Elsevier

Neurocomputing

Volume 219, 5 January 2017, Pages 163-173
Neurocomputing

On the periodic dynamics of memristor-based neural networks with leakage and time-varying delays

https://doi.org/10.1016/j.neucom.2016.09.029Get rights and content

Abstract

In this paper, the periodic dynamics have been studied for a general kind of memristor-based neural networks with leakage and time-varying delays. Some new sufficient conditions have been derived ensuring that the existence, uniqueness and globally exponential stability of the periodic solution for the neural network by using differential inclusions theory, the topological degree theory in set-valued analysis and Lyapunov function technique and so on. As a special case, we have shown that the existence, uniqueness and global exponential stability of equilibrium point for the autonomous neural networks with leakage delays.

Introduction

A new concept and a new circuit element, memristor, postulated by Chua [1] in 1971 and a team at HP Labs [2] in 2008, respectively. From then on, the memristor behavior plays a prominent part in the design of integrated circuit [3], [4], where the memristor may be used as a nonvolatile memory switch [5], [6], [7]. Recently, there are many more physical realization of memristors and memristive systems [8], the modeling of basic memristor circuits [9], the designs and analysis of memristor-based application circuits [10], etc., have attracted much attention in the electrical and electronic engineering communities due to the potential applications of this device in next generation computer and powerful brain-like neural computer.

The memristor-based neural networks is a new kind of neural networks model, where the connection weights change according to its state, for example, a state-dependent switching neural networks. The analysis of the memristor-based neural networks has been found useful to address a number of interesting engineering tasks, such as dry friction, impacting machines, systems oscillating under the effect of an earthquake, power circuits, switching in electronic circuits and many others, and therefore have received a great deal of attention in the literature [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28] and the references therein.

In many applications, knowing the property of periodic oscillatory solutions is very interesting and valuable. For example, as many biological and cognitive activities (e.g., heartbeat, respiration, mastication, locomotion, and memorization) require repetition. An equilibrium point can be viewed as a special case of periodic solution with an arbitrary period or zero amplitude. In this sense, the analysis of periodic oscillation of neural networks is more general than the stability analysis of equilibrium points. Meanwhile, the delays are actually encountered in practical implementation, due to the finite switching speed of the neuron amplifiers and the finite signal propagation speed. There have been many results on the stability and the periodicity analysis of recurrent neural networks with and without delays in [29], [30], [31] and references therein. It is necessary to point out that the global periodicity and global stability of the delayed memristor-based recurrent neural networks (DMNN) plays also important roles in many potential applications, for example, super-dense nonvolatile computer memory and neural synapses. In very recently, results on the periodicity and the almost periodicity of the memristor-based recurrent neural networks with time-varying delays have been obtained in the literature [25], [28]. On the other hand, a typical time delay called leakage (or forgetting) delay has a great impact on the dynamical behavior of neural networks [32], [33], [34]. Especially in a real nervous system, a typical time delay in the negative feedback terms which is known as leakage delays has a tendency to destabilize the system [35]. Since leakage delays have a destabilizing influence on the dynamical behaviors of neural networks, it is necessary and important to consider the leakage delay effects on the study of state estimation of neural networks. At present there have been more works on neural networks with leakage delays in the literature [36], [37], [38], [39], [40], [41], [42], [43], [44], [45]. However to the best of our knowledge, relative works on DMNN with leakage delay has not been investigated until now.

Based on the above discussions and the works in [25], our objective in this paper is to study the existence and exponential stability of the periodic solutions for a memristor-based neural networks with leakage and time-varying delays. The main advantages are highlighted as follows:

  • The periodic behaviors of DMNN with time delays in the leakage term is firstly studied in the present paper.

  • By employing the topological degree theory in set-valued analysis, differential inclusions theory and new Lyapunov function techniques, we derive some new sufficient conditions ensuring the existence, uniqueness and exponential stability of the periodic solution for DMNN with leakage and time-varying.

  • Our result is more general and it is valid for the usual neural networks model. Many relative works are included in this paper. As a special case, it improves and generalizes some existence works in [25], [28] and references therein.

The rest of the paper is organized as follows: a DMNN with leakage delays and time-varying delays is introduced and some necessary definitions are given in Section 2. The main results are shown in Section 3. The existence of periodic solutions of the system are investigated in Section 3.1, the uniqueness and global exponential stability of the ω-periodic solution for the neural networks are investigated in Section 3.2. In Section 4 two examples and simulations are obtained to show the effectiveness of the theoretical results given in Section 3. Finally, the paper is concluded in Section 5.

