Elsevier

Neurocomputing

Volume 363, 21 October 2019, Pages 295-305
Neurocomputing

Synchronization control for memristive high-order competitive neural networks with time-varying delay

https://doi.org/10.1016/j.neucom.2019.06.049Get rights and content

Abstract

This paper concerns the synchronization problem of memristive high-order competitive neural networks with time-varying delay. First, a novel control scheme with a linear term and a discontinuous term is proposed. Then, based on the Lyapunov stability theory, several criteria with algebraic form or matrix form are derived to ensure global exponential synchronization of the networks by adopting some inequality techniques. Finally, two numerical examples are presented to substantiate the effectiveness of the results.

Introduction

In recent decades, neural networks have been widely applied in many areas because of their associative memory function and adaptive capacity. However, many scholars mainly focus on the study of neural networks with only one type of state variable, such as some classical neural networks including Hopfield neural network, BAM neural network and BP neural network. Actually, there also exist some neural networks with more than one type of state variable, such as competitive neural network which was put forward by Cohen and Grossberg [1] in 1983. About thirteen years later, Meyer-Bäse first proposed a competitive neural network model with different time scales [2], which has two different types of state variables, i.e., short-term memory and long-term memory. Moreover, he pointed out that short-term memory describes the fast neural activity level and long-term memory describes the slow unsupervised synaptic modifications. Some self-organizing networks can be constructed from competitive neural network because it is usually unsupervised learning, such as self-organizing map networks, counterpropagation networks.

Synchronization has attracted considerable attention of researchers due to its wide potential applications, such as secure communication [3], [4], [5], chemical reactions [6] and information science [7]. Recently, the related results on synchronization of competitive neural networks with different types of time delays have been obtained [8], [9], [10], [11]. In [8], the author designed an adaptive feedback controller to ensure that competitive neural networks with stochastic perturbation are synchronized. In [9], the authors studied synchronization problem of the competitive neural network model with mixed delays subjected to unknown hybrid perturbations. In [10], the LMI approach was adopted to achieve global exponential synchronization for the switched competitive neural networks with mixed delays subjected to stochastic disturbance by designing a delay-dependent controller. In [11], an adaptive controller was employed to ensure that delayed competitive neural networks are globally synchronized.

Compared with traditional neural networks, neural networks with high-order terms, called high-order neural networks, have better performances [12], [13], [14], such as greater storage capacity, faster convergence rate, higher fault tolerance and stronger approximation property. Obviously, it is necessary to investigate the synchronization of high-order neural networks. However, there exist few results about the synchronization of high-order neural networks [12]. In [12], some sufficient conditions of global exponential synchronization for second-order recurrent neural networks with mixed delays were derived. In [15], the high-order term is introduced into competitive neural network to built a new model, which is called high-order competitive neural networks.

In the circuit implementation for neural networks, we generally use resistors to simulate synapse among the neurons [16]. We know that the synapse plays a key role in the formation of memory, but the resistor has no memory function. Due to the complexity of the human brain, in order to better describe the human brain, we should replace resistor with memristor, which was first put forward by Chua [17] in 1971. Unfortunately, the memristor failed to attract much attention, until 2008 when the researchers of Hewlett-Packard laboratory built the first memristor device [18]. Since the memristor has the memory function, memristive neural networks can better simulate the human brain [19], [20], [21], [22], [23], [24]. As far as we know, almost no results on the synchronization problem of memristive high-order competitive neural networks have been obtained. Then, we cannot help asking how to solve this problem.

Based on the above discussion, the object of this paper is to study the synchronization problem of memristive high-order competitive neural network with time delay. Some novel criteria for global exponential synchronization of MHCNNs are obtained by utilizing Lyapunov function method, which can be divided into algebraic form and matrix form. Moreover, the existence of Fillipov solution of the drive system and the response system is also proved by using differential inclusion theory.

The structure of this paper is organized as follows. The model of delayed MHCNN is formulated and some preliminaries are presented in Section 2. In Section 3, the sufficient conditions for global exponential synchronization of the drive-response system are derived. In Section 4, two examples and numerical simulations are given to substantiate the effectiveness of our results. Finally, the conclusions are drawn in Section 5.

Section snippets

Preliminaries

First, the following memristive high-order competitive neural network (MHCNN) with time-varying delay is considered:{STM:εdxi(t)dt=aixi(t)+j=1n[cij(xi(t))hj(xj(t))+dij(xi(tτ(t)))hj(xj(tτ(t)))]+j=1nk=1nWijk(xi(t))hj(xj(t))hk(xk(t))+j=1nk=1nRijk(xi(tτ(t)))hj(xj(tτ(t)))×hk(xk(tτ(t)))+bis=1pvis(t)ςs+Ii,LTM:dvisdt=civis(t)+ςshi(xi(t)),where i=1,2,,n, s=1,2,,p, xi(t) is the neuron state, ai > 0 represents the decay of the neuron, bi is the strength of the external stimulus, ci > 0

Main results

By using Lyapunov method, the following criteria for global exponential synchronization of systems (2) and (6) are presented in this section.

