Spatial Temporal Dynamic of a Coupled Reaction-Diffusion Neural Network with Time Delay

Abstract

In neural networks, the diffusion effect cannot be avoided due to the electrons diffuse from the high region to low region. However, the spatial temporal dynamic of neural network with diffusion and time delay is not well understood. The goal of this paper is to study the spatial temporal dynamic of a coupled neural network with diffusion and time delay. Based on the eigenvalue of the Laplace operator, the characteristic equation is obtained. By analyzing the characteristic equation, some conditions for the occurrence of Turing instability and Hopf bifurcations are obtained. Moreover, normal form theory and center manifold theorem of the partial differential equation are used to analyze the period and direction of Hopf bifurcation. It found that the diffusion coefficients can lead to the diffusion-driven instability, and time delay can give rise to the periodic solution. Near the Turing instability point, there exist some spatially non-homogeneous patterns such as spike, spiral wave, and zebra-stripe. Near the Hopf bifurcation point, the spatial temporal dynamic can be divided into four types: the stable zero equilibrium, the two distinct stripe patterns, and the irregular pattern. The effects of diffusion and time delay on the spatial temporal dynamic of a coupled reaction-diffusion neural network with time delay are investigated. It is found that the diffusion coefficients have a marked impact on selection of the type and characteristics of the emerging pattern. The results obtained in this paper are novel and supplement some existing works.