A Structure Preserving Numerical Scheme for Fokker--Planck Equations of Structured Neural Networks with Learning Rules
Qing He, Jingwei Hu, Zhennan Zhou · SIAM Journal on Scientific Computing · 2022
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 multiscale 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 we explore the solution's behavior when the neural network is excitatory.