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Stochastic chains with unbounded memory and application in neuroscience

Grant number: 16/12918-0
Support type:Scholarships in Brazil - Doctorate
Effective date (Start): September 01, 2016
Effective date (End): February 28, 2019
Field of knowledge:Physical Sciences and Mathematics - Probability and Statistics - Probability
Principal Investigator:Alexsandro Giacomo Grimbert Gallo
Grantee:Ricardo Felipe Ferreira
Home Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
Associated research grant:13/07699-0 - Research, Innovation and Dissemination Center for Neuromathematics - NeuroMat, AP.CEPID

Abstract

Stochastic chains with unbounded memory are a natural generalization of Markov chains, in the sense that the transition probabilities may depend on the whole past. These process, introduced by Onicescu and Mihoc (1935) and Doeblin and Fortet (1937), have been receiving increasing attention in the probabilistic literature, because they form a class richer than the Markov chains and have practical capabilities modelling of scientific data in several areas, from biology to linguistics. In this work, we intend to use them to model properties of brain activity such as spike trains. The main objective in this project is to develop new mathematical results about stochastic chains with unbounded memory. First, we intend to study the sufficient conditions that guarantee the existence and uniqueness of invariant measure for such chains and on this occasion, addressing issues such as law of large numbers (ergodicity of chain), central limit theorem, large deviations and concentration inequality. Then, we will address the understanding of the phenomenology of spike trains and their modelling via stochastic chains with unbounded memory.