  We propose a mean-field model of interacting point processes where each
process has a memory of the time elapsed since its last event (age) and its
recent past (leaky memory), generalizing Age-dependent Hawkes processes. The
model is motivated by interacting nonlinear Hawkes processes with Markovian
self-interaction and networks of spiking neurons with adaptation and short-term
synaptic plasticity. By proving propagation of chaos and using a path integral
representation for the law of the limit process, we show that, in the
mean-field limit, the empirical measure of the system follows a
multidimensional nonlocal transport equation.
