The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 25, Number 4 (2015), 2096-2133.
Global solvability of a networked integrate-and-fire model of McKean–Vlasov type
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.
Ann. Appl. Probab., Volume 25, Number 4 (2015), 2096-2133.
Received: April 2014
First available in Project Euclid: 21 May 2015
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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