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August 2015 Global solvability of a networked integrate-and-fire model of McKean–Vlasov type
François Delarue, James Inglis, Sylvain Rubenthaler, Etienne Tanré
Ann. Appl. Probab. 25(4): 2096-2133 (August 2015). DOI: 10.1214/14-AAP1044


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.


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François Delarue. James Inglis. Sylvain Rubenthaler. Etienne Tanré. "Global solvability of a networked integrate-and-fire model of McKean–Vlasov type." Ann. Appl. Probab. 25 (4) 2096 - 2133, August 2015.


Received: 1 April 2014; Published: August 2015
First available in Project Euclid: 21 May 2015

zbMATH: 1322.60085
MathSciNet: MR3349003
Digital Object Identifier: 10.1214/14-AAP1044

Primary: 60H10
Secondary: 60J75, 60K35, 92C20

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.25 • No. 4 • August 2015
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