  We study two population models describing the dynamics of interacting
neurons, initially proposed by Pakdaman, Perthame, and Salort (2010, 2014). In
the first model, the structuring variable $s$ represents the time elapsed since
its last discharge, while in the second one neurons exhibit a fatigue property
and the structuring variable is a generic "state". We prove existence of
solutions and steady states in the space of finite, nonnegative measures.
Furthermore, we show that solutions converge to the equilibrium exponentially
in time in the case of weak nonlinearity (i.e., weak connectivity). The main
innovation is the use of Doeblin's theorem from probability in order to show
the existence of a spectral gap property in the linear (no-connectivity)
setting. Relaxation to the steady state for the nonlinear models is then proved
by a constructive perturbation argument.
