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Bifurcation Analysis of Two-Neuron Networks with Discrete and Distributed Delays

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Abstract

Background

Although the amount of research work on bifurcation analysis of neural networks is growing explosively, the majority of such research effort is devoted to bifurcation of neural networks with discrete delays, while efficient detection of Hopf bifurcation phenomena in neural networks with distributed delays remains a challenging task. The aim of this study is to determine the effect of time delay on the dynamical behavior of two-neuron networks with discrete and distributed delays.

Methods

Using the Routh–Hurwitz method and analyzing the characteristic equation, we determine the existence and stability of Hopf bifurcation for the studied system. Normal form theory and the center manifold argument are used to analyze the stability, direction, and period of Hopf bifurcation for the system.

Results

The research shows that, if the time delay falls in a certain interval, the system is asymptotically stable, whereas if the time delay exceeds a critical value, the system loses stability and Hopf bifurcation appears. In addition, we obtain some explicit formulas which determine the direction, stability, and period of bifurcating periodic solutions.

Conclusions

The effects of time delay on the dynamical behavior of neural networks were investigated in this study. Modulation of the coefficients and time delays of neural networks may prove beneficial for controlling the stability, direction, and amplitude of bifurcating periodic solutions of the system.

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (Nos. 11261010, 11201138, and 11101126), Natural Science and Technology Foundation of Guizhou Province (J[2015]2025), 125 Special Major Science and Technology of Department of Education of Guizhou Province ([2012]011), and Scientific Research Fund of Hunan Provincial Education Department (no. 12B034).

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Authors and Affiliations

  1. Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang, 550004, People’s Republic of China

    Changjin Xu

  2. College of Science, Hunan University of Technology, Zhuzhou, 412007, People’s Republic of China

    Qiming Zhang

  3. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023, People’s Republic of China

    Yusen Wu

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  1. Changjin Xu
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Correspondence to Changjin Xu.

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Changjin Xu, Qiming Zhang, and Yusen Wu declare that they have no conflict of interest.

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Xu, C., Zhang, Q. & Wu, Y. Bifurcation Analysis of Two-Neuron Networks with Discrete and Distributed Delays. Cogn Comput 8, 1103–1118 (2016). https://doi.org/10.1007/s12559-016-9422-8

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