Anti-synchronization control of a class of memristive recurrent neural networks

Abstract

In this paper, we formulate and investigate a class of memristive recurrent neural networks. Two different types of anti-synchronization algorithms are derived to achieve the exponential anti-synchronization of the coupled systems based on drive–response concept, differential inclusions theory and Lyapunov functional method. The proposed anti-synchronization algorithms are simple and can be easily realized. The analysis in the paper employs results from the theory of differential equations with discontinuous right-hand side as introduced by Filippov. The obtained results extend some previous works on conventional recurrent neural networks.

Introduction

Memristor, the fourth fundamental passive circuit element, was originally predicted by Chua [1]. He reasoned that the memristor was a similarly fundamental device for providing conceptual symmetry with resistor, inductor and capacitor. The symmetry follows from the description of basic passive circuit elements as defined by a relation between two of the four fundamental circuit variables, namely current, voltage, charge and magnetic flux. Almost forty years after Chua’s proposal, memristor was fabricated by scientists at the Hewlett–Packard (HP) research team, with an official publication in Nature [2], [3]. This new circuit element of memristor has generated unprecedented worldwide interest because of its potential applications [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. For example, by using memristor memory, new type of computers will have no boosting time, battery of cell phone could last one to two months rather than several days, brain-like computer can be made by using memristive synapses etc. So it is necessary to investigate the properties or characteristics of memristor because of its many key applications.

Recently, many researchers concentrate on the dynamical nature of memristor, which can be used to ultra-dense, seminon-volatile memories and learning networks. And some representative works [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] described a series of properties of the memristor and showed its usefulness in the modeling and understanding of various physical systems. According to these related researches, we can know that the memristor exhibits the feature of pinched hysteresis, which means that a lag occurs between the application and the removal of a field and its subsequent effect, just as the neurons in the human brain have. Because of this feature, broad potential applications of the memristor have been identified, one of which is to apply this device to build a new model of neural networks to emulate the human brain [5], [6], [7], [12], which have received widespread concern. Hu and Wang [6] consider the dynamic behaviors for the memristive recurrent neural networks. Pershin and Di Ventra [7] have shown that the electronic (memristive) synapses and neurons can represent important functionalities of their biological counterparts. According to the works in [5], [6], [7], [12], one can see that the memristive recurrent neural networks, which reproduce the characteristic time hysteresis behavior of memristor devices, can mimick the functionalities of the human brain, and provide an in-depth understanding of key design implications of memristor-based memories.

We describe a general class of recurrent neural networks with the architecture as shown in Fig. 1, the Kirchoff’s current law of the ith subsystem is described by the following equation:

$${\dot{x}}_i(t)=-x_i(t)+\frac{1}{{\mathcal{C}}_i}{\displaystyle \sum _{j=1}^n}\frac{g_j(x_j(t))}{{\mathcal{R}}_{\mathit{ij}}}\times {\mathrm{sgin}}_{\mathit{ij}}+\frac{1}{{\mathcal{C}}_i}{\displaystyle \sum _{j=1}^n}\frac{g_j(x_j(t-{\tau }_j))}{{\mathcal{F}}_{\mathit{ij}}}\times {\mathrm{sgin}}_{\mathit{ij}}\text{,}t⩾0\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}$$

where $x_i(t)$ is the voltage of the capacitor ${\mathcal{C}}_i$, ${\mathcal{R}}_{\mathit{ij}}$ denotes the resistor through the feedback function $g_i(x_i(t))$ and $x_i(t)\text{,}{\mathcal{F}}_{\mathit{ij}}$ denotes the resistor through the feedback function $g_i(x_i(t-{\tau }_i))$ and $x_i(t)\text{,}{\tau }_i$ corresponds to the transmission delay and satisfies $0⩽{\tau }_i⩽\tau$ (τ is a constant), and

$${\mathrm{sgin}}_{\mathit{ij}}=\left\{\begin{array}{cc}1\text{,}\hfill & i\ne j\text{,}\hfill \\ -1\text{,}\hfill & i=j\text{.}\hfill \end{array}\right.$$

By replacing the resistors ${\mathcal{R}}_{\mathit{ij}}$ and ${\mathcal{F}}_{\mathit{ij}}$ in the primitive recurrent neural networks (1) with memristors, whose memductances ${\mathcal{W}}_{\mathit{ij}}$ and ${\mathcal{M}}_{\mathit{ij}}$, respectively, then we can construct the memristive recurrent neural networks of the form

$${\dot{x}}_i(t)=-x_i(t)+{\displaystyle \sum _{j=1}^n}\left(\frac{{\mathcal{W}}_{\mathit{ij}}}{{\mathcal{C}}_i}\times {\mathrm{sgin}}_{\mathit{ij}}\times g_j(x_j(t))+\frac{{\mathcal{M}}_{\mathit{ij}}}{{\mathcal{C}}_i}\times {\mathrm{sgin}}_{\mathit{ij}}\times g_j(x_j(t-{\tau }_j))\right)\text{,}t⩾0\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{.}$$

