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

F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré

Stochastic Processes and their Applications, 2015, vol. 125, issue 6, 2451-2492

Abstract: 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.

Keywords: McKean nonlinear diffusion process; Counting process; Propagation of chaos; Integrate-and-fire network; Skorohod M1 topology; Neuroscience (search for similar items in EconPapers)
Date: 2015
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (12)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304414915000186
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:125:y:2015:i:6:p:2451-2492

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spa.2015.01.007

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().