Synchronization of a Class of Memristive Stochastic Bidirectional Associative Memory Neural Networks with Mixed Time-Varying Delays via Sampled-Data Control

1School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing 100083, China 2Beijing Key Laboratory of Knowledge Engineering for Materials Science, Beijing 100083, China 3North China University of Science and Technology, Tangshan 063009, China 4Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China 5Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany 6Department of Electrical Engineering and Computer Science, Cleveland State University, Cleveland, OH 44115, USA


Introduction
Associate memory is one of the most significant activities of human brain, which can be applied in study of brain-like systems [1], intelligent thinking for intelligent robots [2], and so on.Since Kosko discussed the concept of bidirectional associative memory neural networks (BAMNNs) [3] in 1988, BAMNNs occupied the great researchers' time and have been studied for several years.Nowadays, due to the wide applications in signal processing, associative memory, pattern recognition, and so on [4], chaos control and synchronization of BAMNNs have been intensively investigated.Owing to the special characters of the memristor [5], researchers have replaced resistor with memristor in large scale integration circuits to construct the MBAMNNs [6].MBAMNNs are more suitable for mimicking the associative memory process of human brain contrast with the BAMNNs.Thus, more and more researchers build the MBAMNNs models for investigating a variety of applications [7][8][9][10].
In practical MBAMNNs systems, an ever-present phenomenon is the threshold of the sensitive memristor with a nonlinear drift effect [11], such as the voltages, current, and magnetic flux.It is well known that the presence of the threshold input may drastically deteriorate the desired performance, even inducing the inaccuracy of closed-loop systems under investigation.Hence, it is important to take the nonlinear characteristic of the MBAMNNs into consideration in the dynamical systems.In general, two common methods are utilized to cope with the nonlinear characteristic of the MBAMNNs.One is treating the parameters as constants [12,13], and the other is making a study of antisynchronization [14].Recently, the problem of nonlinear characteristic has also been considered to the field of memristive neural networks (MNNs) [15][16][17][18][19][20][21][22].Although the importance of nonlinear characteristic has been rather well recognized, but few related results have been reported on the synchronization of MBAMNNs.This is the first motivation of the present paper.
Synchronization is an elementary collective phenomenon that enables coherent behavior in neural networks (NNs), where neurons coact with each other and achieve a common dynamic behavior.Synchronization is of great significance for its potential applications in many areas, including harmonic oscillation generation, biology systems, and secure communication.Looking through the literatures on the synchronization of MBAMNNs, one can find that most of the results are based on the two kinds of continuous-time control strategies: the state feedback control [23][24][25][26][27][28] and adaptive control [29].A prerequisite of these approaches is that the controllers must obtain signals from sensors in a continuous way [30].This will increase the control cost heavily and cause a waste of communication bandwidth [31].
In contrast to the continuous-time control, the sampleddata control merely makes use of the sampling signals at discrete time instants.Consequently, the sampled-data control can eliminate the continuous monitoring of system states as well as the continuous information transmission.Therefore, the sampled-data control is a more managing choice in applications.Till now, numerous results have been reported in this aspect [30][31][32][33].However, to the best of our knowledge, there are few relevant achievements that consider the sampled-data synchronization of MBAMNNs.This is the second motivation of the present paper.
Owing to the limited speed of signal transmission between the neurons, the finite switching speed between different circuit elements in hardware implementations of NNs, and the viscosity of synapses triggered by biological NNs, time delay is an inevitable phenomenon in NNs.There are many types of delays, like discrete delay, leakage time delay, distributed delay, neutral-type delay, and so on.These delays are the main factors that contribute to the oscillation, instability, and the performance degradation to the dynamical systems [34].Therefore, the dynamic systems with time delay becomes a hot topic in the theoretical and application realms.
Besides, the actual communication between real systems is usually disturbed by a stochastic perturbation from various uncertainties.In secure communication systems, the digital signal is transmitted by switching forth and back continuously between synchronization.The stochastic perturbation will probably lead to package losses or influences the signal transmission.Hence, it is important to discuss the effect of probabilistic delays, stochastic perturbations, and so on [35][36][37][38].Therefore, it is valuable and practical to discuss the effect of the stochastic perturbations and time-varying delays on MBAMNNs.
Motivated by the foregoing discussions, this paper aims at investigating the synchronization of MBAMNNs with mixed time-varying delays and stochastic perturbations by designing a suitable sampled-data controller.The main contributions of this paper are summarized as follows.
(1) We first investigate the globally asymptotic synchronization of MBAMNNs with mixed time-varying delays and stochastic perturbations.
(2) According to the characters of memristor, we consider the parameters mismatch between the drive-response systems and design a suitable sampled-data controller to fit the features of the memristor.
(3) We also analyze the feasible region of the sampling period according to simulations, which is significant to some potential future research.Owing to the sampled-data synchronization analysis, Lyapunov functional method, and stochastic analysis theory, the synchronization criteria of the parameters mismatched MBAMNNs are derived.
The rest of this paper is organized as follows.The systems and problems formulation are presented in Section 2. In Section 3, based on Lyapunov functional method, stochastic analysis theory, and inequality techniques, sufficient criteria that depend on such system for synchronization are obtained.Numerical simulations are demonstrated to verify the effectiveness of the obtained results in Section 4. Finally, conclusions are given in Section 5.

