A multivariate time-frequency based phase synchrony measure and applications to dynamic brain network analysis

Irregular, non-stationary, and noisy multichannel data are abound in many fields of research. Observations of multichannel data in nature include changes in weather, the dynamics of satellites in the solar system, the time evolution of the magnetic field of celestial bodies, population growth in ecology and the dynamics of the action potentials in neurons.;One particular application of interest is the functional integration of neuronal networks in the human brain. Human brain is known to be one of the most complex biological systems and quantifying functional neural coordination in the brain is a fundamental problem. It has been recently proposed that networks of highly nonlinear and non-stationary reciprocal interactions are the key features of functional integration. Among many linear and nonlinear measures of dependency, time-varying phase synchrony has been proposed as a promising measure of connectivity. Current state-of-the-art in time-varying phase estimation uses either the Hilbert transform or the complex wavelet transform of the signals. Both of these methods have some major drawbacks such as the assumption that the signals are narrowband for the Hilbert transform and the non-uniform time-frequency resolution inherent to the wavelet analysis. Furthermore, the current phase synchrony measures are limited to quantifying bivariate relationships and do not reveal any information about multivariate synchronization patterns which are important for understanding the underlying oscillatory networks.;In this dissertation, a new phase estimation method based on the Rihaczek distribution and Reduced Interference Rihaczek distribution belonging to Cohen's class is proposed. These distributions offer phase estimates with uniformly high time-frequency resolution which can be used for defining time and frequency dependent phase synchrony within the same frequency band as well as across different frequency bands. Properties of the phase estimator and the corresponding phase synchrony measure are evaluated both analytically and through simulations showing the effectiveness of the new measures compared to existing ones. The proposed distribution is then extended to quantify the cross-frequency phase synchronization between two signals across different frequencies. In addition, a cross frequency-spectral lag distribution is introduced to quantify the amount of amplitude modulation between signals. Furthermore, the notion of bivariate synchrony is extended to multivariate synchronization to quantify the relationships within and across groups of signals. Measures of multiple correlation and complexity are used as well as a more direct multivariate synchronization measure, `Hyperspherical Phase Synchrony', is proposed. This new measure is based on computing pairwise phase differences to create a multidimensional phase difference vector and mapping this vector to a high dimensional space. Hyperspherical phase synchrony offers lower computational complexity and is more robust to noise compared to the existing measures. Finally, a subspace analysis framework is proposed for studying time-varying evolution of functional brain connectivity. The proposed approach identifies event intervals accounting for the underlying neurophysiological events and extracts key graphs for describing the particular intervals with minimal redundancy. Results from the application to EEG data indicate the effectiveness of the proposed framework in determining the event intervals and summarizing brain activity with a few number of representative graphs.