**1. Introduction**

26 Atrial Fibrillation

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Analysis of cardiac time series by nonlinear metrics has recently gained great interest, because the latter observations suggest that the mechanisms involved in cardiovascular regulation likely interact with each other in a nonlinear way [1]. Furthermore, chaotic behavior can be appreciated in the diseased heart with atrial fibrillation (AF) at cellular level and atrial electrophysiological remodeling during this arrhythmia is a far-from-linear process [2]. Hence, the purpose of this chapter is to review the use of nonlinear methods in the analysis of AF, highlighting the clinically useful revealed information that can improve the understanding of this arrhythmia mechanisms and the existing treatments.

Considering that the atrial activity (AA) can be viewed as uncoupled to the ventricular activity (VA) during AF [3], the applications of nonlinear metrics to AA and VA are addressed separately. Regarding the AA study, different measures of irregularity, chaos and complexity of time series have provided a successful assessment of the fibrillatory (*f*) wave regularity from both single-lead invasive and surface recordings. This evaluation of temporal organization of AF has been directly associated with the number of active reentries wandering throughout the atrial tissue [4], which maintain and can perpetuate the arrhythmia [5, 6]. In agreement with this relation, nonlinear metrics have shown powerful prognostic information in the prediction of AF organization-dependent events, including spontaneous termination of paroxysmal AF, successful electrical cardioversion (ECV) of persistent AF patients, atrial remodeling time course during the arrhythmia or infusion effects of different drugs. In addition, these nonlinear analysis methods have also been applied to every signal collected by basked catheters, thus providing an estimation of spatial organization of AF by comparing different atrial sites. On the other hand, the application of nonlinear coupling approaches to intraatrial electrograms (EGMs) recorded simultaneously from different atrial places has reveal differences in the spatio-temporal organization of AF consistent with clinical studies [7]. Thus, differences between paroxysmal

Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Alcaraz and Joaquín Rieta; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Alcaraz and Rieta, licensee InTech. This is an open access chapter distributed under the terms of the

and persistent AF episodes and among patients with different organization degree, classified by following Wells' criteria [8], have been statistically detected. Moreover, patients successfully cardioverted making use of anti-arrhythmic drugs or ECV have been appropriately identified. Finally, with regard to the VA analysis, ventricular response has been widely characterized by quantifying nonlinear dynamics in interval series between successive R peaks, i.e., RR-interval series [9]. In this respect, multiple measures of fractal fluctuations, irregularity and geometric structure of time series have shown ability to evaluate the cardiovascular autonomic regulation before, during and after AF onset and characterize the main electrophysiological characteristics of the atrioventricular (AV) node.

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mapping [15]. Although recordings from each one of these modalities have their own characteristics, unipolar recordings are generally characterized by a substantial far-field contamination, such as VA, whereas bipolar recordings contains local atrial activations of the place in which the electrodes are located. Nonetheless, these recordings are also affected by ventricular interference, especially in recording sites closer to the ventricles, even if its effect is less evident than on unipolar EGMs and surface ECG recordings. Thereby, for the VA cancellation both from unipolar and bipolar recordings, an averaged ventricular interference complex, as in ABS, is usually computed and subtracted from each atrial signal [16, 17]. Only remark that the ventricular activations are habitually detected from a surface ECG recording

The Contribution of Nonlinear Methods in the Understanding of Atrial Fibrillation

On the other hand, given that atrial dynamics can be analyzed both from simple EGMs and local atrial period (LAP) series, i.e., the sequence of temporal distances between two consecutive local atrial activations, the appropriate identification of these points is a important task in this context. For this purpose, EGMs are habitually high-pass filtering (40–250 Hz) to remove baseline shifts and high-frequency noise [18]. The filtered signal is then rectified, introducing low-frequency components related to the amplitude of the high-frequency oscillations of the original signal. The modulus of the filtered signal is further low-pass filtered (cut-off at 20 Hz) to extract a waveform proportional to the amplitude of the components of occurring at 40–250 Hz. The atrial activations are then detected by threshold crossing and their occurrence time can be identified by different methods, including the local

maximum peak, maximum slope of the atrial depolarization or their barycenter [19].

The dynamics of a time series can be explored through its correlation properties, or in other words, the time ordering of the series. Fractal analysis is an appropriate method to characterize complex time series by focusing on the time-evolutionary properties on the data series and on their correlation properties. In this context, the *detrended fluctuation analysis* (DFA) method was developed specifically to distinguish between intrinsic fluctuations generated by complex systems and those caused by external or environmental stimuli acting on the system [20]. The DFA method can quantify the temporal organization of the fluctuations in a given non-stationary time series by a single scaling exponent *α*, a self-similarity parameter that represents the long-range power-law correlation properties of the signal. The scaling exponent *α* is obtained by computing the root-mean-square fluctuation *F*(*n*) of integrated and detrended time series at different observation windows of size *n* and plotting *F*(*n*) against *n* on a log-log scale. Fractal signals are characterized by a power law relation between the average magnitudes of the fluctuations *F*(*n*) and the number of points *n*, *F*(*n*) ∼ *nα*. The slope of the regression line relating log(*F*(*n*)) to log(*n*) determines the

The principle of chaos analysis is to transform the properties of a time series into the topological properties of a geometrical object (attractor) constructed out of a time series, which is embedded in a *state/phase space*. The concept of phase space reconstruction is central

simultaneously acquired for more accuracy.

**3. Nonlinear time series analysis**

**3.1. Fractal fluctuations quantification**

scaling exponent *α*.

**3.2. Chaos degree quantification**
