Stability of Stochastic Reaction-Diffusion Recurrent Neural Networks with Unbounded Distributed Delays

Abstract

Stability of reaction-diffusion recurrent neural networks (RNNs) with continuously distributed delays and stochastic influence are considered. Some new sufficient conditions to guarantee the almost sure exponential stability and mean square exponential stability of an equilibrium solution are obtained, respectively. Lyapunov's functional method, M-matrix properties, some inequality technique, and nonnegative semimartingale convergence theorem are used in our approach. The obtained conclusions improve some published results.

1. Introduction

For decades, studies have been intensively focused on recurrent neural networks (RNNs) because of the successful hardware implementation and their various applications such as classification, associative memories, parallel computation, optimization, signal processing and pattern recognition, see, for example, [1–3]. These applications rely crucially on the analysis of the dynamical behavior of neural networks. Recently, it has been realized that the axonal signal transmission delays often occur in various neural networks and may cause undesirable dynamic network behaviors such as oscillation and instability. Consequently, the stability analysis problems resting with delayed recurrent neural networks (DRNNs) have drawn considerable attention. To date, a great deal of results on DRNNs have been reported in the literature, see, for example, [4–9] and references therein. To a large extent, the existing literature on theoretical studies of DRNNs is predominantly concerned with deterministic differential equations. The literature dealing with the inherent randomness associated with signal transmission seems to be scarce; such studies are, however, important for us to understand the dynamical characteristics of neuron behavior in random environments 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 [10]; (ii) it has been realized that a neural network could be stabilized or destabilized by certain stochastic effects [11, 12].

Although systems are often perturbed by various types of environmental “noise”, 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 [12, 13]. More detailed mechanism of the stochastic effects on the interaction of neurons and analog circuit implementing can be found in [13, 14]. However, because the Brownian motion 𝜔(𝑡) is nowhere differentiable, the derivative of Brownian motion d𝜔(𝑡)/d𝑡 can not be defined in the ordinary way, the stability analysis for stochastic neural networks is difficult. Some initial results have just appeared, for example, [11, 15–23]. In [11, 15], Liao and Mao discussed the exponentially stability of stochastic recurrent neural networks (SRNNs); in [16], the authors continued their research to discuss almost sure exponential stability for a class of stochastic CNN with discrete delays by using the nonnegative semimartingale convergence theorem; in [18], exponential stability of SRNNs via Razumikhin-type was investigated; in [17], Wan and Sun investigated mean square exponential stability of stochastic delayed Hopfield neural networks(HNN); in [19], Zhao and Ding studied almost sure exponential stability of SRNN; in [20], Sun and Cao investigated 𝑝th moment exponential stability of stochastic recurrent neural networks with time-varying delays.

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 [24], 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 [5, 25, 26]. In [27], Wang et al. developed a linear matrix inequality (LMI) approach to study the stability of SRNNs with mixed delays. To the best of the authors' knowledge, few authors investigated the convergence dynamics of SRNNs with unbounded distributed delays. On the other hand, if the RNNs depend on only time or instantaneously time and time delay, the model is in fact an ordinary differential equation or a functional differential equation. In the factual operations, however, the diffusion phenomena could not be ignored in neural networks and electric circuits once electrons transport in a nonuniform electromagnetic field. Hence, it is essential to consider the state variables varying with the time and space variables. The neural networks with diffusion terms can commonly be expressed by partial differential equations [28–30].

Keeping this in mind, in this paper, we consider the SRNNs described by the following stochastic reaction-diffusion RNNs d𝑦_𝑖(𝑡,𝑥)=^𝑚_𝑘=1𝜕𝜕𝑥_𝑘𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘+d𝑡−𝑐_𝑖ℎ_𝑖𝑦_𝑖(𝑡,𝑥)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡,𝑥)+^𝑛_𝑗=1𝑏_𝑖𝑗^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑔_𝑗𝑦_𝑗()+𝑠,𝑥d𝑠d𝑡_∞𝑙=1𝜎_𝑖𝑙𝑦_𝑖(𝑡,𝑥)d𝜔_𝑖𝑙(𝑡),𝑥∈𝑋,𝜕𝑦_𝑖𝜕𝑛∶=𝜕𝑦_𝑖𝜕𝑥₁,…,𝜕𝑦_𝑖𝜕𝑥_𝑚^𝑇𝑦=0,𝑡≥0,𝑖=1,…,𝑛,𝑥∈𝜕𝑋,_𝑖=𝑦_𝑖(𝑠,𝑥)=𝜉_𝑖(𝑠,𝑥),−∞<𝑠≤0,𝑥∈𝑋. (1.1) In the above model, 𝑛≥2 is the number of neurons in the network, 𝑥_𝑖 is space variable, 𝑦_𝑖(𝑡,𝑥) is the state variable of the 𝑖th neuron at time 𝑡 and in space variable 𝑥, 𝑓_𝑗(𝑦_𝑗(𝑡,𝑥)), and 𝑔_𝑗(𝑦_𝑗(𝑡,𝑥)) denote the output of the 𝑗th unit at time 𝑡 and in space variable 𝑥; 𝑐_𝑖, 𝑎_𝑖𝑗, 𝑏_𝑖𝑗 are constants: 𝑐_𝑖 represents the rate with which the 𝑖th unit will reset its potential to the resting state in isolation when disconnected from the network and the external stochastic perturbation, and is a positive constant; 𝑎_𝑖𝑗 and 𝑏_𝑖𝑗 weight the strength of the 𝑗th unit on the 𝑖th unit at time 𝑡. Moreover, {𝑤_𝑖𝑙(𝑡):𝑖=1,…,𝑛,𝑙∈𝑁} are independent scalar standard Wiener processes on the complete probability space (Ω,ℱ,P) with the natural filtration {ℱ_𝑡}_𝑡≥0 generated by the standard Wiener process {𝑤(𝑠):0≤𝑠≤𝑡} which is independent of 𝑤_𝑖𝑙(𝑡), where we associate Ω with the canonical space generated by 𝑤(𝑡), and denote by ℱ the associated 𝜎-algebra generated by 𝑤(𝑡) with the probability measure 𝑃; Furthermore, we assume the following boundary condition(𝐴₀)smooth function 𝐷_𝑖𝑘=𝐷_𝑖𝑘(𝑡,𝑥,𝑦)≥0 is a diffusion operator, 𝑋 is a compact set with smooth boundary 𝜕𝑋 and measure mes𝑋>0 in 𝑅^𝑚. 𝜕𝑦_𝑖/𝜕𝑛|_𝜕𝑋=0, 𝑡≥0 and 𝜉_𝑖(𝑠,𝑥) are the boundary value and initial value, respectively.

