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In this work, we are concerned with the Fokker-Planck equations associatedwith the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks.Due to the jump mechanism at the microscopic level, such Fokker-Planckequations are endowed with an unconventional structure: transporting theboundary flux to a specific interior point. While the equations exhibitdiversified solutions from various numerical observations, the properties ofsolutions are not yet completely understood, and by far there has been norigorous numerical analysis work concerning such models. We propose aconservative and conditionally positivity preserving scheme for theseFokker-Planck equations, and we show that in the linear case, the semi-discretescheme satisfies the discrete relative entropy estimate, which essentiallymatches the only known long time asymptotic solution property. We also provideextensive numerical tests to verify the scheme properties, and carry outseveral sets of numerical experiments, including finite-time blowup,convergence to equilibrium and capturing time-period solutions of the variantmodels.

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