Or equivalently,

$${\dot{x}}_i(t)=-x_i(t)+{\displaystyle \sum _{j=1}^n}\left(a_{\mathit{ij}}(x_i(t))g_j(x_j(t))+b_{\mathit{ij}}(x_i(t))g_j(x_j(t-{\tau }_j))\right)\text{,}t⩾0\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}$$

where

$$a_{\mathit{ij}}(x_i(t))=\frac{{\mathcal{W}}_{\mathit{ij}}}{{\mathcal{C}}_i}\times {\mathrm{sgin}}_{\mathit{ij}}\text{,}{b}_{\mathit{ij}}(x_i(t))=\frac{{\mathcal{M}}_{\mathit{ij}}}{{\mathcal{C}}_i}\times {\mathrm{sgin}}_{\mathit{ij}}\text{.}$$

According to the feature of the memristor and the current–voltage characteristic given in Fig. 2, as a matter of convenience, in this paper we discuss a simplified mathematical model of the memristive recurrent neural networks (3) as follows:

$${\dot{x}}_i(t)=-x_i(t)+{\displaystyle \sum _{j=1}^n}\left(a_{\mathit{ij}}(x_i)g_j(x_j(t))+b_{\mathit{ij}}(x_i)g_j(x_j(t-{\tau }_j))\right)\text{,}t⩾0\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}$$

where

$$a_{\mathit{ij}}(x_i)=a_{\mathit{ij}}(x_i(t))=\left\{\begin{array}{cc}{\hat{a}}_{\mathit{ij}}\text{,}\hfill & \left|x_i(t)\right|<{\mathrm{T}}_i\text{,}\hfill \\ {\breve{a}}_{\mathit{ij}}\text{,}\hfill & \left|x_i(t)\right|>{\mathrm{T}}_i\text{,}\hfill \end{array}\right.$$

$$b_{\mathit{ij}}(x_i)=b_{\mathit{ij}}(x_i(t))=\left\{\begin{array}{cc}{\hat{b}}_{\mathit{ij}}\text{,}\hfill & \left|x_i(t)\right|<{\mathrm{T}}_i\text{,}\hfill \\ {\breve{b}}_{\mathit{ij}}\text{,}\hfill & \left|x_i(t)\right|>{\mathrm{T}}_i\text{,}\hfill \end{array}\right.$$

with switching jumps ${\mathrm{T}}_i>0\text{,}{\hat{a}}_{\mathit{ij}}$, ${\breve{a}}_{\mathit{ij}}\text{,}{\hat{b}}_{\mathit{ij}}\text{,}{\breve{b}}_{\mathit{ij}}$, $i\text{,}j=1\text{,}2\text{,}\dots \text{,}n$, are constant numbers.

In fact, even $a_{\mathit{ij}}(x_i)$ and $b_{\mathit{ij}}(x_i)$ appear other types of memristive twinkling changes, the main results in this paper still can be made some parallel promotions.

In addition, we note that numerous complex nonlinear behaviors including chaos appear even in a simple memristive system [4], [10], [11], a detailed analytical study of synchronization or anti-synchronization control of the basic oscillator is necessary. Moreover, we also notice that, in recent years, the anti-synchronization control has been success fully applied to many areas for image processing, secure communication, information science, and harmonic oscillation generation [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. Using anti-synchronization to lasers, one may generate not only drop-outs of the intensity (as with ordinary low frequency fluctuations) but also short pulses of high intensity, which offers new ways for generating pulses of special shapes. Using anti-synchronization to communication systems, one may transmit digital signals by the transform between synchronization and anti-synchronization continuously, which will strengthen the security and secrecy. And therefore, due to many important applications in engineering fields, the anti-synchronization problem has become an important area of study. However, to our knowledge, there are very few works dealing with the anti-synchronization control of the memristive recurrent neural networks. And the anti-synchronization control of memristive recurrent neural networks plays important roles in many potential applications, e.g., non-volatile memories, neuromorphic devices to simulate learning, adaptive and spontaneous behavior. Moreover, the anti-synchronization analysis for memristive recurrent neural networks can provide the designer with an exciting variety of properties, richness of flexibility, and opportunities. Motivated by the above discussions, based on the works in [5], [6], [7], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], our objective in this paper is to study the exponential anti-synchronization for memristive recurrent neural networks (4). By applying the drive–response concept, differential inclusions theory and Lyapunov functional method, two different types of controller designs are proposed to ensure the exponential stability for the anti-synchronization error system, and thus the drive system anti-synchronizes with the response system. Also, the controller gains can be easily obtained. The analysis in the paper employs results from the theory of differential equations with discontinuous right-hand side as introduced by Filippov. It is believed that the paper provides some new and efficacious methods for the qualitative analysis of memristive recurrent neural networks. These methods may be applied for analyzing other classes of memristive neural networks or some other complex nonlinear memristive systems. These issues will be the topics of future research.

The structure of this paper is outlined as follows. In Section 2, some preliminaries are introduced. In Section 3, two different kinds of anti-synchronization algorithms are derived to achieve the exponential anti-synchronization of the drive–response-based coupled systems. In Section 4, a numerical example is presented to illustrate the results, and finally, concluding remarks will be drawn in Section 5.