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 xed nearest-neighbour connection topology accompanied by a suitable scaling of the diusion coe- cients; 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-diusion equation; convection accounts for the possible lack of symmetry in the connection topology. Several convergence issues are discussed, both theoretically and numerically.
|Titolo:||The derivation of continuum limits of neuronal networks with gap-junction couplings|
|Data di pubblicazione:||2014|
|Digital Object Identifier (DOI):||10.3934/nhm.2014.9.111|
|Appare nelle tipologie:||1.1 Articolo in rivista|
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