Asymptotic synchronization for stochastic memristor-based neural networks with noise disturbance

Abstract

In this paper, globally asymptotical synchronization for stochastic memristor-based neural networks with random noise disturbance is investigated. Under the framework of differential inclusions theory and set-valued maps, a state feedback controller and an adaptive updated law are designed by constructing a suitable Lyapunov functional. By using Itô formula and some significant inequality techniques, sufficient conditions for the global synchronization of the stochastic memristor-based neural networks which are more general are obtained. Finally, numerical simulations are provided to illustrate the theoretical results.

Introduction

In 1971, Chua firstly introduced the memristor as the fourth ideal electrical circuit element besides the resistor, capacitor and inductor to describe the relationship between electric charge and magnetic flux in [1]. Until 2008, the Hewlett-Packard research team [2] successfully obtained a practical memristor device. Because the memristor has many distinct properties, such as nanoscale, low energy dissipation and the memory ability, increasing research from different branches of science and application fields pay more attention to memristor. It has been shown that memristors can be proposed to work as synaptic weights in artificial neural networks in [3]. Due to these properties, memristor can replace resistor to model a new neural network that is memristor-based neural networks (MNNs) which simulate the human brain [4]. The studies of MNNs would benefit many practical applications such as neural learning circuits [5], associative memories [6], new classes of artificial neural systems [7], and so on.

Recently, synchronization or anti-synchronization of memristor-based neural networks have received great attention because of their potential applications such as secure communication, information science and biological technology [8]. But the networks are not always able to synchronize by themselves. Then, various effective control approaches and techniques have been proposed for synchronization or anti-synchronization, such as impulsive [9], feedback [10], [11], [12], [13], adaptive [14], [15], intermittent control [16], [17] and so on. Li and Cao [18] have proposed new synchronization criteria on adaptive control and feedback control for memristor-based networks. On the other hand, stochastic effects usually are unavoidable in real nervous systems and synaptic transmission is a noisy process brought on by random disturbance from the release of neurotransmitters and other probabilistic causes. When the inaccurate information with noises propagates in the network, it is more difficult to synchronization all the agents and it might lead to a divergence. So we should devote a great effort to study the problem of synchronization under noisy environment. Song and Wen [19] have proposed a class of stochastic memristor-based neural networks model with discrete and distributed delays and have established some new criteria to guarantee the exponential synchronization in the pth moment. However, to the best of our knowledge, there is little research on adaptive control for memristor-based neural networks with noise disturbance, despite the effect of noise disturbance is inevitable.

Motivated by the aforementioned discussions, a state feedback controller and an adaptive updated law are designed to deal with the synchronization problem of stochastic memristor-based neural networks with noise disturbance and time-varying delays. By combining differential inclusions theory with set-valued maps, we construct a suitable Lyapunov functional. By using Itô formula and some significant inequality techniques, sufficient conditions for the global synchronization of the stochastic MNNs are obtained. Moreover, in the most previous work [17], [19], some basic assumptions are always required. But these assumptions are not always satisfied. Our constraints on the neuron activation functions are less conservative and our synchronization criteria are more general than the existing literature.

This paper focuses on the asymptotic synchronization problem for stochastic memristor-based neural networks with noise disturbance. The main contributions of this paper lie in the following aspects. Firstly, the basic assumptions in previous references [17], [19], [20] are removed and some less conservative conclusions are obtained. Secondly, the time-varying delay is considered and a new mathematical model of stochastic memristor-based neural networks is established, which takes the random disturbances into consideration. So the model is more close to the actual model. Additionally, a simple feedback controller and a novel adaptive control law are designed to study the globally asymptotically synchronization for stochastic memristor-based neural networks with random noise disturbances. The two controllers are different from the techniques employed in [17], [21], [22]. The new proposed results in this paper are simple to verify and they achieve a valuable improvement, and extend the previous results.

The organization of this brief is as follows. Some preliminaries and the model formulation are introduced in Section 2. In Section 3, feedback controller and adaptive synchronization criteria for the stochastic memristor-based neural networks are designed. Then, numerical simulations are given to demonstrate the effectiveness of the proposed approach in Section 4. Finally, conclusion is given in Section 5.

Notation: Let $\mathbb{R}$ and ${\mathbb{R}}^n$ be the real numbers and the n-dimensional Euclidean space, respectively. For $\tau >0$, $\mathcal{C}\left(\right[-\tau ,0],{\mathbb{R}}^n)$ denotes the family of continuous functions ϕ from $[-\tau ,0]$ to ${\mathbb{R}}^n$ with the norm $\parallel \varphi \parallel ={({\sum }_{i=1}^n{\varphi }_i^p)}^{\frac{1}{p}}$. $\mathrm{co}\left(Q\right)$ denotes the closure of the convex hull of Q. $\mathbb{E}[·]$ is the mathematical expectation. $D^{+}$ denotes the upper right Dini derivation.