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

Poincar\'e type inequalities for compact degenerate pure jump Markov processes