For the sake of convenience, some of the standing definitions and assumptions are formulated below:(𝐴₁)ℎ_𝑖∶𝑅→𝑅 is differentiable and 𝛾_𝑖=inf_𝑥∈𝑅{ℎ^′_𝑖(𝑥)}>0, ℎ_𝑖(0)=0 for 𝑖∈{1,…,𝑛},(𝐴₂)𝑓_𝑗, 𝑔_𝑗, and 𝜎_𝑖𝑙 are Lipschitz continuous with positive Lipschitz constants 𝛼_𝑗, 𝛽_𝑗, 𝐿_𝑖𝑙, respectively, and 𝑓_𝑗(0)=𝑔_𝑗(0)=𝜎_𝑖𝑙(0)=0, ∑^∞_𝑙=1𝐿_𝑖𝑙<∞ for 𝑖,𝑗∈{1,…,𝑛}, 𝑙∈ℕ,(𝐴₃)For 𝑖,𝑗∈{1,…,𝑛}, there is a positive constant 𝜇, such that ^∞₀𝐾_𝑖𝑗(𝑠)d𝑠=1,^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜇𝑠d𝑠<∞. (1.2)

Remark 1.1. The authors in [28] (see (𝐻₁), (𝐻₂), (𝐻₃)) and [30] (see (𝐻₁), (𝐻₂), (𝐻^′₃)) also studied the convergence dynamics of (1.1) under the following foundational conditions:(𝐻₁)the kernels 𝐾_𝑖𝑗, 𝑖,𝑗=1,…,𝑛 are real-valued nonnegative piecewise continuous functions defined on [0,∞) and satisfy: ∫^∞₀𝐾_𝑖𝑗(𝑡)d𝑡=1,(𝐻₂)for each 𝑖,𝑗, we have ∫^∞₀𝑡𝐾_𝑖𝑗(𝑡)d𝑡<∞,(𝐻₃)for each 𝑖,𝑗, there is a positive constant 𝜇 such that ^∞₀𝑡𝑒^𝜇𝑡𝐾_𝑖𝑗(𝑡)d𝑡=1 (1.3)(𝐻^′₃)for each 𝑖,𝑗, there is a positive constant 𝜇 such that ^∞₀𝑡𝑒^𝜇𝑡𝐾_𝑖𝑗(𝑡)d𝑡<∞. (1.4) Just take the widely applied delay kernels as mentioned in [31–33] as an example, which given by 𝐾_𝑖𝑗(𝑠)=(𝑠^𝑟/𝑟!)𝛾^𝑟+1_𝑖𝑗𝑒^{−𝛾𝑖𝑗𝑠} for 𝑠∈[0,∞), where 𝛾_𝑖𝑗∈[0,∞), 𝑟∈{0,1,…,𝑛}. One can easy to find out that the conditions about the kernels in [28, 30] are not satisfy at the same time. Therefore, the applications of the main results in [28, 30] appears to be somewhat certain limits because of the obviously restrictive assumptions on the kernels. The main purpose of this paper is to further investigate the convergence of stochastic reaction-diffusion RNNs with more general kernels.

