Time evolution of a mean-field generalized contact process

TL;DR

This paper studies the dynamic behavior and stationary states of a mean-field generalized contact process inspired by neuron models, providing complete solutions in uniform cases and exploring fixed points and wave solutions.

Contribution

It offers a comprehensive analysis of the time evolution and stationary states of a nonlinear integral-differential model for neuron-inspired contact processes.

Findings

Complete solutions for spatially uniform case

Existence of fixed points and traveling wave solutions

Partial solutions in the general case

Abstract

We investigate the macroscopic time evolution and stationary states of a mean field generalized contact process in $\mathbb{R}^d$. The model is described by a coupled set of nonlinear integral-differential equations. It was inspired by a model of neurons with discrete voltages evolving by a stochastic integrate and fire mechanism. We obtain a complete solution in the spatially uniform case and partial solutions in the general case. The system has one or more fixed points and also traveling wave solutions.