Elsevier

Neurocomputing

Volume 168, 30 November 2015, Pages 1157-1163
Neurocomputing

Stability of bidirectional associative memory neural networks with Markov switching via ergodic method and the law of large numbers

https://doi.org/10.1016/j.neucom.2015.05.007Get rights and content

Abstract

This paper devotes to stability analysis of continuous time and discrete time bidirectional associative memory (BAM) neural networks whose parameters are randomly varying in a finite state Markov chain sense. Based on the ergodic theory of continuous time Markov chain, the matrix measure approach and Lyapunov theory, almost sure stability and exponential stability in the mean square for continuous time BAM neural networks are derived. We also present some new stability results for discrete time BAM neural networks with the help of the law of large numbers. Meanwhile, some examples with numerical simulations are given to show that the Markov chain plays an important role in stability of neural networks.

Introduction

Bidirectional associative memory (BAM) neural networks have been introduced by Kosko [1], [2], [3]. BAM neural network is a special class of recurrent neural network which can be formed by neurons arranged in the X-layer and Y-layer. This kind of neural networks has been actively studied due to its wide applications in image processing, pattern recognition, automatic control and artificial intelligence. It is worth pointing out that some applications of BAM neural networks require stability of the equilibrium point. Recently, there have been some results about stability analysis of BAM neural networks [4], [5], [6], [7], [8].

It is well known that two kinds of noise disturbances are unavoidable to be considered in the neural networks. One is white noise disturbance, the other is color noise. This is mostly because the synaptic transmission is a noisy process. If white noise disturbance is taken into account to discuss the neural networks, the corresponding systems with white noise are called as stochastic neural networks. Based on the theory of stochastic systems, a lot of stability results of neural networks with white noise have been obtained [9], [10], [11], [12], [13]. However, telegraph noise, which is a simple color noise, can be illustrated as a switching between two or more regimes of environment. If the switching is memoryless and the waiting time for the next switch has an exponential distribution, then we can model the regime switching by a finite-state Markov chain. In [14], [15], [16], Fang studied stochastic stability of jump linear systems with a finite state Markov chain form process. In [17], almost sure exponential synchronization of Hopfield neural networks with stochastic switching weight structures was investigated. By using a parameter-dependent Lyapunov functional and a simple matrix decoupling method, the authors of [18] studied the problem of finite-time synchronization control for uncertain Markov jump neural networks in the presence of constraints on the control input amplitude.

Motivated by the above discussion, we study stability of continuous time and discrete time BAM neural networks with Markov chain which takes value in finite state space. By using ergodic theory of continuous time Markov chain [19] and the matrix measure approach [20], [21], [22], together with Lyapunov theory, some sufficient conditions are derived to ensure that the switched BAM neural networks are almost surely exponentially stable or exponentially stable in the mean square. It is surprised that if subsystems are not stable, but the other subsystems are stable, then the over system will achieve stability in the end. This shows the important role of Markov chain. Meanwhile, by the law of large number [23], we also investigate the stability of discrete time BAM neural networks whose parameters are varying in Markov sense. Finally, some examples with numerical simulations are given to illustrate the applicability of the results. The rest of this paper is organized as follows. In Section 2, continuous time BAM networks model with Markov chain is presented and stability results are derived for switching networks by ergodic theory of continuous time Markov chain. Section 3 devotes to the investigation of almost sure stability of discrete time BAM neural networks. In Section 4, some numerical examples are given to demonstrate that our results. Finally, some conclusions are given in Section 5.

Notations: Throughout this paper, R=(,+), R+=[0,+), Rn denotes the n-dimensional Euclidean space with the Euclidean norm ·. The superscript T represents the transpose. 0M stands for M-dimensional vector with each entry being 0. Ib(x) is the indicator function which is defined as Ib(x)={1ifx=b,0otherwise.

Section snippets

Continuous time BAM neural networks with Markov chain

Let (Ω,F,{Ft},P) be a complete probability space with a natural filtration {Ft}t0 satisfying the usual condition (i.e., it is increasing right continuous and F0 contains all P-null sets), r(t),t0 be a right continuous Markov chain on the probability space taking values in the state space S={1,2,,M} with generator Γ=(γij)M×M, where γii=1jM,jiγij and γij>0(ij) is the transition rate from i to j, i.e., P{r(t+ϵ)=j|r(t)=i}=γijϵ+o(ϵ), where ϵ>0. We see that almost all sample paths of r(t)

Discrete time BAM neural networks with Markov chain

Let r(k) be a finite state Markov chain on the probability space taking values in the state space S={1,2,,M} with transition probability matrix P=(pij)M×M, where 1jMpij=1,pij0. The Markov chain r(k) possesses a single ergodic class. Let Π=(π1,π2,,πM) be unique invariant probability distribution satisfying ΠP=Π and i=1Mπi=1. We consider discrete time BAM neural network with Markov switching:{xi(k+1)=c^i(r(k))xi(k)+j=1mb^ij(r(k))fj(yj(k))+j=1ml=1me^ijl(r(k))fj(yj(k))fl(yl(k)),yj(k+1)=d^j

Examples and simulations

In this section, some examples and numerical simulation are provided to illustrate our results.

