Mean Square Exponential Stability of Stochastic Cohen-Grossberg Neural Networks with Unbounded Distributed Delays

Abstract

This paper addresses the issue of mean square exponential stability of stochastic Cohen-Grossberg neural networks (SCGNN), whose state variables are described by stochastic nonlinear integrodifferential equations. With the help of Lyapunov function, stochastic analysis technique, and inequality techniques, some novel sufficient conditions on mean square exponential stability for SCGNN are given. Furthermore, we also establish some sufficient conditions for checking exponential stability for Cohen-Grossberg neural networks with unbounded distributed delays.

1. Introduction

Consider the Cohen-Grossberg neural networks (CGNN) described by a system of ordinary differential equations𝑥̇_𝑖(𝑡)=−𝑎_𝑖(𝑥_𝑖(𝑡))[𝑏̂_𝑖(𝑥_𝑖(𝑡))−^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡))], (1.1) where 𝑡≥0, 𝑛≥2; 𝑛 corresponds to the number of units in a neural network; 𝑥_𝑖(𝑡) denotes the potential (or voltage) of cell 𝑖 at time 𝑡; 𝑓_𝑗(⋅) denotes a non-linear output function between cell 𝑖 and 𝑗; 𝑎_𝑖(⋅)>0 represents an amplification function; 𝑏̂_𝑖(⋅) represents an appropriately behaved function; the 𝑛×𝑛 connection matrix 𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛 denotes the strengths of connectivity between cells, and if the output from neuron 𝑗 excites (resp., inhibits) neuron 𝑖, then 𝑐_𝑖𝑗≥0 (resp., 𝑐_𝑖𝑗≤0).

During hardware implementation, time delays do exist due to finite switching speed of the amplifiers and communication time and, thus, it is important to incorporate delays in the neural networks. Just take the delayed cellular neural network as an example, which had been successfully applied to solve some moving image processing problem [1]. For model (1.1), Ye et al. [2] introduced delays by considering the following delay differential equations:𝑥̇_𝑖(𝑡)=−𝑎_𝑖(𝑥_𝑖(𝑡))[𝑏̂_𝑖(𝑥_𝑖(𝑡))−^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡))−^𝐾𝑛_𝑘=1𝑗=1𝑑^𝑘_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡−𝜏_𝑘))],𝑖=1,…,𝑛. (1.2) Guo and Huang [3] generalized model (1.1) as the following delay differential equations:𝑥̇_𝑖=−𝑎_𝑖(𝑥_𝑖)[𝑏̂_𝑖(𝑥_𝑖)−^𝑛_𝑗=1𝑑_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡−𝜏_𝑖𝑗))+𝐼_𝑖]. (1.3) Some other more detailed justifications for introducing delays into model equations of neural networks can be found in [4, 5] and references therein.

The delays in all above mentioned papers have been largely restricted to be discrete. As is well known, the use of constant fixed delays in models of delayed feedback provides of a good approximation in simple circuits consisting of a small number of cells. However, neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. Thus there will be a distribution of conduction velocities along these pathways and a distribution of propagation delays. In these circumstances, the signal propagation is not instantaneous and cannot be modeled with discrete delays and a more appropriate way is to incorporate continuously distributed delays. For instance, in [6], Tank and Hopfield designed an analog neural circuit with distributed delays, which can solve a general problem of recognizing patterns in a time-dependent signal. A more satisfactory hypothesis is that to incorporate continuously distributed delays, we refer to [7–11]. Then model (1.3) can be modified as a system of integro-differential equations of the formd𝑥_𝑖(𝑡)d𝑡=𝑎_𝑖(𝑥_𝑖(𝑡))[−𝑏̃_𝑖(𝑥_𝑖(𝑡))+^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡))+^𝑛_𝑗=1𝑑_𝑖𝑗𝑔_𝑗(^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑥_𝑗(𝑠)d𝑠)+𝐼_𝑖], (1.4) with initial values given by 𝑢_𝑖(𝑠)=𝜓_𝑖(𝑠) for 𝑠∈(−∞,0], where each 𝜓_𝑖(⋅) is bounded and continuous on (−∞,0].

In the past few years, the dynamical behaviors of stochastic neural networks have emerged as a new subject of research mainly for two reasons: (i) in real nervous systems and in the implementation of artificial neural networks, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes, hence, noise is unavoidable and should be taken into consideration in modeling [12–14]; (ii) it has been realized that a neural network could be stabilized or destabilized by certain stochastic effects [15–17]. Although systems are often perturbed by various types of environmental “noise” [12–14, 18], it turns out that one of the reasonable interpretation for the “noise” perturbation is the so-called white noise d𝜔(𝑡)/d𝑡, where 𝜔(𝑡) is the Brownian motion process, also called as Wiener process [17, 19]. More detailed mechanism of the stochastic effects on the interaction of neurons can be found in [19]. However, because the Brownian motion 𝜔(𝑡) is nowhere differentiable, the derivative of Brownian motion d𝜔(𝑡)/d𝑡 cannot be defined in the ordinary way, the stability analysis for stochastic neural networks is difficult. In [12], through constructing a novel Lyapunov-Krasovskii functional, Zhu and Cao obtain several novel sufficient conditions to ensure the exponential stability of the trivial solution in the mean square. In [13], using linear matrix inequality (LMI) approach, Zhu et al. investigated the asymptotical mean square stability of Cohen-Grossberg neural networks with random delay, In [14], by utilizing Poincaré inequality, Pan and Zhong derived some sufficient conditions to check the almost sure exponential stability and mean square exponential stability of stochastic reaction-diffusion Cohen-Grossberg neural networks. In [20], Wang et al. developed a linear matrix inequality (LMI) approach to study the stability of SCGNN with mixed delays. To the best of the authors' knowledge, the convergence dynamics of stochastic Cohen-Grossberg neural networks with unbounded distributed delays have not been studied yet, and still remain as a challenging task.

