Elsevier

Neural Networks

Volume 84, December 2016, Pages 172-180
Neural Networks

Coexistence and local μ-stability of multiple equilibrium points for memristive neural networks with nonmonotonic piecewise linear activation functions and unbounded time-varying delays

https://doi.org/10.1016/j.neunet.2016.08.006Get rights and content

Abstract

In this paper, the coexistence and dynamical behaviors of multiple equilibrium points are discussed for a class of memristive neural networks (MNNs) with unbounded time-varying delays and nonmonotonic piecewise linear activation functions. By means of the fixed point theorem, nonsmooth analysis theory and rigorous mathematical analysis, it is proven that under some conditions, such n-neuron MNNs can have 5n equilibrium points located in n, and 3n of them are locally μ-stable. As a direct application, some criteria are also obtained on the multiple exponential stability, multiple power stability, multiple log-stability and multiple log–log-stability. All these results reveal that the addressed neural networks with activation functions introduced in this paper can generate greater storage capacity than the ones with Mexican-hat-type activation function. Numerical simulations are presented to substantiate the theoretical results.

Introduction

The research on memristive neural networks (MNNs) began in the early 1970s. Based on physical symmetry arguments, Chua postulated in 1971 that besides the resistor, capacitor and inductor, there should be a fourth fundamental two-terminal circuit element called memristor (an abbreviation for memory and resistor) (Chua, 1971). However, no much attention was paid to Chua’s theory until nearly forty years later. In 2008, the researchers at Hewlett-Packard Laboratories announced that they had built successfully such a practical memristor device (Strukov, Snider, Stewart, & Williams, 2008). In this memristor, its value (memristance) depends on the magnitude and polarity of the voltage applied to it and the length of the time that the voltage has been applied. When the voltage is turned off, the memristor remembers its most recent value until next time it is turned on. It should be pointed out that the memristor exhibits the feature of pinched hysteresis just as the neurons in the human brain have (Pershin & Ventra, 2010). Owing to such important feature, this device can be applied to design a new model of neural networks to emulate the human brain, and its potential applications are in next generation computers and powerful brain-like neural computers (Corinto et al., 2011, Itoh and Chua, 2008).

In the past few years, valuable results on dynamical behaviors of MNNs have been reported. Wu and Zeng (2012a) studied the exponential stabilization of MNNs and derived some sufficient conditions in terms of linear matrix inequality (LMI). Based on local inhibition, Wu and Zeng (2012b) investigated further the issue of nondivergence and global attractivity of MNNs. Dealt with in Chandrasekar, Rakkiyappan, Cao, and Lakshmanan (2014), Wen, Bao, Zeng, Chen, and Huang (2013), Yang, Cao, and Yu (2014), Zhang and Shen (2013) and Zhang and Shen (2014) was the synchronization control problem of memristive Hopfield neural networks and memristive Cohen–Grossberg neural networks. By constructing proper Lyapunov functions and using the M-matrix theory and LaSalle invariant principle, the global exponential dissipativity and stabilization of MNNs with time-varying delays were considered in Guo, Wang, and Yan (2013). Based on the matrix measure approach, generalized Halanay inequality and fixed point theorem, the dissipativity of solutions and existence of positive periodic solution were addressed in Cai and Huang (2014) for a class of memristive BAM (bidirectional associative memory) neural networks with time-varying delays. By virtue of the theory of fractional-order differential equations with discontinuous right-hand sides, a class of fractional-order MNNs was introduced in Chen, Zeng, and Jiang (2014) and the issue of global Mittag-Leffler stability and synchronization was analyzed.

A neural network is said to be multistable if it possesses multiple equilibrium points which are locally stable. Multistability is necessary when neural networks are applied in image processing, pattern recognition, associative memory and many other fields. For this reason, during the past decade considerable efforts have been devoted to studying the multistability of conventional recurrent neural networks (Cao et al., 2008, Cheng et al., 2006, Cheng and Shih, 2009, Di Marco et al., 2012, Huang et al., 2014, Huang et al., 2010, Kaslik and Sivasundaram, 2011, Nie and Cao, 2009, Nie and Cao, 2011, Nie et al., 2013, Zhang et al., 2009). We remark that these multistability works were obtained based on the assumption that the time delays involved in delayed neural networks are bounded. A new concept of multiple μ-stability, which is a generalization of multiple exponential stability, multiple power stability, multiple log-stability and multiple log–log-stability, was introduced in a recent paper (Wang & Chen, 2014) to investigate the multistability of conventional recurrent neural networks with unbounded time-varying delays. Wang and Chen (2014) showed that under some conditions, there exist 3n equilibrium points for the n-neuron neural networks with nondecreasing activation functions, and 2n of them are locally μ-stable, based on the geometrical configuration and rigorous mathematical analysis. However, to the best of our knowledge, the research on multiple μ-stability of MNNs is still an open problem that deserves attention and investigation. Compared with conventional recurrent neural networks which are continuous dynamical systems, MNNs are state-dependent switching systems which are discontinuous dynamical systems. Therefore, the study on multiple μ-stability of MNNs is much complicated and challenging.

