# American Institute of Mathematical Sciences

March  2014, 9(1): 111-133. doi: 10.3934/nhm.2014.9.111

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

 1 Department of Mathematical Sciences, Corso Duca degli Abruzzi 29, 10129 Torino, Italy, Italy

Received  April 2013 Revised  March 2014 Published  April 2014

We consider an idealized network, formed by $N$ neurons individually described by the FitzHugh-Nagumo equations and connected by electrical synapses. The limit for $N \to \infty$ 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.
Citation: Claudio Canuto, Anna Cattani. The derivation of continuum limits of neuronal networks with gap-junction couplings. Networks & Heterogeneous Media, 2014, 9 (1) : 111-133. doi: 10.3934/nhm.2014.9.111
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