## AbstractSynchronization, i.e. the self-organized adjustment of rhythmic activity in oscillating systems, is a universal phenomenon in nature. It can be observed in systems as diverse as coupled lasers, pacemaker cells in the heart, electrochemical oscillators, spiking neurons in the brain or in synchronous flashing in a population of fireflies. While these examples might seem unrelated on the surface, the underlying mechanisms of synchronization are very similar. In all of these examples, many individual oscillators communicate and reach an agreement upon a common rhythm. Analyzing synchronous activity can therefore reveal important insights regarding the interaction between these oscillators. However, the standard methods for synchronization analysis are severely distorted if the true dynamics of the individual oscillators is not available, but only mixtures of their signals. A prime example is the cortical Electroencephalogram (EEG): each electrode on the scalp measures a mixture of many different cortical sources. This thesis contributes to the understanding --both theoretically and from a data analysis perspective-- of interactions in such situations. First, the distorting effect of the mixing is demonstrated: the usual synchronization indices yield arbitrary results, indicating spurious synchronization even for completely independent signals. Inspired by a simple geometric insight, a new synchronization measure, termed Interaction Evidence, is proposed which is unaffected by linear mixtures and the presence of independent signals. This interaction evidence can be represented as a skew-symmetric matrix (or a set of such matrices). Based on this interaction evidence and in formal analogy to existing variance-based methods, three different linear projection algorithms are derived: -- Principal Interacting Component Analysis (PICA) allows one to find directions in the data space that contain the most evident synchronization. -- Given two data sets that represent different classes, Common Spatial Interaction Patterns (CSIP) allows one to find the subspace in which the synchronization is most discriminative between those classes. -- Pairwise Interacting Source Analysis (PISA) separates a multivariate time series into pairs of synchronized sources. The proposed algorithms are theoretically and empirically analyzed and applied to EEG experiments where it can be shown that CSIP leads to features that can distinguish between different mental states and that the PISA decomposition leads to physiologically plausible and interpretable interaction patterns and -spectra.
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