Keeping this in mind, in this paper, we consider the SCGNN described by the following stochastic nonlinear integro-differential equations:d𝑥_𝑖(𝑡)=𝑎_𝑖(𝑥_𝑖(𝑡))[−𝑏̃_𝑖(𝑥_𝑖(𝑡))+^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡))+^𝑛_𝑗=1𝑑_𝑖𝑗𝑔_𝑗(^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑥_𝑗(𝑠)d𝑠)+𝐼_𝑖+]d𝑡_𝑛𝑗=1𝜎_𝑖𝑗(𝑥_𝑗(𝑡))d𝜔_𝑗(𝑡), (1.5) where 𝜎(𝑡)=(𝜎_𝑖𝑗(𝑡))_𝑛×𝑛 is the diffusion coefficient matrix and 𝜔(𝑡)=(𝜔₁(𝑡),…,𝜔_𝑛(𝑡))^𝑇 is an 𝑛-dimensional Brownian motion defined on a complete probability space (Ω,ℱ,𝐏) with a natural filtration {ℱ_𝑡}_𝑡≥0 (i.e., ℱ_𝑡=𝜎{𝑤(𝑠)∶0≤𝑠≤𝑡}).

Obviously, model (1.5) is quite general and it includes several well-known neural networks models as its special cases such as Hopfield neural networks, cellular neural networks, and bidirectional association memory neural networks [21, 22].

The remainder of this paper is organized as follows. In Section 2, the basic notations and assumptions are introduced. In Section 3, some criteria are proposed to determine mean square exponential stability for (1.5). Furthermore, we also establish some sufficient conditions for checking exponential stability for Cohen-Grossberg neural networks with unbounded distributed delays in this section. In Section 4, an illustrative examples is given. We conclude this paper in Section 5.

2. Preliminaries

Noting that a vector 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈𝑅^𝑛 usually can be equipped with the common norms ‖𝑥‖ as ‖𝑥‖=(^𝑛_𝑖=1|𝑥_𝑖|²)^1/2. (2.1) For the sake of convenience, some of the standing definitions and assumptions are formulated below.

Definition 2.1 (see [17]). The trivial solution of (1.5) is said to be mean square exponentially stable if there is a pair of positive constants 𝜆 and 𝐺 such that 𝐸‖𝑥(𝑡;𝑡₀,𝑥₀)‖<𝐺‖𝑥₀‖𝑒^{−𝜆(𝑡−𝑡0)},on𝑡≥𝑡₀ (2.2) for all 𝑥₀∈𝑅^𝑛, where 𝜆 also called as convergence rate.
One also assumes that(𝐻₁) there exist positive constants 𝐿_𝑗,𝐺_𝑗,𝑗=1,…,𝑛, such that |𝑓_𝑗(𝑢)−𝑓_𝑗(𝑣)|≤𝐿_𝑗|𝑔|𝑢−𝑣|,∀𝑢,𝑣∈𝑅,_𝑗(𝑢)−𝑔_𝑗(𝑣)|≤𝐺_𝑗|𝑢−𝑣|,∀𝑢,𝑣∈𝑅; (2.3)(𝐻₂)There exist positive constants 𝑏_𝑗, such that (𝑢−𝑣)[𝑏̃_𝑗(𝑢)−𝑏̃_𝑗(𝑣)]≥𝑏_𝑗(𝑢−𝑣)²,∀𝑢,𝑣∈𝑅; (2.4)(𝐻₃) There exist positive constants 𝛼_𝑖,𝛼_𝑖, such that 𝛼_𝑖≤𝑎_𝑖(𝑥_𝑖(𝑡))≤𝛼_𝑖; (2.5)(𝐻₄) Assume 𝜎_𝑖𝑗(𝑥^∗_𝑗)=0 (𝑥^∗=(𝑥^∗₁,…,𝑥^∗_𝑛)^𝑇 to be determined later), and there exist positive constants 𝑀_𝑖,𝑗,𝑖,𝑗=1,…,𝑛, such that |𝜎_𝑖𝑗(𝑢)−𝜎_𝑖𝑗(𝑣)|≤𝑀_𝑖𝑗|𝑢−𝑣|,∀𝑢,𝑣∈𝑅. (2.6)

Remark 2.2. The activation functions are typically assumed to be continuous, bounded, differentiable, and monotonically increasing, such as the functions of sigmoid type; these conditions are no longer needed in this paper. For example, when neural networks are designed for solving optimization problems in the presence of constraints (linear, quadratic, or more general programming problems), unbounded activations modelled by diode-like exponential-type functions are needed to impose constraints satisfaction [3]. In this paper, the activation functions 𝑓_𝑖(⋅),𝑔_𝑖(⋅) also including some kinds of typical functions widely used in the circuit designs, such as nondifferentiable piecewise linear output functions of the form 𝑓(𝑢)=(1/2)(|𝑢−1|−|𝑢+1|), nonmonotonically increasing functions of the form Gaussian and inverse Gaussian functions, see [4, 5, 23] and references therein.

Using variable substitution, (𝑡−𝑠)↦𝑠^′, we get ^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑥_𝑗(𝑠)d𝑠=^∞₀𝐾_𝑖𝑗(𝑠^′)𝑥_𝑗(𝑡−𝑠^′)d𝑠^′, (2.7) therefore, system (1.5) for convenience can be put in the formd𝑥_𝑖(𝑡)=𝑎_𝑖(𝑥_𝑖(𝑡))[−𝑏̃_𝑖(𝑥_𝑖(𝑡))+^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡))+^𝑛_𝑗=1𝑑_𝑖𝑗𝑔_𝑗(^∞₀𝐾_𝑖𝑗(𝑠)𝑥_𝑗(𝑡−𝑠)d𝑠)+𝐼_𝑖+]d𝑡_𝑛𝑗=1𝜎_𝑖𝑗(𝑥_𝑗(𝑡))d𝜔_𝑗(𝑡). (2.8) As usual, the initial conditions for system (2.8) are 𝑥(𝑡)=𝜑(𝑡), −∞<𝑡≤0, 𝜑∈𝐿²_ℱ0((−∞,0],𝑅^𝑛), here 𝐿²_ℱ0((−∞,0],𝑅^𝑛) is the family of all ℱ₀-measurable 𝑅^𝑛-valued random variables satisfying that 𝐸{sup_{−∞≤𝑠≤0}|𝜑(𝑠)|²}<∞.

The conditions (𝐻₁) and (𝐻₄) imply that (2.8) has a unique global solution on 𝑡≥0 for the initial conditions[17].

If 𝑉∈𝐶^1,2(𝑅×𝑅^𝑛;𝑅₊), define an operator ℒ𝑉 associated with (2.8) asℒ𝑉(𝑡,𝑥)=𝑉_𝑡(𝑡,𝑥)−^𝑛_𝑖=1𝜕𝑉(𝑡,𝑥)𝜕𝑥_𝑛𝑎_𝑖(𝑥_𝑖𝑏̃(𝑡))×[−_𝑖(𝑥_𝑖(𝑡))+^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡))+^𝑛_𝑗=1𝑑_𝑖𝑗𝑔_𝑗(^∞₀𝐾_𝑖𝑗(𝑠)𝑥_𝑗(𝑡−𝑠)d𝑠)+𝐼_𝑖]+12trace[𝜎^𝑇𝑉_𝑥𝑥(𝑡,𝑥)𝜎], (2.9) where 𝑉_𝑡(𝑡,𝑥)=𝜕𝑉(𝑡,𝑥)/𝜕𝑡, 𝑉_𝑥𝑥(𝑡,𝑥)=(𝜕²𝑉(𝑡,𝑥)/𝜕𝑥_𝑖𝜕𝑥_𝑗)_𝑛×𝑛.

We always assume that the delay kernels 𝐾_𝑖𝑗,𝑖,𝑗=1,…,𝑛 to be real-valued nonnegative functions defined on [0,∞) and satisfy^∞₀𝐾_𝑖𝑗(𝑠)d𝑠=1,^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜇𝑠d𝑠<∞, (2.10) for some positive constant 𝜇. A typical example of such delay kernel function is given by 𝐾_𝑖𝑗(𝑠)=(𝑠^𝑟/𝑟!)𝛾^𝑟+1_𝑖𝑗𝑒^{−𝛾𝑖𝑗𝑠} for 𝑠∈[0,∞), where 𝛾_𝑖𝑗∈[0,∞),𝑟∈{0,1,…,𝑛}. These kernels have been used in [6, 9, 24] for various stability investigations on neural network models. In [24], these kernels are called as the Gamma Memory Filter.

3. Main Results

A point 𝑥^∗=(𝑥^∗₁,…,𝑥^∗_𝑛)^𝑇 in 𝑅^𝑛 is called to be an equilibrium (or trivial solution) of system (1.4) if this point 𝑥^∗ satisfies the following: −𝑏̃_𝑖(𝑥^∗_𝑖)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑥^∗_𝑗)+^𝑛_𝑗=1𝑑_𝑖𝑗𝑔_𝑗(𝑥^∗_𝑗)+𝐼_𝑖=0. (3.1) Different from the bounded activation function case where the existence of an equilibrium point is always guaranteed [25], for unbounded activations, it may happen that there is no equilibrium point [26]. In fact, we have the following theorem.

Theorem 3.1. Under the assumptions (𝐻₁),(𝐻₂), if there exist a positive diagonal matrix 𝑄=diag(𝑞₁,…,𝑞_𝑛), such that Ξ_𝑖=𝑝𝑞_𝑖𝑏_𝑖−[𝑞_𝑖^𝑛_𝑗=1|𝑐_𝑖𝑗|𝐿_𝑗+^𝑛_𝑗=1𝑞_𝑗|𝑐_𝑗𝑖|𝐿_𝑖+𝑞_𝑖^𝑛_𝑗=1|𝑑_𝑖𝑗|𝐺_𝑗+^𝑛_𝑗=1𝑞_𝑗|𝑑_𝑗𝑖|𝐺_𝑖]>0, (3.2) then for every input 𝐼=(𝐼₁,…,𝐼_𝑛)^𝑇, system (1.4) has a unique equilibrium 𝑥^∗=(𝑥^∗₁,…,𝑥^∗_𝑛)^𝑇.

