Adaptive Synchronization of Reaction Diffusion Neural Networks With Infinite Distributed Delays and Stochastic Disturbance

In this paper, the diffusion effect, distributed delays and stochastic disturbance are involved in constructing the model of neural networks. Then, the global exponential synchronization problem is investigated for a class of reaction diffusion neural networks (RDNNs) with infinite distributed delays and stochastic disturbance. By employing the stochastic analysis method and Lyapunov functional theory, an adaptive controller is designed to guarantee the exponential synchronization of the drive and response RDNNs. The derived synchronization conditions are simple and the theoretical results can be directly extended to other RDNNs with or without distributed delays and stochastic disturbance. Finally, one example is provided to verify the effectiveness of the theoretical results and adaptive control approach.


I. INTRODUCTION
During the past several decades, various models of neural networks (NNs) have been put forward and applied in different fields, such as the optimized calculation, associative memory and image processing [1]- [4]. These successful applications are heavily dependent on the dynamical behaviors of NNs, which the main cases are the stability and synchronization. In addition, as a typical cluster discharge activity of neurons, the synchronous behavior is closely related to the neural information processing function [5]- [8]. Thus, it is critical and necessary to study the dynamics, especially the synchronization of NNs.
The diffusion effects with spatial dynamic characteristics should be considered in circuit design of NNs [9]. The main reason is that the magnetic field environment of the electronic circuit simulating the NNs is often nonuniform, which leads to the phenomenon of spatial diffu-The associate editor coordinating the review of this manuscript and approving it for publication was Rajeeb Dey . sion of its electrons [10]- [12]. Then, a new kind of reaction diffusion NNs (RDNNs) whose neuron states vary with time and space simultaneously, is proposed [13], [14]. Correspondingly, to sufficiently reflect this time-space feature, the mathematical model of RDNNs can be described by a partial differential equation. Therefore, dynamical analysis of RDNNs is necessary and difficult, deserving to be further investigated. Recently, great efforts have been devoted to analyze dynamical behaviors of various RDNNs, especially the synchronization problem via different control approaches [15]- [19].
Time delays are inevitable in light of the finite switching limits of neuron amplifiers and the signal transmission [20], [21]. Generally, they are unavoidably existed in circuits of NNs and may result in undesirable dynamics, such as instability and chaotic behavior [22]- [24]. Furthermore, owing to the existence of massive parallel pathways with extensive axon sizes and lengths, the delays are not sufficiently described in the discrete form but in the distributed form [25]. In other words, the distributed delays are more VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ general and should be considered in models of RDNNs.
In [26], [27], the periodic solution and its stability were discussed for RDNNs with continuous distributed delays.
In [28], [29], the stability was analyzed for RDNNs with S-type distributed delays. In [30], [31], the stabilization problem was addressed for memristor-based RDNNs with finite distributed delays via state feedback and intermittent control methods. As is well known, the current behaviors of a neuron relate to its entire past states, which gives rise to the existence of the infinite distributed delays [32]. Thus, it is more reasonable to consider the infinite distributed delays in model of RDNNs.
On the other hand, in practical applications, the stochastic disturbance is always encountered because of the complicated environmental noise and interference [33]- [36]. Regarding the RDNNs with stochastic disturbance, lots of research works have been reported [37]- [41]. Moreover, by taking both the distributed delays and stochastic disturbance into consideration, authors in [42]- [44] studied the stability and synchronization of RDNNs via the stochastic analysis method. It is noted that the distributed delays in above papers [42]- [44] are restricted to be finite and bounded. Up to now, there is little work on the dynamics of RDNNs with infinite distributed delays and stochastic disturbance, which deserves further investigation and motivates the synchronization study of this paper.
Considering the RDNNs with infinite distributed delays and stochastic disturbance, the model is represented as a class of partial differential stochastic systems with mixed delays. Thus, it brings great difficulties to realize the synchronization of this kind of systems. In this paper, we aim to investigate the drive-response synchronization for RDNNs with infinite distributed delays and stochastic disturbance. The contributions are listed as follows.
1) This paper addresses the synchronization problem for a class of stochastic RDNNs with infinite distributed delays. Since the model studied involves the diffusion effect, stochastic disturbance and infinite distributed delays, it is generalized compared to the stochastic RDNNs with finite distributed delays in [42]- [44].
2) By employing the theory of partial differential system and stochastic analysis method, the simple and easily verified synchronization criteria are derived via a designed adaptive controller. The comparisons over existing work are provided in Remarks 4-6, which shows that the synchronization results effectually complement or improve the existing results of RDNNs with or without stochastic disturbance. Moreover, the superiority of the adaptive control approach compared with the linear one is also presented in Corollary 1.
The rest of this paper is organized as follows. The model and problem descriptions are given in Section II. The main synchronization results of RDNNs via the adaptive controller are given in Section III. Sections IV and V show the numerical simulations and conclusions, respectively.

