A Spectral Method for a Fokker-Planck Equation in Neuroscience with Applications in Neuron Networks with Learning Rules (Communications in Computational Physics, 2024)
In this work, we consider the Fokker-Planck equation of the Nonlinear NoisyLeaky Integrate-and-Fire (NNLIF) model for neuron networks. Due to the firing eventsof neurons at the microscopic level, this Fokker-Planck equation contains dynamicboundary conditions involving specific internal points. To efficiently solve this problem and explore the properties of the unknown, we construct a flexible numericalscheme for the Fokker-Planck equation in the framework of spectral methods that canaccurately handle the dynamic boundary condition. This numerical scheme is stablewith suitable choices of test function spaces, and asymptotic preserving, and it is easily extendable to variant models with multiple time scales. We also present extensivenumerical examples to verify the scheme properties, including order of convergenceand time efficiency, and explore unique properties of the model, including blow-upphenomena for the NNLIF model and learning and discriminative properties for theNNLIF model with learning rules.