Fonte, Claudia;
Schmutz, Valentin;
(2022)
Long Time Behavior of an Age- and Leaky Memory-Structured Neuronal Population Equation.
SIAM Journal on Mathematical Analysis
, 54
(4)
pp. 4721-4756.
10.1137/21m1428571.
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Abstract
We study the asymptotic stability of a two-dimensional mean-field equation, which takes the form of a nonlocal transport equation and generalizes the time-elapsed neuron network model by the inclusion of a leaky memory variable. This additional variable can represent a slow fatigue mechanism, such as spike-frequency adaptation or short-term synaptic depression. Even though two-dimensional models are known to have emergent behaviors, such as population bursts, which are not observed in standard one-dimensional models, we show that in the weak connectivity regime, two-dimensional models behave like one-dimensional models, i.e., they relax to a unique stationary state. The proof is based on an application of Harris's ergodic theorem and a perturbation argument, both adapted to the case of a multidimensional equation with delays.
| Type: | Article |
|---|---|
| Title: | Long Time Behavior of an Age- and Leaky Memory-Structured Neuronal Population Equation |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1137/21m1428571 |
| Publisher version: | https://doi.org/10.1137/21m1428571 |
| Language: | English |
| Additional information: | © 2022, Society for Industrial and Applied Mathematics. This work is licensed under a Creative Commons Attribution 4.0 International License. |
| Keywords: | long time behavior, nonlocal transport equation, mean-field equation, Doeblin's and Harris's theorems, piecewise deterministic Markov process, spiking neuron, spike-frequency adaptation, short-term synaptic plasticity |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Brain Sciences UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Brain Sciences > UCL Queen Square Institute of Neurology UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Brain Sciences > UCL Queen Square Institute of Neurology > Imaging Neuroscience |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10178710 |
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