Time evolution of a mean-field generalized contact process

Abstract

<jats:title>Abstract</jats:title> <jats:p>We investigate the macroscopic time evolution and stationary states of a mean field discrete voltage neuron model, or equivalently, a generalized contact process in <jats:inline-formula> <jats:tex-math></jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="jstatac4985ieqn1.gif" xlink:type="simple" /> </jats:inline-formula>. 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.</jats:p>