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Stability Switches and Hopf Bifurcation of a Neuron System with both Leakage and Distributed Delays

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Abstract

This paper focuses on the effect of leakage delays on the dynamics of nonlinear delay systems based on a neuron system with discrete and distributed delay. By introducing a virtual neuron, we can eliminate the impact of distributed delay and transform the original neuron system with mixed delays to an equivalent new system involving only discrete delay. This paper extends the existing works on neural networks to the case of leakage delays. We choose the leakage delay as the bifurcation parameter and analyze the role of leakage delays on the stability and Hopf bifurcation. Sufficient conditions for ensuring the neuron system to be stable, and undergoing the Hopf bifurcation are derived. Finally, by using the software package DDE-BIFTOOL, we provide the simulation results to substantiate our theoretical analysis, and give the relationship between the onset of Hopf bifurcation and the system parameters by drawing the bifurcation curves.

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Acknowledgements

This work is supported in part by the National Natural Science Foundation of China (Nos. 61573194, 61833005, 61773004), and the 1311 Talents Project through the Nanjing University of Posts and Telecommunications.

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Authors and Affiliations

  1. School of Mathematics, Nanjing Normal University, Nanjing, 210023, China

    Yi Yao

  2. College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China

    Min Xiao

  3. School of Mathematics, Southeast University, Nanjing, 210096, China

    Jinde Cao

  4. School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, China

    Chengdai Huang

  5. Department of Mathematics, Chongqing Jiaotong University, Chongqing, 400074, China

    Qiankun Song

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  1. Yi Yao
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  2. Min Xiao
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  3. Jinde Cao
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  4. Chengdai Huang
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  5. Qiankun Song
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Correspondence to Yi Yao.

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We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, and there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in the review of the manuscript entitled “Stability switches and Hopf bifurcation of a neuron system with both leakage and distributed delays”.

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Yao, Y., Xiao, M., Cao, J. et al. Stability Switches and Hopf Bifurcation of a Neuron System with both Leakage and Distributed Delays. Neural Process Lett 50, 341–355 (2019). https://doi.org/10.1007/s11063-018-9916-3

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