In addition, it is well known that the type of activation functions plays a crucial role in multistability analysis of neural networks. It is worth mentioning that most of the activation functions employed in multistability analysis are restricted in sigmoidal activation functions (Cheng et al., 2006, Cheng and Shih, 2009, Huang et al., 2010, Nie and Cao, 2009), nondecreasing saturated activation functions (Cao et al., 2008, Cheng and Shih, 2009, Nie and Cao, 2009, Nie and Cao, 2011), and piecewise linear activation functions (Di Marco et al., 2012, Huang et al., 2014, Nie et al., 2013, Zhang et al., 2009), which are all monotonically increasing. Recent works in Nie, Cao, and Fei (2014) and Wang and Chen (2012) introduced a class of nonmonotonic piecewise linear activation function which is called Mexican-hat-type activation function, and investigated respectively the multistability of Hopfield neural networks and competitive neural networks. The Mexican-hat-type activation function has the following form (see Fig. 1): f(x)={1,<x<1,x,1x1,x+2,1<x3,1,3<x<+. In order to increase the storage capacity of neural networks, in this paper we consider another class of continuous nonmonotonic piecewise linear activation functions employed in Nie and Zheng (2015) (see Fig. 2): fi(x)={ui,<x<pi,ki,1x+ci,1,pixqi,ki,2x+ci,2,qi<x<ri,ki,3x+ci,3,rixsi,vi,si<x<+, where pi,qi,ri,si,ui,vi,ki,1,ki,2,ki,3,ci,1,ci,2,ci,3 are constants with <pi<qi<ri<si<+, ki,1>0, ki,2<0, ki,3>0, ui=fi(ri), and vi>fi(qi),i=1,2,,n. We remark here that the activation functions introduced in Nie and Zheng (2016) may be considered as the corresponding discontinuous case of activation functions (2). It is easy to see that activation functions (2) are Lipschitz continuous, i.e., x,y, there exists positive number ρi=max{|ki,1|,|ki,2|,|ki,3|} such that |fi(x)fi(y)|ρi|xy|.

Motivated by the above discussions, our main goal in this paper is to investigate the multiple μ-stability of MNNs with both activation functions (2) and unbounded time-varying delays. More precisely, the main contributions of this paper lie in the following aspects. Firstly, under the framework of Filippov’s solution, we present sufficient conditions under which the n-neuron MNNs with activation functions (2) can have 5n equilibrium points located in n, by applying the well-known fixed point theorem. Secondly, based on rigorous mathematical analysis and the theories of set-valued maps and differential inclusions, we analyze the dynamical behaviors of MNNs with both activation functions (2) and unbounded time-varying delays, and show that the addressed MNNs can have 5n equilibrium points, and 3n of them are locally μ-stable. Thirdly, the above-obtained results can also be applied to several special cases as so to get the multiple exponential stability, multiple power stability, multiple log-stability and multiple log–log-stability of the addressed MNNs, correspondingly. Lastly, when compared with the neural networks with Mexican-hat-type activation function (1), the MNNs with activation functions (2) can generate both more total equilibrium points and more locally stable equilibrium points.

The rest of this paper is organized as follows. In Section  2, some preliminaries, model description and necessary definitions are presented. Section  3 discusses the coexistence and local μ-stability of MNNs. Then numerical simulations are given in Section  4. Finally, some conclusions are made in Section  5.

Section snippets

Model

In this paper, based on the previous works in Wu and Zeng (2012a), Zhang and Shen, 2013, Zhang and Shen, 2014, we consider a class of memristor-based neural networks with time-varying delays described by dxi(t)dt=dixi(t)+j=1naij(xj(t))fj(xj(t))+j=1nbij(xj(tτij(t)))fj(xj(tτij(t)))+Ii,i=1,2,,n, where xi(t) is the voltage of the capacitor Ci, fj() is the activation function defined in (2), aij(xj(t)) and bij(xj(tτij(t))) represent memristor synaptic connection weights, and aij(xj(t))=MijCi×

Main results

In this section, the multiple μ-stability of MNNs (3) with both activation functions (2) and unbounded time-varying delays is investigated. First of all, we give the following theorem on the coexistence of multiple equilibrium points for MNNs (3) by applying the known fixed point theorem under the framework of Filippov’s solution.

Theorem 1

If the following conditions hold for all i=1,2,,n:dipi+max{(a¯ii+b¯ii)ui,(āii+b̄ii)ui}+ji,j=1nmax{(a¯ij+b¯ij)uj,(a¯ij+b¯ij)vj,(āij+b̄ij)uj,(āij+b̄ij)vj}+Ii<0,d

An illustrative example

Example 1

Consider the following two-dimensional memristor-based neural networks with unbounded time-varying delays: dxi(t)dt=dixi(t)+j=12aij(xj(t))fj(xj(t))+j=12bij(xj(tτij(t)))fj(xj(tτij(t)))+Ii,i=1,2, where d1=d2=1.5,I1=I2=0.6,τ11(t)=τ21(t)=0.2t,τ12(t)=τ22(t)=0.3t and a11(x1(t))={2,|x1(t)|1,2.2,|x1(t)|>1,a12(x2(t))={0.08,|x2(t)|1,0.1,|x2(t)|>1,a21(x1(t))={0.07,|x1(t)|1,0.09,|x1(t)|>1,a22(x2(t))={2.3,|x2(t)|1,2,|x2(t)|>1,b11(x1(tτ11(t)))={0.15,|x1(tτ11(t))|1,0,|x1(tτ11(t))|>1,b12(x2(tτ

Conclusions

Under the framework of Filippov’s solution, the problem of multiple μ-stability has been studied in this paper for a class of MNNs with both nonmonotonic piecewise linear activation functions and unbounded time delays. The new sufficient conditions have been presented to ensure the coexistence of 5n equilibrium points and the local μ-stability of 3n equilibrium points. For different cases of time-varying delays, we have obtained the multiple exponential stability, multiple power stability,

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants nos. 61673111, 61573096 and 61272530, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20120092120029, the Natural Science Foundation of Jiangsu Province of China under Grant BK2012319, the Qing Lan Project of Jiangsu Province of China, the 333 Engineering Foundation of Jiangsu Province of China under Grant no. BRA2015286, and the Australian Research Council

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