Towards a proper estimation of phase synchronization from time series

In experimental synchronization studies a continuous phase variable is commonly estimated from a scalar time series by means of its representation on the complex plane. The aim is to obtain a pair of functions {A(t), φ(t)} defining its instantaneous amplitude and phase, respectively. However, any arbitrary pair of functions cannot be considered as the amplitude and the phase of the real observable. Here, we point out some criteria that the pair {A(t), φ(t)} must observe to unambiguously define the instantaneous amplitude and phase of the observed signal. In this work, we illustrate how the complex representation may fail if the signal possesses a multi-component or a broadband spectra. We also point out a practical procedure to test whether a signal, not displaying a single oscillation at a unique frequency, has a narrow-band behavior. Implications for the study of phase interdependencies are illustrated and discussed. Phase dynamics estimated from electric brain activities recorded from an epileptic patient are also discussed.