Global solvability of a networked integrate-and-fire model of McKean-Vlasov type

François Delarue; James Inglis; Sylvain Rubenthaler; Etienne Tanré

Global solvability of a networked integrate-and-fire model of McKean-Vlasov type

François Delarue ¹ James Inglis ² Sylvain Rubenthaler ¹ Etienne Tanré ²

1 JAD - Laboratoire Jean Alexandre Dieudonné

2 INRIA Sophia Antipolis / INRIA Nancy - Grand Est/ IECN - TOSCA

Abstract : We here investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the potential of a neuron increases when some of the others fire: precisely, the kick it receives is proportional to the instantaneous proportion of firing neurons at the same time. From a mathematical point of view, the coefficient of proportionality is of great importance as the resulting system is known to blow-up as this becomes large. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when the coefficient of proportionality is small enough.

Keywords : McKean nonlinear diffusion process renewal process first hitting time density estimates integrate-and-fire network nonhomogeneous diffusion process neuroscience

Document type :

Journal articles

Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2015, pp.33

Domain :

Mathematics / Probability

Cognitive science / Neuroscience

https://hal.inria.fr/hal-00747565
Contributor : James Inglis <>
Submitted on : Monday, June 2, 2014 - 12:09:07 PM
Last modification on : Thursday, December 18, 2014 - 10:10:05 AM

Global solvability of a networked integrate-and-fire model of McKean-Vlasov type

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