Fokker-Planck Solution for a Neuronal Spiking Model

Abstract

The stochastic dynamics of a neuronal spiking model in neuroscience, when viewed as a large simulated network, are known to reduce to the classic problem of solving the Fokker-Planck equation, or the equivalent Kolmogorov differential equation in probability theory, for the numerical evaluation of the statistical properties of neurons as a random injection of ion currents. The problem here, however, is that the initial condition for the Fokker-Planck equation is a Dirac delta function so the actual implementation of Delta functions that at the same time can attain numerical stability can become problematic in computational neuroscience. Therefore, in this brief communication, a computational method for implementing such an initial condition is suggested, which itself has led to an exact solution for this problem.