Model Description and Preliminaries
2.1.Model Description.In order to better understand the MBAMNNs, firstly we describe the circuit of a general class of BAMNNs with the architecture as shown in Figure 1.Take the th subsystem and the th subsystem as the unit of analysis so as to simplify illustration [39]; one can clearly see that the Kirchhoff 's current law (KCL) of the subsystems of BAMNNs [3] is described as the following differential equation: where   () and   () are the voltages of capacitors C  and C , respectively.And F  presents the resistor between the feedback function   (  ( − ())) and   (); R  depicts the resistor between the feedback function   (  ( − ())) and   ().Then the transmission time-varying delay is illustrated by (), I  () is the bias function or external input on the th subsystem at time , J  () denotes the bias function or external input on the th subsystem at time , and Remark 1. Enlightened by [23][24][25][26][27][28][29], especially for [39], we proposed the following system which contains not only discrete time-varying delays () and (), but also distributed time-varying delays () and ().And self-inhibition weights   (  ()) and   (  ()) are also time-varying.Therefore, the obtained results are more general and practical than some existing results.
Based on the physical properties of a memristor, the proposed MBAMNNs with mixed time-varying delays are described by the following differential equations: where   () and   () denote the voltages of capacitors   and C at time , for  ≥ 0,  =

Mathematical Problems in Engineering
According to the current-voltage characteristic and the property of a memristor, the memristive connection weights of system (3) can be modeled as in which switching jumps   > 0, ď  > 0, d > 0 and ǎ  , â , b  , b , č , ĉ , for  = 1, 2, . . ., ,  = 1, 2, . . .,  are constants, so do   , p , p , m , m , q  , q , ň  , and n .In this paper, we treat system (3) as the drive system; then the corresponding response system with stochastic perturbations is described as () and   () are the appropriate controllers that will be proposed in order to gain the certain control objectives.Consider the following state feedback sampled-data controllers: where   and   are the sampled-data controllers gain matrices to be designed.  and   are positive scalars.Then   ( 1 ) and   ( 2 ) are discrete measurements of   () and   () at the sampling instant  1 and  2 , respectively.And the sampling instants satisfy the following conditions: Remark 3. Some of the published papers studied the synchronization of MNNs [24,25] through the following assumption to design the error system: Let co[, ] denote the closure of convex hull generated by real numbers  and  or real matrices  and .However this assumption has not always been proved to be correct.In [26], authors tried to deal with the synchronization issue of MABMNNs, but the results are unreasonable without taking the switching jumps into account.
The antisynchronization can avoid the problems mentioned above by constructing the error system as the following assumption: But it still has limitations in practical applications, such as associative memory and associative learning.It is worth mentioning that we fully consider the complex property of the inner connection and take the switching jumps into consideration.Therefore, the obtained results are more practical and less conservative than some published literatures.

Definitions and Assumptions.
In order to get our primary conclusions in the next section, we make the following assumptions.For the sampling interval, one has the following.Assumption 4 (see [40]).It is supposed that the interval between any two sampling instants is bounded by  ( > 0): The constants  1 ,  2 denote the maximum time span between  1+1 and  1 ,  2+1 and  2 .The state is sampled, and  1+1 ,  2+1 are the next update reaching the destination.
Due to the discrete terms   ( 1 ) and   ( 2 ), it is difficult to analyze the synchronization of MBAMNNs directly.Therefore, according to the input delay approach,  1 and  2 are defined as where 0 ≤  1 () ≤  1 , 0 ≤  2 () ≤  2 , and the controllers can be designed as Remark 5.It should be noted that [41] has made use of the sampled-data control to the MNNs, but to the best of our knowledge little attention has been paid to the MBAMNNs with mixed time-varying delays and stochastic perturbations based on sampled-data control theory, which motivates our present study.

