#### Abstract

The global exponential antisynchronization in mean square of memristive neural networks with stochastic perturbation and mixed time-varying delays is studied in this paper. Then, two kinds of novel delay-dependent and delay-independent adaptive controllers are designed. With the ability of adapting to environment changes, the proposed controllers can modify their behaviors to achieve the best performance. In particular, on the basis of the differential inclusions theory, inequality theory, and stochastic analysis techniques, several sufficient conditions are obtained to guarantee the exponential antisynchronization between the drive system and response system. Furthermore, two numerical simulation examples are provided to the validity of the derived criteria.

#### 1. Introduction

Memristor is a nonlinear resistor with memory function. Because of the nonlinear nature, it has been found that memristor responds well to the variable strength of synapses in the human brain. Therefore, we use memristor instead of resistor to reform new neural networks model named memristive neural networks (MNNs). In recent decades, dynamic behaviors analysis of MNNs has been attracting increasing attentions [1–3].

As a typical dynamic behavior, the stability and synchronization problems of MNNs have been widely discussed, including exponential synchronization [4, 5], lag synchronization [6], finite time synchronization [7, 8], and antisynchronization [9, 10]. In communication system, the digital signal is transmitted by switching back and forth continuously between synchronization and antisynchronization, which strengthen security and secrecy. Besides that, the antisynchronization analysis of MNNs can provide the designers with some amazing properties, extensive flexibility, and opportunities. Thus, there are a few articles dealing with the antisynchronization issues of MNNs [11–15]. In [11], they studied the exponential antisynchronization in mean square of MNNs with the uncertain terms which include nonmodeled dynamics with boundary and stochastic perturbations. By an intermittent sampled-data controller [12] and utilizing matrix measure approach and Halanay inequality [13], the antisynchronization control of MNNs with time-varying delays has been addressed. Wang et al. discussed the antisynchronization control of MNNs with multiple proportional delays [14] and mixed time-varying delays [15]. Recently, the antisynchronization control has been widely employed in many fields, for example, secure communication, harmonic oscillation generation, and information science [16–18].

In practice, because of certain switching speeds of the amplifiers, time delays are often encountered in hardware implementation [19], which may affect the stability of the system, even result in oscillation, divergence, and instability phenomena. Consequently, a great deal of effort has been devoted to the stability analysis of MNNs with various types of time delays [20–22]. Moreover, since an amount of parallel pathways of multiple axon sizes and lengths reside in the MNNs, such a unique nature can be appropriately modeled by distributed delay which means the signal transmission is distributed over a specific period of time. Hence, the authors in [23–25] have concentrated on the mixed delays. Meng and Xiang [24] considered a class of recurrent MNNs with mixed time-varying delays, in which the passivity and exponential passivity were investigated.

Furthermore, in [25], Yang et al. studied the th moment exponential stochastic synchronization for MNNs with discrete and distributed time-varying delays. However, the discrete delay in [25] must be differentiable. Clearly, such a constraint on the delay term was relatively strong. So, we discuss two kinds of discrete time-varying delay in this paper. One is that its derivative is bounded; the other is that it is not differentiable, or the derivative is unknown or unrestricted.

On the other hand, as a result of random fluctuations from the release of neurotransmitters or other probabilistic causes in the nervous system, synaptic transmission is indeed a process accompanied by noise. The stability analysis of MNNs with stochastic perturbation has aroused great interests of many researchers [26–28]. For example, Song and Wen [27] investigated the stochastic MNNs with mixed delays and proposed the exponential synchronization criteria in the th moment. Subsequently, Bao et al. [28] discussed a class of coupled stochastic MNNs with probabilistic time-varying delay in order to achieve exponential synchronization. However, to the best of our knowledge, the research on the exponential antisynchronization analysis of stochastic MNNs with mixed delays is still an open problem.

