Abstract
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.
Highlights
The activity of a cortical column may be mathematically described through the model developed by Wilson and Cowan [1, 2]
A comprehensive paper has been done by Destexhe and Sejnowski [3] which summarized all important development and theoretical results for Wilson-Cowan networks
The main technique is based on the theory of time scales, the contraction mapping principle, and the Lyapunov functional method
Summary
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]. AP(t) > 0 and aN(t) > 0 represent the function of the excitatory and inhibitory neurons with natural decay over time, respectively. 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 solution on time scales. The main technique is based on the theory of time scales, the contraction mapping principle, and the Lyapunov functional method
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