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

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.


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 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 A 0 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 mes X > 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: A 2 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 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 ij e −γ ij s 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.

Preliminaries
Let u u 1 , . . ., u n T and L 2 X is the space of scalar value Lebesgue measurable functions on X which is a Banach space for the L 2 -norm: R n be an F 0 -measurable R n valued random variable, where, for example, F 0 F s restricted on −∞, 0 , and C −∞, 0 ; L 2 X, R n is the space of all continuous R nvalued 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.
where B ⊂ D a.s.means 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 1 a ij r 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 satisfies Z z ij , z ij ≤ 0, for i / j. 2.6

Main Results
Theorem 3.1.Under the assumptions A 0 , A 1 , A 2 , A 3 , and

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 A 4 and Lemma 2.4, there exist constants q i > 0, 1 ≤ i ≤ n such that b ji β i q j > 0.

3.2
That also can be expressed as follows: From the assumption A 3 , one can choose a constant 0 < λ μ such that ∞ 0 K ji s e λs sds < ∞, 3.4

3.5
Consider the following Lyapunov functional: Using the It ô formula, for T > 0, we have q i σ 2 il y i t, x dt using inequality 3.7 x e λs ds.

3.9
Submitting inequality 3.9 to 3.8 , it is easy to calculate that q i y i t, x σ il y i t, x dω il t using inequality 3.5 q i y i t, x σ il y i t, x dω il t .

3.10
On the other hand, we observe that V 0, y 0,

3.11
From inequality 3.4 , we have Hence, V 0, u 0 is bounded.Integrate both sides of 3.10 with respect to x, we have q i y i t, x σ il y i t, x dxdω il t .

3.13
Therefore, we have min 1≤i≤n q i e 2λt y T, x q i y i t, x σ il y 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

3.16
On the other hand, since E T 0 2e 2λt n i 1 ∞ l 1 X q i y 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 E y t; t 0 , x 2 ≤ K x 2 e −2λt , on t ≥ 0, K max 1≤i≤n q i min 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: σ il y 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.

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: 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 ∈ AE.
The derived conditions for exponential stability of system 3.20 can be obtained directly from of our results from Corollary 3.2.

3.21
then the trivial solution of system 3.20 is exponentially stable.

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:

4.3
With the help of Matlab, one can get the eigenvalues of matrix Q quickly, which are λ 1 0.5361, λ 2 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

Conclusions
In this paper, stochastic recurrent neural networks with unbounded distributed delays and reaction-diffusion have been investigated.All features of stochastic systems, reactiondiffusion 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

Figure 3 :
Figure 3: Numerical simulation for the mean square of y 1 t .
Remark 1.1.The authors in 28 see H 1 , H 2 , H 3 and 30 see H 1 , H 2 , H 3 also studied the convergence dynamics of 1.1 under the following foundational conditions: μs ds < ∞. 1.2 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 t 0 and ξ, such that 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 F 0 -measurable random variable.Define 2 ≤ K ξ 2 e −λ t−t 0 , on t ≥ t 0 .2.2