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A peculiar application of Atangana–Baleanu fractional derivative in neuroscience: Chaotic burst dynamics

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  • Doungmo Goufo, Emile F.
  • Mbehou, Mohamed
  • Kamga Pene, Morgan M.

Abstract

Recent discussions on the non validity of index law in fractional calculus have shown the amazing filtering feature of Mittag–Leffler function foreseing Atangana–Baleanu derivative as one of reliable mathematical tools for describing some complex world problems, like problems of neuronal activities. In this paper, neuronal dynamics described by a three dimensional model of Hindmarsh–Rose nerve cells with external current are analyzed analytically and numerically. We make use of the Atangana–Baleanu fractional derivative in Caputo sense (ABC derivative) and asses its impact on the dynamic, especially the role played by its derivative order in combination with another control parameter, the intensity of the applied external current. Our analysis proves existence of equilibria whose some are unstable of type saddle point, paving the ways for possible bifurcations in the process. Numerical approximations of solutions reveal a system with initially regular bursts that evolve into period-adding chaotic bifurcations as the control parameters change, with precisely the Atangana–Baleanu fractional derivative’s order decreasing from 1 down to 0.5.

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  • Doungmo Goufo, Emile F. & Mbehou, Mohamed & Kamga Pene, Morgan M., 2018. "A peculiar application of Atangana–Baleanu fractional derivative in neuroscience: Chaotic burst dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 170-176.
    • Handle: RePEc:eee:chsofr:v:115:y:2018:i:c:p:170-176
    DOI: 10.1016/j.chaos.2018.08.003
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    References listed on IDEAS

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    1. Atangana, Abdon & Koca, Ilknur, 2016. "Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 447-454.
    2. Nikos K. Logothetis & Jon Pauls & Mark Augath & Torsten Trinath & Axel Oeltermann, 2001. "Neurophysiological investigation of the basis of the fMRI signal," Nature, Nature, vol. 412(6843), pages 150-157, July.
    3. Doungmo Goufo, Emile Franc, 2017. "Solvability of chaotic fractional systems with 3D four-scroll attractors," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 443-451.
    4. Atangana, Abdon, 2018. "Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 688-706.
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    Cited by:

    1. Atangana, Abdon & Khan, Muhammad Altaf, 2019. "Validity of fractal derivative to capturing chaotic attractors," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 50-59.
    2. Atangana, Abdon & Araz, Seda İğret, 2019. "Analysis of a new partial integro-differential equation with mixed fractional operators," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 257-271.
    3. Turkyilmazoglu, Mustafa & Altanji, Mohamed, 2023. "Fractional models of falling object with linear and quadratic frictional forces considering Caputo derivative," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    4. Panda, Sumati Kumari & Abdeljawad, Thabet & Ravichandran, C., 2020. "A complex valued approach to the solutions of Riemann-Liouville integral, Atangana-Baleanu integral operator and non-linear Telegraph equation via fixed point method," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).

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