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Synchronization is a fundamental component of a musical ensemble. Recent advancements in behavioral time series analyses have produced increasing evidence of global coordination, matching in statistical structures of fluctuation in long timescales, separate from synchronization as local coordination. Previous studies have demonstrated that strong interactions in local timescales and interactions in multiple scales with weak local interactions can be sources of global coordination. However, knowledge of global coordination in human rhythmic movement is still limited. The present study investigated fluctuation and coordination dynamics for inter-tap intervals (ITI) in a paired rhythmic tapping task, imitating duo performances, from local to global timescales. In addition, we examined parameter dynamics for a coupled oscillator model to test whether the model could replicate the task dynamics and integrate sources of global coordination. Measured ITI series demonstrated globally coordinated hybrid dynamics: persistent fluctuation and positively correlated coordination in global scales and anti-persistent fluctuation and negatively correlated coordination in local scales. Positive lag ± 1 and negative lag 0 local cross-correlation suggested that the global coordination was emerged from local mutual timing adjustment processes. The coupled oscillator model for the task replicated these dynamics. Moreover, the model could generate both types of global coordination, which emerged from strong and weak local interactions. These findings can help to improve knowledge of interpersonal coordination in musical ensembles, allowing a better understanding, for example, of the basis of temporal harmony and individual differences in the coordination of each pair and their source.

Optimization and control of collective dynamics in rhythmic systems have attracted increasing interest, and various methods have been developed on the basis of the phase reduction framework for limit cycle oscillators. The phase reduction relies on the notion of the isochron, which represents the set of system states that share the same asymptotic phase. However, the phase reduction does not take into account the amplitude degrees of freedom representing deviations of the system state from the limit cycle attractor, to which some rich and nontrivial transient behaviors of the oscillators are attributed. In this chapter, after a brief introduction of the phase reduction framework, a phase-amplitude reduction framework that is applicable to transient dynamics far from the limit cycle attractor is formulated. A rigorous theoretical background for defining the amplitudes is provided by the notion of the isostable, which naturally complements the isochron in the sense that both of them can be understood from a unified viewpoint of the spectral properties of the Koopman operator. The utility of the proposed phase-amplitude reduction framework is illustrated by evaluating the optimal injection timing of a weak control input that efficiently suppresses deviations of the system state from the limit cycle attractor.

Phase reduction framework for limit-cycling systems based on isochrons has been used as a powerful tool for analyzing rhythmic phenomena. Recently, the notion of isostables, which complements the isochrons by characterizing amplitudes of the system state, i.e., deviations from the limit-cycle attractor, has been introduced to describe transient dynamics around the limit cycle [Wilson and Moehlis, Phys. Rev. E 94, 052213 (2016)]. In this study, we introduce a framework for a reduced phase-amplitude description of transient dynamics of stable limit-cycling systems. In contrast to the preceding study, the isostables are treated in a fully consistent way with the Koopman operator analysis, which enables us to avoid discontinuities of the isostables and to apply the framework to system states far from the limit cycle. We also propose a new, convenient bi-orthogonalization method to obtain the response functions of the amplitudes, which can be interpreted as an extension of the adjoint covariant Lyapunov vector to transient dynamics in limit-cycling systems. We illustrate the utility of the proposed reduction framework by estimating optimal injection timing of external input that efficiently suppresses deviations of the system state from the limit cycle in a model of a biochemical oscillator., 19 pages, 2 figures

We propose a method for controlling synchronization patterns of limit-cycle oscillators by common noisy inputs, i.e., by utilizing noise-induced synchronization. Various synchronization patterns, including fully synchronized and clustered states, can be realized by using linear filters that generate appropriate common noisy signals from given noise. The optimal linear filter can be determined from the linear phase response property of the oscillators and the power spectrum of the given noise. The validity of the proposed method is confirmed by numerical simulations., 6 pages, 4 figures

The phase reduction method is a dimension reduction method for weakly driven limit-cycle oscillators, which has played an important role in the theoretical analysis of synchro- nization phenomena. Recently, we proposed a generalization of the phase reduction method [W. Kurebayashi et al., Phys. Rev. Lett. 111, 2013]. This generalized phase reduction method can robustly predict the dynamics of strongly driven oscillators, for which the conventional phase reduction method fails. In this generalized method, the external input to the oscillator should be properly decomposed into a slowly varying component and remaining weak fluctua- tions. In this paper, we propose a simple criterion for timescale decomposition of the external input, which gives accurate prediction of the phase dynamics and enables us to systematically apply the generalized phase reduction method to a general class of limit-cycle oscillators. The validity of the criterion is confirmed by numerical simulations., Comment: 10 pages, 2 figures

The phase reduction method for limit cycle oscillators subjected to weak perturbations has significantly contributed to theoretical investigations of rhythmic phenomena. We here propose a generalized phase reduction method that is also applicable to strongly perturbed limit cycle oscillators. The fundamental assumption of our method is that the perturbations can be decomposed into a slowly varying component as compared to the amplitude relaxation time and remaining weak fluctuations. Under this assumption, we introduce a generalized phase parameterized by the slowly varying component and derive a closed equation for the generalized phase describing the oscillator dynamics. The proposed method enables us to explore a broader class of rhythmic phenomena, in which the shape and frequency of the oscillation may vary largely because of the perturbations. We illustrate our method by analyzing the synchronization dynamics of limit cycle oscillators driven by strong periodic signals. It is shown that the proposed method accurately predicts the synchronization properties of the oscillators, while the conventional method does not.

Hybrid dynamical systems characterized by discrete switching of smooth dynamics have been used to model various rhythmic phenomena. However, the phase reduction theory, a fundamental framework for analyzing the synchronization of limit-cycle oscillations in rhythmic systems, has mostly been restricted to smooth dynamical systems. Here we develop a general phase reduction theory for weakly perturbed limit cycles in hybrid dynamical systems that facilitates analysis, control, and optimization of nonlinear oscillators whose smooth models are unavailable or intractable. On the basis of the generalized theory, we analyze injection locking of hybrid limit-cycle oscillators by periodic forcing and reveal their characteristic synchronization properties, such as ultrafast and robust entrainment to the periodic forcing and logarithmic scaling at the synchronization transition. We also illustrate the theory by analyzing the synchronization dynamics of a simple physical model of biped locomotion., Comment: 37 pages, 5 figures