Nonparametric estimation of the jump rate in mean field interacting systems of neurons
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909667455664128 |
|---|---|
| author | Duarte, Aline Laxa, Kadmo Löcherbach, Eva Loukianova, Dasha |
| author_facet | Duarte, Aline Laxa, Kadmo Löcherbach, Eva Loukianova, Dasha |
| contents | We consider finite systems of $N$ interacting neurons described by non-linear Hawkes processes in a mean field frame. Neurons are described by their membrane potential. They spike randomly, at a rate depending on their potential. In between successive spikes, their membrane potential follows a deterministic flow. We estimate the spiking rate function based on the observation of the system of $N$ neurons over a fixed time interval $[0,t]$. Asymptotic are taken as $N,$ the number of neurons, tends to infinity. We introduce a kernel estimator of Nadaraya-Watson type and discuss its asymptotic properties with help of the deterministic dynamical system describing the mean field limit. We compute the minimax rate of convergence in an $L^2 -$error loss over a range of Hölder classes and obtain the classical rate of convergence $ N^{ - 2β/ ( 2 β+ 1)} , $ where $ β$ is the regularity of the unknown spiking rate function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_24065 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonparametric estimation of the jump rate in mean field interacting systems of neurons Duarte, Aline Laxa, Kadmo Löcherbach, Eva Loukianova, Dasha Probability 60F15, 60G55, 62M05, 62M20 We consider finite systems of $N$ interacting neurons described by non-linear Hawkes processes in a mean field frame. Neurons are described by their membrane potential. They spike randomly, at a rate depending on their potential. In between successive spikes, their membrane potential follows a deterministic flow. We estimate the spiking rate function based on the observation of the system of $N$ neurons over a fixed time interval $[0,t]$. Asymptotic are taken as $N,$ the number of neurons, tends to infinity. We introduce a kernel estimator of Nadaraya-Watson type and discuss its asymptotic properties with help of the deterministic dynamical system describing the mean field limit. We compute the minimax rate of convergence in an $L^2 -$error loss over a range of Hölder classes and obtain the classical rate of convergence $ N^{ - 2β/ ( 2 β+ 1)} , $ where $ β$ is the regularity of the unknown spiking rate function. |
| title | Nonparametric estimation of the jump rate in mean field interacting systems of neurons |
| topic | Probability 60F15, 60G55, 62M05, 62M20 |
| url | https://arxiv.org/abs/2506.24065 |