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

TL;DR

This paper establishes Poincaré inequalities for a class of degenerate pure jump Markov processes inspired by neural models, highlighting their mathematical properties and potential implications for understanding neural dynamics.

Contribution

It introduces novel Poincaré inequalities for degenerate jump processes with memory loss, extending previous models of neural interactions.

Findings

Proved Poincaré inequalities for the class of processes

Characterized the behavior of neurons with degenerate jumps

Linked mathematical properties to neural system dynamics

Abstract

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.