"Contemporary neuroimaging techniques provide neural activity recordings with increasing spatial and temporal resolution yielding rich multichannel datasets that can be exploited for detailed description of anatomical and functional connectivity patterns in the brain. Studies indicate that the changes in functional connectivity patterns across spatial and temporal scales play an important role in a wide range of cognitive and executive processes such as memory and attention as well as in the understanding the causes of many neural diseases and psychopathologies such as epilepsy, Alzheimers, Parkinsons and schizophrenia. Early work in the area was limited to the analysis of static brain networks obtained through averaging long-term functional connectivity, thus neglecting possible time-varying connections. There is growing evidence that functional networks dynamically reorganize and coordinate on millisecond scale for the execution of mental processes. Functional networks consist of distinct network states, where each state is defined as a period of time during which the network topology is quasi-stationary. For this reason, there has been an interest in characterizing the dynamics of functional networks using high temporal resolution electroencephalogram recordings. In this thesis, dynamic functional connectivity networks are represented by multiway arrays, tensors, which are able to capture the complete topological structure of the networks. This thesis proposes new methods for both tracking the changes in these dynamic networks and characterizing or summarizing the network states. In order to achieve this goal, a Tucker decomposition based approach is introduced for detecting the change points for task-based electroencephalogram (EEG) functional connectivity networks through calculating the subspace distance between consecutive time steps. This is followed by a tensor-matrix projection based approach for summarizing multiple networks within a time interval. Tensor based summarization approaches do not necessarily result in sparse network and succinct states. Moreover, subspace based summarizations tend to capture the background brain activity more than the low energy sparse activations. For this reason, we propose utilizing the sparse common component and innovations (SCCI) model which simultaneously finds the sparse common component of multiple signals. However, as the number of signals in the model increases, this becomes computationally prohibitive. In this thesis, a hierarchical algorithm to recover the common component in the SCCI model is proposed for large number of signals. The hierarchical recovery of SCCI model solves the time and memory limitations at the expense of a slight decrease in the accuracy. This hierarchical model is used to separate the common and innovation components of functional connectivity networks across time. The innovation components are tracked over time to detect the change points, and the common component of the detected network states are used to obtain the network summarization. SCCI recovery algorithm finds the sparse representation of the common and innovation components of signals with respect to pre-determined dictionaries. However, input signals are not always well-represented by pre-determined dictionaries. In this thesis, a structured dictionary learning algorithm for SCCI model is developed. The proposed method is applied to EEG data collected during a study of error monitoring where two different types of brain responses are elicited in response to the stimulus. The learned dictionaries can discriminate between the response types and extract the error-related potentials (ERP) corresponding to the two responses."--Pages ii-iii.
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