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Numerical approximation of the time fractional cable model arising in neuronal dynamics

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Abstract

The cable equation is one useful description for modeling phenomena such as neuronal dynamics and electrophysiology. The time-fractional cable model (TFCM) generalizes the classical cable equation by considering the anomalous diffusion that occurs in the ionic motion present for example in the neuronal system. This paper proposes a novel meshless numerical procedure, the radial basis function-generated finite difference (RBF-FD), to approximate the TFCM involving two fractional temporal derivatives. The time discretization of the TFCM is performed based on the Grünwald–Letnikov expansion. The spatial derivatives are discretized using the RBF-FD. The pattern of data distribution in the support domain is assumed as having a fixed number of points. The RBF-FD is based on the local support domain that leads to a sparsity system and also tackles the ill-conditioning problem caused by global collocation method. The theoretical stability and the convergence analysis of the scheme are also discussed in detail. It is shown that the proposed method is efficient and that the numerical results confirm the theoretical formulation.

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Acknowledgements

The authors express their gratitude to the Editor-in-Chief (Professor Mark Shephard), Associate Editor and both the reviewers for their valuable suggestions and useful comments to make this paper better.

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Authors and Affiliations

  1. School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran

    O. Nikan, A. Golbabai & T. Nikazad

  2. Department of Electrical Engineering, ISEP-Institute of Engineering, Polytechnic of Porto, Porto, Portugal

    J. A. Tenreiro Machado

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  1. O. Nikan
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  2. A. Golbabai
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  3. J. A. Tenreiro Machado
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  4. T. Nikazad
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Correspondence to O. Nikan.

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Nikan, O., Golbabai, A., Machado, J.A.T. et al. Numerical approximation of the time fractional cable model arising in neuronal dynamics. Engineering with Computers 38, 155–173 (2022). https://doi.org/10.1007/s00366-020-01033-8

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