Scaling Limit of a Generalized Contact Process

Abstract

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.