  We aim in proving Poincar\'e inequalities for a class of pure jump Markov
processes inspired by the model introduced in \cite{G-L} by Galves and
L\"ocherbach to describe the behaviour of interacting brain neurons. In
particular, we consider neurons with degenerate jumps, i.e. that lose their
memory when they spike, while the probability of a spike depends on the actual
position and thus the past of the whole neural system.
