# Spatially extended networks with singular multi-scale connectivity patterns

Abstract : The cortex is a very large network characterized by a complex connectivity including at least two scales: a microscopic scale at which the interconnections are non-specific and very dense, while macroscopic connectivity patterns connecting different regions of the brain at larger scale are extremely sparse. This motivates to analyze the behavior of networks with multiscale coupling, in which a neuron is connected to its $v(N)$ nearest-neighbors where $v(N)=o(N)$, and in which the probability of macroscopic connection between two neurons vanishes. These are called singular multi-scale connectivity patterns. We introduce a class of such networks and derive their continuum limit. We show convergence in law and propagation of chaos in the thermodynamic limit. The limit equation obtained is an intricate non-local McKean-Vlasov equation with delays which is universal with respect to the type of micro-circuits and macro-circuits involved.
Keywords :
Type de document :
Article dans une revue
Journal of Statistical Physics, Springer Verlag, 2014, pp.27. 〈10.1007/s10955-014-1015-x〉

https://hal.inria.fr/hal-00942201
Contributeur : Jonathan Touboul <>
Soumis le : mardi 4 février 2014 - 18:14:38
Dernière modification le : jeudi 11 janvier 2018 - 06:23:24

### Citation

Jonathan Touboul. Spatially extended networks with singular multi-scale connectivity patterns. Journal of Statistical Physics, Springer Verlag, 2014, pp.27. 〈10.1007/s10955-014-1015-x〉. 〈hal-00942201〉

### Métriques

Consultations de la notice