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Current Vascular Pharmacology, 2004, 2, 149-162

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Applications of Fractal and Non-linear Time Series Analysis to the Study of Short-term Cardiovascular Control Julián J. González a,* and Ernesto Peredab a

Laboratory of Biophysics, Department of Physiology, College of Medicine, University of La Laguna, Tenerife, Spain, bDepartment of Basic Physics, College of Physics, University of La Laguna, Tenerife, Spain Abstract: The short-term cardiovascular control system is reviewed from the analysis of the heart rate, respiration and blood pressure beat-to-beat variability signals. The present state of the art concerning fractal and non-linear techniques as applied to the cardiovascular system and the differences between both approaches are highlighted. We present results obtained in mammals from statistics, such as the fractal exponent, the correlation dimension or the maximal Lyapunov exponent and discuss the convenience of these indexes for characterizing the irregularity present in the signals. Finally, the interdependence between the systems involved in the cardiovascular control is addressed. Recent results obtained from interdependence indexes between the cardio, respiratory and vascular signals are discussed and their convenience in physiological studies and clinical applications are stressed.

1. FROM CARDIOVASCULAR VARIABLES TO INTERBEAT VARIABILITY SIGNALS Short-term cardiovascular control is usually studied from the beat-to-beat variability signals of different cardiovascular variables. The transition from these variables to the variability signals normally takes place by calculating a series of consecutive RR time intervals (or its inverse) from a digitized ECG. From this series, the cardiac interval variability signal (RRV) or the heart-rate (HR) variability signal (HRV) are built (see Fig. 1). We can analyze these variability signals to assess how the autonomic nervous system (ANS) mediates the heart-rate or blood pressure (BP). Also using BP or respiratory digitized recordings in addition to the ECG, we can construct a beat to beat variability signal of BP (BPV) (systolic SP, diastolic DP or mean BP, (Fig. 4, top-left) and respiration (RSPV). In this way, we can analyze other aspects of cardiovascular control, such as the respiratory modulation of HR or the BP control of the HR (baroreflex) in different physiological situations.

time between each sample is the value of the successive RR intervals, which are not constant. A typical way of circumventing such problem consists of taking the average RR value as sampling interval, which is a good approximation as long as this interval does not change greatly along the record. Another possibility involves making use of filtering or interpolation procedures to get a signal sampled at a constant sampling interval. As an example, the spectral power of a RRV signal obtained in such a way is shown in (Fig. 2) in both linear and logarithmic scales. However useful and promising these methods may be, they suffer from a fundamental drawback: they are unable to handle in a proper way the irregularity present in most of the signals. Instead, they simply disregard it or consider it as coming from some external source random in nature. Recent results strongly suggests that such irregularity actually reveals a more complex behavior of the system -even in the basal state- than that proposed by the traditional point of view of homeostatic equilibrium (see for instance [3, 4]). The study of this behavior calls for different methodologies that consider this irregularity from a different standpoint.

2. WHAT LINEAR ANALYSIS CANNOT DO Traditional approaches for the analysis of the cardiovascular variability signals usually include the application of time series analysis methods to characterize them. In the time domain, statistical methods were first used to describe the amplitude distribution of signals, such as the HRV or the BPV [1]. Later, researchers used spectral analysis methods for this purpose (for a review see [2]). In this case, it is necessary to take into account that the variability signals are not real digitized signals, because the *Address correspondence to this author at the Departamento de Fisiología, Facultad de Medicina, 38071 Universidad de La Laguna, Tenerife, Spain; Tel: +34-922-31-9357; Fax: +34-922-31-9397; Email: [email protected] 1570-1611/04 $45.00+.00

3. DIFFERENCES BETWEEN FRACTALITY AND NON-LINEARITY There exist at present two different strategies to analyze the irregular components of a physiological time series: they are the fractal and the non-linear approach. Although very different to each other, they are often mixed up. This is actually a serious mistake. On the one hand, fractality refers to the feature of some stochastic time series that present temporal self-similarity. Formally, it is said that a time series is self-similar if its amplitude distribution remain unchanged by a constant factor even when one changes the sampling rate [5]. In practical terms, this feature manifests itself in the time domain as the existence of similar patterns at different © 2004 Bentham Science Publishers Ltd.

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Fig. (1). The raw time series of cardiac interbeat intervals of an adult male Sprague-Dawley rat (500 g. of weight) in basal state.

Fig. (2). Power spectrum of the signal in figure 1 in both linear (left) and logarithmic (right) scale.

time scales (see Fig. 3). In the frequency domain, the hallmark of fractal time series is their power law spectrum (see also Fig. 4). The use of non-linear methods, on the other hand, presumes that the signal under study comes from a non-linear system and therefore, possesses some deterministic component that may be analyzed and characterized from it [7, 8]. Henceforth, we will review the major contributions of both analyses to the understanding of the cardiovascular system, with special focus on the effect of different drugs on the fractal and non-linear properties of the signals characterizing the cardiovascular system. 4. FRACTAL ANALYSIS Early evidence of the applicability of fractal analysis to cardiovascular time series goes back to the beginning of the

nineties, when it was first suggested that the HRV and BPV signals have a significant fractal component [9], because of its power law spectrum as expressed by f −β in log frequency (f) vs. log power plots (Fig. 2), right; (Fig. 4). Peng et al. further verified the fractal nature of the human heartbeat, by using two distinct measures of the heart rate variability. In subsequent works [11, 12, 13], Yamamoto et al. carried out a comprehensive analysis of the fractal nature of heart rate variability in humans by making use of the Coarse-Graining Spectral Analysis (CGSA) technique [6]. By means of the CGSA it is possible to split the total spectral power in two components. The harmonic one corresponds to the welldefined, distinct power peaks of the spectrum (and whose origin in the cardiovascular signals, though well studied, is not yet altogether clear). The non-harmonic component, on the other hand, corresponds to the rest of the spectral power, normally spread over a broad frequency band. This

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Fig. (3). An example of a fractal time series showing similar structures at different time scales.

Fig. (4). Top left : an example of systolic BPV signal from a rat at baseline. Top right: the (total) power spectrum of the raw signal. Bottom: the fractal (left) and the harmonic (right) power spectra obtained via CGSA method.

component, traditionally regarded as noise, was shown to be sensitive to physiological changes in the system that produces the signals [11, 12, 13]. Fig. (4) shows an example of both kinds of powers. Thus, there is evidence that the fractal power is mediated mainly by the parasympathetic nervous system, because the fractal exponent β changed after the administration of atropine [13], whereas neither an non-specific betaadrenergic antagonist, such as propranolol, nor changes in

the respiratory components produced any effect on it [11, 12]. It is noteworthy that these results do not agree with those from a later work carried out also in humans [14], where the fractal exponent did not change even after a double blockade with atropine and propranolol. In this latter work, however, the spectral exponent was calculated from the total power rather than from the fractal power, and it is likely that changes in the total power masked the effect of the drugs on the fractal part of the spectra, thus supporting

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the use of CGSA methodology in cardiovascular studies. However, the appropriateness of the spectral methodology to assess the fractal characteristics of cardiovascular signals has been recently questioned, because of the intrinsic nonstationarity of these time series [15]. Instead, the use of wavelet analysis has been favored as a more suitable alternative for this purpose [16]. Wavelets constitute an extension of the classical Fourier spectral analysis in which the basic function are of finite, scalable size, thus making it possible to scrutinize the features of the signal at different time scales. In a wavelet spectrum, self similarity manifests itself by the appearance of tree-like branching patterns that resemble the fractal objects, in contrast with the regular periodic patterns of a non-fractal signal (see Fig. 3 in [15]). While wavelets can be used as a tool to highlight the qualitative characteristics of fractal signals, the proper quantitative estimation of these characteristics, particularly its fractal exponent, can also be carried out in the timedomain. These methods must assess the scaling properties of the fractal signal, typically by quantifying the relationship among the fluctuations of the signal at different scales. There are a number of methods to carry out this in point processes such as the cardiac interbeat time series. For instance, we can name the Allan and Fano factors and the so called detrended fluctuation analysis (DFA) [18, 19], which is doubtless one of the most frequently used. In brief, DFA works by dividing the time series into boxes of equal length, n and calculating for each box the rms deviation between the samples within each box and the trend of this box as assessed by least squares fit. Self-similarity implies that fluctuations in small boxes are not independent to that of larger boxes, which in practical terms produces a linear relationship in a log-log plot of the rms vs. n, whose slope is the fractal exponent α, which usually conveys the same information than its “brother” β. The use of this algorithm, whose detailed description and source code is publicly available at the Internet site www.physionet.org, has demonstrated that fractality is a ubiquitous property of the cardiac time series of healthy subjects, whereas any disturbance of the autonomic control on the heart rate is linked to changes in the value of the fractal exponent so obtained [15]. 5. THE IMPORTANCE OF BEING NON-LINEAR 5.1. The Concept of System While the study of the statistical and/or fractal properties of a given signal may be useful from both the theoretical and the practical point of view, the assessment of its dynamical properties goes one step beyond. It aims at studying not only the behavior of the signal, but also how the system that produces it works. Thereby, one establishes a connection between the measured signal and its source [7]. In general terms, we may define a system as any real process capable of producing a measurable signal. In this context, our goal is to be able to gather information about how the former works by analyzing the latter. One may divide systems in two groups: linear and nonlinear. The fundamental distinction between them is made according to how the inputs to the system interact with each other to produce the output(s). A system is non-linear if the response to one of its inputs depends on the values of the

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others, the most relevant consequence being that, contrary to what happens in linear systems, in non-linear ones the value of the outputs may greatly change even by the slightest variations of the inputs. Linear systems can be well studied by means of spectral methods, such as the Fourier Transform. However, the study of non-linear systems is a far less developed field, since it involves a higher level of analytical difficulty. For example, the concept of transfer function, which is a very useful way of characterizing the relationship between inputs and outputs in linear systems in the frequency domain, is not applicable to the case of non-linear systems, where this relationship is not unique but depends on the values of the inputs. 5.2. Why Using Non-Linear Methods in Cardiovascular Physiology? As commented above, from the point of view of a linear world, in which small causes must have small effects, all of the system's irregular behavior is regarded as due to some external source random in nature. The theory of non-linear dynamic systems allows an alternative explanation for this irregularity in terms of what is known as chaos theory [20]. There exists a major conceptual difference between the treatment of the irregularity as a random variable or by means of the chaos theory. Both cases require certain assumptions to be made about the nature of the system whose signals are being recorded. It seems evident that any biological system responds better to the description of a non-linear system. For example, the response of the cardiovascular system to an increase in BP depends on the values of other variables, such as the heart rate, the oxygen demand in certain parts of the organism, etc. In generating the corresponding output, all these variables must be taken into account together and not in isolation, so that the influence of each variable will be strongly mediated by the values of the others, giving rise to a typically non-linear behavior of this system. Further evidence of this non-linearity is the similarity between the power spectra of typical non-linear systems and those from many biological systems, which are normally broadband spectra with the power spread over a wide range of frequency with few definite spectral peaks (see Fig. 5). Therefore, it seems appropriate to study biological systems by making use of mathematical tools that are able to take into account their intrinsically non-linear nature. Another important advantage of the non-linear analysis over the fractal methodology is that the very concept of system invites the researcher to investigate the relationships among the different variables that compose it even though, this adds a further degree of difficulty to the problem. Indeed, it normally implies the simultaneous recording of different related signals and the analysis of their mutual interdependencies (multivariate analysis, as opposed to the univariate analysis where a single signal is studied in isolation). The advances obtained in latter years in the field of non-linear multivariate analysis of experimental signals [21, 22] represent an important step in this direction. Consequently, in this review we will also cover the more challenging problem of characterizing the interdependencies

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Fig. (5). Temporal course of the variable x(t) from the Lorentz system (left) and its power spectral density (right).