Proof of Theorem 3.1. A point 𝑥^∗=(𝑥^∗₁,…,𝑥^∗_𝑛)^𝑇 is said to be an equilibrium of system (1.5) if this point 𝑥^∗ satisfies the following: −𝑏̃_𝑖(𝑥^∗_𝑖)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑥^∗_𝑗)+^𝑛_𝑗=1𝑑_𝑖𝑗𝑔_𝑗(𝑥^∗_𝑗)+𝐼_𝑖=0. (3.3) Let 𝐻_𝑖(𝑥)=−𝑏̃_𝑖(𝑥_𝑖)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑥_𝑗)+^𝑛_𝑗=1𝑑_𝑖𝑗𝑔_𝑗(𝑥_𝑗)+𝐼_𝑖,𝐻(𝑥)=(𝐻₁(𝑥),…,𝐻_𝑛(𝑥))^𝑇. (3.4) Similar to the proofs of Theorems 1, 2, and 3 of [25], one can get that 𝐻 is injective on 𝑅^𝑛, and lim_{‖𝑦‖→∞}‖𝐻(𝑦)‖→∞. Then, 𝐻 is homeomorphism on 𝑅^𝑛, therefore, 𝐻 has a unique root on 𝑅^𝑛. This completes the proof.

Obviously, the following inequality (3.5) impling the inequality (3.2) holds if the assumptions (𝐻₁),(H₂),(𝐻₄) are true, then system (2.8) admits an equilibrium solution 𝑥(𝑡)≡𝑥^∗. We have the following theorems on the stochastic stability of the unique equilibrium 𝑥^∗=(𝑥^∗₁,…,𝑥^∗_𝑛)^𝑇.

Theorem 3.2. Under the assumptions (𝐻₁)– (𝐻₄), if there exist a positive diagonal matrix 𝑄=diag(𝑞₁,…,𝑞_𝑛), such that 2𝑞_𝑖𝛼_𝑖𝑏_𝑖>𝑞_𝑖𝛼_𝑖^𝑛_𝑗=1|𝑐_𝑖𝑗|𝐿_𝑗+^𝑛_𝑗=1𝑞_𝑗𝛼_𝑗|𝑐_𝑗𝑖|𝐿_𝑖+𝑞_𝑖𝛼_𝑖^𝑛_𝑗=1|𝑑_𝑖𝑗|𝐺_𝑗+^𝑛_𝑗=1𝑞_𝑗𝛼_𝑗|𝑑_𝑗𝑖|𝐺_𝑖+^𝑛_𝑗=1𝑞_𝑗𝛼_𝑗|𝑀_𝑗𝑖|², (3.5) then the trivial solution of system (2.8) is mean square exponentially stable.

