  We generalize multivariate Hawkes processes mainly by including a dependence
with respect to the age of the process, i.e. the delay since the last point.
Within this class, we investigate the limit behaviour, when n goes to infinity,
of a system of n mean-field interacting age-dependent Hawkes processes. We
prove that such a system can be approximated by independent and identically
distributed age dependent point processes interacting with their own mean
intensity. This result generalizes the study performed by Delattre, Fournier
and Hoffmann (2015). In continuity with the work of Chevallier et al. (2015),
the second goal of this paper is to give a proper link between these
generalized Hawkes processes as microscopic models of individual neurons and
the age-structured system of partial differential equations introduced by
Pakdaman, Perthame and Salort (2010) as macroscopic model of neurons.
