Exponential Stability and Numerical Methods of Stochastic Recurrent Neural Networks with Delays

Shifang Kuang,Feiqi Deng,Wenhua Gao,Yunjian Peng

doi:10.1155/2013/761237

Abstract

Exponential stability in mean square of stochastic delay recurrent neural networks is investigated in detail. By using Itô’s formula and inequality techniques, the sufficient conditions to guarantee the exponential stability in mean square of an equilibrium are given. Under the conditions which guarantee the stability of the analytical solution, the Euler-Maruyama scheme and the split-step backward Euler scheme are proved to be mean-square stable. At last, an example is given to demonstrate our results.

Highlights

It is well known that neural networks have wide range of applications in many fields, such as signal processing, pattern recognition, associative memory, and optimization problems
Under the conditions which guarantee the stability of the analytical solution, the Euler-Maruyama scheme and the split-step backward Euler scheme are proved to be mean-square stable
There has been a few literatures about the exponential stability of numerical methods for stochastic delay neural networks

Summary

Introduction

It is well known that neural networks have wide range of applications in many fields, such as signal processing, pattern recognition, associative memory, and optimization problems. It is very useful to establish numerical methods for studying the properties of stochastic delay neural networks. There are many papers concerned with the stability of numerical solutions for stochastic delay differential equations ([19–29] and references therein). There has been a few literatures about the exponential stability of numerical methods for stochastic delay neural networks. To the best of the authors knowledge, only [30–32] studied the exponential stability of numerical methods for stochastic delay Hopfield neural networks. The stability of numerical methods for stochastic delay recurrent neural networks remains open, which motivates this paper. The main aim of the paper is to investigate the mean-square stability (MS stability) of the Euler-Maruyama (EM) method and the split-step backward Euler (SSBE) method for stochastic delay recurrent neural networks.

Model Description and Analysis of Analytical Solution

Stability of EM Numerical Solution

Stability of SSBE Numerical Solution

Example

Conclusions