Mathematical Problems in Engineering
Assumption 8 (see [37]).There exist constants For the stochastic system [42], where () is the Brownian motion and it is clearly E() = 0.
L is the operator defined as follows: where Assumption 9.The time-varying delays (), () in this paper are differential functions, where for all  ≥ 0.

Main Results
In this section, we get some new sufficient conditions to ensure the synchronization of MBAMNNs by the designed sampled-data controller.
Theorem 11.Assume that Assumption 7 is satisfied; then error system (9) achieves global stable situation under the designed sampled-data feedback controller (7) with the control law as follows: where and  1 ,  2 ,  1 ,  2 are all positive constants.
Proof.Consider the following Lyapunov function for synchronization error system (9): where at time , according to the jumping rules, systems (3) and ( 5) are reduced to systems ( 28) and ( 29), respectively: Then we define the corresponding response system And error system (9) can be rewritten: Then We conclude that where From all the above discussion, we get and then According to (31), one has Due to Assumptions 7 and 8, we obtain the following inequality: and then Based on Lemma 10, we get the inequalities as follows: Thus, we infer that Mathematical Problems in Engineering After organizing the above formulas, we deduce that With similar process of V  (), we obtain V  () as follows: ]  2  () . ( So according to (25) and the compilation of the above equations, we have According to the definitions of  2  and  2  , one has V() ≤ 0. Then Then the corresponding error system can be defined as follows: (47) The following line is similar to V  () of Case 1, and we obtain V  () and V() as follows: According to the definitions of  2  and  2  , one has V() ≤ 0. Thus  (52) We infer the following inequality through (26): According to (52), we have Through Assumptions 7 and 8 and Lemma 10, we get ] Then, (55) can be written as ] According to a similar process of V  (), we have V() be extended as follows: According to the definitions of  2  ,  2  ,   , and   , one has V() ≤ 0. Then we get Similarly, evaluating V  () along the trajectories of (60), we get V  () and V().
According to the definitions of  2  ,  2  ,   , and   , we obtain the following estimation V() ≤ 0. So we get The proof of Theorem 11 is completed.
Remark 12.In some previous studies [23][24][25][26][27]44] the authors treated the self-inhibition   (  ()) = 1 or   (  ()) =   > 0. It should be mentioned that the above researches cannot refer to the case that the self-inhibition switches at two states.Because the discontinuous self-inhibition can make the system illustrate the complicated nonlinear dynamic behaviors, thus we considered the self-inhibitions   (  ()) and   (  ()) as the state switching parameters.
Corollary 13.If Assumption 7 holds, systems (31) and (32) without perturbations are asymptotically stable when the following conditions are satisfied: where Proof.The proof process is similar to Theorem 11.So the proof is omitted.
Remark 14. Usually, we observe that the actual communication between subsystems of MBAMNNs is inevitably disturbed by the time-varying leakage delays from various uncertainties.Thus, we considered the time-varying delays in the leakage term, and then we get the asymptotic synchronization criterion for the drive and response systems.Under Assumption 9, the drive system of MBAMNNs with mixed time-varying delays can be described by And the corresponding response system can be defined as (66) Corollary 15.Suppose Assumption 7 is satisfied, then systems ( 65) and ( 66) globally achieve synchronization under designed sampled-data feedback controller (7) with the control law as follows: (68) Proof.The proof process is similar to Theorem 11.So the proof is omitted.
Remark 16.Some existing researches show the criterion for systems with differentiable delays, bounded in the in the leakage term.However, when the time-varying delays () and () are not differentiable or their derivatives are unknown or no bounded, the criterion cannot be applicable any more.Under these circumstances, the conclusion we obtained is more valid and effective than the exciting results.
Corollary 17. Assume that Assumption 7 is satisfied; then systems ( 65) and ( 66) with constant delays in the leakage term (() = , () = ) will achieve asymptotic synchronization under the same criteria.Remark 18.There is no extra restraint on activation functions demanding that they are bounded and the time-varying delays are mixed.Furthermore, overall consideration of our obtained results with sampled-data control schemes is shown, which can be expected to have a powerful potential application in areas like associative memory, image encryption, digital processing, and so on.

