Weak formulation and spectral approximation of a Fokker-Planck equation for neural ensembles

TL;DR

This paper develops a spectral Galerkin method for efficiently solving the Fokker-Planck equation related to neural network models, handling complex boundary conditions and extending to more detailed neuron population dynamics.

Contribution

It introduces a variational spectral discretization that naturally incorporates boundary conditions and extends to complex neural population models with delays and refractory states.

Findings

The numerical scheme is consistent and accurate.

The method captures blow-up events and oscillations.

Experimental validation confirms the scheme's effectiveness.

Abstract

In this paper, we focus on efficiently and flexibly simulating the Fokker-Planck equation associated with the Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) model, which reflects the dynamic behavior of neuron networks. We apply the Galerkin spectral method to discretize the spatial domain by constructing a variational formulation that satisfies complex boundary conditions. Moreover, the boundary conditions in the variational formulation include only zeroth-order terms, with first-order conditions being naturally incorporated. This allows the numerical scheme to be further extended to an excitatory-inhibitory population model with synaptic delays and refractory states. Additionally, we establish the consistency of the numerical scheme. Experimental results, including accuracy tests, blow-up events, and periodic oscillations, validate the properties of our proposed method.