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Inhibitor-Induced Wavetrains and Spiral Waves in an Extended FitzHugh–Nagumo Model of Nerve Cell Dynamics

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Abstract

It is well known that the FitzHugh–Nagumo model is one of the simplified forms of the four-variable Hodgkin–Huxley model that can reflect most of the significant phenomena of nerve cell action potential. However, this model cannot capture the irregular action potentials of sufficiently large periods in a one-parameter family of solutions. Motivated by this, we propose a modified FitzHugh–Nagumo reaction-diffusion system by changing its recovery kinetics. First, we investigate the parameter regime to know the existence of the wavetrains. Second, we conceive the occurrence of Eckhaus bifurcations of solutions that divide the solution region into two parts. The essential spectra at different grid points explore the occurrence of bifurcations of the waves. We find that the wavetrains of sufficiently large periods cross the stability boundary. This characteristic phenomenon is absent in the standard FitzHugh–Nagumo model. Finally, we observe a reasonable agreement between the direct PDE simulations and the solutions in the traveling wave ODEs. Furthermore, the model exhibits spiral wave for monotone and non-monotone cases that agrees with the waves observed in cellular activity.

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Acknowledgements

It is acknowledged the support provided by the GCOE program entitled “Formation and Development of Mathematical Sciences Based on Modeling and Analysis”, of the Meiji University, Japan, where this work was carried out.

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Authors and Affiliations

  1. Mathematical and Computational Biology (MCB) Research Group, Department of Mathematics, Jahangirnagar University, Dhaka, 1342, Bangladesh

    M. Osman Gani & M. Humayun Kabir

  2. Center for Mathematical Modeling and Applications (CMMA), Meiji University, Tokyo, 164-8525, Japan

    M. Osman Gani & M. Humayun Kabir

  3. Graduate School of Advanced Mathematical Sciences, Meiji University, 4-21-1, Nakano-ku, Tokyo, 164-8525, Japan

    Toshiyuki Ogawa

  4. Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525, Japan

    Toshiyuki Ogawa

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  1. M. Osman Gani
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  2. M. Humayun Kabir
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  3. Toshiyuki Ogawa
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Correspondence to M. Osman Gani.

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Gani, M.O., Kabir, M.H. & Ogawa, T. Inhibitor-Induced Wavetrains and Spiral Waves in an Extended FitzHugh–Nagumo Model of Nerve Cell Dynamics. Bull Math Biol 84, 145 (2022). https://doi.org/10.1007/s11538-022-01100-9

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