Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions

TL;DR

This paper establishes local existence of solutions for certain renormalised stochastic PDE-ODE systems in three dimensions, including the FitzHugh-Nagumo model, using advanced regularity structures and explicit renormalisation constants.

Contribution

It extends Hairer's regularity structures to non-regularising semigroups and provides explicit renormalisation constants for complex nonlinearities in high-dimensional SPDEs.

Findings

Proved local existence for FitzHugh-Nagumo SPDEs in 3D.

Extended regularity structures to non-regularising semigroups.

Derived explicit renormalisation constants for cubic nonlinearities.

Abstract

We prove local existence of solutions for a class of suitably renormalised coupled SPDE-ODE systems driven by space-time white noise, where the space dimension is equal to 2 or 3. This class includes in particular the FitzHugh-Nagumo system describing the evolution of action potentials of a large population of neurons, as well as models with multidimensional gating variables. The proof relies on the theory of regularity structures recently developed by M. Hairer, which is extended to include situations with semigroups that are not regularising in space. We also provide explicit expressions for the renormalisation constants, for a large class of cubic nonlinearities.