Luca Faes

Surrogate data analysis for assessing the significance of the coherence function

In cardiovascular variability analysis, the significance of the coupling between two time series is commonly assessed by setting a threshold level in the coherence function. While traditionally used statistical tests consider only the parameters of the adopted estimator, the required zero-coherence level may be affected by some features of the observed series. In this study, three procedures, based on the generation of surrogate series sharing given properties with the original but being structurally uncoupled, were considered: independent identically distributed (IID), Fourier transform (FT), and autoregressive (AR). IID surrogates maintained the distribution of the original series, while FT and AR surrogates preserved the power spectrum. The ability of the three methods to define the threshold for zero coherence was validated and compared by computer simulations reproducing typical cardiovascular interactions. While the IID threshold depended only on record length and design parameters of the coherence estimator, FT and AR thresholds were frequency-dependent with peaks corresponding to the local maxima of the estimated coherence. FT and AR surrogates were able to compensate spurious coherence peaks due to equal-frequency but independent oscillations in the two series. The benefit of frequency-dependent thresholds was evident for short series with narrow-band oscillations. Thus, surrogates preserving the power spectrum of the original series are recommended to avoid false coupling detections in the presence of oscillations occurring at nearby frequencies but produced by different mechanisms, as may frequently happen in cardiovascular and cardiorespiratory regulation.

A method for quantifying atrial fibrillation organization based on wave-morphology similarity

A new method for quantifying the organization of single bipolar electrograms recorded in the human atria during atrial fibrillation (AF) is presented. The algorithm relies on the comparison between pairs of local activation waves (LAWs) to estimate their morphological similarity, and returns a regularity index (/spl rho/) which measures the extent of repetitiveness over time of the detected activations. The database consisted of endocardial data from a multipolar basket catheter during AF and intraatrial recordings during atrial flutter. The index showed maximum regularity (/spl rho/=1) for all atrial flutter episodes and decreased significantly when increasing AF complexity as defined by Wells (type I: /spl rho/=0.75/spl plusmn/0.23; type II: /spl rho/=0.35/spl plusmn/0.11; type III: /spl rho/=0.15/spl plusmn/0.08; P<0.01). The ability to distinguish different AF episodes was assessed by designing a classification scheme based on a minimum distance analysis, obtaining an accuracy of 85.5%. The algorithm was able to discriminate among AF types even in presence of few depolarizations as no significant /spl rho/ changes were observed by reducing the signal length down to include five LAWs. Finally, the capability to detect transient instances of AF complexity and to map the local regularity over the atrial surface was addressed by the dynamic and multisite evaluation of /spl rho/, suggesting that our algorithm could improve the understanding of AF mechanisms and become useful for its clinical treatment.

An integrated approach based on uniform quantization for the evaluation of complexity of short-term heart period variability : Application to 24 h holter recordings in healthy and heart failure humans

We propose an integrated approach based on uniform quantization over a small number of levels for the evaluation and characterization of complexity of a process. This approach integrates information-domain analysis based on entropy rate, local nonlinear prediction, and pattern classification based on symbolic analysis. Normalized and non-normalized indexes quantifying complexity over short data sequences ( approximately 300 samples) are derived. This approach provides a rule for deciding the optimal length of the patterns that may be worth considering and some suggestions about possible strategies to group patterns into a smaller number of families. The approach is applied to 24 h Holter recordings of heart period variability derived from 12 normal (NO) subjects and 13 heart failure (HF) patients. We found that: (i) in NO subjects the normalized indexes suggest a larger complexity during the nighttime than during the daytime; (ii) this difference may be lost if non-normalized indexes are utilized; (iii) the circadian pattern in the normalized indexes is lost in HF patients; (iv) in HF patients the loss of the day-night variation in the normalized indexes is related to a tendency of complexity to increase during the daytime and to decrease during the nighttime; (v) the most likely length L of the most informative patterns ranges from 2 to 4; (vi) in NO subjects classification of patterns with L=3 indicates that stable patterns (i.e., those with no variations) are more present during the daytime, while highly variable patterns (i.e., those with two unlike variations) are more frequent during the nighttime; (vii) during the daytime in HF patients, the percentage of highly variable patterns increases with respect to NO subjects, while during the nighttime, the percentage of patterns with one or two like variations decreases.

