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Pattern formation by super-diffusion in FitzHugh–Nagumo model

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  • Iqbal, Naveed
  • Wu, Ranchao
  • Liu, Biao

Abstract

The aim of this paper is to examine the Turing instability and pattern formation in the FitzHugh–Nagumo model with super-diffusion in two dimensional numerical simulation. We also studied the effects of the super-diffusive exponent on pattern formation concluding that with the presence of super-diffusion the stable homogenous steady state becomes unstable. By using the stability analysis of local equilibrium point, we procure the conditions which ensure that the Turing and Hopf bifurcations occur. For pattern selection, the weak nonlinear multi-scale analysis is used to derive the amplitude equations of the stationary patterns. We then apply amplitude equations and observe that this model has very rich dynamical behaviors, such as stripes, spots and hexagon patterns. The complexity of the dynamics in this system is theoretically discussed and graphically displayed in numerical simulation. The simulation helps us to show the effectiveness of theoretical analysis and patterns which appear numerically.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:313:y:2017:i:c:p:245-258
    DOI: 10.1016/j.amc.2017.05.072
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    References listed on IDEAS

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    1. Perc, Matjaž, 2007. "Effects of small-world connectivity on noise-induced temporal and spatial order in neural media," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 280-291.
    2. Perc, Matjaž, 2007. "Fluctuating excitability: A mechanism for self-sustained information flow in excitable arrays," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 1118-1124.
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    1. Zheng, Qianqian & Shen, Jianwei, 2020. "Turing instability induced by random network in FitzHugh-Nagumo model," Applied Mathematics and Computation, Elsevier, vol. 381(C).
    2. Mondal, Argha & Hens, Chittaranjan & Mondal, Arnab & Antonopoulos, Chris G., 2021. "Spatiotemporal instabilities and pattern formation in systems of diffusively coupled Izhikevich neurons," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    3. Tah, Forwah Amstrong & Tabi, Conrad Bertrand & Kofane, Timoléon Crépin, 2021. "Pattern formation in the Fitzhugh–Nagumo neuron with diffusion relaxation," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
    4. Fateev, I. & Polezhaev, A., 2024. "Chimera states in a lattice of superdiffusively coupled neurons," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    5. 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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