Modified Log-Sobolev Inequality for a Compact Pure Jump Markov Process with Degenerate Jumps

We study the modified log-Sobolev inequality for a class of pure jump Markov processes that describe the interactions between brain neurons. As a result, we obtain concentration properties for empirical approximations of the process. In particular, we focus on a finite and compact process with degenerate jumps inspired by the model introduced by Galves and Löcherbach (J Stat Phys 151:896–921, 2013).