Proof of Theorem 3.2. Let 𝑦_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥^∗_𝑖, 𝜙_𝑖(𝑡)=𝜑_𝑖(𝑡)−𝑥^∗_𝑖, 𝑦(𝑡)=(𝑦₁(𝑡),…,𝑦_𝑛(𝑡))^𝑇, 𝜙(𝑡)=(𝜙₁(𝑡),…,𝜙_𝑛(𝑡))^𝑇, using (3.3), model (2.8) take the following form: d𝑦_𝑖(𝑡)=−𝑎_𝑖(𝑦_𝑖(𝑡))[𝑏̂_𝑖(𝑦_𝑖(𝑡))−^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑦_𝑗(𝑡))−^𝑛_𝑗=1𝑑_𝑖𝑗𝑔_𝑗(^∞₀𝐾_𝑖𝑗(𝑠)𝑦_𝑗+(𝑡−𝑠)d𝑠)]d𝑡_𝑛𝑗=1𝜎_𝑖𝑗(𝑦_𝑗(𝑡))d𝜔_𝑗(𝑡), (3.6) with the initial condition 𝑦(𝑡)=𝜙(𝑡),𝑡∈(−∞,0], where 𝑎_𝑖(𝑦_𝑖(𝑡))=𝑎_𝑖(𝑥_𝑖(𝑡)+𝑥^∗_𝑖), 𝑏̂_𝑖(𝑦_𝑖(𝑡))=𝑏̃_𝑖(𝑥_𝑖(𝑡))−𝑏̃_𝑖(𝑥^∗_𝑖), 𝑓_𝑗(𝑦_𝑗(𝑡))=𝑓_𝑗(𝑥_𝑗(𝑡))−𝑓_𝑗(𝑥^∗_𝑗), 𝑓_𝑗(𝑦_𝑗(𝑡))=𝑔_𝑗(𝑥_𝑗(𝑡))−𝑔_𝑗(𝑥^∗_𝑗), and 𝜎_𝑖𝑗(𝑦_𝑗(𝑡))=𝜎_𝑖𝑗(𝑥_𝑗(𝑡))−𝜎_𝑖𝑗(𝑥^∗_𝑗).
Then the assumptions (𝐻₁), (𝐻₃), (𝐻₄) can be transformed as follows: (𝐻^′₁)𝑓_𝑗(0)=𝑔_𝑗(0)=0 and there exist positive constants 𝐿_𝑗,𝐺_𝑗,𝑗=1,…,𝑛, such that |𝑓_𝑗(𝑢)−𝑓_𝑗(𝑣)|≤𝐿_𝑗|𝑢−𝑣|,∀𝑢,𝑣∈𝑅,|𝑔_𝑗(𝑢)−𝑔_𝑗(𝑣)|≤𝐺_𝑗|𝑢−𝑣|,∀𝑢,𝑣∈𝑅; (3.7)(𝐻^′₃)There exist positive constants 𝑏_𝑗, such that 𝑦_𝑗(𝑡)𝑏̂_𝑗(𝑦_𝑗(𝑡))≥𝑏_𝑗𝑦²_𝑗(𝑡); (3.8)(𝐻^′₄)𝜎_𝑖𝑗(0)=0 and there exist positive constants 𝑀_𝑖,𝑗,𝑖,𝑗=1,…,𝑛, such that |𝜎_𝑖𝑗(𝑢)−𝜎_𝑖𝑗(𝑣)|≤𝑀_𝑖𝑗|𝑢−𝑣|,∀𝑢,𝑣∈𝑅. (3.9)From assumptions (2.10) and (3.5), we pick a constant 𝜆: 0<𝜆<𝜇 satisfying the following requirements: ^∞₀𝐾_𝑗𝑖(𝑠)𝑒^𝜆𝑠𝑠d𝑠<∞,2𝑞_𝑖𝛼_𝑖𝑏_𝑖>2𝜆𝑞_𝑖+𝑞_𝑖𝛼_𝑖^𝑛_𝑗=1|𝑐_𝑖𝑗|𝐿_𝑗+^𝑛_𝑗=1𝑞_𝑗𝛼_𝑗|𝑐_𝑗𝑖|𝐿_𝑖+𝑞_𝑖𝛼_𝑖^𝑛_𝑗=1|𝑑_𝑖𝑗|𝐺_𝑗^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠+d𝑠_𝑛𝑗=1𝑞_𝑗𝛼_𝑗|𝑑_𝑗𝑖|𝐺_𝑖^∞₀𝐾_𝑗𝑖(𝑠)𝑒^𝜆𝑠d𝑠+^𝑛_𝑗=1𝑞_𝑗𝛼_𝑗|𝑀_𝑗𝑖|². (3.10)(3.11) Define the following Lyapunov functional 𝑉(𝑡,𝑥(𝑡))=𝑉₁+𝑉₂=𝑒^{𝑛2𝜆𝑡}_𝑖=1𝑞_𝑖|𝑥_𝑖(𝑡)|²+^𝑛𝑛_𝑖=1𝑗=1𝑞_𝑖𝛼_𝑖|𝑑_𝑖𝑗|𝐺_𝑗^∞₀[𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠(^𝑡_𝑡−𝑠𝑒^2𝜆𝜉|𝑥_𝑗(𝜉)|²d𝜉)]d𝑠. (3.12) As 𝑉 at most has one point 𝑦=0 where it is not differentiable, we can calculate 𝑉_𝑦,𝑉_𝑦𝑦 by instead using Dini derivative. Obviously, calculate the operator ℒ𝑉₁ associated with (3.6) as follows: ℒ𝑉₁=2𝜆𝑒^{𝑛2𝜆𝑡}_𝑖=1𝑞_𝑖|𝑦_𝑖(𝑡)|²d𝑡+2𝑒^{𝑛2𝜆𝑡}_𝑖=1𝑞_𝑖𝑦_𝑖(𝑡)d𝑦_𝑖(𝑡)=𝑒^2𝜆𝑡{^𝑛_𝑖=12𝜆𝑞_𝑖|𝑦_𝑖(𝑡)|²−^𝑛_𝑖=1𝑝𝑞_𝑖𝑦_𝑖(𝑡)𝑎_𝑖(𝑦_𝑖𝑏̂(𝑡))×[_𝑖(𝑦_𝑖(𝑡))−^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑦_𝑗(𝑡))−^𝑛_𝑗=1𝑑_𝑖𝑗𝑔_𝑗(^∞₀𝐾_𝑖𝑗(𝑠)𝑦_𝑗+(𝑡−𝑠)d𝑠)]_𝑛𝑖=1𝑞_𝑖^𝑛_𝑗=1𝑀²_𝑖𝑗𝑦²_𝑗(𝑡)}d𝑡≤𝑒^2𝜆𝑡{^𝑛_𝑖=12𝜆𝑞_𝑖|𝑦_𝑖(𝑡)|²−^𝑛_𝑖=1𝑞_𝑖𝛼_𝑖2𝑏_𝑖|𝑦_𝑖(𝑡)|²+^𝑛𝑛_𝑖=1𝑗=1𝑞_𝑖𝛼_𝑖2|𝑐_𝑖𝑗|𝐿_𝑗|𝑦_𝑖(𝑡)||𝑦_𝑗+(𝑡)|_{𝑛𝑛𝑖=1𝑗=1}𝑞_𝑖𝛼_𝑖^∞₀𝐾_𝑖𝑗(𝑠)2|𝑑_𝑖𝑗|𝐺_𝑗|𝑦_𝑖(𝑡)𝑦_𝑗+(𝑡−𝑠)|d𝑠_{𝑛𝑛𝑖=1𝑗=1}𝑞_𝑖𝛼_𝑖𝑀²_𝑖𝑗𝑦²_𝑗(𝑡)}d𝑡(usingconditions(𝐻^′₁),(𝐻₂),(𝐻^′₃),(𝐻^′₄)). (3.13) On