Detecting nonlinear causal interactions between dynamical systems by non-uniform embedding of multiple time series

This study introduces a new approach for the detection of nonlinear Granger causality between dynamical systems. The approach is based on embedding the multivariate (MV) time series measured from the systems X and Y by means of a sequential, non-uniform procedure, and on using the corrected conditional entropy (CCE) as unpredictability measure. The causal coupling from X to Y is quantified as the relative decrease of CCE measured after allowing the series of X to enter the embedding procedure for the description of Y. The ability of the approach to quantify nonlinear causality is assessed on MV time series measured from simulated dynamical systems with unidirectional coupling (the Rössler-Lorenz deterministic system) and bidirectional coupling (two coupled stochastic systems). The method is then applied to real magnetoencephalographic data measured during a visuo-tactile cognitive experiment, showing values of causal coupling consistent with the hypothesis of a cross-processing of different sensory modalities.