We consider multi-class systems of interacting nonlinear Hawkes processes modeling several large families
of neurons and study their mean field limits. As the total number of neurons goes to infinity we prove
that the evolution within each class can be described by a nonlinear limit differential equation driven by
a Poisson random measure, and state associated central limit theorems. We study situations in which the
limit system exhibits oscillatory behavior, and relate the results to certain piecewise deterministic Markov
processes and their diffusion approximations.