2. Preliminaries

Let 𝑢=(𝑢₁,…,𝑢_𝑛)^𝑇 and 𝐿²(𝑋) is the space of scalar value Lebesgue measurable functions on 𝑋 which is a Banach space for the 𝐿₂-norm: ‖𝜈‖₂=(∫_𝑋|𝜈(𝑥)|²d𝑥)^1/2, 𝜈∈𝐿²(𝑋), then we define the norm ‖𝑢‖ as ‖𝑢‖=(∑^𝑛_𝑖=1‖𝑢_𝑖‖²₂)^1/2. Assume 𝜉={(𝜉₁(𝑠,𝑥),…,𝜉_𝑛(𝑠,𝑥))^𝑇∶−∞<𝑠≤0}∈𝐶((−∞,0];𝐿²(𝑋,𝑅^𝑛)) be an ℱ₀-measurable 𝑅^𝑛 valued random variable, where, for example, ℱ₀=ℱ_𝑠 restricted on (−∞,0], and 𝐶((−∞,0];𝐿²(𝑋,𝑅^𝑛)) is the space of all continuous 𝑅^𝑛-valued functions defined on (−∞,0]×𝑋 with a norm ‖𝜉‖_𝑐=sup_{−∞<𝑠≤0}{‖𝜉(𝑠,𝑥)‖₂}. Clearly, (1.1) admits an equilibrium solution 𝑥(𝑡)≡0.

Definition 2.1. Equation (1.1) is said to be almost surely exponentially stable if there exists a positive constant 𝜆 such that for each pair of 𝑡₀ and 𝜉, such that limsup_𝑡→∞1𝑡ln𝑦𝑡;𝑡_𝑜,𝜉<−𝜆. (2.1)

Definition 2.2. Equation (1.1) is said to be exponentially stable in mean square if there exists a pair of positive constants 𝜆 and 𝐾 such that 𝐸𝑦(𝑡;𝑡₀,𝜉)²≤𝐾𝜉²𝑒^{−𝜆(𝑡−𝑡0)},on𝑡≥𝑡₀. (2.2)

Lemma 2.3 (semimartingale convergence theorem [12]). Let 𝐴(𝑡) and 𝑈(𝑡) be two continuous adapted increasing processes on 𝑡≥0 with 𝐴(0)=𝑈(0)=0 a.s. Let 𝑀(𝑡) be a real-valued continuous local martingale with 𝑀(0)=0 a.s. Let 𝜉 be a nonnegative ℱ₀-measurable random variable. Define 𝑋(𝑡)=𝜉+𝐴(𝑡)−𝑈(𝑡)+𝑀(𝑡),for𝑡≥0. (2.3) If 𝑋(𝑡) is nonnegative, then lim_𝑡→∞𝐴(𝑡)<∞⊂lim_𝑡→∞𝑋(𝑡)<∞∩lim_𝑡→∞𝑈(𝑡)<∞,a.s., (2.4) where (2.4)𝐵⊂𝐷 a.s. means (2.4)𝑃(𝐵∩𝐷^𝑐)=0. In particular, If (2.4)lim_𝑡→∞𝐴(𝑡)<∞ a.s., then for almost all (2.4)𝜔∈Ωlim_𝑡→∞𝑋(𝑡)<∞,lim_𝑡→∞𝑈(𝑡)<∞, (2.4) that is both 𝑋(𝑡) and 𝑈(𝑡) converge to finite random variables.

Lemma 2.4 (see [34]). A nonsingular 𝑀-matrix 𝐴=(𝑎_𝑖𝑗) is equivalent to the following properties: there exists 𝑟_𝑗>0, such that ∑^𝑛_𝑗=1𝑎_𝑖𝑗𝑟_𝑗>0, 1≤𝑖≤𝑛.

Furthermore, an 𝑀-matrix is a 𝑍-matrix with eigenvalues whose real parts are positive. In mathematics, the class of 𝑍-matrices are those matrices whose off-diagonal entries are less than or equal to zero, that is, a 𝑍-matrix 𝑍 satisfies𝑍=𝑧_𝑖𝑗,𝑧_𝑖𝑗≤0,for𝑖≠𝑗. (2.6)

3. Main Results

Theorem 3.1. Under the assumptions (𝐴₀), (𝐴₁), (𝐴₂), (𝐴₃), and (𝐴₄), 𝐶𝛾−𝐶−𝐴⁺𝛼−𝐵⁺𝛽 is a nonsingular 𝑀-matrix, where 𝐶=diag𝑐₁,…,𝑐_𝑛,𝛾=diag𝛾₁,…,𝛾_𝑛,𝐴⁺=𝑎_{𝑖𝑗𝑛×𝑛},𝛼=diag𝛼₁,…,𝛼_𝑛,𝛽=diag𝛽₁,…,𝛽_𝑛,𝐵⁺=𝑏_{𝑖𝑗𝑛×𝑛},𝐶=diag𝑐₁,…,𝑐_𝑛,𝑐_𝑖∶=𝑐_𝑖𝛾_𝑖+^𝑛_𝑗=1𝑎_𝑖𝑗𝛼_𝑗+^𝑛_𝑗=1𝑏_𝑖𝑗𝛽_𝑗+^∞_𝑙=1𝐿²_𝑖𝑙, (3.1) then the trivial solution of system (1.1) is almost surely exponentially stable and also is exponential stability in mean square.