Motivated by the above discussion, we focus our minds on the exponential antisynchronization problem of the MNNs with stochastic perturbation and mixed time-varying delays. Compared with other existing articles [11, 28, 29], our model is more complex and closer to the actual system; the obtained results are less conservative. The main contributions of this paper can be summarized in the following:(1)The presented MNNs model contains not only stochastic perturbation but also two types of time-varying delays, namely, discrete and distributed time-varying delays.(2)It is known that the time delays have a great influence on the stability of the MNNs, so the time delays cannot be neglected. As far as we know, many articles are based on the assumption that the time delay is bounded and derivable. Actually, it may happen that the time delay is not differentiable or its derivative is unknown or unrestricted. Therefore, we study two kinds of discrete time-varying delay.(3)Under two new types of controllers with delay-dependent and delay-independent, the obtained criteria which need no excessive numerical calculation can be extended to other general MNNs models.

The rest structure of this paper is outlined as follows. In Section 2, the models of the stochastic MNNs with mixed time-varying delays and some preliminaries are introduced. In Section 3, the main results on exponential antisynchronization of the stochastic MNNs are derived. In Section 4, some numerical simulations are presented to demonstrate the efficiency of the theoretical results. In Section 5, the conclusion is showed.

#### 2. Model Description and Preliminaries

On the basis of the above discussion, we propose a class of MNNs with discrete and distributed time-varying delays described by the following differential equations: where corresponds to the number of neurons in system (1). denotes the voltage of the capacitor at time . The self-inhibition connection weight is . The neuron activation functions and are continuous and bounded. and are two time-varying delays satisfying , , (, , and are positive constants). represents the continuous external inputs on the th neuron at time . , , and are the memristive connection weights: in which , , and denote the memductances of memristors , , and , respectively. In addition, represents the memristor between the activation function and , represents the memristor between the activation function and , and represents the memristor between the activation function and . As far as we know, capacitor is constant while memductances of memristors , , and respond to change in pinched hysteresis loops [30]. As a consequence, , , and will change with time. Based on the current-voltage characteristic of the memristor, we let where the switching jumps , for . Then , , , , , and are known constants with respect to the memristance. Moreover, the initial condition of system (1) is given to be , .

*Remark 1. *According to the above explanation, , , and in system (1) are changed as the state of the memristance is switching or the state has switched. Therefore, the MNNs is considered as a time-varying system with state-dependent switching. When , , and are constants, system (1) becomes the general recurrent neural networks.

Since , , and are discontinuous, in this paper, the solutions of all the following systems are illustrated in Filippov’s sense. [] represents the interval. Set , , , , , and for . denotes closure of the convex hull generated by real numbers and . Considering system (1), we define the following set-valued maps:

Obviously, , , and , for . By the theory of differential inclusions, system (1) can be written as follows: or equivalently, for , there exist , , and , such that In this paper, consider system (5) or (6) as the drive system; then the corresponding response system is designed by the following form: or equivalently, for , there exist , , , such that where , , are the appropriate adaptive controllers designed to achieve the exponential antisynchronization stability. denotes the stochastic perturbation, where is an -dimensional Wiener process. It is defined on a complete probability space with a natural filtration (i.e., ).

*Remark 2. *Compared with the articles already published [28, 29], the proposed system contains not only discrete time-varying delay but also distributed time-varying delay , while the stochastic perturbation is also taken into account. Therefore, the obtained results have a high value of practical application in this paper.

As a matter of convenience, we will take advantage of the following assumptions.

*Assumption 3. *For , , we assume the activation functions and are odd and satisfy the Lipschitz continuous condition with positive constants and , such that

*Assumption 4. * is locally Lipschitz continuous and meets the linear growth condition with . There exist nonnegative constants , , such that, for ,

*Assumption 5. *For ,

*Remark 6. *In [4], however, the error system was defined based on the following assumption: In fact, this assumption has been proved not always to be correct in [5]. Recently, many researchers have tried to explore a novel and appropriate way to solve this problem. So far, there are two kinds of convincing method. One is, in [31], according to the switching parameter ; the authors discussed the parameter in four cases and obtained several synchronization conditions of the chaotic MNNs with time delays. The other is, in [11, 15–17], by Assumption 5; the authors turned to study the antisynchronization issues of MNNs.

Now, we define the following error system: According to the theories of set-valued maps, stochastic differential inclusions, and Assumption 5, we derive the error dynamical system as follows: or equivalently, for , there exist , , and , such that where , , and .

