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High order WENO finite volume approximation for population density neuron model

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  • Kumar, Santosh
  • Singh, Paramjeet

Abstract

In this article, we study the excitatory and inhibitory population density model based on leaky integrate-and-fire neuron model. The time evolution of population density is determined by a hyperbolic partial differential equation. There are two factors which might cause difficulties to find the solution. First one is the presence of point-wise delta type source term and second one is the presence of the non-local terms of the voltage of a neuron. We design a high accurate scheme based on WENO-finite volume approximation for spatial discretization. The time evolution is done by SSP-Runge Kutta scheme. To check the efficiency of the designed scheme, we compare the results with the existing scheme in literature. Finally, we remove the non-local terms by diffusion approximation and simulate the model equation. The diffusion approximation is tackled with the Strang splitting method. Some test examples are reported. The performance of the designed scheme is shown by the convergence results.

Suggested Citation

  • Kumar, Santosh & Singh, Paramjeet, 2019. "High order WENO finite volume approximation for population density neuron model," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 173-189.
    • Handle: RePEc:eee:apmaco:v:356:y:2019:i:c:p:173-189
    DOI: 10.1016/j.amc.2019.03.020
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    References listed on IDEAS

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    1. Delarue, F. & Inglis, J. & Rubenthaler, S. & Tanré, E., 2015. "Particle systems with a singular mean-field self-excitation. Application to neuronal networks," Stochastic Processes and their Applications, Elsevier, vol. 125(6), pages 2451-2492.
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