II. PRELIMINARIES
In this paper, R n , R and R + denote the set of n-dimensional Euclidean space, real numbers and nonnegative numbers, respectively. For a constant n, let n = {1, 2, . . . , n}. ( , , P) is a completed probability space with a natural filtration { t } t≥0 satisfying the usual condition. E is the mathematical expectation operator concerning the probability P.
Consider the following RDNNs model with infinite distributed delays is the state at space u and time t, and u = (u 1 , u 2 , . . . , u I ) T ∈ ⊂ R I . α si ≥ 0 is the transmission diffusion parameter. d s > 0 is the self feedback connection weight, the parameters a sk , b sk and c sk are the connection weight coefficients, where b sk and c sk correspond to the delayed and distributed delayed ones. Discrete delay λ k (t) is bounded and the delay kernel K sk ∈ R + is a continuous function. The activations f jk ∈ R(j = 1, 2, 3) are continuous functions.
Remark 1: Compared with the stochastic RDNNs model with finite distributed delays in [42]- [44], the distributed delays in our model are infinite, which implies wider range of applications in practice.
The following assumptions are given for the activation functions and delays in system (1). H1: For each k ∈ n , there exist positive constants for all x, y ∈ R. H2: For each k ∈ n , there exist positive constants λ 1 and λ 2 such that H3: For each s, k ∈ n , there exists positive constant ϑ sk such that Consider system (1) as the drive system, and involve the stochastic disturbance, then the corresponding response system is where e is the base of the natural logarithm, π, γ s , p ≥ 2 are positive constants for s ∈ n . Then the following assumption is presented for the noise intensity functions.
H4: For each s, k ∈ n , there exist τ sk > 0, η sk > 0 such that for allê 1 ,ě 1 ,ê 2 ,ě 2 ∈ R. Remark 2: The noise intensity functions sk (s, k ∈ n ) are of multiplicative case. The stochastic Brownian noise with these functions can be regarded as a result from the occurrence of the internal error of NNs circuits and random fluctuation. Therefore, the noise intensity functions sk (s, k ∈ n ) also rely on the state variables x k (u, t), y k (u, t), x k (u, t − λ k (t)), y k (u, t − λ k (t)) of systems (1) and (5).
Remark 3: The control gains of the adaptive controller can be adjusted to be optimized, which shows the superiority compared with the linear state feedback control whose feedback control strength is fixed.
Then the following error system is derived from systems (1) and (5).
Definition 1: The drive and response systems (1) and (5) are said to be pth moment globally exponentially synchronized, if there exist positive constants µ > 0 and δ ≥ 1 such that for any Lemma 1 [10]:
Remark 5: Our results show the superiority of the ones in [32], [41]- [44]. On the one hand, the synchronization criteria in this paper are more simpler compared to the ones in [32], [41], [43], [44] on account of the complicated calculation for the criteria in [32], [43], [44]. On the other hand, simple synchronization conditions are obtained for stochastic RDNNs via adaptive control approach in [42]. It is worth noting that our criteria are derived with infinite distributed delays while the ones in [42] are with the finite distributed delays.
If the control gain β s (u, t) = β s , then the adaptive control turns out to be the linear one. Then we can get the following Corollary 1 from Theorem 1.
Corollary 1: Suppose that assumptions H1-H4 hold, if there exists constant β s < 0 such that then systems (1) and (5) are globally exponentially synchronized via the controller s (u, t) = β s v s (u, t). Remark 6: It is obvious from Corollary 1 that the criteria via the adaptive control approach is simpler and less conservative compared to the criteria via the linear control approach. In the following part, we will show that our results still hold for stochastic RDNNs with finite distributed delays.
If the delay kernel function satisfies where ρ is a positive constant. Then the drive and response RDNNs turn out to be the following two systems. and with the control input where the parameters are similar defined as in (5)-(7). Corollary 2: Suppose that assumptions H1, H2 and H4 hold, then systems (28) and (29) are globally exponentially synchronized via the adaptive controller (30).
Remark 7: Authors in [42] investigated the synchronization problem for RDNNs with finite distributed delays and stochastic disturbance via adaptive control approach. The results in [42] can be obtained directly from Corollary 2.

