NUMERICAL APPROXIMATIONS TO THE TRANSPORT EQUATION ARISING IN NEURONAL VARIABILITY

This paper studies some finite difference approximations to find the numerical solution of first-order hyperbolic partial differential equation of mixed type, i.e., transport equation with point-wise delay and advance. We are interested in the challenging issues in neuronal sciences stemming from the mod- eling of neuronal variability. The resulting mathematical model is a first-order hyperbolic partial differential equation involving point-wise delay and advance which models the distribution of time intervals between successive neuronal firings. We construct, analyze, and implement explicit numerical schemes for solving such type of initial and boundary-interval problems. Analysis shows that numerical approximations are conditionally stable, consistent and conver- gent in discrete L 1 norm. Numerical approximations works irrespective the size of point-wise delay and advance. Some numerical tests are reported to validate the computational efficiency of the numerical approximations.

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