Relaxation in time elapsed neuron network models in the weak connectivity regime

Stéphane Mischler (1) , Q Weng (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In order to describe the firing activity of a homogenous assembly of neurons, we consider time elapsed models, which give mathematical descriptions of the probability density of neurons structured by the distribution of times elapsed since the last discharge. Under general assumption on the firing rate and the delay distribution, we prove the uniqueness of the steady state and its nonlinear exponential stability in the weak connectivity regime. The result generalizes some similar results obtained in [10] in the case without delay. Our approach uses the spectral analysis theory for semigroups in Banach spaces developed recently by the first author and collaborators.

Mots clés: Neuron networks; time elapsed dynamics; semigroup spectral analysis; weak connectivity; long time asymptotic AMS Subject Classification 35B40; 35F15; 35F20; 92B20

Domaines: Equations aux dérivées partielles [math.AP]

Stéphane Mischler, Q Weng. Relaxation in time elapsed neuron network models in the weak connectivity regime. 2015. ⟨hal-01148645v1⟩