the other hand, by directly applying the elementary inequality 2𝑎𝑏≤𝑎²+𝑏², we have ℒ𝑉₁≤𝑒^2𝜆𝑡{^𝑛_𝑖=12𝜆𝑞_𝑖|𝑦_𝑖(𝑡)|²−^𝑛_𝑖=1𝑞_𝑖𝛼_𝑖2𝑏_𝑖|𝑦_𝑖(𝑡)|²+^𝑛𝑛_𝑖=1𝑗=1𝑞_𝑖𝛼_𝑖|𝑐_𝑖𝑗|𝐿_𝑗(|𝑦_𝑖(𝑡)|²+|𝑦_𝑗(𝑡)|²)+^𝑛𝑛_𝑖=1𝑗=1𝑞_𝑖𝛼_𝑖^∞₀𝐾_𝑖𝑗(𝑠)|𝑑_𝑖𝑗|𝐺_𝑗(|𝑦_𝑖(𝑡)|²+|𝑦_𝑗(𝑡−𝑠)|²+)d𝑠_{𝑛𝑛𝑖=1𝑗=1}𝑞_𝑖𝛼_𝑖𝑀²_𝑖𝑗𝑦²_𝑗(𝑡)}d𝑡=−𝑒^{𝑛2𝜆𝑡}_𝑖=1|𝑦_𝑖(𝑡)|²{2𝑞_𝑖𝛼_𝑖𝑏_𝑖−2𝜆𝑞_𝑖−^𝑛_𝑗=1𝑞_𝑖𝛼_𝑖|𝑐_𝑖𝑗|𝐿_𝑗−^𝑛_𝑗=1𝑞_𝑗𝛼_𝑗|𝑐_𝑗𝑖|𝐿_𝑖−^𝑛_𝑗=1𝑞_𝑗𝛼_𝑗𝑀²_𝑗𝑖−^𝑛_𝑗=1𝑞_𝑖𝛼_𝑖|𝑑_𝑖𝑗|𝐺_𝑗^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠d𝑠}d𝑡+𝑒^2𝜆𝑡[^𝑛𝑛_𝑖=1𝑗=1𝑞_𝑖𝛼_𝑖|𝑑_𝑖𝑗|𝐺_𝑗^∞₀𝐾_𝑖𝑗(𝑠)|𝑦_𝑗(𝑡−𝑠)|²d𝑠]d𝑡. (3.14) At the same time, calculate the operator ℒ𝑉₂ associated with (3.6) as follows: ℒ𝑉₂=^𝑛𝑛_𝑖=1𝑗=1𝑞_𝑖𝛼_𝑖|𝑑_𝑖𝑗|𝐺_𝑗^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠[𝑒^2𝜆𝑡|𝑦_𝑗(𝑡)|²−𝑒^{2𝜆(𝑡−𝑠)}|𝑦_𝑗(𝑡−𝑠)|²]d𝑠d𝑡≤𝑒^2𝜆𝑡[^𝑛𝑛_𝑖=1𝑗=1𝑞_𝑗𝛼_𝑗|𝑑_𝑗𝑖|𝐺_𝑖|𝑦_𝑖(𝑡)|^2∞₀𝐾_𝑗𝑖(s)𝑒^𝜆𝑠d𝑠]d𝑡−𝑒^2𝜆𝑡[^𝑛𝑛_𝑖=1𝑗=1𝑞_𝑖𝛼_𝑖|𝑑_𝑖𝑗|𝐺_𝑗^∞₀𝐾_𝑖𝑗(𝑠)|𝑦_𝑗(𝑡−𝑠)|²d𝑠]d𝑡. (3.15) From (3.14) and (3.15), it is easy to get ℒ𝑉≤−𝑒^{𝑛2𝜆𝑡}_𝑖=1|𝑦_𝑖(𝑡)|²{2𝑞_𝑖𝛼_𝑖𝑏_𝑖−2𝜆𝑞_𝑖−𝑞_𝑖𝛼_i^𝑛_𝑗=1|𝑐_𝑖𝑗|𝐿_𝑗−^𝑛_𝑗=1𝑞_𝑗𝛼_𝑗|𝑐_𝑗𝑖|𝐿_𝑖−𝑞_𝑖𝛼_𝑖^𝑛_𝑗=1|𝑑_𝑖𝑗|𝐺_𝑗^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠d𝑠−^𝑛_𝑗=1𝑞_𝑗𝛼_𝑗|𝑑_𝑗𝑖|𝐺_𝑖×^∞₀𝐾_𝑗𝑖(𝑠)𝑒^𝜆𝑠d𝑠−^𝑛_𝑗=1𝑞_𝑗𝛼_𝑗|𝑀_𝑗𝑖|²}d𝑡. (3.16) Using the Ito formula, for 𝛿>0, from inequality (3.11) and inequality (3.16), we have 𝑉(𝛿,𝑦(𝛿))≤𝑉(0,𝑦(0))+𝑝^𝛿₀^𝑛𝑛_𝑖=1𝑗=1𝑞_𝑖𝑒^𝑝𝜆𝑡|𝑦_𝑖(𝑡)𝜎_𝑖𝑗(𝑦_𝑗(𝑡))|d𝑤_𝑗(𝑡). (3.17) On the other hand, we observe that 𝑉(0,𝑦(0))=^𝑛_𝑖=1𝑞_𝑖|𝑦_𝑖(0)|²+^𝑛𝑛_𝑖=1𝑗=1𝑞_𝑖𝛼_𝑖|𝑑_𝑖𝑗|𝐺_𝑗^∞₀[𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠(⁰_−𝑠𝑒^2𝜆𝜉|𝑥_𝑗(𝜉)|²≤d𝜉)]d𝑠_𝑛𝑖=1𝑞_𝑖[1+^𝑛_𝑗=1𝑞_𝑗𝑞_𝑖𝛼_𝑗|𝑑_𝑗𝑖|𝐺_𝑖^∞₀𝐾_𝑗𝑖(𝑠)𝑒^𝜆𝑠𝑠d𝑠]sup_{−∞≤𝑠≤0}{|𝑦_𝑖(𝑠)|²}. (3.18) From inequality (3.10), we have 𝛼_𝑗|𝑑_𝑗𝑖|𝐺_𝑖^∞₀𝐾_𝑗𝑖(𝑠)𝑒^𝜆𝑠𝑠d𝑠<∞. (3.19) Hence, 𝑉(0,𝑦(0)) is bounded. According to stochastic analysis theory [17], we get 𝐸(^𝛿₀𝑉_𝑦(𝑡,𝑦(𝑡))𝜎(𝑦(𝑡))d𝑤(𝑡))=0. (3.20) Take expectation on both sides of (3.17) yields 𝐸(𝑉(𝛿,𝑦(𝛿)))≤𝐸(𝑉(0,𝑦(0)))<∞. (3.21) That is to say 𝑒^{𝑛𝑝𝜆𝛿}_𝑖=1𝑞_𝑖𝐸(|𝑦_𝑖(𝑡)|^𝑝)≤𝐸(𝑉(𝛿,𝑦(𝛿)))<∞. (3.22) Therefore, there exist a positive constant 𝐺, such that 𝐸‖𝑦(𝑡;𝑡₀,𝑦₀)‖<𝐺‖𝑦₀‖𝑒^−2𝜆𝑡,on𝑡≥0, (3.23) then the trivial solution of system (3.6) is mean square exponentially stable that is to say, the trivial solution of system (2.8) is mean square exponentially stable. This completes the proof.