between different parts of the cardiovascular system by the joint analysis of two or more of its characteristic signals. 5.3. Univariate Analysis: What should We Look At? The key question to answer when we try to characterize the complexity of any physiological signal and in particular of any cardiovascular one from the point of view of nonlinear analysis is: what should we try to assess? Whereas in fractal analysis the main feature is the degree of correlation among the fluctuations at different scales, something that is well characterized by the fractal exponent, in non-linear analysis it is first necessary to establish which of the signal’s features we are interested in even before choosing the optimal method of calculation. Furthermore, non-linear methodology poses an additional problem for the researcher. It turns out that the features of the non-linear system that generates a signal cannot be directly studied on the raw signal. Similarly to what happens with the spectral methods, it is first necessary to carry out a transformation of the data with the aim of obtaining the state space of the system under study [23]. The problem of properly reconstructing the space state of a system is not an easy one and actually represents a whole branch of applied mathematics, called embedology [24]. The interested reader may refer to the abundant literature on the subject for further details (see for instance [7, 8, 24]). Here we will restrict ourselves to some practical ‘recipes’ and, once the state space of the system (see below) is reconstructed, we will briefly sketch how to characterize its most significant properties. 5.3.1. Reconstructing the State Space In order to reconstruct the state space, we have to first obtain the system’s space vectors. For this purpose, the most used technique by far is delay embedding [23, 24], in which m-dimensional vectors are constructed by using the consecutive samples of the time series:

Yk=(y(k), y(k-τ), .., y(k-(m-1)τ))

(1)

where y(k) is the k-th sample of the time series, m is the so-called embedding dimension and τ is the delay time. New difficulties arise when one tries to establish the correct values for m and τ. Despite the considerable effort devoted to this issue [7, 8] there is at present a surprising lack of general agreement about this issue. Roughly speaking, both parameters should be great enough to avoid spurious results due to the temporal correlations present in the data because of oversampling, but not so large that the number of vectors is greatly reduced for statistical reasons. Normally, it is possible to take τ as the first minimum of the mutual information function of the data [7], although one also obtains good results by using the first zero of its autocorrelation function [8]. As for the embedding dimension, there is a good practical method for the assessment of its proper value, namely the false nearest neighbour method, whose details can be found elsewhere [25]. Once we have the vectors, its evolution along time, also called the trajectories of the system, reveals its internal dynamics. Fig. (6) shows an example of the original and the reconstructed trajectories of the well-known Lorentz system [27], where the similarity between the original and the reconstructed state spaces is apparent. The next step is the computation of some statistics that quantify the different properties of the non-linear systems. We can name three of them as the most remarkable ones and those to which researchers have paid more attention so far. 5.3.2. The Complexity of the System The trajectories of some non-linear systems -those undergoing a chaotic regime- possess the striking property of being able to remain confined to a finite region of the state space from some given time on [20]. These regions are the attractors of the system (see Fig. 6). Attractors are actually common to both linear and non-linear systems, but those

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Fig. (6). Trajectories in the state space of the Lorentz system [23] at two different scales (left and right). Up: Original state vectors. Down: Reconstructed state vectors using time delay embedding.

from non-linear chaotic ones, often called strange attractors, are the most interesting ones. The bulk or size of an attractor is termed its dimension, which represents a rough estimation of the effective number of variables of the system, often also called its degrees of freedom [8, 20]. One of the simplest and most used methods to estimate the dimension of an attractor from the reconstructed state space makes use of the correlation integral C(ε) [26]. In practical terms, C(ε) is a function of the intervector distance in the state space of a system (ε), which represents the probability of finding a state vector closer to a reference vector than a given distance ε. This function is therefore bounded between 0 and 1 and more importantly, its scaling with the distance is a good estimation of the system’s dimension. Thus, for such attractors and within certain range of distance, it is verified that: C(ε)=a ε D2

