  The elapsed time equation is an age-structured model that describes dynamics
of interconnected spiking neurons through the elapsed time since the last
discharge, leading to many interesting questions on the evolution of the system
from a mathematical and biological point of view. In this work, we first deal
with the case when transmission after a spike is instantaneous and the case
when there exists a distributed delay that depends on previous history of the
system, which is a more realistic assumption. Then we study the well-posedness
and the numerical analysis of the elapsed time models. For existence and
uniqueness we improve the previous works by relaxing some hypothesis on the
nonlinearity, including the strongly excitatory case, while for the numerical
analysis we prove that the approximation given by the explicit upwind scheme
converges to the solution of the non-linear problem. We also show some
numerical simulations to compare the behavior of the system in the case of
instantaneous transmission with the case of distributed delay under different
parameters, leading to solutions with different asymptotic profiles.
