Spatial Temporal Dynamic of a Coupled Reaction-Diffusion Neural Network with Time Delay
- 87 Downloads
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.
KeywordsReaction-diffusion neuron network Turing instability Hopf bifurcation Pattern formation
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.
Compliance with Ethical Standards
Conflict of Interest
The authors declare that they have no conflict of interest.
This article does not contain any studies with human participants or animals performed by any of the authors..
- 3.Li C, Liu C, Deng K, Yu X. Data-driven charging strategy of PEVs under transformer aging risk. IEEE Trans Neural Netw Learn Syst. 2018. https://doi.org/10.1109/TCST.2017.2713321.
- 5.Liu H, Fang J, Xu X, Sun F. Surface material recognition using active multi-modal extreme learning machine. Cogn Comput. 2018;8:78–104.Google Scholar
- 7.Li C, Yu X, Yu W. Distributed optimal consensus over resource allocation network and its application to dynamical economic dispatch. IEEE Transactions on Neural Networks and Learning Systems. 2018. https://doi.org/10.1109/TNNLS.2017.2691760.
- 10.Wu Y, Liu L, Hu J, Feng G. Adaptive antisynchronization of multilayer reaction-diffusion neural networks. IEEE Trans Neural Netw Learn Syst. 2017;25:1–12.Google Scholar
- 12.He X, Li C, Huang T, Li C. Bogdanov–Takens singularity in tri-neuron network with time delay. IEEE Trans Neural Netw Learn Syst. 2013;46:1001–7.Google Scholar
- 18.Tian X, Xu R, Gan Q. Hopf bifurcation analysis of a BAM neural network with multiple time delays and diffusion. Appl Math Comput. 2015;266:909–26.Google Scholar
- 21.X. Tian, R. Xu, and Q. Gao, Hopf bifurcation analysis of a BAM neural network with multiple time delays and diffusion, Appl Math Comput, vol. 134, pp. 909–925, 2015.Google Scholar
- 22.Tian X, Xu R. Hopf bifurcation analysis of a reaction-diffusion neural network with time delay in leakage terms and distributed delays. Neural. Process. Lett. 2016;73(3):115–24.Google Scholar
- 23.Dong T, Xu W, Liao X. Hopf bifurcation analysis of reaction–diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay. Nonlinear Dynamics. 2017:1–17.Google Scholar
- 25.Ruan S, Wei J. On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. DynContin Discrete Impuls Syst Ser A Math Anal. 2003;10:863–73.Google Scholar