(2)

where a is some constant [8, 20, 26]. Thus, if one plots log C(ε) vs. log ε, the slope of the resulting straight line is D2, the correlation dimension one is interesting in (Fig. 7). The suffix accounts for the fact that D2 is only one of the members of a whole family of dimensions that is possible to obtain from the system (the interested reader can find additional information in ref. [8]). If it has received the preference among others options it is only due to be the one easiest to calculate in those cases in which the formal equations of the system are totally unknown, as happens with experimental signals. The value of D2 is normally noninteger but fractional, which linked to the fact that the strange attractors possess normally fractal geometry makes that people sometimes refer to this parameter as the fractal

dimension of the signal, thereby the confusion between fractality and non-linearity, something that must be avoided at all cost. 5.3.3. Sensitive Dependence on Initial Conditions The hallmark of chaotic systems is clearly the fact that small changes in the values of the inputs might largely vary, as time goes by, the future evolution of the system. This dependence of the system on initial conditions was actually the feature that allowed Lorentz, one of the first researchers who dealt with chaos in natural systems, to discover this phenomenon almost by chance [27]. Due to this property, two state vectors initially very close might produce completely different trajectories. Contrary to stochastic time series, where the closeness of reconstructed state vectors might randomly occur so that successive vectors are quickly far from each other, in (partially) deterministic systems both trajectories remain close to each other during a certain time, after which they separate. The rate at which this separation takes place is proportional to the largest Lyapunov exponent of the system, λ1 [20]. Again, the suffix indicates that λ1 is not alone, but the elder member of a whole spectrum of exponents, the Lyapunov spectrum, which has as many members as the embedding dimension of the reconstructed state space [8, 20, 28]. Its meaning might be better understood once one knows the way in which it is calculated. Take a reference vector and pick the n vectors in the state space that are closer to it. Then calculate the average distance between the trajectory of the reference vector and that of their n nearest neighbours as time evolves. Such average distance scales with time according to the expression:

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Fig. (7) . Left : Correlation integral vs. distance in log-log scale for different embedding dimensions for a human HRV signal during slow wave sleep. Right: The estimated correlation dimension as a function of the distance.

S(t) = ke tλ1

(3)

Therefore, a log-log plot of S(t) vs. t produces a straight line whose slope is an estimation of λ1, as shown in (Fig. 8). The larger its value, the more sensitive the system is to small changes in its initial conditions. This statistic is greater than zero for non-linear chaotic system and lower than zero otherwise, except for the case of a stochastic signal, where it should be, in theory, infinite. 5.3.4. The System’s Predictability Closely linked to the sensitive dependence on initial conditions is the lack of predictability of the evolution of the trajectories. This concept is usually assessed in terms of the information that one loses from the system as the trajectories evolve. This loss is called the entropy of the system, and can be quantified by making use of the correlation integral (see for instance [29, 30]). If we take the value of C(ε) at a given εo for state spaces of dimension m and m+1, we are somehow measuring how the probability of finding a vector closer to the reference one than this given distance decreases as we include an additional component (i.e. a new consecutive value of the time series). The greater the decrease, the more difficult will be to predict the future values of the system, even if we know its entire past story. The entropy so estimated is the Kolmogorov entropy K2 [29]: K2 =

I ln C(ε, m) T C(ε, m + 1)

(4)

where T is the sampling time, i.e. the period of time between two consecutive samples in the time series. 5.3.5. The Graphical Approach In those practical applications where one has to deal with nearly periodic signals, it is possible to make use of a graphical approach that allows visualizing in a convenient way the non-linear features listed above. This approach is called the recurrence quantification analysis (RQA, [31]). The key factor to understand is the concept of recurrent pairs

of vectors. A pair of vector (Yi, Yj) is said to be recurrent for a certain εo if the distance between both vectors is lower than εo. If we plot such recurrence by placing a dot in a square matrix in the (i, j) as well as the (j, i) position, we obtain a recurrence plot of the time series [32]. Nevertheless, if we extend the analysis by taking different distances from zero to ε max – the maximum distance between two vectors in the state space-, and assigning a different colour to each distance, we obtain (Fig. 9). The appearing of recurrent coherence structures in this figure at different times strongly suggests the existence of long-term correlations, whether deterministic or stochastic, in the signal’s dynamics. The quantification of the signal’s properties is carried out by selecting one single distance (usually a short one). The square root of the embedding dimension m times the standard deviation of the signal seems to perform well in practical applications [33, 34]) and calculating the percentage of recurrence (%RC), the percentage of determinism (%DT) and the length of the largest diagonal line (LT). The first index, %RC, is the ratio, expressed as a percentage, between the number of recurrent vectors and the number of total vectors, which is actually the value of the correlation integral C(ε) at this given distance. The second index, %DT, is the number of recurrent vectors that are not isolated, i.e. those pairs (i, j) for which the corresponding pair (i+1, j+1) is also recurrent. Both indices give a hint of a signal’s complexity. Finally, LT is the longest number of consecutive recurrent vectors, and is reportedly inversely proportional to a signal’s largest Lyapunov exponent [31]. This approach performs especially well in the case of periodic or nearly periodic point processes, such as the cardiac interval intervals [31, 35], but has also been used to study the sequence of interspike intervals in neuronal records [36]. 5.4. Multivariate Analysis: Two is Company Undoubtedly, the most important advantage of the nonlinear approach as compared to the fractal one is that the

