2013/07/10 by Claudio Canuto, Canuto, Claudio, Anna Cattani +1
Biochemistry, Genetics and Molecular Biology · Neuroscience · Physics and Astronomy · #05C90 #34C60 #35K57 #92C42 #Dynamical Systems (math.DS) #FOS: Biological sciences #FOS: Mathematics #Gene Regulatory Network Analysis #Neural dynamics and brain function #Neurons and Cognition (q-bio.NC) #Numerical Analysis (math.NA) #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1307.2730
openalex publication_date 2013/07/10 · openalex created_date 2022/08/15 · openalex updated_date 2026/07/28
We consider an idealized network, formed by N neurons individually described\nby the FitzHugh-Nagumo equations and connected by electrical synapses. The\nlimit for N to infinity of the resulting discrete model is thoroughly\ninvestigated, with the aim of identifying a model for a continuum of neurons\nhaving an equivalent behaviour. Two strategies for passing to the limit are\nanalysed: i) a more conventional approach, based on a fixed nearest-neighbour\nconnection topology accompanied by a suitable scaling of the diffusion\ncoefficients; ii) a new approach, in which the number of connections to any\ngiven neuron varies with N according to a precise law, which simultaneously\nguarantees the non-triviality of the limit and the locality of neuronal\ninteractions. Both approaches yield in the limit a pde-based model, in which\nthe distribution of action potential obeys a nonlinear\nreaction-convection-diffusion equation; convection accounts for the possible\nlack of symmetry in the connection topology. Several convergence issues are\ndiscussed, both theoretically and numerically.\n