  The elapsed time model has been widely studied in the context of mathematical
neuroscience with many open questions left. The model consists of an
age-structured equation that describes the dynamics of interacting neurons
structured by the elapsed time since their last discharge. Our interest lies in
highly connected networks leading to strong nonlinearities where perturbation
methods do not apply. To deal with this problem, we choose a particular case
which can be reduced to delay equations. We prove a general convergence result
to a stationary state in the inhibitory and the weakly excitatory cases.
Moreover, we prove the existence of particular periodic solutions with jump
discontinuities in the strongly excitatory case. Finally, we present some
numerical simulations which ilustrate various behaviors, which are consistent
with the theoretical results.
