumption and Lemmas are needed:

Let

Fix

On the other hand, Lemma

By Gronwall inequality

Furthermore, for

Assume that there is a function

For convenience, we adopt the following notation:

Assume that (H1)–(H5) hold and, furthermore, that the following inequality is satisfied:

We define a Lyapunov function

For

Suppose that (H1)–(H3) and (H5) hold true. Assume, furthermore, that the following inequality is satisfied:

Let

Theorem

In the following, we will give an example to illustrate our results.

Consider the following model:

Let

It is easy to check that

When

It is obvious that

When

This is the first time that stochastic cellular neural networks with piecewise constant argument of generalized type are considered. Sufficient conditions on existence, uniqueness, and the

The author declares that there is no conflict of interests regarding the publication of this paper.

The author would like to thank Dr. Yanchun Zhang, Dr. Yeqing Ren, the editor, and the anonymous referees for their detailed comments and valuable suggestions which considerably improved the presentation of this paper. This work was supported by the Project of Humanities and Social Sciences of Ministry of Education of China under Grant no. 13YJC630232.