Strongly nonlinear age structured equation,time-elapsed model and large delays
Abstract
The time-elapsed model for neural networks is a nonlinear age structured equationwhere the renewal term describes the network activity and influences the dischargerate, possibly with a delay due to the length of connections.We solve a long standing question, namely that an inhibitory network withoutdelay will converge to a steady state and thus the network is desynchonised. Ourapproach is based on the observation that a non-expansion property holds true.However a non-degeneracy condition is needed and, besides the standard one, weintroduce a new condition based on strict nonlinearity.When a delay is included, and following previous works for Fokker-Planck models,we prove that the network may generate periodic solutions. We introduce a newformalism to establish rigorously this property for large delays.The fundamental contraction property also holds for some other age structuredequations and systems.
Cite
@article{arxiv.2503.09157,
title = {Strongly nonlinear age structured equation,time-elapsed model and large delays},
author = {Benoît Perthame and Delphine Salort and Clément Rieutord},
journal= {arXiv preprint arXiv:2503.09157},
year = {2025}
}