Complex networks, ranging from gene regulatory networks in biology to social networks in sociology, havereceived growing attention from the scientific community. The analysis of complex networks employs techniquesfrom graph theory, machine learning and signal processing. In recent years, complex network analysis tools havebeen applied to neuroscience and neuroimaging studies to have a better understanding of the human brain. In thisthesis, we focus on inferring and analyzing the complex functional brain networks underlying multichannelelectroencephalogram (EEG) recordings. Understanding this complex network requires the development of a measureto quantify the relationship between multivariate time series, algorithms to reconstruct the network based on thepairwise relationships, and identification of functional modules within the network.Functional and effective connectivity are two widely studiedapproaches to quantify the connectivity between two recordings.Unlike functional connectivity which only quantifies the statisticaldependencies between two processes by measures such as crosscorrelation, phase synchrony, and mutual information (MI), effectiveconnectivity quantifies the influence one node exerts on anothernode. Directed information (DI) measure is one of the approachesthat has been recently proposed to capture the causal relationshipsbetween two time series. Two major challenges remain with theapplication of DI to multivariate data, which include thecomputational complexity of computing DI with increasing signallength and the accuracy of estimation from limited realizations ofthe data. Expressions that can simplify the computation of theoriginal definition of DI while still quantifying the causalityrelationship are needed. In addition, the advantage of DI overconventionally causality measures such as Granger causality has notbeen fully investigated. In this thesis, we propose time-laggeddirected information and modified directed information to addressthe issue of computational complexity, and compare the performanceof this model free measure with model based measures (e.g. Grangercausality) for different realistic signal models.Once the pairwise DI between two random processes is computed,another problem is to infer the underlying structure of the complexnetwork with minimal false positive detection. We propose to useconditional directed information (CDI) proposed by Kramer to addressthis issue, and introduce the time-lagged conditional directedinformation and modified conditional directed information to lowerthe computational complexity of CDI. Three network inferencealgorithms are presented to infer directed acyclic networks whichcan quantify the causality and also detect the indirect couplingssimultaneously from multivariate data.One last challenge in the study of complex networks, specifically in neuroscience applications, is to identifythe functional modules from multichannel, multiple subject recordings. Most research on community detection inthis area so far has focused on finding the association matrix based on functional connectivity, instead ofeffective connectivity, thus not capturing the causality in the network. In addition, in order to find a modularstructure that best describes all of the subjects in a group, a group analysis strategy is needed. In thisthesis, we propose a multi-subject hierarchical community detection algorithm suitable for a group of weightedand asymmetric (directed) networks representing effective connectivity, and apply the algorithm to multichannelelectroencephalogram (EEG) data.
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