Asymptotic dynamics of inhibitory networks for the NNLIF Model in the large-delay limit

We investigate the impact of large synaptic delays on the emergence of periodic dynamics in inhibitory neuronal networks, within the framework of the NNLIF model. Inspired by the work of [11] where the notion of pseudo-equilibria was introduced and developed, and by our earlier analysis in [14], we show that, as the delay tends to infinity, solutions of sufficitently inhibitory networks oscillate between distinct pseudo-equilibria over any finite time interval. Employing the Doeblin-Harris method, we rigorously establish a local convergence in the Ces{à}ro mean toward a limit function determined solely by these pseudo-equilibria.