Finite Difference Approximation for First-Order Hyperbolic Partial Differential Equation Arising in Neuronal Variability with Shifts

Vinit Chauhan, Nagesh Kumar Singh

Abstract: This paper studies some finite difference approximations to find the numerical solution of first-order hyperbolic partial differential equation of mixed type. We are interested in the challenging issues in neuronal science stemming from the modeling of neuronal variability based on Stein?s Model [8]. The resulting mathematical model is a first order hyperbolic partial differential equation having point-wise delay and advance which models the distribution of time intervals between successive neuronal firings. We construct, analyze and implement explicit numerical scheme for solving such type of initial and boundary-interval problems. Analysis shows that numerical scheme is conditionally stable, consistent and convergent in discrete norm. Some numerical tests are reported to validate the computational efficiency of the numerical approximation

Keywords: hyperbolic partial differential equation, neuronal firing, point-wise delay and advance, finite difference method, Lax-Friedrichs scheme