  In the present work, we introduce a general class of mean-field interacting
nonlinear Hawkes processes modelling the reciprocal interactions between two
neuronal populations, one excitatory and one inhibitory. The model incorporates
two features: inhibition, which acts as a multiplicative factor onto the
intensity of the excitatory population and additive retroaction from the
excitatory neurons onto the inhibitory ones. We give first a detailed analysis
of the well-posedness of this interacting system as well as its dynamics in
large population. The second aim of the paper is to give a rigorous analysis of
the longtime behavior of the mean-field limit process. We provide also
numerical evidence that inhibition and retroaction may be responsible for the
emergence of limit cycles in such system.
