A structure preserving numerical scheme for Fokker-Planck equations of structured neural networks with learning rules

TL;DR

This paper develops a numerical scheme for a Fokker-Planck equation modeling structured neural networks with learning rules, ensuring mass conservation, positivity, and asymptotic accuracy, and investigates the network's learning and excitatory behavior.

Contribution

It introduces a novel structure-preserving numerical scheme for multi-scale Fokker-Planck equations in neural network models with Hebbian learning.

Findings

The scheme is mass conservative and positivity preserving.

Numerical experiments demonstrate the scheme's effectiveness.

The model's learning ability and excitatory behavior are analyzed.

Abstract

In this work, we are concerned with a Fokker-Planck equation related to the nonlinear noisy leaky integrate-and-fire model for biological neural networks which are structured by the synaptic weights and equipped with the Hebbian learning rule. The equation contains a small parameter $\varepsilon$ separating the time scales of learning and reacting behavior of the neural system, and an asymptotic limit model can be derived by letting $\varepsilon\to 0$, where the microscopic quasi-static states and the macroscopic evolution equation are coupled through the total firing rate. To handle the endowed flux-shift structure and the multi-scale dynamics in a unified framework, we propose a numerical scheme for this equation that is mass conservative, unconditionally positivity preserving, and asymptotic preserving. We provide extensive numerical tests to verify the schemes' properties and carry out a set of numerical experiments to investigate the model's learning ability, and explore the solution's behavior when the neural network is excitatory.