Proof. From (𝐴₄) and Lemma 2.4, there exist constants 𝑞_𝑖>0, 1≤𝑖≤𝑛 such that 𝑞_𝑖𝑐_𝑖𝛾_𝑖−𝑐_𝑖−^𝑛_𝑗=1𝑎_𝑗𝑖𝛼_𝑖𝑞_𝑗−^𝑛_𝑗=1𝑏_𝑗𝑖𝛽_𝑖𝑞_𝑗>0. (3.2) That also can be expressed as follows: 2𝑐_𝑖𝛾_𝑖𝑞_𝑖>^𝑛_𝑗=1𝑎_𝑖𝑗𝛼_𝑗𝑞_𝑖+^𝑛_𝑗=1𝑎_𝑗𝑖𝛼_𝑖𝑞_𝑗+^𝑛_𝑗=1𝑏_𝑖𝑗𝛽_𝑗𝑞_𝑖+^𝑛_𝑗=1𝑏_𝑗𝑖𝛽_𝑖𝑞_𝑗+^∞_𝑙=1𝐿²_𝑖𝑙𝑞_𝑖. (3.3) From the assumption (𝐴₃), one can choose a constant 0<𝜆≪𝜇 such that ^∞₀𝐾_𝑗𝑖(𝑠)𝑒^𝜆𝑠𝑠d𝑠<∞,2𝑐_𝑖𝛾_𝑖𝑞_𝑖−2𝜆𝑞_𝑖>^𝑛_𝑗=1𝑎_𝑖𝑗𝛼_𝑗𝑞_𝑖+^𝑛_𝑗=1𝑎_𝑗𝑖𝛼_𝑖𝑞_𝑗+^𝑛_𝑗=1𝑏_𝑖𝑗^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠d𝑠𝛽_𝑗𝑞_𝑖+^𝑛_𝑗=1𝑏_𝑗𝑖^∞₀𝐾_𝑗𝑖(𝑠)𝑒^𝜆𝑠d𝑠𝛽_𝑖𝑞_𝑗+^∞_𝑙=1𝐿²_𝑖𝑙𝑞_𝑖. (3.4)(3.5) Consider the following Lyapunov functional: 𝑉𝑡,𝑦(𝑡,𝑥)=𝑒^{𝑛2𝜆𝑡}_𝑖=1𝑞_𝑖𝑦²_𝑖(𝑡,𝑥)+^𝑛𝑛_𝑖=1𝑗=1𝑏_𝑖𝑗𝑞_𝑖𝛽_𝑗^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠𝑡_𝑡−𝑠𝑒^2𝜆𝜉𝑦²_𝑗(𝜉,𝑥)d𝜉d𝑠. (3.6) From the boundary condition (𝐴₀), we get ^𝑚_𝑘=1𝑋𝑦_𝑖𝜕𝜕𝑥_𝑘𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘d𝑥=−^𝑚_𝑘=1𝑋𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘²d𝑥<0. (3.7) Using the Itô formula, for 𝑇>0, we have 𝑉()()=𝑇,𝑦𝑇,𝑥−𝑉0,𝑦0,𝑥_𝑇02𝜆𝑒^{𝑛2𝜆𝑡}_𝑖=1𝑞_𝑖𝑦²_𝑖(𝑡,𝑥)d𝑡+^𝑇₀^𝑛𝑛_𝑖=1𝑗=1𝑏_𝑖𝑗𝑞_𝑖𝛽_𝑗^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠×𝑒^2𝜆𝑡𝑦²_𝑗(𝑡,𝑥)−𝑒^{2𝜆(𝑡−𝑠)}𝑦²_𝑗(𝑡−𝑠,𝑥)d𝑠d𝑡+^𝑇₀2𝑒^{𝑛2𝜆𝑡}_𝑖=1𝑞_𝑖𝑦_𝑖()×𝑡,𝑥_𝑚𝑘=1𝜕𝜕𝑥_𝑘𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘−𝑐_𝑖ℎ_𝑖𝑦_𝑖(𝑡,𝑥)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑦_𝑗()+𝑡,𝑥_𝑛𝑗=1𝑏_𝑖𝑗^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑔_𝑗𝑦_𝑗(𝑠,𝑥)d𝑠d𝑡+^𝑇₀2𝑒^{𝑛2𝜆𝑡∞}_𝑖=1𝑙=1𝑞_𝑖𝑦_𝑖(𝑡,𝑥)𝜎_𝑖𝑙𝑦_𝑖(𝑡,𝑥)d𝜔_𝑖𝑙(𝑡)+^𝑇₀𝑒^{𝑛2𝜆𝑡∞}_𝑖=1𝑙=1𝑞_𝑖𝜎²_𝑖𝑙𝑦_𝑖()()≤𝑡,𝑥d𝑡usinginequality3.7_𝑇02𝜆𝑒^{𝑛2𝜆𝑡}_𝑖=1𝑞_𝑖𝑦²_𝑖(𝑡,𝑥)d𝑡+^𝑇₀𝑒^{𝑛2𝜆𝑡𝑛}_𝑖=1𝑗=1𝑏_𝑖𝑗𝑞_𝑖𝛽_𝑗^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠×𝑦²_𝑗(𝑡,𝑥)−𝑒^−2𝑠𝑦²_𝑗(𝑡−𝑠,𝑥)d𝑠d𝑡+^𝑇₀2𝑒^2𝜆𝑡×^𝑛_𝑖=1𝑞_𝑖𝑦_𝑖(𝑡,𝑥)−𝑐_𝑖ℎ_𝑖𝑦_𝑖(𝑡,𝑥)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑦_𝑗()+𝑡,𝑥_𝑛𝑗=1𝑏_𝑖𝑗^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑔_𝑗𝑦_𝑗()+𝑠,𝑥d𝑠d𝑡_𝑇02𝑒^{𝑛2𝜆𝑡∞}_𝑖=1𝑙=1𝑞_𝑖𝑦_𝑖(𝑡,𝑥)𝜎_𝑖𝑙𝑦_𝑖(𝑡,𝑥)d𝜔_𝑖𝑙(𝑡)+^𝑇₀𝑒^{𝑛2𝜆𝑡∞}_𝑖=1𝑙=1𝑞_𝑖𝜎²_𝑖𝑙𝑦_𝑖(𝑡,𝑥)d𝑡usingassumptions𝐴₁,𝐴₂,𝐴₄≤^𝑇₀2𝜆𝑒^{𝑛2𝜆𝑡}_𝑖=1𝑞_𝑖𝑦²_𝑖(𝑡,𝑥)d𝑡+^𝑇₀𝑒^{𝑛2𝜆𝑡𝑛}_𝑖=1𝑗=1𝑏_𝑖𝑗𝑞_𝑖𝛽_𝑗^∞₀𝐾_𝑖𝑗(𝑠)×𝑒^𝜆𝑠𝑦²_𝑗(𝑡,𝑥)−𝑒^−𝑠𝑦²_𝑗(𝑡−𝑠,𝑥)d𝑠d𝑡+^𝑇₀2𝑒^𝜆𝑡×^𝑛_𝑖=1𝑞_𝑖−2𝑐_𝑖𝛾_𝑖𝑦²_𝑖(𝑡,𝑥)+^𝑛_𝑗=12𝑎_𝑖𝑗𝛼_𝑗𝑦_𝑖(𝑡,𝑥)𝑦_𝑗(+𝑡,𝑥)_𝑛𝑗=1𝑏_𝑖𝑗𝛽_𝑗^𝑡_−∞2𝐾_𝑖𝑗(𝑡−𝑠)𝑦_𝑖(𝑡,𝑥)𝑦_𝑗(+𝑠,𝑥)d𝑠d𝑡_𝑇02𝑒^{𝑛2𝜆𝑡∞}_𝑖=1𝑙=1𝑞_𝑖𝑦_𝑖(𝑡,𝑥)𝜎_𝑖𝑙𝑦_𝑖(𝑡,𝑥)d𝜔_𝑖𝑙(𝑡)d𝑡+^𝑇₀𝑒^{𝑛2𝜆𝑡∞}_𝑖=1𝑙=1𝑞_𝑖𝐿²_𝑖𝑙𝑦²_𝑖(𝑡,𝑥)d𝑡. (3.8) Notice that ^𝑡_−∞2𝐾_𝑖𝑗(𝑡−𝑠)𝑦_𝑖(𝑡,𝑥)𝑦_𝑗(=𝑠,𝑥)d𝑠_𝑡−∞𝐾_𝑖𝑗(𝑡−𝑠)2𝑦_𝑖(𝑡,𝑥)𝑒^{𝜆(𝑡−𝑠)/2}𝑦_𝑗(𝑠,𝑥)𝑒^{−𝜆(𝑡−𝑠)/2}≤d𝑠_𝑡−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑦²_𝑖(𝑡,𝑥)𝑒^{𝜆(𝑡−𝑠)}d𝑠+^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑦²_𝑗(𝑠,𝑥)𝑒^{−𝜆(𝑡−𝑠)}d𝑠=𝑦²_𝑖(𝑡,𝑥)^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑒^{𝜆(𝑡−𝑠)}d𝑠+^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑦²_𝑗(𝑠,𝑥)𝑒^{−𝜆(𝑡−𝑠)}d𝑠=𝑦²_𝑖(𝑡,𝑥)^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠d𝑠+^∞₀𝐾_𝑖𝑗(𝑠)𝑦²_𝑗(𝑡−𝑠,𝑥)𝑒^𝜆𝑠d𝑠. (3.9) Submitting inequality (3.9) to (3.8), it is easy to calculate that 𝑉()()≤𝑇,𝑦𝑇,𝑥−𝑉0,𝑦0,𝑥_𝑇0𝑒^{𝑛2𝜆𝑡}_𝑖=12𝜆𝑞_𝑖𝑦²_𝑖(𝑡,𝑥)+^𝑛_𝑗=1𝑏_𝑗𝑖𝑞_𝑗𝛽_𝑖𝑦²_𝑖(𝑡,𝑥)^∞₀𝐾_𝑗𝑖(𝑠)𝑒^𝜆𝑠−d𝑠_𝑛𝑗=1𝑏_𝑗𝑖𝑞_𝑗𝛽_𝑖^∞₀𝐾_𝑗𝑖(𝑠)𝑒^−𝜆𝑠𝑦²_𝑖(𝑡−𝑠,𝑥)d𝑠−2𝑐_𝑖𝛾_𝑖𝑞_𝑖𝑦²_𝑖()+𝑡,𝑥_𝑛𝑗=1𝑎_𝑖𝑗𝛼_𝑗𝑞_𝑖𝑦²_𝑖(𝑡,𝑥)+^𝑛_𝑗=1𝑎_𝑗𝑖𝛼_𝑖𝑞_𝑗𝑦²_𝑖()+𝑡,𝑥_𝑛𝑗=1𝑏_𝑖𝑗𝛽_𝑗𝑞_𝑖𝑦²_𝑖(𝑡,𝑥)^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠d𝑠+^∞_𝑙=1𝑞_𝑖𝐿²_𝑖𝑙𝑦²_𝑖()+𝑡,𝑥_𝑛𝑗=1𝑏_𝑖𝑗𝑞_𝑗𝛽_𝑖𝑦²_𝑖(𝑡,𝑥)^∞₀𝐾_𝑗𝑖(𝑠)𝑒^𝜆𝑠+d𝑠d𝑡_𝑇02𝑒^{𝑛2𝜆𝑡∞}_𝑖=1𝑙=1𝑞_𝑖𝑦_𝑖(𝑡,𝑥)𝜎_𝑖𝑙𝑦_𝑖(𝑡,𝑥)d𝜔_𝑖𝑙(𝑡)=^𝑇₀𝑒^{𝑛2𝜆𝑡}_𝑖=1𝑦²_𝑖()×𝑡,𝑥_𝑛𝑗=1𝑎_𝑖𝑗𝛼_𝑗𝑞_𝑖+^𝑛_𝑗=1𝑎_𝑗𝑖𝛼_𝑖𝑞_𝑗+^𝑛_𝑗=1𝑏_𝑖𝑗^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠d𝑠𝛽_𝑗𝑞_𝑖+^𝑛_𝑗=1𝑏_𝑗𝑖^∞₀𝐾_𝑗𝑖(𝑠)𝑒^𝜆𝑠d𝑠𝛽_𝑖𝑞_𝑗+^∞_𝑙=1𝐿²_𝑖𝑙𝑞_𝑖−2𝑐_𝑖𝛾_𝑖𝑞_𝑖−2𝜆𝑞_𝑖+d𝑡_𝑇02𝑒^{𝑛2𝜆𝑡∞}_𝑖=1𝑙=1𝑞_𝑖𝑦_𝑖(𝑡,𝑥)𝜎_𝑖𝑙𝑦_𝑖(𝑡,𝑥)d𝜔_𝑖𝑙(𝑡(≤)usinginequality3.5)_𝑇02𝑒^{𝑛2𝜆𝑡∞}_𝑖=1𝑙=1𝑞_𝑖𝑦_𝑖(𝑡,𝑥)𝜎_𝑖𝑙𝑦_𝑖(𝑡,𝑥)d𝜔_𝑖𝑙(𝑡). (3.10) On the other hand, we observe that 𝑉0,𝑦(0,𝑥)=^𝑛_𝑖=1𝑞_𝑖𝑦²_𝑖(0,𝑥)+^𝑛𝑛_𝑖=1𝑗=1𝑏_𝑖𝑗𝑞_𝑖𝛽_𝑗^∞₀𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠0_−𝑠𝑒^2𝜆𝜉𝑦²_𝑗()≤𝑞𝜉,𝑥d𝜉d𝑠_𝑖+^𝑛_𝑗=1𝑏_𝑗𝑖𝑞_𝑗𝛽_𝑖^∞₀𝐾_𝑗𝑖(𝑠)𝑠𝑒^𝜆𝑠d𝑠sup_{−∞≤𝜉≤0}𝑦²_𝑖(𝜉,𝑥). (3.11) From inequality (3.4), we have 𝑞_𝑖+^𝑛_𝑗=1𝑏_𝑗𝑖𝑞_𝑗𝛽_𝑖^∞₀𝐾_𝑗𝑖(𝑠)𝑠𝑒^𝜆𝑠<∞. (3.12) Hence, 𝑉(0,𝑢(0)) is bounded. Integrate both sides of (3.10) with respect to 𝑥, we have _𝑋𝑉𝑇,𝑦(𝑇,𝑥)d𝑥≤_𝑋𝑉0,𝑦(0,𝑥)d𝑥+^𝑇₀2𝑒^{𝑛2𝜆𝑡∞}_{𝑖=1𝑙=1𝑋}𝑞_𝑖𝑦_𝑖(𝑡,𝑥)𝜎_𝑖𝑙𝑦_𝑖(𝑡,𝑥)d𝑥d𝜔_𝑖𝑙(𝑡). (3.13) Therefore, we have min_{1≤𝑖≤𝑛}𝑞_𝑖𝑒^2𝜆𝑡𝑦(𝑇,𝑥)²₂≤^𝑛_𝑖=1𝑞_𝑖𝑦_𝑖(0,𝑥)²₂+^𝑇₀2𝑒^{𝑛2𝜆𝑡∞}_{𝑖=1𝑙=1𝑋}𝑞_𝑖𝑦_𝑖(𝑡,𝑥)𝜎_𝑖𝑙𝑦_𝑖(𝑡,𝑥)d𝑥d𝜔_𝑖𝑙(𝑡). (3.14) It is obvious that the right-hand side of (3.14) is a nonnegative martingale, From Lemma 2.3, it can be easily seen that lim_𝑇→∞𝑒^𝑛𝜆𝑡_𝑖=1𝑦(𝑇,𝑥)<∞a.s., (3.15) that is limsup_𝑇→∞1𝑇ln𝑦(𝑇,𝑥)<−𝜆. (3.16) On the other hand, since 𝐸(∫^𝑇₀2𝑒^2𝜆𝑡∑^𝑛_𝑖=1∑^∞_𝑙=1∫_𝑋𝑞_𝑖𝑦_𝑖(𝑡,𝑥)𝜎_𝑖𝑙(𝑦_𝑖(𝑡,𝑥))d𝑥d𝜔_𝑖𝑙(𝑡)d𝑡)=0, taking expectation on both sides of the equality (3.14) yields 𝐸𝑦(𝑡;𝑡₀,𝑥)²≤𝐾‖𝑥‖²𝑒^−2𝜆𝑡,on𝑡≥0,𝐾=max_{1≤𝑖≤𝑛}𝑞_𝑖min_{1≤𝑖≤𝑛}𝑞_𝑖. (3.17) From (3.16) and (3.17), the trivial solution of system (1.1) is almost surely exponentially stable, which is also exponential stability in mean square. This completes the proof.