Example 1

Let r(t) be a right continuous Markov chain taking values in S={1,2} with generator Γ=-1313, then γ12=1,γ21=3,π1=34,π2=14. Consider the following BAM neural networks with Markov switching:{xi(t)=ci(r(t))xi(t)+j=12bij(r(t))fj(yj(t))+j=12l=12eijl(r(t))fj(yj(t))fl(yl(t)),yj(t)=dj(r(t))yj(t)+i=12hji(r(t))gi(xi(t))+i=12l=12sjil(r(t))gi(xi(t))gl(xl(t)),where c1(1)=6,c2(1)=5, c1(2)=0.8,c2(2)=0.5, d

Conclusions

In this paper, stability analysis of continuous time and discrete time BAM neural networks with Markov switching has been studied. Markov chain takes values in finite state space. By using matrix measure method and the ergodic theory of Markov chain, together the law of large number, some new stability criteria have been derived. Finally, some standard numerical packages imply that the developed results are practical. Our future work will focus on the stability of delayed neural networks with

Lijun Pan was born in Ganzhou, China, 1977. He received the B.S. degree from Zhanjiang Normal University, Zhanjiang, China, the M.S. degree from South China Normal University, Guangzhou, China, and the Ph.D. degree from Southeast University, Nanjing, China, in 1999, 2006, and 2013, respectively. Since 2006, he has been with the School of Mathematics, Jiaying University, Meizhou, China, where he is an associate professor. His research interests include neural networks, complex networks,

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  • Cited by (0)

    Lijun Pan was born in Ganzhou, China, 1977. He received the B.S. degree from Zhanjiang Normal University, Zhanjiang, China, the M.S. degree from South China Normal University, Guangzhou, China, and the Ph.D. degree from Southeast University, Nanjing, China, in 1999, 2006, and 2013, respectively. Since 2006, he has been with the School of Mathematics, Jiaying University, Meizhou, China, where he is an associate professor. His research interests include neural networks, complex networks, impulsive systems, stochastic systems. He is a reviewer of the journal Mathematical Reviews.

    Jinde Cao received the B.S. degree from Anhui Normal University, Wuhu, China, the M.S. degree from Yunnan University, Kunming, China, and the Ph.D. degree from Sichuan University, Chengdu, China, all in Mathematics/Applied Mathematics, in 1986, 1989, and 1998, respectively. From March 1989 to May 2000, he was with Yunnan University. In May 2000, he joined the Department of Mathematics, Southeast University, Nanjing, China. From July 2001 to June 2002, he was a post-doctoral research fellow at the Department of Automation and Computer Aided Engineering, Chinese University of Hong Kong, Shatin, Hong Kong. In the period from 2006 to 2008, he was a visiting research fellow and a visiting professor with the School of Information Systems, Computing and Mathematics, Brunel University, Middlesex, UK. On August 2014, he was a visiting professor at the School of Electrical and Computer Engineering, RMIT University, Australia. Currently, he is a distinguished professor and doctoral advisor at the Southeast University and also distinguished adjunct professor at the King Abdulaziz University, prior to which he was a professor at Yunnan University from 1996 to 2000. He is the author or co-author of more than 300 research papers and five edited books. His research interests include nonlinear systems, neural networks, complex systems and complex networks, stability theory, and applied mathematics. Dr. Cao was an associate editor of the IEEE Transactions on Neural Networks, Journal of the Franklin Institute and Neurocomputing. He is an associate editor of the IEEE Transactions on Cybernetics, Differential Equations and Dynamical Systems, Mathematics and Computers in Simulation, and Neural Networks. He is a reviewer of Mathematical Reviews and Zentralblatt-Math. He is an ISI highly-cited researcher in Mathematics and Engineering listed by Thomson Reuters.

    This work was supported by National Natural Science Foundation of China under Grant 60874088, and the Natural Science Foundation of Jiangsu Province of China under Grant BK2009271.

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