New tricks to an old dog: advances on the assessment of synchronization from time series in reconstructed state spaces

The study of synchronization between dynamical systems is a well stablished field of research in theoretical and applied physics. Should the state equations be available, the analysis of the spectrum of Lyapunov exponents or the auxiliary system approach are useful methods to study this phenomenon. Yet in many experimental systems, such equations (and even the exact number of state variables) are unknown. In these cases, it is possible, however, to reconstruct the state space of the systems by making use of the time series of one or many of the signals recorded from the systems. In this talk, we review the classical results on the field of the assessment of synchronization (interdependence) between dynamical systems the state spaces reconstructed from their time series. We present the pros and cons of this methodology as well as its limitations, and how advanced time series methods such as the use of bivariate surrogate data may help to obtain relevant information in this framework. We then introduce the concept of multivariate embedding and show that it allows for a more accurate assessment of the true underlying interdependence among multivariate time series. We finally present a recent application of this new concept to the experimental data in systems neuroscience.