Numerical Simulation
In this section, numerical examples are presented to demonstrate the results for plausibility and validity.Based on the following parameters, consider two-dimensional MBAMNNs with mixed time-varying delays and stochastic perturbations as follows:   x The dynamic behaviors of drive-response systems are given in Figure 2, and the following simulations are conducted on the basis of this situation.In order to verify Theorem 11, we take systems (69) and (70) as example.Figure 3 shows the state trajectories of such systems without control; then it illustrates the synchronization errors correlation between the drive-response systems without and under control.From the above illustration, we conclude that systems (69) and (70) will achieve globally asymptotic synchronization with the help of the proposed controller (14).
Based on Theorem 11, we indicate the considered systems without perturbations.Under these circumstances, define the errors as   () =  x () −   () and   () =  ŷ () −   (); the error states without control and under control are depicted in Figure 4.The comparison between (a) and (b) provides the clearest significant role of the controller played in the synchronization control.
According to Corollary 15, for given time-varying delays () and (), systems (65) and (66) are globally asymptotically synchronized under controller (14).In this simulation, we take () = () = 0.15, and Figure 5 depicts the trajectories of error states.In order to testify Corollary 17, we make  =  = 0.15 in the leakage terms; Figure 6 illustrates the error states of the proposed systems.From the comparison results we conclude the sampled-data control inputs which contribute to the effective chaos synchronization.
The error system becomes unstable when we choose sampling period  1 =  2 = 100.Under these circumstances, it means the sampled-data controller with the feedback gains   and   is not enough to guarantee the stability of the error systems.For comparing and analyzing the influence of the sampling period relationship, we make the state feedback controller with the same feedback gains.Figure 7 depicts the trajectories of error states.From Figure 8 we detect that the error system becomes stable at 11.921.So we conclude that when we take the sampled-data controller, the effect on error convergence is not only relevant to the feedback gains but also related to the sampling period.Table 1 illustrates that the best sampling period for our simulations is  1 () =  2 () = 0.01 sin .

Remark 19.
Comparing with the exciting researches on MBAMNNs, we investigate the relationship between the sampling period and convergence time.It can be found that even the longer sampling period will cause the lower communication channel occupation, less packet transmission, and little actuation of the controller; it also sacrifices the accuracy of controller at the same time.Thus the suitable sampling period has important significance to the stable of error systems.So our conclusion is greatly helpful to some potential future research topics including synchronization or stability on sampled-data synchronization strategy to NNs.

Conclusion
In this paper, a synchronization problem has been investigated for MBAMNNs with stochastic perturbations and mixed time-varying delays.The dynamics of the synaptic weights between the drive and response systems were considered and analyzed rather than treating them as consonants.Firstly, both various mixed time-varying delays and stochastic perturbations are considered in this paper, which include no-delay, finite distributed time-varying delays, discrete time-varying delays, and leakage time-varying delays.Secondly, we have proposed a sampled-data synchronization strategy for each node of the MBAMNNs.By utilizing the sign function and the definition of asymptotic stability, a suitable nonlinear state feedback sampled-data controller is designed.In addition, we verify the sampling period effect on the convergence of error system according to simulations.By utilizing the Lyapunov functional method, stochastic analysis theory, and inequality techniques, some sufficient conditions are derived to guarantee synchronization of the MBAMNNs model.Simulation examples have been presented to validate the theoretical results.

Figure 3 :
Figure 3: (a) and (b) show the time response curves of drive-response systems (69) and (70); (c) and (d) show the curves of error systems without or with delay-dependent controller (14), respectively.

Figure 4 :Figure 5 :
Figure 4: (a) The synchronization error of drive-response systems (69) and (70) without perturbations and control; (b) the synchronization error of drive-response systems (69) and (70) without perturbations but under control.

Figure 6 :
Figure 6: (a) The synchronization error of drive-response systems (65) and (66) with constant leakage delays but without control; (b) the synchronization error of drive-response systems (65) and (66) with constant leakage delays and control.

Table 1 :
The relationship between convergence time and sampling period.