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ORIGINAL ARTICLE

Global exponential stability of a class of memristive neural

networks with time-varying delays

Xin Wang

•

Chuandong Li

•

Tingwen Huang

•

Shukai Duan

Received: 11 November 2012 / Accepted: 28 February 2013 / Published online: 16 April 2013

Ó Springer-Verlag London 2013

Abstract This paper studies the uniqueness and global

exponential stability of the equilibrium point for memris-

tor-based recurrent neural networks with time-varying

delays. By employing Lyapunov functional and theory of

differential equations with discontinuous right-hand side,

we establish several sufﬁcient conditions for exponential

stability of the equilibrium point. In comparison with the

existing results, the proposed stability conditions are milder

and more general, and can be applied to the memristor-

based neural networks model whose connection weight

changes continuously. Numerical examples are also pre-

sented to show the effectiveness of the theoretical results.

Keywords Memristive neural network  Exponential

stability  Time delay  Lyapunov functional

1 Introduction

In 1971, Chua [1] predicted that, besides the resistor,

capacitor and inductor, there should be the fourth circuit

element which is now called the memristor (contraction of

memory resistor). However, the great ﬁnding has not caused

the attention of the scientists until a group of scientists from

Hewlett-Packard Laboratory announced that they had build

a prototype of the memristor in 2008 [2]. This new circuit

element shares many properties of resistors and shares the

same unit of measurement (i.e., ohm). Recently, the

researchers showcased a number of promising applications

of memristive devices [3–7]. Memristor behavior is more

and more noticeable as new technology process nodes are

introduced in integrated circuit design, where the memristor

may be used as a nonvolatile memory switch. Because of

this feature, the new models of networks based on memr-

istor have been designed and analyzed [8–11, 24–26].

As we know, neural networks have found many important

applications in the ﬁelds of associative memory, pattern rec-

ognition, signal processing, systems control and optimization

problem [12–22]. During the past few years, the problem of

memristor-based neural networks has been one of the most

active research areas and has attracted the attention of many

researchers. However, the existing memristor-based networks

have been found to be computationally restrictive. The

applicability of these memristor-based networks is strongly

restricted. So some researchers turn their attentions to the

general memristor-based neural network which was ﬁrstly

introduced in [8]. Furthermore, an interesting issue is to

investigate the dynamic behavior of memristor-based recur-

rent neural networks, an ideal model for the case where the

memristor-based circuit networks exhibit complex switching

phenomena. Hu and Wang [8] considered the global asymp-

totic stability of memristor-based recurrent neural networks.

Guo et al. [10] presented some sufﬁcient conditions for

exponential stability for memristor-based recurrent neural

networks. However, these criteria only suits for the binary

memristor-based connection weight. The wilder exponential

stability criteria for multiple-valued cases are expected.

Motivated by the aforementioned discussion, in this paper,

we address the issue of global exponential stability for the

X. Wang  C. Li (&)

College of Computer Science, Chongqing University,

Chongqing 400044, China

e-mail: licd@cqu.edu.cn

T. Huang

Texas A&M University at Qatar,

Doha 23874, Qatar

S. Duan

School of Electronic and Information Engineering,

Southwest University, Chongqing 400715, China

123

Neural Comput & Applic (2014) 24:1707–1715

DOI 10.1007/s00521-013-1383-1

equilibrium point of the memristor-based recurrent neural

networks with time-varying delays. Several novel exponen-

tial stability criteria for memristor-based recurrent neural are

derived via Lyapunov functional and Filipov theory. The

existence and uniqueness of the equilibrium of memristor-

based recurrent neural networks are also analyzed.

The organization of the paper is as follows. In the next

section, the problem investigated in this paper is formu-

lated, and some preliminaries are presented. The existence

and uniqueness of the equilibrium of memristor-based

recurrent neural networks are analyzed in Sect. 3. In Sect.

4, new sufﬁcient conditions for global exponential stability

of memristor-based recurrent neural networks are pre-

sented. In Sect. 5, two examples and some comparisons

with previous results are given to illustrate our results.

