Noisy integrate-and-fire equation: continuation after blow-up
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912040496398336 |
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| author | Dou, Xu'An Perthame, Benoît Salort, Delphine Zhou, Zhennan |
| author_facet | Dou, Xu'An Perthame, Benoît Salort, Delphine Zhou, Zhennan |
| contents | The integrate and fire equation is a classical model for neural assemblies which can exhibit finite time blow-up. A major open problem is to understand how to continue solutions after blow-up. Here we study an approach based on random discharge models and a change of time which generates a classical global solution to the expense of a strong absorption rate 1/$ε$. We prove that in the limit $ε$ $\rightarrow$ 0 + , a global solution is recovered where the integrate and fire equation is reformulated with a singular measure. This describes the dynamics after blow-up and also gives information on the blow-up phenomena itself.The major difficulty is to handle nonlinear terms. To circumvent it, we establish two new estimates, a kind of equi-integrability of the discharge measure and a L 2 estimate of the density. The use of the new timescale turns out to be fundamental for those estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14749 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Noisy integrate-and-fire equation: continuation after blow-up Dou, Xu'An Perthame, Benoît Salort, Delphine Zhou, Zhennan Analysis of PDEs The integrate and fire equation is a classical model for neural assemblies which can exhibit finite time blow-up. A major open problem is to understand how to continue solutions after blow-up. Here we study an approach based on random discharge models and a change of time which generates a classical global solution to the expense of a strong absorption rate 1/$ε$. We prove that in the limit $ε$ $\rightarrow$ 0 + , a global solution is recovered where the integrate and fire equation is reformulated with a singular measure. This describes the dynamics after blow-up and also gives information on the blow-up phenomena itself.The major difficulty is to handle nonlinear terms. To circumvent it, we establish two new estimates, a kind of equi-integrability of the discharge measure and a L 2 estimate of the density. The use of the new timescale turns out to be fundamental for those estimates. |
| title | Noisy integrate-and-fire equation: continuation after blow-up |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.14749 |