Synchronization stability in complex interconnected neural networks with nonsymmetric coupling

Abstract

Some global stability criteria for arrays of linearly coupled delayed neural networks with nonsymmetric coupling are established on the basis of linear matrix inequality (LMI) method, in which the coupling configuration matrix may be arbitrary matrix with appropriate dimensions. Meanwhile, the relations of coupling matrix to stability and synchronization are discussed in detail, and some analysis methods to the synchronization stability of complex networks are also argued. The difficulty arising from the nonsymmetry of the coupling matrix has been overcome in this work. Numerical simulation is also given to verify the effectiveness of the theoretical results.

Introduction

There exists increasing interest in the study of dynamical properties of delayed recurrent neural networks due to its potential applications in various fields, including online optimization, pattern recognition, signal and image processing, and associative memories [1], [2], [3]. Most of the previous studies mainly concentrated on stability analysis, periodic or almost periodic attractors, and dissipativity of recurrent neural networks with or without delays [4], [5], [6], [7], [8], [17], [18]. Since it has been reported that there are synchronization phenomena in many real systems, such as in an array composing of identical delayed neural networks, it is important to study the synchronization problem in coupled networks and systems in engineering applications, such as secure communication and signal generators design [9], [10], [11], [12], [13], [14], [16]. Therefore, the study of synchronization of coupled neural networks is an important step for both understanding brain science and designing coupled neural networks for practical use.

Nowadays, there are mainly two kinds of coupling matrix structure to study the synchronization in an array composing of identical delayed neural networks:

(1) Symmetric coupling matrix $G=(G_{\mathit{ij}})\in {\mathfrak{R}}^{N\times N}$, which means that for two connected nodes the influences to each other are the same. That is, $G_{\mathit{ij}}=G_{\mathit{ji}}\ge 0$ for $i\ne j$ and $G_{\mathit{ii}}=-{\sum }_{j=1,j\ne i}^N{G}_{\mathit{ij}}$.

(2) Nonsymmetric coupling matrix, $G_{\mathit{ij}}\ge 0$ for $i\ne j$ and $G_{\mathit{ii}}=-{\sum }_{j=1,j\ne i}^N{G}_{\mathit{ij}}$, $i,j=1,\dots ,N,$ N is the number of coupling term, $G=(G_{\mathit{ij}})\in {\mathfrak{R}}^{N\times N}$ is an irreducible coupling matrix.

For the case of symmetric coupling matrix, the synchronization problems have been investigated in [9], [15], [16], [19], [20], [22], [23], [24], and some synchronization criteria have been derived based on LMI method or other methods. All the LMI-based synchronization results are on the basis of Kronecker product expression, in which the symmetric and irreducible feature of coupling matrix plays an important role in the derivation.

For the case of nonsymmetric coupling matrix, the synchronization problems have been studied in [25], [26], [27], [28], [29], [30], [31], in which the coupling matrix G is irreducible. In [26], the coupling matrix elements $G_{\mathit{ij}}\ge 0$ for $i\ne j$, and $G_{\mathit{ii}}=-{\sum }_{j=1,j\ne i}{G}_{\mathit{ij}}$, and G is irreducible. The methods in [27], [28], [29] are to use the Jacobian matrix of nonlinear function at synchronization state, which can only ensure the local synchronization. Meanwhile, all the results in [27], [28], [29], [30] are in the Kronecker form, which enhance the difficulty to check. The results in [31] are on the basis of eigenvalue approach, in which some relations between the coupling matrix of linear couplig term and the parameters of isolated system are established.

In this paper, we will extend the requirement condition of the coupling matrix $G=(G_{\mathit{ij}})$, and study an array of linearly delay coupled system consisting of N identical delayed neural networks with each network being an n-dimensional dynamical system. Synchronization stability problem of the coupled interconnected large-scale system is first studied on the basis of LMI, and some discussions on the coupling matrix G are compared with the synchronization problem of the existing complex networks.

The main contributions of the paper are as follows: (1) Without the requirement of symmetric and irreducible conditions on coupling matrix G, some global asymptotical synchronization stability criteria are established for an array of linearly delay coupled neural networks, which make the synchronization criteria of complex networks more flexible. (2) The relations between the stability of isolated neural networks and the synchronization of an array of linearly delay coupled neural networks are discussed, which present a deep insight into the research on the dynamics of complex systems. (3) Some discussions are made on the coordinate transformation method in dealing with synchronization problem of complex networks, which reveal the necessity of the uniqueness assumption on the equilibrium point as required in [19].

The paper is organized as follows. Section 2 presents some problem formulations and preliminaries. Some discussions on the different requirement of coupling matrix G are presented. Meanwhile, some remarks are also made on the coordinate transformation method in dealing with synchronization problem of complex networks. Section 3 presents the main results and some remarks on the relations between the stability of isolated neural networks and the synchronization of an array of linearly delay coupled neural networks are made. Section 4 presents some simulations with different kinds of coupling matrix G to verify the effectiveness of the proposed results, and conclusions are drawn in Section 5.