  In the last decade, Hawkes processes have received a lot of attention as good
models for functional connectivity in neural spiking networks. In this paper we
consider a variant of this process, the Age Dependent Hawkes process, which
incorporates individual post-jump behaviour into the framework of the usual
Hawkes model. This allows to model recovery properties such as refractory
periods, where the effects of the network are momentarily being suppressed or
altered. We show how classical stability results for Hawkes processes can be
improved by introducing age into the system. In particular, we neither need to
a priori bound the intensities nor to impose any conditions on the Lipschitz
constants. When the interactions between neurons are of mean field type, we
study large network limits and establish the propagation of chaos property of
the system.
