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Article

Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes

1
Institut National de la Recherche Agronomique (INRA), MaIAGE, Allee de Vilvert, 78352 Jouy-en-Josas, France
2
Neuromat, Instituto de Matematica e Estatistica, Universidade de Sao Paulo, Sao Paulo SP-CEP 05508-090, Brasil
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(6), 518; https://doi.org/10.3390/math7060518
Received: 9 May 2019 / Revised: 31 May 2019 / Accepted: 1 June 2019 / Published: 6 June 2019
(This article belongs to the Special Issue Stochastic Processes in Neuronal Modeling)
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. View Full-Text
Keywords: Poincaré inequality; brain neuron networks; Galves-Löcherbach model Poincaré inequality; brain neuron networks; Galves-Löcherbach model
MDPI and ACS Style

Hodara, P.; Papageorgiou, I. Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes. Mathematics 2019, 7, 518. https://doi.org/10.3390/math7060518

AMA Style

Hodara P, Papageorgiou I. Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes. Mathematics. 2019; 7(6):518. https://doi.org/10.3390/math7060518

Chicago/Turabian Style

Hodara, Pierre, and Ioannis Papageorgiou. 2019. "Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes" Mathematics 7, no. 6: 518. https://doi.org/10.3390/math7060518

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