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Turing instability induced by random network in FitzHugh-Nagumo model

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  • Zheng, Qianqian
  • Shen, Jianwei

Abstract

Although there is general agreement that the network plays an essential role in the biological system, how the connection probability of network affects the natural model(Especially neural network) is poorly understood. In this paper, we show the impact of the network on Turing instability in the FitzHugh-Nagumo(FN) model. Then we obtain the condition of how the Turing bifurcation, saddle-node bifurcation, and Turing instability occur. By using the Gershgorin circle theorem, we investigate the relationship between degree and eigenvalue of the network matrix, and obtain the approximate range of eigenvalue of the network matrix. Also, We derive the instability condition about diffusion and the connection probability in the network-organized system. And then we obtain the estimated range of connection probability. Meanwhile we apply these results to explaining the spiking of neuron and find this system has rich dynamics behavior. Finally, the numerical simulation verifies our analytical results.

Suggested Citation

  • Zheng, Qianqian & Shen, Jianwei, 2020. "Turing instability induced by random network in FitzHugh-Nagumo model," Applied Mathematics and Computation, Elsevier, vol. 381(C).
    • Handle: RePEc:eee:apmaco:v:381:y:2020:i:c:s0096300320302708
    DOI: 10.1016/j.amc.2020.125304
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    References listed on IDEAS

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    1. Iqbal, Naveed & Wu, Ranchao & Liu, Biao, 2017. "Pattern formation by super-diffusion in FitzHugh–Nagumo model," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 245-258.
    2. Malbor Asllani & Joseph D. Challenger & Francesco Saverio Pavone & Leonardo Sacconi & Duccio Fanelli, 2014. "The theory of pattern formation on directed networks," Nature Communications, Nature, vol. 5(1), pages 1-9, December.
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    1. Zheng, Qianqian & Shen, Jianwei & Pandey, Vikas & Guan, Linan & Guo, Yantao, 2023. "Turing instability in a network-organized epidemic model with delay," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    2. Zheng, Qianqian & Shen, Jianwei & Xu, Yong & Pandey, Vikas & Guan, Linan, 2022. "Pattern mechanism in stochastic SIR networks with ER connectivity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 603(C).
    3. Chen, Mengxin & Wu, Ranchao & Liu, Hongxia & Fu, Xiaoxue, 2021. "Spatiotemporal complexity in a Leslie-Gower type predator-prey model near Turing-Hopf point," Chaos, Solitons & Fractals, Elsevier, vol. 153(P1).
    4. Mondal, Arnab & Upadhyay, Ranjit Kumar & Mondal, Argha & Sharma, Sanjeev Kumar, 2022. "Emergence of Turing patterns and dynamic visualization in excitable neuron model," Applied Mathematics and Computation, Elsevier, vol. 423(C).

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