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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.
Eilif Benjamin Muller, Michael Reimann, James Gonzalo King, Marwan Muhammad Ahmed Abdellah, Pramod Shivaji Kumbhar, András Ecker, Sirio Bolaños Puchet, James Bryden Isbister, Daniela Egas Santander, Jorge Blanco Alonso, Giuseppe Chindemi, Ioannis Magkanaris