Wave Similarity Mapping Shows the Spatiotemporal Distribution of Fibrillatory Wave Complexity in the Human Right Atrium During Paroxysmal and Chronic Atrial Fibrillation

Introduction: The complexity of waveforms during atrial fibrillation may reflect critical activation patterns for the arrhythmia perpetuation. In this study, we introduce a novel concept of map, based on the analysis of the wave morphology, which gives a direct evidence in the human right atrium on the spatiotemporal distribution of fibrillatory wave complexity in paroxysmal (PAF) and chronic (CAF) atrial fibrillation.

Measuring Connectivity in Linear Multivariate Processes: Definitions, Interpretation, and Practical Analysis

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.

Extended causal modeling to assess Partial Directed Coherence in multiple time series with significant instantaneous interactions

The Partial Directed Coherence (PDC) and its generalized formulation (gPDC) are popular tools for investigating, in the frequency domain, the concept of Granger causality among multivariate (MV) time series. PDC and gPDC are formalized in terms of the coefficients of an MV autoregressive (MVAR) model which describes only the lagged effects among the time series and forsakes instantaneous effects. However, instantaneous effects are known to affect linear parametric modeling, and are likely to occur in experimental time series. In this study, we investigate the impact on the assessment of frequency domain causality of excluding instantaneous effects from the model underlying PDC evaluation. Moreover, we propose the utilization of an extended MVAR model including both instantaneous and lagged effects. This model is used to assess PDC either in accordance with the definition of Granger causality when considering only lagged effects (iPDC), or with an extended form of causality, when we consider both instantaneous and lagged effects (ePDC). The approach is first evaluated on three theoretical examples of MVAR processes, which show that the presence of instantaneous correlations may produce misleading profiles of PDC and gPDC, while ePDC and iPDC derived from the extended model provide here a correct interpretation of extended and lagged causality. It is then applied to representative examples of cardiorespiratory and EEG MV time series. They suggest that ePDC and iPDC are better interpretable than PDC and gPDC in terms of the known cardiovascular and neural physiologies.

Assessment of Granger Causality by Nonlinear Model Identification: Application to Short-term Cardiovascular Variability

A method for assessing Granger causal relationships in bivariate time series, based on nonlinear autoregressive (NAR) and nonlinear autoregressive exogenous (NARX) models is presented. The method evaluates bilateral interactions between two time series by quantifying the predictability improvement (PI) of the output time series when the dynamics associated with the input time series are included, i.e., moving from NAR to NARX prediction. The NARX model identification was performed by the optimal parameter search (OPS) algorithm, and its results were compared to the least-squares method to determine the most appropriate method to be used for experimental data. The statistical significance of the PI was assessed using a surrogate data technique. The proposed method was tested with simulation examples involving short realizations of linear stochastic processes and nonlinear deterministic signals in which either unidirectional or bidirectional coupling and varying strengths of interactions were imposed. It was found that the OPS-based NARX model was accurate and sensitive in detecting imposed Granger causality conditions. In addition, the OPS-based NARX model was more accurate than the least squares method. Application to the systolic blood pressure and heart rate variability signals demonstrated the feasibility of the method. In particular, we found a bilateral causal relationship between the two signals as evidenced by the significant reduction in the PI values with the NARX model prediction compared to the NAR model prediction, which was also confirmed by the surrogate data analysis. Furthermore, we found significant reduction in the complexity of the dynamics of the two causal pathways of the two signals as the body position was changed from the supine to upright. The proposed is a general method, thus, it can be applied to a wide variety of physiological signals to better understand causality and coupling that may be different between normal and diseased conditions.

Causal transfer function analysis to describe closed loop interactions between cardiovascular and cardiorespiratory variability signals

