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

Claudio Canuto; Anna Cattani; Claudio Canuto; Anna Cattani

doi:10.3934/nhm.2014.9.111

Networks and Heterogeneous Media

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

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

Claudio Canuto ^{1
,},
Anna Cattani ^{1
,}

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

Received: 01 April 2013 Revised: 01 March 2014
Primary: 34C60, 35K57; Secondary: 92C42, 05C90.

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.
- Neuronal networks,
- FitzHigh-Nagumo model,
- electrical synapses,
- discrete-to-continuum limit,
- reaction-convection-diffusion equations
Citation: Claudio Canuto, Anna Cattani. The derivation of continuum limits of neuronal networks with gap-junction couplings[J]. Networks and Heterogeneous Media, 2014, 9(1): 111-133. doi: 10.3934/nhm.2014.9.111

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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.

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