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On the Chaotic Pole of Attraction with Nonlocal and Nonsingular Operators in Neurobiology

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Fractional Derivatives with Mittag-Leffler Kernel

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 194))

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Abstract

Until the neurologists J.L. Hindmarsh and R.M. Rose improved the Hodgkin–Huxley model to provide a better understanding on the diversity of neural response, features like pole of attraction unfolding complex bifurcation for the membrane potential was still a mystery. This work explores the possible existence of chaotic poles of attraction in the dynamics of Hindmarsh–Rose neurons with external current input. Combining with fractional differentiation, the model is generalized with introduction of an additional parameter, the non-integer order of the derivative \(\sigma \) and solved numerically thanks to the Haar Wavelets. Numerical simulations of the membrane potential dynamic show that in the standard case the control parameter is \(\sigma =1,\) the nerve cell’s behavior seems irregular with a pole of attraction generating a limit cycle. This irregularity accentuates as \(\sigma \) decreases (\(\sigma =0.8\) and \(\sigma =0.5\)) with the pole of attraction becoming chaotic.

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Acknowledgements

The work of EF Doungmo Goufo was partially supported by the grant No: 105932 from the National Research Foundation (NRF) of South Africa.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of South Africa, Florida, 0003, South Africa

    Emile F. Doungmo Goufo & Melusi Khumalo

  2. Institute for Groundwater Studies, University of the Free State, Bloemfontein, 9300, South Africa

    Abdon Atangana

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  1. Emile F. Doungmo Goufo
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  2. Abdon Atangana
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Correspondence to Emile F. Doungmo Goufo .

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Editors and Affiliations

  1. CONACYT-Tecnológico Nacional de México, Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, Morelos, Mexico

    José Francisco Gómez

  2. CONACYT-Instituto de Ingeniería, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Lizeth Torres

  3. Tecnológico Nacional de México, Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, Morelos, Mexico

    Ricardo Fabricio Escobar

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Doungmo Goufo, E.F., Atangana, A., Khumalo, M. (2019). On the Chaotic Pole of Attraction with Nonlocal and Nonsingular Operators in Neurobiology. In: Gómez, J., Torres, L., Escobar, R. (eds) Fractional Derivatives with Mittag-Leffler Kernel. Studies in Systems, Decision and Control, vol 194. Springer, Cham. https://doi.org/10.1007/978-3-030-11662-0_8

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  • DOI: https://doi.org/10.1007/978-3-030-11662-0_8

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