Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes

Pierre Hodara and Ioannis Papageorgiou
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Pierre Hodara: Institut National de la Recherche Agronomique (INRA), MaIAGE, Allee de Vilvert, 78352 Jouy-en-Josas, France
Ioannis Papageorgiou: Neuromat, Instituto de Matematica e Estatistica, Universidade de Sao Paulo, Sao Paulo SP-CEP 05508-090, Brasil

Mathematics, 2019, vol. 7, issue 6, 1-18

Abstract: We aim to prove Poincaré inequalities for a class of pure jump Markov processes inspired by the model introduced by Galves and Löcherbach to describe the behavior of interacting brain neurons. In particular, we consider neurons with degenerate jumps, i.e., which 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. The process studied by Galves and Löcherbach is a point process counting the spike events of the system and is therefore non-Markovian. In this work, we consider a process describing the membrane potential of each neuron that contains the relevant information of the past. This allows us to work in a Markovian framework.

Keywords: Poincaré inequality; brain neuron networks; Galves-Löcherbach model (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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