abstract: The goal of these two lectures is to address mean field models for neuronal networks with excitatory interactions. A specific feature of some of these models is to exhibit different behaviors depending on the intensity of the excitation.
We shall focus on a model for which existence and uniqueness of a solution are challenging. As a noticeable fact, a singularity may emerge when the excitation parameter driving the interactions between the neurons grows up. Conversely, we shall prove there is no singularity when the excitation parameter is small enough.
Finally, we shall justify the passage from the finite to the limiting model.
Possibly, we shall also discuss extensions and related models in the literature.
The lecture will be mostly based on the following two papers:
Delarue F., Inglis J., Rubenthaler R., Tanré, E. (2015). Global solvability of a networked integrate-and-fire model of McKean-Vlasov type. Annals of Applied Probability, 2015, 2096--2133.
Delarue F., Inglis J., Rubenthaler R., Tanré E. (2015). Particle systems with a singular mean-field self-excitation. Application to neuronal networks. Stochastic Processes and their Applications, 125, pp.2451-2492
We will also refer to:
Nadtochiy S. and Shkolnikov, M. (2017). Particle systems with singular interaction through hitting times: application in systemic risk modeling.
Hambly, B. and Ledger, S. (2017). A stochastic McKean Vlasov equation for absorbing diffusions on the half line. To appear in Annals of App. Probability.