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EnglishWe 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.
The derivation of continuum limits of neuronal networks with gap-junction couplings / Claudio Canuto; Anna Cattani. - In: NETWORKS AND HETEROGENEOUS MEDIA. - ISSN 1556-1801. - 9(2014), pp. 111-133. [10.3934/nhm.2014.9.111]
| Titolo: | The derivation of continuum limits of neuronal networks with gap-junction couplings | |
| Autori: | ||
| Data di pubblicazione: | 2014 | |
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| Digital Object Identifier (DOI): | http://dx.doi.org/10.3934/nhm.2014.9.111 | |
| Appare nelle tipologie: | 1.1 Articolo in rivista |
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