%0 Journal Article
%T Exponential Stability of Periodic Solution to Wilson-Cowan Networks with Time-Varying Delays on Time Scales
%A Jinxiang Cai
%A Zhenkun Huang
%A Honghua Bin
%J Advances in Artificial Neural Systems
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/750532
%X We present stability analysis of delayed Wilson-Cowan networks on time scales. By applying the theory of calculus on time scales, the contraction mapping principle, and Lyapunov functional, new sufficient conditions are obtained to ensure the existence and exponential stability of periodic solution to the considered system. The obtained results are general and can be applied to discrete-time or continuous-time Wilson-Cowan networks. 1. Introduction The activity of a cortical column may be mathematically described through the model developed by Wilson and Cowan [1, 2]. Such a model consists of two nonlinear ordinary differential equations representing the interactions between two populations of neurons that are distinguished by the fact that their synapses are either excitatory or inhibitory [2]. A comprehensive paper has been done by Destexhe and Sejnowski [3] which summarized all important development and theoretical results for Wilson-Cowan networks. Its extensive applications include pattern analysis and image processing [4]. Theoretical results about the existence of asymptotic stable limit cycle and chaos have been reported in [5, 6]. Exponential stability of a unique almost periodic solution for delayed Wilson-Cowan type model has been reported in [7]. However, few investigations are fixed on the periodicity of Wilson-Cowan model [8] and it is troublesome to study the stability and periodicity for continuous and discrete system with oscillatory coefficients, respectively. Therefore, it is significant to study Wilson-Cowan networks on time scales [9, 10] which can unify the continuous and discrete situations. Motivated by recent results [11¨C13], we consider the following dynamic Wilson-Cowan networks on time scale : , where , represent the proportion of excitatory and inhibitory neurons firing per unit time at the instant , respectively. and represent the function of the excitatory and inhibitory neurons with natural decay over time, respectively. and are related to the duration of the refractory period; and are positive scaling coefficients. , , , and are the strengths of connections between the populations. , are the external inputs to the excitatory and the inhibitory populations. is the response function of neuronal activity. , correspond to the transmission time-varying delays. The main aim of this paper is to unify the discrete and continuous Wilson-Cowan networks with periodic coefficients and time-varying delays under one common framework and to obtain some generalized results to ensure the existence and exponential stability of periodic
%U http://www.hindawi.com/journals/aans/2014/750532/