Fonte, ClaudiaSchmutz, Valentin Marc2022-12-122022-12-122022-12-12202210.1137/21M1428571https://infoscience.epfl.ch/handle/20.500.14299/193154We 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.text::journal::journal article::research article