Weak and Strong Connectivity Regimes for a General Time Elapsed Neuron Network Model

For large fully connected neuron networks, we study the dynamics of homogenous assemblies of interacting neurons described by time elapsed models. Under general assumptions on the firing rate which include the ones made in previous works (Pakdaman et al. in Nonlinearity 23(1):55–75, 2010; SIAM J Appl Math 73(3):1260–1279, 2013, Mischler and Weng in Acta Appl Math, 2015), we establish accurate estimate on the long time behavior of the solutions in the weak and the strong connectivity regime both in the case with and without delay. Our results improve (Pakdaman et al. 2010, 2013) where a less accurate estimate was established and Mischler and Weng (2015) where only smooth firing rates were considered. Our approach combines several arguments introduced in the above previous works as well as a slightly refined version of the Weyl’s and spectral mapping theorems presented in Voigt (Monatsh Math 90(2):153–161, 1980) and Mischler and Scher (Ann Inst H Poincaré Anal Non Linéaire 33(3):849–898, 2016).