Analysis of a mean-field limit of interacting two-dimensional nonlinear integrate-and-fire neurons
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908504608997376 |
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| author | Veltz, Romain |
| author_facet | Veltz, Romain |
| contents | We study the solutions of a McKean-Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the mean field limit of a system of $N$ interacting excitatory neurons with $N$ large.
Each neuron spikes randomly with rate depending on its membrane potential. At each spiking time, the neuron potential is reset to the value $\bar v$, its adaptation variable is incremented by $\bar w$ and all other neurons receive an additional amount
$J/N$ of potential after some delay where $J$ is the connection strength. Between jumps, the neurons drift according to some two-dimensional ordinary differential equation with explosive behavior.
We prove the existence and uniqueness of solutions of a heuristically derived mean-field limit of the system when $N\to\infty$. We then study the existence of stationary distributions and provide several properties (regularity, tail decay, etc.) based on a Doeblin estimate using a Lyapunov function. Numerical simulations are provided to assess the hypotheses underlying the results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_19134 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analysis of a mean-field limit of interacting two-dimensional nonlinear integrate-and-fire neurons Veltz, Romain Probability 60J75 (Primary) 60K35, 60G55, 45A05 (Secondary) We study the solutions of a McKean-Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the mean field limit of a system of $N$ interacting excitatory neurons with $N$ large. Each neuron spikes randomly with rate depending on its membrane potential. At each spiking time, the neuron potential is reset to the value $\bar v$, its adaptation variable is incremented by $\bar w$ and all other neurons receive an additional amount $J/N$ of potential after some delay where $J$ is the connection strength. Between jumps, the neurons drift according to some two-dimensional ordinary differential equation with explosive behavior. We prove the existence and uniqueness of solutions of a heuristically derived mean-field limit of the system when $N\to\infty$. We then study the existence of stationary distributions and provide several properties (regularity, tail decay, etc.) based on a Doeblin estimate using a Lyapunov function. Numerical simulations are provided to assess the hypotheses underlying the results. |
| title | Analysis of a mean-field limit of interacting two-dimensional nonlinear integrate-and-fire neurons |
| topic | Probability 60J75 (Primary) 60K35, 60G55, 45A05 (Secondary) |
| url | https://arxiv.org/abs/2508.19134 |