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

François Delarue 1 James Inglis 2, 3 Sylvain Rubenthaler 1 Etienne Tanré 2
2 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
3 NEUROMATHCOMP - Mathematical and Computational Neuroscience
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
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.
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Stochastic Processes and their Applications, Elsevier, 2015, 125, pp.2451--2492. 〈10.1016/j.spa.2015.01.007〉
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Dernière modification le : lundi 4 décembre 2017 - 15:14:19
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François Delarue, James Inglis, Sylvain Rubenthaler, Etienne Tanré. Particle systems with a singular mean-field self-excitation. Application to neuronal networks.. Stochastic Processes and their Applications, Elsevier, 2015, 125, pp.2451--2492. 〈10.1016/j.spa.2015.01.007〉. 〈hal-01001716v3〉

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