Particle systems with a singular mean-field self-excitation. Application to neuronal networks. - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Stochastic Processes and their Applications Année : 2015

Particle systems with a singular mean-field self-excitation. Application to neuronal networks.

Résumé

We discuss the construction and approximation of solutions to a nonlinear McKean-Vlasov equation driven by a singular self-excitatory interaction of the mean-field type. Such an equation is intended to describe an infinite population of neurons which interact with one another. Each time a proportion of neurons 'spike', the whole network instantaneously receives an excitatory kick. The instantaneous nature of the excitation makes the system singular and prevents the application of standard results from the literature. Making use of the Skorohod M1 topology, we prove that, for the right notion of a 'physical' solution, the nonlinear equation can be approximated either by a finite particle system or by a delayed equation. As a by-product, we obtain the existence of 'synchronized' solutions, for which a macroscopic proportion of neurons may spike at the same time.
Fichier principal
Vignette du fichier
Delarue_Inglis_Rubenthaler_Tanre_SPA.pdf (444.94 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-01001716 , version 1 (04-06-2014)
hal-01001716 , version 2 (06-06-2014)
hal-01001716 , version 3 (23-01-2015)

Identifiants

Citer

François Delarue, James Inglis, Sylvain Rubenthaler, Etienne Tanré. Particle systems with a singular mean-field self-excitation. Application to neuronal networks.. Stochastic Processes and their Applications, 2015, 125, pp.2451--2492. ⟨10.1016/j.spa.2015.01.007⟩. ⟨hal-01001716v3⟩
562 Consultations
556 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More