Furthermore, if we remove noise from the system, then system (2.8) turns out to be system (1.4), the derived conditions for exponential stability of system (1.4) can be viewed as byproducts of our results from general SRNN. For convenience of reading, we provide the definition of the exponential stability.

Definition 3.3. The trivial solution of CGNN model (1.4) is said to be globally exponentially stable if there exist positive constants 𝐺,𝜆, which are independent of the initial values of the system, such that ‖𝑥(𝑡,(0,𝜑))‖≤𝐺‖𝜙‖𝑒^−𝜆𝑡,for𝑡>0. (3.24) Where, the initial values take the following continuous function 𝑥_𝑖(𝑡)=𝜑_𝑖(𝑡),−∞≤𝑡≤0,‖𝜙‖=[^𝑛_𝑖=1(sup_{−∞≤𝑡≤0}|𝜑_𝑖(𝑡)|²)]^1/2. (3.25)

We have the following Corollary 3.4 for system (1.4). As the main idea come from Theorem 3.2, so the proof is trivial, we omit it.

Corollary 3.4. Under the assumptions (𝐻₁),(𝐻₂),(𝐻₃), if there exist a positive diagonal matrix 𝑄=diag(𝑞₁,…,𝑞_𝑛), such that 2𝑞_𝑖𝛼_𝑖𝑏_𝑖>𝑞_𝑖𝛼∑_𝑖^𝑛_𝑗=1|𝑐_𝑖𝑗|𝐿_𝑗+∑^𝑛_𝑗=1𝑞_𝑗𝛼_𝑗|𝑐_𝑗𝑖|𝐿_𝑖+𝑞_𝑖𝛼∑_𝑖^𝑛_𝑗=1|𝑑_𝑖𝑗|𝐺_𝑗+∑^𝑛_𝑗=1𝑞_𝑗𝛼_𝑗|𝑑_𝑗𝑖|𝐺_𝑖, (3.26) then the trivial solution of system (1.5) is globally exponentially stable.

Remark 3.5. To the best of our knowledge, few authors have considered exponential stability for recurrent neural networks with unbounded distributed delays. We can find paper [27] in this direction. However, it is assumed in [27] that 𝑘_𝑖𝑗(⋅) satisfies: (i) ∫^∞₀𝐾_𝑖𝑗(𝑠)d𝑠=1; (ii) ∫^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜇𝑠d𝑠=1; (iii) ∫^∞₀𝑠𝐾_𝑖𝑗(𝑠)d𝑠<∞. Obviously, these requirements are strong, and our paper relax assumptions on 𝑘_𝑖𝑗(⋅). In fact, using assumptions (2.10) instead of assumptions (i), (ii), (iii), and choosing the same Lyapunov functionals in Theorem 3.2 of this paper, one can get the same results as those in [27].

Remark 3.6. We notice that Wang et al. developed a linear matrix inequality (LMI) approach to study the stability of SCGNN with mixed delays and obtained some novel results in a very recent paper [20], where the considered model including both discrete and distributed delays. Yet the distributed delays are bounded. In fact, a neural network model with unbounded distributed delay is more general than that with discrete and bounded distributed delays, this is because the distributed delay becomes a discrete delay when the delay kernel is a 𝛿 function at a certain time.

Remark 3.7. The results in this paper show that, the convergence criteria on SGNN with unbounded distributed delays are independent of delays, but are dependent of the magnitude of noise, and therefore, noisy fluctuations should be regarded adequately when design SGNN.

Remark 3.8. Comparing our results with the previous results derived in the literature for the usual continuously distributed delays CNN without stochastic perturbation, by Corollary 3.4, we can find that the corresponding main results obtained in [8–10] are trivial.

4. Illustrative Examples

In this section, an example is presented to demonstrate the correctness and effectiveness of the main obtained results in this paper.

Example 4.1. Consider the following stochastic neural networks with distributed delays: d𝑥₁(𝑡)=𝑎₁(𝑥₁(𝑡)){−5𝑥₁1(𝑡)+5𝑓₁(𝑥₁2(𝑡))+5𝑓₂(𝑥₂+4(𝑡))5𝑔₁(^∞₀𝐾₁₁(𝑠)𝑥₁1(𝑡−𝑠)d𝑠)+5𝑔₂(^∞₀𝐾₁₂(𝑠)𝑥₂+1(𝑡−𝑠)d𝑠)}d𝑡𝑥10¹(𝑡)d𝑤₁1(𝑡)+𝑥10²(𝑡)d𝑤₂(𝑡),d𝑥₂(𝑡)=𝑎₂(𝑥₂(𝑡)){−6𝑥₂3(𝑡)+5𝑓₁(𝑥₁(𝑡))−𝑓₂(𝑥₂(𝑡))+𝑔₁(^∞₀𝐾₂₁(𝑠)𝑥₁+2(𝑡−𝑠)d𝑠)5𝑔₂(^∞₀𝐾₂₂(𝑠)𝑥₂+1(𝑡−𝑠)d𝑠)}d𝑡𝑥10¹(𝑡)d𝑤₁1(𝑡)+𝑥10²(𝑡)d𝑤₂(𝑡),𝑡≥0. (4.1) Figure 1 shows the schematic of the entire delayed neural network, where the nonlinear neuron transfer function (𝑆) is constructed by using the voltage operational amplifiers. The time delay is achieved by using a digital signal processor (DSP) with an analog-to-digital converter (ADC) and a digital-to-analog converter (DAC). In the experiment, the circuit of time delay consists of a TMS320F2812 device and a DAC7724 device. Here, white noise is generated by signal generator Agient E8257D.

Figure 1 

An analog circuit implementing of system (4.1).

In the example, let𝑎₁(𝑥₁(𝑡))=3+sin𝑥₁(𝑡),𝑎₂(𝑥₂(𝑡))=4+cos𝑥₁(𝑡),𝐾(𝑠)=2𝑒^−2𝑠4𝑒^−4𝑠3𝑒^−3𝑠5𝑒^−5𝑠, (4.2) and 𝑓_𝑖(𝑢)=tanh(𝑢),𝑔_𝑖(𝑢)=(1/2)(|𝑢−1|−|𝑢+1|). Notice that each delay kernel 𝐾_𝑖𝑗(⋅) satisfies (2.10), and each 𝑓_𝑗(⋅),𝑔_𝑗(⋅),𝜎_𝑖𝑗(⋅) satisfies assumptions (𝐻₁),(𝐻₄). In the example, let 𝑝=3,𝑞_𝑖=1, by simple computation, one can easily get that20>16.49,36>24.89. (4.3) It follows from Corollary 3.4 that system (4.1) is mean square exponentially stable.

5. Conclusions

In this paper, we studied a nonlinear continuous-time stochastic Cohen-Grossberg neural networks (SCGNN) with unbounded distributed delays. Without assuming the smoothness, monotonicity and boundedness of the activation functions, by applying Lyapunov functional method, stochastic analysis technique, and inequality techniques, some novel sufficient conditions on mean square exponential stability for SCGNN are given. Furthermore, as the byproducts of our main results, we also establish some sufficient conditions for checking exponential stability for Cohen-Grossberg neural networks with unbounded distributed delays. The significant of this paper does offer a wider selection on the networks parameters in order to achieve some necessary convergence in practice.

Acknowledgments

The authors are extremely grateful to Professor Juan J. Nieto and the anonymous reviewers for their constructive and valuable comments, which have contributed a lot to the improved presentation of this paper. This work was supported by the Foundation of Chinese Society for Electrical Engineering (2008), the Excellent Youth Foundation of Educational Committee of Hunan Provincial (10B002), the National Natural Science Funds of China for Distinguished Young Scholar (50925727), and the National Natural Science Foundation of China (60876022).