Section snippets

Preliminaries

Consider a kind of DMNN described by the following differential equations with leakage and time-varying delays: dxi(t)dt=di(t)xi(tηi(t))+j=1naij(t,xj(t))fj(xj(t))+j=1nbij(t,xj(tτij(t)))gj(xj(tτij(t)))+Ii(t),i=1,2,,n,where n in response to the amount of units in a neural network; xi(t) corresponds to the state variable associated with the ith neuron; fj and gj are activation functions of signal transmission; di(t)>0 represents the rate with which the ith unit will reset its potential to

Main results

In this section, we study the existence, uniqueness and global exponential stability of the ω-periodic solution for the DMNN (2.1).

Numerical examples

In this section we consider two numerical examples to show the effectiveness of the theoretical results given in the above section.

Example 4.1

Consider DMNN described by the following differential equations with leakage delays and time-varying delays:{dx1(t)dt=d1x1(tη)+a11a(t,x1(t))f(x1(t))+a12a(t,x2(t))f(x2(t))+b11b(t,x1(tτ(t)))f(x1(tτ(t)))+b12b(t,x2(tτ(t)))f(x2(tτ(t)))+sint,dx2(t)dt=d2x2(tη)+a21a(t,x1(t))f(x1(t))+a22a(t,x2(t))f(x2(t))+b21b(t,x1(tτ(t)))f(x1(tτ(t)))+b22b(t,x2(tτ(t)))f(x2(tτ(t))

Conclusion

As well known, leakage delays have a destabilizing influence on the dynamical behaviors of neural networks, it is necessary and important to consider the leakage delay effects on the study of state estimation of neural networks. The present paper first have considered the periodic dynamics of a memristor-based neural networks with leakage delays and time-varying delays. Some easily testable conditions have been established to check the existence, uniqueness and global exponential stability of

Acknowledgments

This work is supported by the Natural Science Foundation of China under Grant 61125303, National Basic Research Program of China (973 Program) under Grant 2011CB710606, the Program for Science and Technology in Wuhan of China under Grant 2014010101010004, the Program for Changjiang Scholars and Innovative Research Team in University of China under Grant IRT1245.

Ping Jiang received his B.S. degrees from Hubei Normal University, Huangshi, China, and his M.S. degrees in Central south University, Changsha, China, in 2005 and 2008, respectively. He is currently a doctoral candidate in the School of Automation, Huazhong University of Science and Technology, Wuhan, China, and also in the Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan, China. And he is a teacher of School of Computer, Hubei PolyTechnic

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    Ping Jiang received his B.S. degrees from Hubei Normal University, Huangshi, China, and his M.S. degrees in Central south University, Changsha, China, in 2005 and 2008, respectively. He is currently a doctoral candidate in the School of Automation, Huazhong University of Science and Technology, Wuhan, China, and also in the Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan, China. And he is a teacher of School of Computer, Hubei PolyTechnic University, Huangshi, China. His current research interests include stability analysis of landslide and neural networks.

    Zhigang Zeng received the Ph.D. degree in systems analysis and integration from Huazhong University of Science and Technology, Wuhan, China, in 2003. He is currently a Professor with the School of Automation, Huazhong University of Science and Technology, Wuhan, China, and also with the Key Laboratory of Image Processing and Intelligent Control of the Education Ministry of China, Wuhan, China. He has been an Associate Editor of the IEEE Transactions on Neural Networks (2010–2011), IEEE Transactions on Cybernetics (since 2014), IEEE Transactions on Fuzzy Systems (since 2016), and a member of the Editorial Board of Neural Networks (since 2012), Cognitive Computation (since 2010), Applied Soft Computing (since 2013). He has published over 100 international journal papers. His current research interests include theory of functional differential equations and differential equations with discontinuous right-hand sides, and their applications to dynamics of neural networks, memristive systems, and control systems.

    Jiejie Chen received her B.S. degrees from Hubei Normal University, Huangshi, China, and her M.S. degrees in Central south University, Changsha, China, in 2005 and 2008, respectively. She is currently a doctoral candidate in the School of Automation, Huazhong University of Science and Technology, Wuhan, China, and also in the Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan, China. Her current research interests include stability analysis of landslide, neural networks and memristors.

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