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.
The derivation of continuum limits of neuronal networks with gap-junction couplings / Canuto, Claudio; Anna, Cattani. - In: NETWORKS AND HETEROGENEOUS MEDIA. - ISSN 1556-1801. - 9:(2014), pp. 111-133. [10.3934/nhm.2014.9.111]
The derivation of continuum limits of neuronal networks with gap-junction couplings
CANUTO, CLAUDIO;
2014
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 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.Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
https://hdl.handle.net/11583/2582341
Attenzione
Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo