The Annals of Applied Probability

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

François Delarue, James Inglis, Sylvain Rubenthaler, and Etienne Tanré

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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, denoted by $\alpha$, is of great importance as the resulting system is known to blow-up for large values of $\alpha$. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when $\alpha$ is small enough.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 4 (2015), 2096-2133.

Dates
Received: April 2014
First available in Project Euclid: 21 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1432212438

Digital Object Identifier
doi:10.1214/14-AAP1044

Mathematical Reviews number (MathSciNet)
MR3349003

Zentralblatt MATH identifier
1322.60085

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 92C20: Neural biology 60J75: Jump processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

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

Citation

Delarue, François; Inglis, James; Rubenthaler, Sylvain; Tanré, Etienne. Global solvability of a networked integrate-and-fire model of McKean–Vlasov type. Ann. Appl. Probab. 25 (2015), no. 4, 2096--2133. doi:10.1214/14-AAP1044. https://projecteuclid.org/euclid.aoap/1432212438


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