Theorem 1

Assume that Assumptions 1 and 2 hold. Then response MHCNN (6) can be globally exponentially synchronized with drive MHCNN (2) at the rate δ under controller (8), if there exist positive numbers pi and qi, i=1,2,,n such thatkiεδai+εqilipi+j=1npjpi(c¯ji+eδτ1d¯ji)li+j=1nk=1npjpiMk[(W¯jik+W¯jki)+eδτ1(R¯jik+R¯jki)]li,ciδ+pi|bi|εqi,andγi2j=1n[|cijc

Numerical simulation

Consider the two-neuron delayed memrisive high-order competitive neural network{εdxi(t)dt=aixi(t)+j=12cij(xi(t))hj(xj(t))+j=12dij(xi(tτ(t)))hj(xj(tτ(t)))+j=12k=12Wijk(xi(t))hj(xj(t))hk(xk(t))+j=12k=12Rijk(xi(tτ(t)))hj(xj(tτ(t)))×hk(xk(tτ(t)))+bis˜i(t)+Ii,ds˜i(t)dt=cis˜i(t)+hi(xi(t)),where ε=1, I1=I2=0, τ(t)=1, the activation function h(x)=(0.2tanh(x1),0.2cos(x2))T satisfies the Assumption 1 with M1=M2=0.2 and l1=l2=0.2. LetA=[3002],B=[2.5001.8],C=[8006].In addition, the matrices C(

Conclusion

The synchronization problem of memristive high-order competitive neural network with time-varying delay has been discussed. By utilizing Lyapunov method and some inequality techniques, several criteria have been obtained to realize the global exponential synchronization between drive system and response system.

Time-triggered control, event-triggered control, pinning control and impulsive control are becoming more and more popular because of the limitation of storage resources. The further work

Declarations of interest

None.

Shuqing Gong received his B.S. degree from Anyang Normal University, Anying, China in 2013, and was admitted to College of Mathematics and Econometrics, Hunan University, Changsha, China in 2014. His research interests include neural networks, switched systems, memristive systems and networked control systems.

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    Shuqing Gong received his B.S. degree from Anyang Normal University, Anying, China in 2013, and was admitted to College of Mathematics and Econometrics, Hunan University, Changsha, China in 2014. His research interests include neural networks, switched systems, memristive systems and networked control systems.

    Zhenyuan Guo received the B.S. degree in mathematics and applied mathematics and the Ph.D. degree in applied mathematics from the College of Mathematics and Econometrics, Hunan University, Changsha, China, in 2004 and 2009, respectively. He was a joint Ph.D. student with the Department of Applied Mathematics, University of Western Ontario, London, ON, Canada, from 2008 to 2009. He was a Post-Doctoral Research Fellow with the Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, Hong Kong, from 2013 to 2015. He is currently the Professor with the College of Mathematics and Econometrics, Hunan University. His current research interests include the theory of functional differential equations and differential equations with discontinuous right hands, and their applications to the dynamics of neural networks, memristive systems, and control systems.

    Shiping Wen received the M.Eng. degree in control science and engineering from the School of Automation, Wuhan University of Technology, Wuhan, China, in 2010, and the Ph.D degree in control science and engineering from the School of Automation, Huazhong University of Science and Technology, Wuhan, in 2013. He is currently an Associate Professor at School of Automation, Huazhong University of Science and Technology, and also in the Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan, China. His current research interests include memristor-based circuits and systems, neural networks, and deep learning.

    Tingwen Huang received the B.S. degree in mathematics from Southwest Normal University (currently, Southwest University), Chongqing, China, in 1990, the M.S. degree in mathematics from Sichuan University, Chengdu, China, in 1993, and the Ph.D. degree in mathematics from Texas A&M University, College Station, TX, USA, in 2002. He was a Visiting Assistant Professor with Texas A&M University. Then, he joined Texas A&M University at Qatar, Doha, Qatar, as an Assistant Professor in 2003, where he was promoted to a Professor in 2013. He has published over 300 peer-review reputable journal papers, including over 100 papers in IEEE TRANSACTIONS. His current research interests include neural-networks-based computational intelligence, distributed control and optimization, and nonlinear dynamics and applications in smart grids. Prof.Huang serves as an Associate Editor for four journals, including the IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, the IEEE TRANSACTIONS ON CYBERNETICS, and Cognitive Computation.

    Research supported by National Natural Science Foundation of China (61573003), Natural Science Foundation of Hunan (2019JJ40022) and NPRP grants: NPRP 9-466-1-103 from Qatar National Research Fund.

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