Self-sustained oscillations in the full voltage–conductance neuron equation
In the kinetic voltage–conductance model of an excitatory neural network, $F(t, v, y) \ge 0$ is the density of neurons with voltage $v \in (0, v_F)$ and conductance $y > 0$. For parameters $v_E > v_F > 0$ and $y_L, a_*, y_*, c > 0$ put $$J(v, y) = y(v_E - v) - y_L v, \qquad y_F = \frac{y_L v_F}{v_E - v_F}, \qquad N(t) = \int_{y_F}^{\infty} J(v_F, y) F(t, v_F, y)\, dy,$$ $$K_F(t, y) = y_* + cN(t) - y, \qquad a_F(t) = a_* + c^2 N(t),$$ and require, on $(v, y) \in (0, v_F) \times (0, \infty)$, $$\partial_t F + \partial_v(JF) + \partial_y(K_F F) - a_F\, \partial_{yy} F = 0, \qquad \iint F\, dv\, dy = 1.$$ For $0 < y < y_F$ impose $F(t, 0, y) = F(t, v_F, y) = 0$; for $y > y_F$ impose $J(0, y)F(t, 0, y) = J(v_F, y)F(t, v_F, y)$ (neurons that fire at $v_F$ are reset to $0$); at $y = 0$ impose $K_F F - a_F\, \partial_y F = 0$, with vanishing flux as $y \to \infty$. The firing rate $N(t)$ is the flux through $v = v_F$. Solutions are sought in the distributional sense, continuous in time with values in $L^1$, with these boundary traces, finite entropy and second moment in $y$ bounded uniformly over one period. Fonte Sanchez, Mischler and Salort proved that for every $c \ge 0$ and every initial datum $F_0$ of unit mass with $F_0 (1 + y^2 + |\log F_0|) \in L^1$ there is a global weak solution (2026). Cáceres, Carrillo and Tao observed numerically that, as the excitatory coupling increases, the firing rate may converge to a steady state, become periodic, or show damped oscillations (2011).
Problem. For some $v_E > v_F > 0$ and $y_L, a_*, y_*, c > 0$, does this model admit a nonnegative probability density $F(t, v, y)$ that is $T$-periodic in $t$ for some $T > 0$ and has nonconstant firing rate $N(t)$?
Source: Carrillo and Roux (2025), §7.6, for the kinetic voltage–conductance model of Cai, Tao, Shelley and McLaughlin (2004). Stated by AIM — Open Applied Problems in the normalization of Fonte Sanchez, Mischler and Salort (2026).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, kinetic equations, neural networks, periodic solutions
References
@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},
}
@article{cai2004,
author = {Cai, David and Tao, Louis and Shelley, Michael and McLaughlin, David W.},
title = {An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex},
journal = {Proc. Natl. Acad. Sci. USA},
volume = {101},
number = {20},
year = {2004},
pages = {7757--7762},
doi = {10.1073/pnas.0401906101},
}
@misc{fontesanchez2026,
author = {Fonte Sanchez, Claudia and Mischler, St{\'e}phane and Salort, Delphine},
title = {Existence of solutions to the {V}oltage-{C}onductance kinetic equation in a general conductivity regime},
year = {2026},
eprint = {2607.22112},
archivePrefix = {arXiv},
}
@article{caceres2011,
author = {C{\'a}ceres, Mar{\'\i}a J. and Carrillo, Jos{\'e} A. and Tao, Louis},
title = {A numerical solver for a nonlinear {F}okker--{P}lanck equation representation of neuronal network dynamics},
journal = {J. Comput. Phys.},
volume = {230},
number = {4},
year = {2011},
pages = {1084--1099},
doi = {10.1016/j.jcp.2010.10.027},
}
@misc{aim-voltage-conductance-periodic,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Self-sustained oscillations in the full voltage–conductance neuron equation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/354-voltage-conductance-periodic.md},
note = {Entry 354 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00654, "Self-sustained oscillations in the full voltage–conductance neuron equation", clopen.solutions.