Dottorato in AUTOMATICA, BIOINGEGNERIA E RICERCA OPERATIVA -
ABRO

Titolo della tesi: From tensor decomposition to graph signal processing: an algebraic approach for the analysis of functional brain networks in both healthy and pathological individuals

Network theory and its applications nowadays constitute the cutting-edge of different scientific fields, providing scientists with innovative tools for the analysis of complex systems. In this framework, the human brain probably embodies one of the most fascinating examples of complex system in biology, where the interplay of different subunits contributes to characterize the behaviour of the whole system itself. However, despite the methodological boost that network theory experienced in the last decades, brain networks’ analyses still rely on standard frameworks dealing with either single-node or global descriptors. Recent advancements in network theory could therefore be employed to enhance the capabilities of traditional protocols, addressing some of their limitations with the aim to provide an in-dept characterization of brain networks in both healthy and pathological individuals. In this framework stroke probably embodies one of the most representative scenarios where the effects of a lateralized traumatic event are not limited to a circumscribed area but spread over the whole network. A critical aspect of stroke-induced functional alterations thus lies in understanding how stroke affects the communication among clusters of interacting nodes. Furthermore, stroke also induces substantial alterations in both biological and electrical properties of scalp tissues, thus limiting clinical analyses to the investigation of scalp-level signals which are highly susceptible to the effects of volume conduction. Finally, when dealing with large-scale brain data the estimation of the connectivity grand average still represents an open issue. Different approaches have been proposed over the years, but there is currently no consensus on a univocal analysis pipeline for the grand average extraction from a given population. The aim of this Thesis is thus to leverage some of the recent advancements in network theory to support the investigation of open questions in current clinical practice. More in detail, Chapter 1 approaches the issue of estimating the grand average connectivity pattern from a given population by leveraging the properties of the PARAllel FACtorization (PARAFAC) decomposition. On the other hand, Chapter 2 opens on the possibility to exploit Spectral Graph Theory (SGT) as an innovative tool to provide a cluster-level characterization of functional brain networks. This chapter also introduces the SPectral graph theory And Random walK (SPARK) toolbox, an open-source MATLAB framework ad hoc designed to bring spectral graph theory accessible to a broad audience of interested researchers. Chapter 3 expands on the possibility to exploit modern Graph Signal Processing (GSP) techniques to design a tailored graph filter that mitigates the effects of crosstalk in both Power Spectral Density (PSD) scalp maps and functional connectivity estimations. The aim of Chapter 4 is to leverage the tools introduced in Chapter 1, 2 and 3 to provide an in-depth characterization of functional brain networks extracted from a population of post-stroke subjects. Specifically, the analysis was carried out using both resting state (RS) and motor imagery (MI) EEG data retrieved from a population of 48 subacute post-stroke patients. Finally, Chapter 5 explores the possibility to extend the traditional GSP framework to graph signals defined on directed graphs using a perturbative approach to bypass the computation of the Jordan Normal Form for directed GSP applications.