  We study the asymptotic stability of a two-dimensional mean-field equation,
which takes the form of a nonlocal transport equation 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 standard 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 an application of Harris' ergodic theorem and a
perturbation argument, adapted to the case of a multidimensional equation with
delays.
