A structure preserving numerical scheme for Fokker-Planck equations of neuron networks: numerical analysis and exploration

TL;DR

This paper introduces a novel numerical scheme for Fokker-Planck equations in neuron network models, ensuring conservation and positivity, with rigorous analysis and extensive numerical validation of solution behaviors.

Contribution

It presents the first rigorous numerical analysis and a conservative, positivity-preserving scheme for these complex Fokker-Planck equations in neuron networks.

Findings

Scheme satisfies discrete relative entropy estimate in linear case

Numerical tests verify conservation and positivity

Successfully captures blowup, equilibrium, and periodic solutions

Abstract

In this work, we are concerned with the Fokker-Planck equations associated with the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks. Due to the jump mechanism at the microscopic level, such Fokker-Planck equations are endowed with an unconventional structure: transporting the boundary flux to a specific interior point. While the equations exhibit diversified solutions from various numerical observations, the properties of solutions are not yet completely understood, and by far there has been no rigorous numerical analysis work concerning such models. We propose a conservative and conditionally positivity preserving scheme for these Fokker-Planck equations, and we show that in the linear case, the semi-discrete scheme satisfies the discrete relative entropy estimate, which essentially matches the only known long time asymptotic solution property. We also provide extensive numerical tests to verify the scheme properties, and carry out several sets of numerical experiments, including finite-time blowup, convergence to equilibrium and capturing time-period solutions of the variant models.