Sequence of pseudoequilibria describes the long-time behavior of the nonlinear noisy leaky integrate-and-fire model with large delay
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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_ | 1866917865410527232 |
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| author | Cáceres, María J. Cañizo, José A. Ramos-Lora, Alejandro |
| author_facet | Cáceres, María J. Cañizo, José A. Ramos-Lora, Alejandro |
| contents | There is a wide range of mathematical models that describe populations of large numbers of neurons. In this article, we focus on nonlinear noisy leaky integrate-and-fire (NNLIF) models that describe neuronal activity at the level of the membrane potential. We introduce a sequence of states, which we call pseudoequilibria, and give evidence of their defining role in the behavior of the NNLIF system when a significant synaptic delay is considered. The advantage is that these states are determined solely by the system's parameters and are derived from a sequence of firing rates that result from solving a recurrence equation. We propose a strategy to show convergence to an equilibrium for a weakly connected system with large transmission delay, based on following the sequence of pseudoequilibria. Unlike direct entropy dissipation methods, this technique allows us to see how a large delay favors convergence. We present a detailed numerical study to support our results. This study helps us understand, among other phenomena, the appearance of periodic solutions in strongly inhibitory networks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00971 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sequence of pseudoequilibria describes the long-time behavior of the nonlinear noisy leaky integrate-and-fire model with large delay Cáceres, María J. Cañizo, José A. Ramos-Lora, Alejandro Analysis of PDEs Numerical Analysis There is a wide range of mathematical models that describe populations of large numbers of neurons. In this article, we focus on nonlinear noisy leaky integrate-and-fire (NNLIF) models that describe neuronal activity at the level of the membrane potential. We introduce a sequence of states, which we call pseudoequilibria, and give evidence of their defining role in the behavior of the NNLIF system when a significant synaptic delay is considered. The advantage is that these states are determined solely by the system's parameters and are derived from a sequence of firing rates that result from solving a recurrence equation. We propose a strategy to show convergence to an equilibrium for a weakly connected system with large transmission delay, based on following the sequence of pseudoequilibria. Unlike direct entropy dissipation methods, this technique allows us to see how a large delay favors convergence. We present a detailed numerical study to support our results. This study helps us understand, among other phenomena, the appearance of periodic solutions in strongly inhibitory networks. |
| title | Sequence of pseudoequilibria describes the long-time behavior of the nonlinear noisy leaky integrate-and-fire model with large delay |
| topic | Analysis of PDEs Numerical Analysis |
| url | https://arxiv.org/abs/2403.00971 |