Since the activation functions and are odd and bounded, in the light of the Assumption 3, for , it is so clear that According to the definition of functions and , we deduce

Next, we introduce a definition and some lemmas as preparation.

Lemma 7. *For the stochastic system,where is the Wiener process and it is obvious that . is the operator defined as follows: where*

Lemma 8 (see [32]). *If and are real matrices which have appropriate dimensions, then there exists a constant , such that *

Lemma 9 (see [33], Jensen’s inequality). *For any constant matrix , a scalar , and a vector function , then the following inequality holds:*

*Definition 10. *The error system with mixed delays and stochastic perturbations can exponentially converge to zero in mean square if there exist positive constants and , such that , for , where is called the exponential convergence rate. Systems (6) and (8) are said to be exponential antisynchronization in mean square sense.

#### 3. Main Results

In this section, we get some sufficient criteria to achieve the exponential antisynchronization of the stochastic MNNs with mixed time-varying delays. Then two corollaries are also derived for the stochastic MNNs. The novel delay-dependent adaptive controller is designed as follows: where the constants , , for .

For convenience, some notations are given. Let , , and , for .

Theorem 11. *Under Assumptions 3, 4, 5, and 13, systems (6) and (8) achieve exponential antisynchronization in mean square sense under the delay-dependent adaptive controller (23). For a given constant , if there exist constants , such that the conditions hold then error system (15) can be exponentially converged to zero.*

*Proof. *We choose the following Lyapunov functional: whereThen we have By differential formula, we obtain the stochastic derivative of in the form of From Lemma 7, it is clear that Moreover, from mean value inequality and Assumption 3, we have and together with Lemma 9, it follows that Then we getUnder Assumption 4, one hasthen we obtain the following estimation:Letconsidering the condition of Theorem 11, we get It is easy to derive that Combining with (27), we finally have that Therefore, by Definition 10, we see that systems (6) and (8) can be exponentially synchronized in mean square sense with the exponential convergence rate . The proof of Theorem 11 is completed.

*Remark 12. *In Theorem 11, we assumed that the delay satisfies . Actually, when is not differentiable or its derivatives are unknown or no bounded, Theorem 11 cannot be applicable any more. If the value of is unknown or not restriction, we will obtain the following corollary.

*Assumption 13. *The activation functions and are bounded; namely, there exist constants , such that It is obvious that

Corollary 14. *If the time-varying delay is nondifferentiable, or the value of is unknown or even the derivative is unrestricted. According to Assumptions 3, 4, and 13, system (15) will converge to zero. For a given constant , if there exist constants , and the conditions hold systems (6) and (8) are exponentially antisynchronized in the mean square sense under the following controller: *

*Proof. *We consider the following Lyapunov function: Combining with Assumption 13, the proof of Corollary 14 is analogous to Theorem 11, so we omit it here.

*Remark 15. *Based on the assumption that the delay is bounded and differential, Theorem 11 provides a delay-dependent adaptive controller. It is worth mentioning that the criteria need no excessive numerical calculation such as solving LMIs [11] or computing complex algebraic conditions [28]. In Corollary 14, the delay is not required to be differential but only bounded which makes our result more general. Further, there is an upper limit of the exponential convergence rate . When the exponent rate exceeds the upper limit, system (15) may not be able to converge or even cause unpredictable result.

LetThe vector form of system (15) is given by where , , , , , and .

Moreover, we impose the following assumption and some lemmas.

*Assumption 16. *If there exist appropriate positive constant matrices and , such that the inequality holds for any , and .

Lemma 17 (see [24]). *For any given nonnegative matrix , if there exist a constant and a vector function , the following differential inequality holds: where .*

If there is no time delay item, the delay-independent adaptive controller can be described as follows:

Next, we will discuss the antisynchronization of systems (6) and (8) under the above delay-independent adaptive controller .

Theorem 18. *Under Assumption 13, for a given constant , if there exist constant and matrixes , , the conditions hold as follows: where , , , , and . System (45) will converge to zero with the action of the above delay-independent adaptive controller.*

*Proof. *We construct the following Lyapunov function: whereThen we have According to Lemma 7, we get that