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).

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.

We demonstrate that a time delay in weak coupling between two self-sustained oscillators can be estimated from the observed time series data. We present two methods which are based on the analysis of interrelations between the phases of the signals. We show analytically and numerically that irregularity of the phase dynamics (due to the intrinsic noise or chaos) is essential for determination of the delay. We compare and contrast both methods to the standard cross-correlation analysis.

2003.-Activity of many physiological subsystems has a wellexpressed rhythmic character. Often, a dependency between physiological rhythms is established due to interaction between the corresponding subsystems. Traditional methods of data analysis allow one to quantify the strength of interaction but not the causal interrelation that is indispensable for understanding the mechanisms of interaction. Here we present a recently developed method for quantification of coupling direction and apply it to an important problem. Namely, we study the mutual influence of respiratory and cardiovascular rhythms in healthy newborns within the first 6 mo of life in quiet and active sleep. We find an age-related change of the coupling direction: the interaction is nearly symmetric during the first days and becomes practically unidirectional (from respiration to heart rhythm) at the age of 6 mo. Next, we show that the direction of interaction is mainly determined by respiratory frequency. If the latter is less than Ϸ0.6 Hz, the interaction occurs dominantly from respiration to heart. With higher respiratory frequencies that only occur at very young ages, the dominating direction is less pronounced or even abolished. The observed dependencies are not related to sleep stage, suggesting that the coupling direction is determined by system-inherent dynamical processes, rather than by functional modulations. The directional analysis may be applied to other interacting narrow band oscillatory systems, e.g., in the central nervous system. Thus it is an important step forward in revealing and understanding causal mechanisms of interactions. direction of coupling; development; transmission dynamics INTERACTION BETWEEN PHYSIOLOGICAL (sub-) systems can be addressed by the analysis of the signals they generate. Traditional signal-processing techniques, e.g., cross-spectral (coherence) estimates (20) or computation of mutual information (25), provide symmetric measures of interaction strength. There were also several attempts to address the causality in interaction, based on theoretical information approach (11, 34) and methods of nonlinear dynamics (27, 33). We study directional interrelation using a technique that is particularly suited for the case of weakly interacting rhythmical systems. Thus we exclude from our consideration the input-output systems, i.e., we do not treat the case when, say, signal 1 can be regarded as delayed and/or (nonlinearly) filtered signal 2. We also do not consider the situation when an intervention is possible, i.e., when causal relation can be addressed by application of certain stimuli and analysis of the response to them (for description of such techniques, see, e.g., Ref. 31). In our approach, we restrict ourselves to the case when 1) the system can be only passively observed and 2) it can be modeled by two (or several) weakly coupled self-sustained oscillators, i.e., subsystems, having their own rhythms. Such coupled-oscillator models describe a variety of biological phenomena (9,10,24,38). In the framework...