Networks and Heterogeneous Media (NHM)

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

Pages: 111 - 133, Volume 9, Issue 1, March 2014      doi:10.3934/nhm.2014.9.111

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Claudio Canuto - Department of Mathematical Sciences, Corso Duca degli Abruzzi 29, 10129 Torino, Italy (email)
Anna Cattani - Department of Mathematical Sciences, Corso Duca degli Abruzzi 29, 10129 Torino, Italy (email)

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 $N \to \infty$ 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.

Keywords:  Neuronal networks, FitzHigh-Nagumo model, electrical synapses, discrete-to-continuum limit, reaction-convection-diffusion equations.
Mathematics Subject Classification:  Primary: 34C60, 35K57; Secondary: 92C42, 05C90.

Received: April 2013;      Revised: March 2014;      Available Online: April 2014.