Lead/Lag directionality is not generally equivalent to causality in nonlinear systems: Comparison of phase slope index and conditional mutual information

Applications of causal techniques to neural time series have increased extensively over last decades, including a wide and diverse family of methods focusing on electroencephalogram (EEG) analysis. Besides connectivity inferred in defined frequency bands, there is a growing interest in the analysis of cross-frequency interactions, in particular phase and amplitude coupling and directionality. Some studies show contradicting results of coupling directionality from high frequency to low frequency signal components, in spite of generally considered modulation of a high-frequency amplitude by a low-frequency phase. We have compared two widely used methods to estimate the directionality in cross frequency coupling: conditional mutual information (CMI) and phase slope index (PSI). The latter, applied to infer cross-frequency phase-amplitude directionality from animal intracranial recordings, gives opposite results when comparing to CMI. Both metrics were tested in a numerically simulated example of unidirectionally coupled Rössler systems, which helped to find the explanation of the contradictory results: PSI correctly estimates the lead/lag relationship which, however, is not generally equivalent to causality in the sense of directionality of coupling in nonlinear systems, correctly inferred by using CMI with surrogate data testing.