  We consider spatially extended systems of interacting nonlinear Hawkes
processes modeling large systems of neurons placed in Rd and study the
associated mean field limits. As the total number of neurons tends to infinity,
we prove that the evolution of a typical neuron, attached to a given spatial
position, can be described by a nonlinear limit differential equation driven by
a Poisson random measure. The limit process is described by a neural field
equation. As a consequence, we provide a rigorous derivation of the neural
field equation based on a thorough mean field analysis.
