Scaling limit of a generalized contact process

TL;DR

This paper derives macroscopic equations for a generalized contact process inspired by neuronal models, analyzing the limit as the interaction range grows infinitely large, and extends the model to include different neuron types and mechanisms.

Contribution

It introduces a generalized contact process with multiple states representing neuronal potentials and derives hydrodynamic equations in the large-range limit, including extensions for excitatory and inhibitory neurons.

Findings

Hydrodynamic equations derived in the limit as interaction range goes to zero.

Model extensions include excitatory and inhibitory neuron types.

Framework applicable to biophysical neuronal mechanisms.

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.