The derivation of continuum limits of neuronal networks with gap-junction couplings

TL;DR

This paper derives continuum models for large neuronal networks with gap-junction couplings, analyzing two limit approaches that lead to nonlinear PDEs describing the action potential distribution.

Contribution

It introduces a novel limit approach where the number of connections varies with network size, resulting in a PDE model capturing local neuronal interactions.

Findings

Both limit approaches produce nonlinear reaction-convection-diffusion PDEs.

The new approach ensures non-trivial and local continuum limits.

Numerical and theoretical analysis confirms convergence to the PDE models.

Abstract

We consider an idealized network, formed by N neurons individually described by the FitzHugh-Nagumo equations and connected by electrical synapses. The limit for N to infinity of the resulting discrete model is thoroughly investigated, with the aim of identifying a model for a continuum of neurons having an equivalent behaviour. Two strategies for passing to the limit are analysed: i) a more conventional approach, based on a fixed nearest-neighbour connection topology accompanied by a suitable scaling of the diffusion coefficients; ii) a new approach, in which the number of connections to any given neuron varies with N according to a precise law, which simultaneously guarantees the non-triviality of the limit and the locality of neuronal interactions. Both approaches yield in the limit a pde-based model, in which the distribution of action potential obeys a nonlinear reaction-convection-diffusion equation; convection accounts for the possible lack of symmetry in the connection topology. Several convergence issues are discussed, both theoretically and numerically.