  We derive macroscopic equations for a generalized contact process that is
inspired by a neuronal integrate and fire model on the lattice $\mathbb{Z}^d$.
The states at each lattice site can take values in $0,\ldots,k$. These can be
interpreted as neuronal membrane potential, with the state $k$ corresponding to
a firing threshold. In the terminology of the contact processes, which we shall
use in this paper, the state $k$ corresponds to the individual being infectious
(all other states are noninfectious). In order to reach the firing threshold,
or to become infectious, the site must progress sequentially from $0$ to $k$.
The rate at which it climbs is determined by other neurons at state $k$,
coupled to it through a Kac-type potential, of range $\gamma^{-1}$. The
hydrodynamic equations are obtained in the limit $\gamma\rightarrow 0$.
Extensions of the microscopic model to include excitatory and inhibitory neuron
types, as well as other biophysical mechanisms, are also considered.
