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

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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.

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Funding

This work was supported in part by the National Natural Science Foundation of China under Grant 61503310, in part by the Fundamental Research Funds for the Central Universities under Grant XDJK2016B018, in part by China Postdoctoral Foundation under Grant 2016 M600720, in part by Chongqing Postdoctoral Project under Grant Xm2016003, and in part by the Natural Science Foundation project of CQCSTC under Grant ctsc2014cyjA40053 and Grant cstc2016jcyjA0559.

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Authors and Affiliations

  1. College of Electronics and Information Engineering, Southwest University, Chongqing, 400715, China

    Tao Dong & Linmao Xia

  2. National & Local Joint Engineering Laboratory of Intelligent Transmission and Control Technology, Southwest University, Chongqing, 400715, China

    Tao Dong

  3. Information Technology Center, Chongqing Changan Automobile Company Limited, Chongqing, 401220, China

    Tao Dong

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  1. Tao Dong
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  2. Linmao Xia
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Correspondence to Tao Dong.

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Dong, T., Xia, L. Spatial Temporal Dynamic of a Coupled Reaction-Diffusion Neural Network with Time Delay. Cogn Comput 11, 212–226 (2019). https://doi.org/10.1007/s12559-018-9618-1

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  • DOI: https://doi.org/10.1007/s12559-018-9618-1

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