A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics (Q630334)

The behavior of the considered neural network is assumed to be described by a partial differential equation for the time evolution of the probability density function, the Fokker-Planck equation, supplied by some consistent boundary conditions. The aim of the paper is to develop a finite difference scheme for the solutions of the Fokker-Planck equation and to analyze its performance. Two deterministic methods are coupled. The derivatives with respect to the voltage variable are approximated by means of a fifth order WENO (weighted essentially non-oscillatory) finite difference scheme, originally developed for hyperbolic conservation lows. The derivatives with respect to the conductance variable are approximated by the Chang-Cooper method. For the rest, the evolution in time is implemented using the explicit third-order Runge-Kutta method. The proposed solver is applied to some simple test cases and numerical results are compared to direct Monte-Carlo simulations.