  We study Talagrand concentration and Poincar\'e type inequalities for
unbounded pure jump Markov processes. In particular we focus on processes with
degenerate jumps that depend on the past of the whole system, based on the
model introduced by Galves and L\"ocherbach in \cite{G-L}, in order to describe
the activity of a biological neural network. As a result we obtain exponential
rates of convergence to equilibrium.
