TY - JOUR
T1 - A Spectral Method for a Fokker-Planck Equation in Neuroscience with Applications in Neuron Networks with Learning Rules
AU - Zhang , Pei
AU - Wang , Yanli
AU - Zhou , Zhennan
JO - Communications in Computational Physics
VL - 1
SP - 70
EP - 106
PY - 2024
DA - 2024/01
SN - 35
DO - http://doi.org/10.4208/cicp.OA-2023-0141
UR - https://global-sci.org/intro/article_detail/cicp/22896.html
KW - Integrate-and-Fire model, Fokker-Planck equation, neuron network, spectral methods.
AB - <p style="text-align: justify;">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.</p>