Speaker
Description
Systems of pulse-coupled oscillators model synchronization through singular interactions occurring at discrete times, when particles reach a specific firing phase. They have numerous applications in physics, biology and engineering, for example to cardiac cells, neurons and fireflies. They could also represent a stepping stone for the understanding of networks of voltage-conductance neurons at the mesoscopic scale. In the mean-field limit, the probability density in phase of the population of oscillators satisfies a singular continuity equation prone to finite-time blow-up, for which very few theoretical results are available. With José A. Carrillo, Xu'an Dou and Zhennan Zhou
\cite{CDRZ}, we have introduced a reformulation of the mean-field system based on the inverse distribution function seen in a dilated timescale. It allows to show a hidden contraction/expansion mechanism and to propose simple and rigorous proofs of the long-time behaviour, the existence of steady states, the rates of convergence and the occurence of finite time blow-up for a large class of monotone phase response functions.
Bibliography
@article{CDRZ,
title={Classical solutions of a mean field system for pulse-coupled oscillators: long time asymptotics versus blowup},
author={Carrillo, Jos{\'e} Antonio and Dou, Xu'an and Roux, Pierre and Zhou, Zhennan},
journal={arXiv preprint arXiv:2404.13703},
year={2024}
}