Removing the reaction-diffusion term from the system (1.1), we investigate the following stochastic recurrent neural networks with unbounded distributed delays:d𝑦_𝑖(𝑡)=−𝑐_𝑖ℎ_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑔_𝑗𝑦_𝑗(𝑠)+d𝑠d𝑡_∞𝑙=1𝜎_𝑖𝑙𝑦_𝑖(𝑡)d𝜔_𝑖𝑙(𝑡). (3.18) We have the following Corollary 3.2 for system (3.18). The derived conditions for almost surely exponentially stable and exponential stability in mean square can be viewed as byproducts of our results from Theorem 3.1, so the proof is trivial, we omit it.

Corollary 3.2. Under the assumptions (𝐴₁), (𝐴₂), (𝐴₃), and (𝐴₄), 𝐶𝛾−𝐶−𝐴⁺𝛼−𝐵⁺𝛽 is a nonsingular 𝑀-matrix, where 𝐶=diag𝑐₁,…,𝑐_𝑛,𝛾=diag𝛾₁,…,𝛾_𝑛,𝐴⁺=𝑎_{𝑖𝑗𝑛×𝑛},𝛼=diag𝛼₁,…,𝛼_𝑛,𝛽=diag𝛽₁,…,𝛽_𝑛,𝐵⁺=𝑏_{𝑖𝑗𝑛×𝑛},𝐶=diag𝑐₁,…,𝑐_𝑛,𝑐_𝑖∶=𝑐_𝑖𝛾_𝑖+^𝑛_𝑗=1𝑎_𝑖𝑗𝛼_𝑗+^𝑛_𝑗=1𝑏_𝑖𝑗𝛽_𝑗+^∞_𝑙=1𝐿²_𝑖𝑙, (3.19) then the trivial solution of system (3.18) is almost surely exponentially stable and also is exponential stability in mean square.

