Strongly nonlinear age structured equation,time-elapsed model and large delays
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| Format: | Preprint |
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2025
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| _version_ | 1866916650758963200 |
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| author | Perthame, Benoît Salort, Delphine Rieutord, Clément |
| author_facet | Perthame, Benoît Salort, Delphine Rieutord, Clément |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_09157 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strongly nonlinear age structured equation,time-elapsed model and large delays Perthame, Benoît Salort, Delphine Rieutord, Clément Analysis of PDEs 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. |
| title | Strongly nonlinear age structured equation,time-elapsed model and large delays |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.09157 |