Well-posedness and regularity of solutions to neural field problems with dendritic processing

TL;DR

This paper establishes the well-posedness and regularity of solutions to a novel neural field model incorporating dendritic processing, characterized by anisotropic diffusion and nonlocal synaptic interactions.

Contribution

It proves existence, uniqueness, and regularity of solutions for a neural field model with dendritic structure, extending classical neural field theory.

Findings

Unique solutions exist for the proposed neural field model.

Solutions of the dendritic model and standard neural field stay close over finite times.

Numerical evidence supports the perturbative analysis.

Abstract

We study solutions to a recently proposed neural field model in which dendrites are modelled as a continuum of vertical fibres stemming from a somatic layer. Since voltage propagates along the dendritic direction via a cable equation with nonlocal sources, the model features an anisotropic diffusion operator, as well as an integral term for synaptic coupling. The corresponding Cauchy problem is thus markedly different from classical neural field equations. We prove that the weak formulation of the problem admits a unique solution, with embedding estimates similar to the ones of nonlinear local reaction-diffusion equations. Our analysis relies on perturbing weak solutions to the diffusion-less problem, that is, a standard neural field, for which weak problems have not been studied to date. We find rigorous asymptotic estimates for the problem with and without diffusion, and prove that the solutions of the two models stay close, in a suitable norm, on finite time intervals. We provide numerical evidence of our perturbative results.