Furthermore, if we remove noise from the system, then system (3.18) turns out to be system as the following:d𝑦_𝑖(𝑡)=−𝑐_𝑖ℎ_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑔_𝑗𝑦_𝑗(𝑠)d𝑠d𝑡. (3.20) To investigate the stability of model (3.20), we should modify the assumption (𝐴^′₂) as follows:(𝐴^′₂)𝑓_𝑗, 𝑔_𝑗, and 𝜔_𝑖𝑙 are Lipschitz continuous with positive Lipschitz constants 𝛼_𝑗, 𝛽_𝑗, respectively, and 𝑓_𝑗(0)=𝑔_𝑗(0)=0, for 𝑖,𝑗∈{1,…,𝑛}, 𝑙∈ℕ.

The derived conditions for exponential stability of system (3.20) can be obtained directly from of our results from Corollary 3.2.

Corollary 3.3. Under the assumptions (𝐴₁), (𝐴^′₂), (𝐴₃), and (𝐴₄), 𝐶𝛾−𝐶−𝐴⁺𝛼−𝐵⁺𝛽 is a nonsingular 𝑀-matrix, where 𝐶=diag𝑐₁,…,𝑐_𝑛,𝛾=diag𝛾₁,…,𝛾_𝑛,𝐴⁺=𝑎_{𝑖𝑗𝑛×𝑛},𝛼=diag𝛼₁,…,𝛼_𝑛,𝛽=diag𝛽₁,…,𝛽_𝑛,𝐵⁺=𝑏_{𝑖𝑗𝑛×𝑛},𝐶=diag𝑐₁,…,𝑐_𝑛,𝑐_𝑖̃∶=𝑐_𝑖𝛾_𝑖+^𝑛_𝑗=1𝑎_𝑖𝑗𝛼_𝑗+^𝑛_𝑗=1𝑏_𝑖𝑗𝛽_𝑗, (3.21) then the trivial solution of system (3.20) is exponentially stable.