Abstract.Although the concept of transfer function is intrinsically related to an input–output relationship, the traditional and widely used estimation method merges both feedback and feedforward interactions between the two analyzed signals. This limitation may endanger the reliability of transfer function analysis in biological systems characterized by closed loop interactions. In this study, a method for estimating the transfer function between closed loop interacting signals was proposed and validated in the field of cardiovascular and cardiorespiratory variability. The two analyzed signals x and y were described by a bivariate autoregressive model, and the causal transfer function from x to y was estimated after imposing causality by setting to zero the model coefficients representative of the reverse effects from y to x. The method was tested in simulations reproducing linear open and closed loop interactions, showing a better adherence of the causal transfer function to the theoretical curves with respect to the traditional approach in presence of non-negligible reverse effects. It was then applied in ten healthy young subjects to characterize the transfer functions from respiration to heart period (RR interval) and to systolic arterial pressure (SAP), and from SAP to RR interval. In the first two cases, the causal and non-causal transfer function estimates were comparable, indicating that respiration, acting as exogenous signal, sets an open loop relationship upon SAP and RR interval. On the contrary, causal and traditional transfer functions from SAP to RR were significantly different, suggesting the presence of a considerable influence on the opposite causal direction. Thus, the proposed causal approach seems to be appropriate for the estimation of parameters, like the gain and the phase lag from SAP to RR interval, which have a large clinical and physiological relevance.

Information domain approach to the investigation of cardio-vascular, cardio-pulmonary, and vasculo-pulmonary causal couplings.

The physiological mechanisms related to cardio-vascular (CV), cardio-pulmonary (CP), and vasculo-pulmonary (VP) regulation may be probed through multivariate time series analysis tools. This study applied an information domain approach for the evaluation of non-linear causality to the beat-to-beat variability series of heart period (t), systolic arterial pressure (s), and respiration (r) measured during tilt testing and paced breathing (PB) protocols. The approach quantifies the causal coupling from the series i to the series j (Cij) as the amount of information flowing from i to j. A measure of directionality is also obtained as the difference between two reciprocal causal couplings (Di,j = Cij − Cji). Significant causal coupling and directionality were detected respectively when the median of Cij over subjects was positive (Cij > 0), and when Di,j was statistically different from zero (Di,j > 0 or Di,j < 0). The method was applied on t, s, and r series measured in 15 healthy subjects (22–32 years, 8 males) in the supine (su) and upright (up) positions, and in further 15 subjects (21–29 years, 7 males) during spontaneous (sp) and paced (pa) breathing. In the control condition (su, sp), a significant causal coupling was observed for Crs, Crt, Cst, and Cts, and significant directionality was present only from r to t (Dr,t > 0). During head-up tilt (up, sp), Crs was preserved, Crt decreased to zero median, and Cst and Cts increased significantly; directionality vanished between r and t (Dr,t = 0) and raised from s to t (Ds,t > 0). During PB (su, pa), Crs increased significantly, Crt and Cts were preserved, and Cst decreased to zero median; directionality was preserved from r to t (Dr,t > 0), and raised from r to s (Dr,s > 0). These results suggest that the approach may reflect modifications of CV, CP, and VP mechanisms consequent to altered physiological conditions, such as the baroreflex engagement and the dampening of respiratory sinus arrhythmia induced by tilt, or the respiratory driving on arterial pressure induced by PB. Thus, it could be suggested as a tool for the non-invasive monitoring of CV and cardiorespiratory control systems in normal and impaired conditions.

Quantification of synchronization during atrial fibrillation by Shannon entropy: Validation in patients and computer model of atrial arrhythmias

Atrial fibrillation (AF), a cardiac arrhythmia classically described as completely desynchronized, is now known to show a certain amount of synchronized electrical activity. In the present work a new method for quantifying the level of synchronization of the electrical activity recorded in pairs of atrial sites during atrial fibrillation is presented. A synchronization index (Sy) was defined by quantifying the degree of complexity of the distribution of the time delays between sites by Shannon entropy estimation. The capability of Sy to discriminate different AF types in patients was assessed on a database of 60 pairs of endocardial recordings from a multipolar basket catheter. The analysis showed a progressive and significant decrease of Sy with increasing AF complexity classes as defined by Wells (AF type I Sy = 0.73 +/- 0.07, type II Sy = 0.56 +/- 0.07, type III Sy = 0.36 +/- 0.04, p < 0.001). The extension of Sy calculation to the whole right atrium showed the existence of spatial heterogeneities in the synchronization level. Moreover, experiments simulated by a computer model of atrial arrhythmias showed that propagation patterns with different complexity could be the basis of different synchronization levels found in patients. In conclusion the quantification of synchronization by Shannon entropy estimation of time delay dispersion may facilitate the identification of different propagation patterns associated with AF, thus enhancing our understanding of AF mechanisms and helping in its treatment.