Detecting nonlinearity and phase synchronization with surrogate data

Methods for detecting nonlinearity in univariate and multivariate time series and methods for identification of phase synchronization in bivariate time series are presented. Both techniques utilize the technique of surrogate data. The detection of nonlinearity is based on rejecting a null hypothesis of a linear stochastic process with the same spectra and cross-spectra as the studied series. The phase synchronization is identified especially by rejecting the hypothesis of linear stochastic processes asynchronously oscillating on the same frequencies (spectra) as the series under study. Instantaneous phases are obtained by means of the discrete Hilbert transform. Information-theoretic functionals (redundancies) are used as the test statistics. The methods are illustrated in analysis of bivariate animal cardiorespiratory data reflecting interactions between respiratory movements and heart-rate fluctuations, which are considered to be an important aspect of functional organization in the autonomous nervous system. By means of the methods presented here, certain nonlinearities and synchronization in cardiorespiratory interactions were found in a newborn piglet during quiet sleep.