On the Integrate-and-Fire model with superlinear drift
Abstract
The Nonlinear Integrate-and-Fire equation is a fundamental model for neural assemblies, describing the dynamics of the density of neurons with a given potential. Its standard formulation uses a given firing potential, and is known to undergo finite-time blow-up for excitatory connections. To fit experimental observations, it has been advocated to use superlinear drifts. We show how, in this situation, it makes sense to set the equation on the whole line. The main question we address is whether in these circumstances the solution still exhibits finite-time blow-up. We study the construction of solutions by a fixed point and, using the framework of \cite{carrilloQualitativePropertiesSolutions2015}, we develop novel a priori estimates which use specifically the superlinearity of the drift at +∞. In particular, we obtain Ces{à}ro-mean bounds on the activity, as well as exponential convergence towards the unique steady state in the weakly connected regime. Several critical steps rely on controls depending solely on the L1 norm of the solution or coefficients.
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BibTeX
@misc{perthame2026integrateandfire,
title = {{On the Integrate-and-Fire model with superlinear drift}},
author = {Benoît Perthame and Clément Rieutord and Delphine Salort},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41497},
eprint = {2610.41497},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05843294v1},
}
RIS
TY - UNPB
TI - On the Integrate-and-Fire model with superlinear drift
AU - Benoît Perthame
AU - Clément Rieutord
AU - Delphine Salort
PY - 2026
DA - 2026/10/29
PB - HAL
DO - 10.48550/arxiv.2610.41497
UR - https://hal.science/hal-05843294v1
AB - The Nonlinear Integrate-and-Fire equation is a fundamental model for neural assemblies, describing the dynamics of the density of neurons with a given potential. Its standard formulation uses a given firing potential, and is known to undergo finite-time blow-up for excitatory connections. To fit experimental observations, it has been advocated to use superlinear drifts. We show how, in this situation, it makes sense to set the equation on the whole line. The main question we address is whether in these circumstances the solution still exhibits finite-time blow-up. We study the construction of solutions by a fixed point and, using the framework of \cite{carrilloQualitativePropertiesSolutions2015}, we develop novel a priori estimates which use specifically the superlinearity of the drift at +∞. In particular, we obtain Ces{à}ro-mean bounds on the activity, as well as exponential convergence towards the unique steady state in the weakly connected regime. Several critical steps rely on controls depending solely on the L^1 norm of the solution or coefficients.
ER -