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Fig. (8). Evolution of the average distance between the reference vectors and their closest neighbours along time in logarithmic scale. Three different values of the embedding dimension are plotted. The data correspond to a healthy human HRV signal.

Fig. (9). Recurrence plot of the HRV signal plotted in the bottom. Parts of the time series with similar dynamics have similar colours in the recurrence plot.

former also allows studying the changes, if any, in the interdependence between a signal and others characterizing different systems related to the one under study. Changes in the individual signal alone as a response to changes in the physiological conditions answer the question of how do these changes affect the values of the different univariate

indices. However, the study of the changes in the interdependencies between signals goes one step further. It tries to answer the question of why these indices -and therefore, the characteristics that they represent (signal’s complexity, predictability, etc)- change in such a way. For instance, a decrease in D2 suggests that the number of inputs

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to a system has decreased, and thus, that some variable that was formerly taken into account is no longer influencing the outcome of the signal. By studying the relationship between different signals, multivariate analysis tries to determine which of these variables is no longer considered. Multivariate statistical and spectral analysis pursue the same goal in the time and frequency domain, respectively, by making use of linear tools, such as the cross correlation and the coherence function. However, here again, the non-linear approach has its pluses, because linear indices are symmetric, excluding the possibility of investigating any driver-response relationship, whereas multivariate non-linear indices might take this possibility into account [22]. Furthermore, the non-linear indices are sensitive to any kind of interdependencies between the signals [21, 22], while linear ones cannot deal properly with any interactions beyond the linear ones. Since non-linearity is likely to be present in most of the physiological signals, the later indices are in principle more appropriate for the study of these signals, as was also the case with the univariate approach. In general terms, there are two different ways of undertaking the problem of multivariate analysis of experimental signals beyond the linear approach. We will discuss both of them separately, hereafter for the sake of clarity. 5.4.1. The Analysis of the Joint State Space The most ambitious approach to multivariate non-linear analysis of experimental signals consists of studying the relationship between their reconstructed state spaces. Hitherto, two different ideas have been tested in physiology. The first one implies the construction of the joint space of both signals, and implicitly considers them as representing two different subsystems of the same system [37]. The basic concept here is that if both subsystems are independent, the complexity of the joint system equals to the sum of complexities of both individual subsystems, whereas if both are interdependent this joint complexity should be lower than this sum. Something similar happens for the complexity. Curiously, the formal proof of this -otherwise quite logicalidea only came recently [38], and even then, it was only given for chaotic, noise free systems. But it seems to work equally well for physiological signals [34, 37, 39]. The assessment of the joint complexity and the joint predictability is carried out by calculating the joint correlation integral from delayed vectors whose components are consecutive samples of both signals at the same time [37]: Qk=(y(k), y(k-τ), .., y(k-(m/2-1)τ), x(k), x(k-τ), .., x(k-(m/2-1)τ)) (5) Then, the joint correlation dimension and the joint predictability are compared with the individual ones to produce the independence of complexity and the independence of predictability indices (ICM and IPM, respectively), which range from 0 for totally coupled systems to 1 for completely independent ones [37]. 5.4.2. The Study of Phase Synchronization The second approach consists of assessing the existence of synchronization between the phases of two signals [40,

