We systematically develop a technique for reconstructing the phase dynamics equations for coupled oscillators from data. For autonomous oscillators and for two interacting oscillators we demonstrate how phase estimates obtained from general scalar observables can be transformed to genuine phases. This allows us to obtain an invariant description of the phase dynamics in terms of the genuine, observable-independent phases. We discuss the importance of this transformation for characterization of strength and directionality of interaction from bivariate data. Moreover, we demonstrate that natural (autonomous) frequencies of oscillators can be recovered if several observations of coupled systems at different, yet unknown coupling strengths are available. We illustrate our method by several numerical examples and apply it to a human electrocardiogram and to a physical experiment with coupled metronomes.

We consider the problem of experimental detection of directionality of weak coupling between two self-sustained oscillators from bivariate data. We further develop the method introduced by Rosenblum and Pikovsky [Phys. Rev. E 64, 045202 (2001)], suggesting an alternative approach. Next, we consider another framework for identification of directionality, based on the idea of mutual predictability. Our algorithms provide directionality index that shows whether the coupling between the oscillators is unidirectional or bidirectional, and quantifies the asymmetry of bidirectional coupling. We demonstrate the efficiency of three different algorithms in determination of directionality index from short and noisy data. These techniques are then applied to analysis of cardiorespiratory interaction in healthy infants. The results reveal that the direction of coupling between cardiovascular and respiratory systems varies with the age within the first 6 months of life. We find a tendency to change from nearly symmetric bidirectional interaction to nearly unidirectional one (from respiration to the cardiovascular system).

Advances in technology have enabled us to take a fresh look at data acquired by traditional single experiments and to compare them with genomewide data. The differences can be tremendous, as we show here, in the field of proteomics. We have compared data sets of protein-protein interactions in Saccharomyces cerevisiae that were detected by an identical underlying technical method, the yeast two-hybrid system. We found that the individually identified protein-protein interactions are considerably different from those identified by two genomewide scans. Interacting proteins in the pooled database from single publications are much more closely related to each other with respect to transcription profiles when compared to genomewide data. This difference may have been introduced by two factors: by a selection process in individual publications and by false positives in the whole-genome scans. If we assume that the differences are a result of false positives in the whole-genome data, the scans would contain 47%, 44%, and 91% of false positives for the UETZ, ITO-core, and ITO-full data, respectively. If, however, the true fraction of false positives is considerably lower than estimated here, the data from hypothesis-driven experiments must have been subjected to a serious selection process.

We develop a technique for reconstructing the phase dynamics equations for weakly coupled oscillators from data. We show how, starting from general scalar observables, one can first reconstruct the dynamics in terms of the corresponding protophases, and then, performing a transformation to the genuine, observable-independent phases, obtain an invariant description of the phase dynamics. We demonstrate that natural frequencies of oscillators can be recovered if several observations of coupled systems at different coupling strengths are available. We apply our theory to numerical examples and to a physical experiment with coupled metronomes.