IV. NUMERICAL SIMULATIONS
One example is provided to show the effectiveness of results and the control approach.
Example 1: Consider the RDNNs with infinite distributed delays. Then under the stochastic disturbance and external control input, the response system is a sk f 1k (y k (u, t)) + s (u, t) (32) in which 11 . The other parameters are the same as in the drive system (31). Fig. 1 show the phase plot of drive system (31). It is noted that the chaotic attractors are reflected in this figure. Fig. 2 depicts the phase plot of system (32) with u = −2 and s (u, t) = 0. The trajectories of the state variables y 1 (u, t) and y 2 (u, t) of system (31) without any controller are shown in Figs. 3 and 4. It is obviously that the response system is also a chaotic system, and it is not stable without any controller.
Then to ensure the synchronization of drive and response systems (31) and (32), the following controller is given as   for s = 1, 2. Then the globally exponentially synchronization of systems (31) and (32) (31) and (32) via the controller (33).
as time goes to infinite, which also shows the synchronization performance of the drive and response systems. The control gains β 1 (u, t) and β 2 (u, t) of the adaptive controller (33) are given in Figs. 7 and 8. Moreover, if sk (·, ·) = 0(s, k = 1, 2), then the synchronization of systems (31) and (32) can also be ensured on the strength of Corollary 3. The trajectories of synchronization errors are shown in Figs. 9 and 10. The control gains of the controller are shown in Figs. 11 and 12 on account of the condition sk (·, ·) = 0(s, k = 1, 2).

V. CONCLUSION
This paper has discussed the global exponential synchronization problem for a class of RDNNs with distributed delays. Compared with some existing determined models, the stochastic disturbance has been taken into consideration and the distributed delays are infinite. By employing an adaptive controller, sufficient simple and easily checked criteria have been concluded for the addressed drive and response networks under stochastic disturbance. Finally, one RDNNs model with chaotic attractors has been presented to show the feasibility and validity of the proposed adaptive control approach. Since the parameters of the drive and response systems can not be identical in light of the complicated environment of practical application, future work may study the synchronization problem of RDNNs with mismatched parameters.
XIAOLING WEI received the B.S. degree in biotechnology from Putian University, Fujian, China, in 2018. She is currently pursuing the master's degree with Fujian Agriculture and Forestry University, Fujian. Her research interest includes study of bioinformatics.
LEIMIN WANG (Member, IEEE) received the B.S. degree in mathematics and applied mathematics from South-Central University for Nationalities, Wuhan, China, in 2011, and the Ph.D. degree in control science and engineering from the Huazhong University of Science and Technology, Wuhan, in 2016. He is currently an Associate Professor with the School of Automation, China University of Geosciences, Wuhan. His current research interests include neural networks and nonlinear control.
HUAQIN HE received the B.S. degree in physics from Fujian Normal University, Fujian, China, in 1991, and the M.S. and Ph.D. degrees in agronomy from Fujian Agriculture and Forestry University, Fuzhou, China, in 2000 and 2004, respectively. He is currently a Professor with the College of Life Sciences, Fujian Agriculture and Forestry University. His research interests include neural networks and bioinformatics. VOLUME 8, 2020