A genuinely periodic firing pattern in the delayed noisy integrate-and-fire PDE
The delayed nonlinear noisy leaky integrate-and-fire (NNLIF) model of an inhibitory neural population, with diffusion $a > 0$, coupling $b < 0$, delay $d > 0$, reset voltage $V_R$ and firing voltage $V_F > V_R$, is the equation $$\partial_t p + \partial_v\bigl[(-v + bN(t - d))\,p\bigr] - a\,\partial_{vv} p = N(t)\,\delta_{V_R}, \qquad v < V_F,$$ $$p(t, V_F) = 0, \qquad N(t) = -a\,\partial_v p(t, V_F), \qquad \int_{-\infty}^{V_F} p(t, v)\, dv = 1,$$ for a voltage density $p$ and firing rate $N$. Ikeda, Roux, Salort and Smets approximated its numerically observed periodic solutions formally by a Gaussian wave whose motion is governed by a delay differential equation, and proved that this delay differential equation has a periodic solution.
Problem. Do there exist $a > 0$, $b < 0$, $d > 0$, $V_R < V_F$ and a period $T > 0$ for which this system has a nonnegative $T$-periodic solution with nonconstant firing rate $N$? The density $p$ must be continuous in $v$, classical away from $V_R$, with the derivative jump at $V_R$ prescribed by the equation in the sense of distributions, with vanishing probability flux at $v = -\infty$ and uniformly finite second voltage moment; $p$ and $N$ are defined for all real times, so that the delayed term is unambiguous.
Source: Stated by AIM — Open Applied Problems, after Ikeda, Roux, Salort and Smets (2022) and Carrillo and Roux (2025), §1.7.2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, integrate-and-fire neurons, nonlinear Fokker–Planck equations, periodic solutions
References
@article{ikeda2022,
author = {Ikeda, Kota and Roux, Pierre and Salort, Delphine and Smets, Didier},
title = {Theoretical study of the emergence of periodic solutions for the inhibitory {NNLIF} neuron model with synaptic delay},
journal = {Math. Neurosci. Appl.},
volume = {2},
year = {2022},
pages = {7256},
doi = {10.46298/mna.7256},
}
@article{carrillo2025,
author = {Carrillo, Jos{\'e} A. and Roux, Pierre},
title = {Nonlinear partial differential equations in neuroscience: from modeling to mathematical theory},
journal = {Math. Models Methods Appl. Sci.},
volume = {35},
number = {2},
year = {2025},
pages = {403--584},
doi = {10.1142/S0218202525400044},
eprint = {2501.06015},
archivePrefix = {arXiv},
}
@misc{aim-delayed-nnlif-periodic,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A genuinely periodic firing pattern in the delayed noisy integrate-and-fire PDE}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/353-delayed-nnlif-periodic.md},
note = {Entry 353 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We propose a local branch of nonnegative periodic solutions of the delayed noisy leaky integrate-and-fire reset equation with inhibitory coupling and nonconstant firing rate. Translating both voltage thresholds with the coupling keeps a centered stationary reset profile fixed. An explicit boundary-response function gives a nondegenerate two-parameter reduction at high frequency. Resolvent bounds, time-Sobolev trace estimates, a negative-part energy argument and …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, 353-delayed-nnlif-periodic-branch/aim353_delayed_nnlif_periodic.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00675, "A genuinely periodic firing pattern in the delayed noisy integrate-and-fire PDE", clopen.solutions.