Long time behavior of an age and leaky memorystructured neuronal population equation
Abstract
We study the asymptotic stability of a twodimensional meanfield equation, which takes the form of a nonlocal transport equation and generalizes the timeelapsed neuron network model by the inclusion of a leaky memory variable. This additional variable can represent a slow fatigue mechanism, like spike frequency adaptation or shortterm synaptic depression. Even though twodimensional models are known to have emergent behaviors, like population bursts, which are not observed in standard onedimensional models, we show that in the weak connectivity regime, twodimensional models behave like onedimensional models, i.e. they relax to a unique stationary state. The proof is based on an application of Harris' ergodic theorem and a perturbation argument, adapted to the case of a multidimensional equation with delays.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.11110
 Bibcode:
 2021arXiv210611110F
 Keywords:

 Mathematics  Analysis of PDEs;
 35B40 (primary) 35F15;
 35F20;
 92B20 (secondary)