Finally, some conclusions are drawn in Sect. 6.

2 Preliminaries and problem formulation

We denote by 

the Euclidian norm in R

.Let /

sup

s h 0

/hðÞ

for given continuous function /.

co u

xðÞ

denotes the convexhull of u

xðÞ. Let vectornorms

i¼1

, x

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

i¼1

, x

¼ max

1 i n

According to the feature of memristor and the current

characteristics, we apply a simple mathematical model of

the memristance as follows [8]:

MutðÞðÞ¼

utðÞ[ 0

utðÞ\0

unchanged

utðÞ¼0

ð1Þ

where u is the voltage applied to the memristor,

utðÞis the

derivative of u with respect to time t. When

utðÞ¼0,

‘‘unchanged’’ means that the memristance keeps the current

value. From this representation, we can see that memri-

stance switches between two values according to the volt-

age applied to the memristor. Since the connection weights

can be implemented by using resistors, we can similarly

utilize memristors to build a switching neural network.

In this paper, we further consider the following memr-

istor-based recurrent neural networks model [8]:

tðÞ¼d

tðÞþ

j¼1

 z



tðÞ



j¼1

 z



t  s

tðÞ



þ J

;

i ¼ 1; 2; ...; n ð2Þ

where x

tðÞis the state variable of the ith neuron, d

is the ith

self-feedback connection weight, a

 z



and b

 z



are, respectively, memristor-based connection weights and

those associated with time delays, J

is the external constant

inputs.

ðÞand

ðÞare the ith activation functions and

those associated with time delays, respectively. n denotes

the number of neurons in the indicated neural networks.

System (2) can be rewritten in the following vector form

xtðÞ¼PxðÞ

¼Dx tðÞþAxðÞ

fxtðÞðÞþBxðÞ

gxts tðÞðÞðÞþJ ð3Þ

By applying the theories of set-valued maps and

differential inclusions, the memristor-based recurrent

neural network (3) has the same solution set as the

following differential inclusion [8, 10, 23]:

x2co pxðÞ

¼Dx tðÞþAxðÞ

fxtðÞðÞþBxðÞ

gxts tðÞðÞðÞþJ;

i ¼1;2;...;n ð4Þ

where D ¼diag d

;...;d

ðÞ, A ¼ n

þ 1n





nn

B ¼ n

þ 1n





nn

, n

are arbitrary constants

such that 0n

1 and n

þn

¼1, J ¼ J

;...;J

ðÞ

and

fxtðÞðÞ¼

tðÞðÞ;

tðÞðÞ;...;

tðÞðÞ



gxtðÞðÞ¼

tðÞðÞ;

tðÞðÞ;...;

tðÞðÞðÞ

. Assume pxðÞis locally

bounded, according to the Lemma 2 in [8], the existence of

the solution of (4) is ensured.

The differential inclusion (4) means that there exist

some n

i; j ¼ 1; 2; ...; nðÞsuch that

tðÞ¼d

tðÞþ

j¼1

þ 1 n





tðÞ



j¼1

þ 1 n





t  s

tðÞ



þ J

ð5Þ

Moreover, we assume that the initial conditions of the

system (4) are of the form

tðÞ¼/

tðÞ; t 2s; 0½; t 2max

1i;j n





where /

ðÞ denote real-valued continuous functions

deﬁned on s; 0½.

Suppose that x



¼ x



; x



; ...; x





2 R

is an equilibrium

point of network (2), and let y

tðÞ¼x

tðÞx



i ¼ 1; 2; ...; n. Then, system (4) can be rewritten as

tðÞ¼d

tðÞþ

j¼1

þ 1 n





tðÞðÞ

j¼1

þ 1 n





t  s

tðÞ



;

t 0 ð6Þ

where f

tðÞðÞ¼

tðÞþx











, g

tðÞðÞ¼

tðÞþx











, i ¼ 1; 2; ...; n.

1708 Neural Comput & Applic (2014) 24:1707–1715

123

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