Connectivity between different brain regions generates our minds. Existing work on brain network analysis has mainly been focused on the characterization of connections between the regions in terms of connectivity and causality. Connectivity measures the dependence between regional brain activities, and causality analysis aims to determine the directionality of information flow among the functionally connected brain regions, and find the relationship between causes and effects.Traditionally, the study on connectivity and causality has largely been limited to linear relationships. In this dissertation, as an effort to achieve more accurate characterization of connections between brain regions, we aim to go beyond the linear model, and develop innovative techniques for both non-directional and directional connectivity analysis. Note that due to variability in the brain connectivity of each individual, the connectivity between two brain regions alone may not be sufficient for brain function analysis, in this research, we also conduct network connectivity pattern analysis, so as to reveal more in-depth information.First, we characterize non-directional connectivity using mutual information (MI). In recent years, MI has gradually appeared as an alternative metric for brain connectivity, since it measures both linear and non-linear dependence between two brain regions, while the traditional Pearson correlation only measures the linear dependence. We develop an innovative approach to estimate the MI between two functionally connected brain regions and apply it to brain functional magnetic resonance imaging (fMRI) data. It is shown that: on average, cognitively normal subjects show larger mutual information between critical regions than Alzheimer's disease (AD) patients.Second, we develop new methodologies for brain causality analysis based on directed information (DI). Traditionally, brain causality is based on the well-known Granger Causality (GC) analysis. The validity of GC has been widely recognized. However, it has also been noticed that GC relies heavily on the linear prediction method. When there exists strong nonlinear interactions between two regions, GC analysis may lead to invalid results. In this research, (i) we develop an innovative framework for causality analysis based on directed information (DI), which reflects the information flow from one region to another, and has no modeling constraints on the data. It is shown that DI based causality analysis is effective in capturing both linear and non-linear causal relationships. (ii) We show the conditional equivalence between the DI Framework and Friston's dynamic causal modeling (DCM), and reveal the relationship between directional information transfer and cognitive state change within the brain. Finally, based on brain network connectivity pattern analysis, we develop a robust method for the AD, mild cognitive impairment (MCI) and normal control (NC) subject classification under size limited fMRI data samples. First, we calculate the Pearson correlation coefficients between all possible ROI pairs in the selected sub-network and use them to form a feature vector for each subject. Second, we develop a regularized linear discriminant analysis (LDA) approach to reduce the noise effect. The feature vectors are then projected onto a subspace using the proposed regularized LDA, where the differences between AD, MCI and NC subjects are maximized. Finally, a multi-class AdaBoost Classifier is applied to carry out the classification task. Numerical analysis demonstrates that the combination of regularized LDA and the AdaBoost classifier can increase the classification accuracy significantly.
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