Strongly nonlinear age structured equation,time-elapsed model and large delays

Benoît Perthame; Delphine Salort; Clément Rieutord

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arXiv · 2503.09157

Strongly nonlinear age structured equation,time-elapsed model and large delays

Benoît Perthame·Delphine Salort·Clément Rieutord

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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.

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Benoît Perthame, Delphine Salort, Clément Rieutord. 2025-03-12. Strongly nonlinear age structured equation,time-elapsed model and large delays. https://arxiv.org/abs/2503.09157

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