  For large fully connected neuron networks, we study the dynamics of
homogenous assemblies of interacting neurons described by time elapsed models,
indicating how the time elapsed since the last discharge construct the
probability density of neurons. Through the spectral analysis theory for
semigroups in Banach spaces developed recently in [6, 9], on the one hand, we
prove the existence and the uniqueness of the weak solution in the whole
connectivity regime as well as the parallel results on the long time behavior
of solutions obtained in [10] under general assumptions on the ring rate and
the delay distribution. On the other hand, we extend those similar results
obtained in [11, 12] in the case without delay to the case taking delay into
account and both in the weak and the strong connectivity regime with a
particular step function ring rate. Our approach uses the spectral analysis
theory for semigroups in Banach spaces developed recently by the first author
and collaborators.
