2018/03/31 by Pierre Hodara, Ioannis Papageorgiou · 6 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Neuroscience · #Artificial intelligence #Class (philosophy) #Computer science #Degenerate energy levels #Gene Regulatory Network Analysis #Jump #Markov Chains and Monte Carlo Methods #Markov chain #Markov process #Mathematics #Neural dynamics and brain function #Physics #Position (finance) #Pure mathematics #Quantum mechanics #Spike (software development) #Statistical physics #Statistics #Type (biology) #math.PR #msc:26D10 #msc:60G99 #msc:60K35
paper · pdf · doi:10.3390/math7060518
published in Mathematics 7(6), 518 (Multidisciplinary Digital Publishing Institute)
openalex created_date 2018/09/07 · arxiv created 2019/04/11 · openalex publication_date 2019/06/06 · arxiv updated 2019/06/11 · openalex updated_date 2026/08/06
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.