Stochastic models for neural networks

• Grant number: 16/03988-5
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: June 01, 2016
• End date: May 31, 2017
• Field of knowledge: Physical Sciences and Mathematics - Probability and Statistics - Probability
• Principal Investigator: Jefferson Antonio Galves
• Grantee: Achillefs Tzioufas
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 13/07699-0 - Research, Innovation and Dissemination Center for Neuromathematics - NeuroMat, AP.CEPID

Abstract

Overview Chains with memory of variable length is a family of nite non-Markovian processes have been of great interest for probabilists and have found many applications; we refer to [2] for a review and background on this topic. Interacting particle systems is a class of Markovian processes that has received profound interest as a eld of probability; we refer to [5, 6] for background. Recently an extension of both classes of processes was issued in [4], where the existence of stationary versions of innite non-Markovian interacting particle systems processes has been established. Furthermore, in the case that the interactions between components are given by a critical directed Erdos-Renyi-type random graph, explicit upper-bound for the correlation between successive inter-spike intervals have been provided. We propose to investigate the existence of a phase transition, that is, the uniqueness or not of the invariant measure as a function of the summability or not of the synaptic weights of the model. This is a natural follow-up in the study of the Galves-Locherbach model, which could provide a microscopic interpretation of the existence of dierent macroscopic states. We propose investigating the extension of the existence result for non stationary under strong additional assumptions for a special subclass of these processes. In addition, we examine the related aspect that the interactions between components are given by random regular graphs. Starting with the seminal work of [1] metastability results for interacting particle systems have been of major importance, and metastability for a process on this class of graphs has been investigated recently [7]. We shall investigate the extension of these results for Galves-Locherbach models. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

TZIOUFAS, ACHILLEFS. The Central Limit Theorem for Supercritical Oriented Percolation in Two Dimensions. Journal of Statistical Physics, v. 171, n. 5, p. 802-821, JUN 2018. (16/03988-5)

TZIOUFAS, ACHILLEFS. The randomly fluctuating hyperrectangles are spatially monotone. Statistics & Probability Letters, v. 130, p. 76-79, NOV 2017. (16/03988-5, 13/07699-0)

TZIOUFAS, ACHILLEFS. A note on monotonicity of spatial epidemic models. BRAZILIAN JOURNAL OF PROBABILITY AND STATISTICS, v. 33, n. 3, p. 674-684, AUG 2019. (16/03988-5, 13/07699-0)