  Explicit numerical methods based on Lax-Friedrichs and Leap-Frog finite
difference approximations are constructed to find the numerical solution of the
first-order hyperbolic partial differential equation with point-wise delay or
advance, i.e., shift in space. The differential equation involving point-wise
delay and advance models the distribution of the time intervals between
successive neuronal firings. We construct higher order numerical approximations
and discuss their consistency, stability and convergence. The numerical
approximations constructed in this paper are consistent, stable under CFL
condition, and convergent. We also extend our methods to the higher space
dimensions. Some test examples are included to illustrate our approach. These
examples verify the theoretical estimates and shows the effect of point-wise
delay on the solution.
