A Spectral Method for a Fokker-Planck Equation in Neuroscience with Applications in Neuron Networks with Learning Rules
Year: 2024
Author: Pei Zhang, Yanli Wang, Zhennan Zhou
Communications in Computational Physics, Vol. 35 (2024), Iss. 1 : pp. 70–106
Abstract
In this work, we consider the Fokker-Planck equation of the Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) model for neuron networks. Due to the firing events of neurons at the microscopic level, this Fokker-Planck equation contains dynamic boundary conditions involving specific internal points. To efficiently solve this problem and explore the properties of the unknown, we construct a flexible numerical scheme for the Fokker-Planck equation in the framework of spectral methods that can accurately handle the dynamic boundary condition. This numerical scheme is stable with suitable choices of test function spaces, and asymptotic preserving, and it is easily extendable to variant models with multiple time scales. We also present extensive numerical examples to verify the scheme properties, including order of convergence and time efficiency, and explore unique properties of the model, including blow-up phenomena for the NNLIF model and learning and discriminative properties for the NNLIF model with learning rules.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2023-0141
Communications in Computational Physics, Vol. 35 (2024), Iss. 1 : pp. 70–106
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 37
Keywords: Integrate-and-Fire model Fokker-Planck equation neuron network spectral methods.