We present an approach, framed in information theory, to assess nonlinear causality between the subsystems of a whole stochastic or deterministic dynamical system. The approach follows a sequential procedure for nonuniform embedding of multivariate time series, whereby embedding vectors are built progressively on the basis of a minimization criterion applied to the entropy of the present state of the system conditioned to its past states. A corrected conditional entropy estimator compensating for the biasing effect of single points in the quantized hyperspace is used to guarantee the existence of a minimum entropy rate at which to terminate the procedure. The causal coupling is detected according to the Granger notion of predictability improvement, and is quantified in terms of information transfer. We apply the approach to simulations of deterministic and stochastic systems, showing its superiority over standard uniform embedding. Effects of quantization, data length, and noise contamination are investigated. As practical applications, we consider the assessment of cardiovascular regulatory mechanisms from the analysis of heart period, arterial pressure, and respiration time series, and the investigation of the information flow across brain areas from multichannel scalp electroencephalographic recordings.

In the framework of information dynamics, the temporal evolution of coupled systems can be studied by decomposing the predictive information about an assigned target system into amounts quantifying the information stored inside the system and the information transferred to it. While information storage and transfer are computed through the known self-entropy (SE) and transfer entropy (TE), an alternative decomposition evidences the so-called cross entropy (CE) and conditional SE (cSE), quantifying the cross information and internal information of the target system, respectively. This study presents a thorough evaluation of SE, TE, CE and cSE as quantities related to the causal statistical structure of coupled dynamic processes. First, we investigate the theoretical properties of these measures, providing the conditions for their existence and assessing the meaning of the information theoretic quantity that each of them reflects. Then, we present an approach for the exact computation of information dynamics based on the linear Gaussian approximation, and exploit this approach to characterize the behavior of SE, TE, CE and cSE in benchmark systems with known dynamics. Finally, we exploit these measures to study cardiorespiratory dynamics measured from healthy subjects during head-up tilt and paced breathing protocols. Our main result is that the combined evaluation of the measures of information dynamics allows to infer the causal effects associated with the observed dynamics and to interpret the alteration of these effects with changing experimental conditions.

This tutorial paper introduces a common framework for the evaluation of widely used frequency-domain measures of coupling (coherence, partial coherence) and causality (directed coherence, partial directed coherence) from the parametric representation of linear multivariate (MV) processes. After providing a comprehensive time-domain definition of the various forms of connectivity observed in MV processes, we particularize them to MV autoregressive (MVAR) processes and derive the corresponding frequency-domain measures. Then, we discuss the theoretical interpretation of these MVAR-based connectivity measures, showing that each of them reflects a specific time-domain connectivity definition and how this results in the description of peculiar aspects of the information transfer in MV processes. Furthermore, issues related to the practical utilization of these measures on real-time series are pointed out, including MVAR model estimation and significance assessment. Finally, limitations and pitfalls arising from model mis-specification are discussed, indicating possible solutions and providing practical recommendations for a safe computation of the connectivity measures. An example of estimation of the presented measures from multiple EEG signals recorded during a combined visuomotor task is also reported, showing how evaluation of coupling and causality in the frequency domain may help describing specific neurophysiological mechanisms.

BackgroundThe aims of our study were to investigate the meaning of local atrial activation and its behavior during paroxysmal atrial fibrillation and to study the effect of overdrive pacing on local atrial activity.Methods and ResultsTwenty-five patients with lone paroxysmal atrial fibrillation underwent electrophysiological study. Functional and effective atrial refractoriness was determined. Mean and fifth percentile values of 100 consecutive atrial fibrillation intervals (FF) were evaluated at three atrial sites either at arrhythmia onset or at self-termination (or at minute 5). A high-voltage burst pacing was performed after 6 minutes of stable atrial fibrillation in 10 patients. Mean FF intervals were evaluated 5 seconds before and after atrial pacing. Forty-nine atrial fibrillation episodes were induced: 39 self-terminating within 5 minutes and 10 long-lasting. A significant correlation was found between mean FF and atrial functional refractory period (r=.73,P<.001) and between fifth percentile FF and atrial effective refractory period (r=.57,P<.005). Atrial fibrillation self-termination was associated with significant mean FF prolongation, whereas long-lasting fibrillation behaved the opposite. In 10 patients, burst pacing resulted in significant shortening of the mean FF at the stimulation site; no changes were observed in the two distant recording sites.ConclusionsThe analysis of the FF intervals demonstrates a strict correlation with atrial functional refractoriness. The self-termination of atrial fibrillation is related to a prolongation of the functional refractoriness (mean FF), whereas a shortening of both functional and effective refractoriness (fifth percentile) is associated with atrial fibrillation persistence. The provoked shortening of the mean FF at the stimulation site is consistent with the presence of a gap of excitability during atrial fibrillation in the human atria.