4. An Illustrative Example

In this section, a numerical example is presented to illustrate the correctness of our main result.

Example 4.1. Consider a two-dimensional stochastic reaction-diffusion recurrent neural networks with unbounded distributed delays as follows: d𝑦_𝑖(𝑡,𝑥)=²_𝑘=1𝜕𝜕𝑥_𝑘𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘+d𝑡−𝑐_𝑖ℎ_𝑖𝑦_𝑖(𝑡,𝑥)+²_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡,𝑥)+²_𝑗=1𝑏_𝑖𝑗^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑔_𝑗𝑦_𝑗()+𝑠,𝑥d𝑠d𝑡_∞𝑙=1𝜎_𝑖𝑙𝑦_𝑖(𝑡,𝑥)d𝜔_𝑖𝑙(𝑡),𝑥∈𝑋,𝜕𝑦_𝑖𝜕𝑛∶=𝜕𝑦_𝑖𝜕𝑥₁,𝜕𝑦_𝑖𝜕𝑥₂^𝑇𝑦=0,𝑡≥0,𝑖=1,2,𝑥∈𝜕𝑋,_𝑖=𝑦_𝑖(𝑠,𝑥)=𝜉_𝑖(𝑠,𝑥),−∞<𝑠≤0,𝑥∈𝑋, (4.1) where, 𝑐₁=10, 𝑐₂=15; ℎ₁(𝑦)=ℎ₂(𝑦)=𝑦; 𝑎₁₁=1/5, 𝑎₁₂=2/5, 𝑎₂₁=3/5, 𝑎₂₂=1; 𝑏₁₁=4/5, 𝑏₁₂=1/5, 𝑏₂₁=1, 𝑏₂₂=2/5; 𝑓₁(𝑢)=𝑓₂(𝑢)=tanh(𝑢); 𝑔₁(𝑢)=𝑔₂(𝑢)=1/2(|𝑢−1|−|𝑢+1|)𝑘₁₁(𝑠)=2𝑒^−2𝑠, 𝑘₁₂(𝑠)=4𝑒^−4𝑠, 𝑘₂₁(𝑠)=3𝑒^−3𝑠, 𝑘₂₂(𝑠)=5𝑒^−5𝑠; 𝜎₁₁(𝑦)=𝑦, 𝜎₁₂(𝑦)=𝑦/2, 𝜎₂₁(𝑦)=𝜎₂₂(𝑦)=3𝑦/2, 𝜎_𝑖𝑗(𝑦)=0 for 𝑖,𝑗≠1,2. In the example, let 𝐷_𝑖𝑘 is a positive constant, according to Theorem 3.1, by simple computation, we get _Ω𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘²_𝑘=1⋅∇𝑦_𝑖²⁻¹d𝑥=²_𝑘=1Ω𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘²()()d𝑥>0,𝐶=diag10,15,𝛾=diag1,1,𝐴⁺=𝑎_𝑖𝑗2×2,𝛼=diag(1,1),𝛽=diag(1,1),𝐵⁺=𝑏_𝑖𝑗2×2,𝐶=diag(7.85,13.5). (4.2) Therefore, 𝑄=𝐶𝛾−𝐶−𝐴⁺𝛼−𝐵⁺𝑞𝛽=¹¹,𝑞₁₂𝑞₂₁,𝑞₂₂=1.15,−0.6−1.6,2.1. (4.3) With the help of Matlab, one can get the eigenvalues of matrix 𝑄 quickly, which are 𝜆₁=0.5361, 𝜆₂=2.7139, by Lemma 2.4, 𝑄 is a nonsingular 𝑀-matrix, therefore, all conditions in Theorem 3.1 are hold, that is to say, the trivial solution of system (4.1) is almost surely exponentially stable and also is exponential stability in mean square. Just choose 𝑥≡constant, then these conclusions can be verified by the following numerical simulations (Figures 1–4).

Figure 1 

Numerical simulation for 𝑦1(𝑡).

Figure 2 

Numerical simulation for 𝑦2(𝑡).

Figure 3 

Numerical simulation for the mean square of 𝑦1(𝑡).

Figure 4 

Numerical simulation for the mean square of 𝑦2(𝑡).

5. Conclusions

In this paper, stochastic recurrent neural networks with unbounded distributed delays and reaction-diffusion have been investigated. All features of stochastic systems, reaction-diffusion systems have been taken into account in the neural networks. The proposed results generalized and improved some of the earlier published results. The results obtained in this paper are independent of the magnitude of delays and diffusion effect, which implies that strong self-regulation is dominant in the networks. If we remove the noise and reaction-diffusion terms from the system, the derived conditions for stability of general deterministic neural networks can be viewed as byproducts of our results.

Acknowledgments

The authors are extremely grateful to Professor Binggen Zhang 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 Key Project of Chinese Ministry of Education (2011), the Foundation of Chinese Society for Electrical Engineering (2008), the Excellent Youth Foundation of Educational Committee of Hunan Province of China (10B002), the National Natural Science Funds of China for Distinguished Young Scholar (50925727), the National Natural Science Foundation of China (60876022), and the Scientific Research Fund of Yunnan Province of China (2010ZC150).