Long time behavior of an age and leaky memory-structured neuronal population equation
Résumé
We study the asymptotic stability of a two-dimensional mean-field neuronal population equation, which is explicitly linked to spiking neuron models, 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, like spike frequency adaptation or short-term synaptic depression. Even though two-dimensional models are known to have emergent behaviors, like population bursts, which are not observed in 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 a version of Harris' ergodic theorem and a perturbation argument. The multidimensionality of the model and delay effects require careful consideration.
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