Towards a Quantitative Theory of Digraph-Based Complexes and its Applications in Brain Network Analysis

• Ano de defesa: 2024
• Autor(a) principal: Baldo, Heitor
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Brain Connectivity; Coerência Parcial Direcionada; Complexos de Caminhos; Complexos de Cliques Direcionados; Conectividade Cerebral; Conectividade de Ordem Superior Direcionada; Directed Clique Complexes; Directed Higher-Order Connectivity; Electroencephalography; Eletroencefalografia; Epilepsia; Epilepsy; Graph Theory; Partial Directed Coherence; Path Complexes; Teoria dos Grafos
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/95/95131/tde-04072024-124243/

The development of mathematical methods for studying the structure, organization, and functioning of the brain has become increasingly important, especially in the context of brain connectivity networks studies, highlighting, in recent decades, methods associated with graph theory, network science, and computational (algebraic) topology. In particular, these methods have been used to study neurological disorders associated with abnormal structural and functional properties of brain connectivity, such as epilepsy, Alzheimer\'s disease, Parkinson\'s disease, and multiple sclerosis. In this work, we developed new mathematical methods for analyzing network topology and we applied these methods to the analysis of brain networks. More specifically, we rigorously developed quantitative methods based on complexes constructed from digraphs (digraph-based complexes), such as path complexes and directed clique complexes (alternatively, we refer to these complexes as \"higher-order structures,\" or \"higher-order topologies,\" or \"simplicial structures\"), and, in the case of directed clique complexes, also methods based on the interrelations between the directed cliques, what we called \"directed higher-order connectivities.\" This new quantitative theory for digraph-based complexes can be seen as a step towards the formalization of a \"quantitative simplicial theory.\" Subsequently, we used these new methods, such as characterization measures and similarity measures for digraph-based complexes, to analyze the topology of digraphs derived from brain connectivity estimators, specifically the estimator known as information partial directed coherence (iPDC), which is a multivariate estimator that can be considered a representation of Granger causality in the frequency-domain, particularly estimated from electroencephalography (EEG) data from patients diagnosed with left temporal lobe epilepsy, in the delta, theta and alpha frequency bands, to try to find new biomarkers based on the higher-order structures and connectivities of these digraphs. In particular, we attempted to answer the following questions: How does the higher-order topology of the brain network change from the pre-ictal to the ictal phase, from the ictal to the post-ictal phase, at each frequency band and in each cerebral hemisphere? Does the analysis of higher-order structures provide new and better biomarkers for seizure dynamics and also for the laterality of the seizure focus than the usual graph theoretical analyses? We found that all simplicial characterization measures considered in the study showed statistically significant increases in their magnitudes from the pre-ictal phase to the ictal phase, for several higher orders, for both cerebral hemispheres, particularly in the delta and theta bands but no statistically significant changes were observed from the ictal to the post-ictal phase, which may suggest that several topological and functional aspects of brain networks change from the pre-ictal to the ictal phase, at various levels of higher-order topological organization. Regarding the laterality of the seizure focus, the analysis based on simplicial similarities found no statistically significant difference between the clique topology of the left and right hemispheres in the ictal phase. We conclude from this study that, despite a number of limitations, there may be evidence supporting the viability and reliability of using higher-order structures associated with digraphs to identify biomarkers associated with epileptic networks. However, further research is needed, and the applicability of the newly introduced methods to other disorders of brain connectivity networks will also depend on future studies.