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41]. This approach requires first the calculation of the phases of each individual signal, and then the quantification of the degree of synchronization between these phases. The details on how to perform such calculations for different kinds of physiological signals can be found elsewhere [40]. Compared to the other approach, the assessment of phase synchronization has the advantage of being applicable even to stochastic signals, because no hypothesis is made about the nature of the signals. Additionally, and although both the indices from the joint space approach and those based on the concept of phase synchronization are symmetric, recent results suggest the possibility of deriving asymmetric indices of phase synchronization for cardiovascular signals [41]. 6. NOT ALL THAT GLISTENS IS GOLD However appealing the abovementioned methodology may seem, we must be careful, because appearances might be deceptive and this is not an exception. For instance, in the univariate case, the scaling of the correlation integral and the rate of average distances with the distance and the time (eq. 2 and 3, respectively) is, strictly speaking, only valid in the limits of infinite length, noiseless stationary time series [8, 20], which is nowhere near the case for physiological signals. Besides, in many practical applications it is often impossible to know whether the signal under study comes from a non-linear system. Unfortunately, the values of the statistics, such as the D2, K2 and λ1 are neither a definitive sign of chaos, since several studies have demonstrated that fractal signals can fool the non-linear algorithms and mimic the results obtained for chaotic signals [42, 43, 44]. Therefore, additional methods are necessary, which allow the researcher to discriminate a priori between fractal and nonlinear signals lest the results obtained are misinterpreted and lead to wrong conclusions. Such methods are the nonlinearity tests. They work by scrutinizing the signal in search of some of the abovementioned characteristics. For instance, non-linear prediction algorithms might be applied to forecast the future values of the signal and their results compared with linear algorithms [45]. A lower forecasting error in the non-linear case suggests the existence of non-linear structure in the signal. However, the most popular test for nonlinearity is that of the surrogate data [46, 47]. In this case, a set of signals is constructed, which shares with the original all its properties, except those whose existence in the data we want to check. Then a statistic is calculated, whose value should be sensitive to such property, and the result from the original data is compared with those from the surrogates. If the difference between the original and the surrogates as assessed by this index is greater than expected by random, then one can assert that the property under question is indeed present in the original data. Additional difficulties appear when the signal presents a power law spectrum, because the long term correlations in the data are often taken as a sign of determinism if the test is not carefully applied [48], which calls for new refinements in the methodology, making things even more complicated. There is an extensive survey from Schreiber and Schmidt where the interested reader can become acquainted with all the nuances regarding the surrogate data test [47]. Overall, however, it is possible though not always easy- to circumvent such difficulties and profit from the non-linear methodology for the univariate as well as the multivariate analysis of experimental signals.

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7. NON-LINEAR ANALYSIS OF CARDIOVASCULAR SIGNALS 7.1. The Univariate Approach Unlike the case of other physiological signals, such as the electroencephalogram, there was a certain reluctance to apply the new non-linear methodology to cardiovascular signals. A variety of reasons contributed to this reluctance, among them the lack of theoretical justification until the nineties (in the case of the HRV) to characterize a system from a series of intervals between events [49]. In addition, good results were being obtained in studying cardiovascular variables by means of classical spectral techniques (see for example [1, 2, 50]). The earliest applications of nonlinear methods in the field date from the mid-1980s, when Glass and co-workers [51-53], working with cultured heart-cell aggregates, used Poincaré maps to show the existence of a phase transition in the response of the cells to periodic stimuli. The response passed from a regular to a highly irregular pattern (which the authors denominated chaotic), something that was very similar dynamically to that occurring in cardiac arrhythmias. Similar results were obtained in studies performed by means of the periodic stimulation of cardiac Purkinje fibres [54, 55]. Shortly afterwards, Skinner and co-workers reported the first evidence of chaos in the cardiac interbeat signal and characterized it from the perspective of its non-linearity indices [56, 57]. According to their results, the correlation dimension of the signal decreased in response to myocardial ischemia. Since then, both the correlation dimension and the largest Lyapunov exponent have been extensively used for the analysis of cardiovascular signals, mainly the HRV [14, 58-60] and the BPV signals [59, 61-67]. Unfortunately, many of these studies took a finite value of D2 or a λ1 as an indicative of chaos, without performing any subsequent test to check the nature of the signal. It is therefore, hardly surprising that the results were contradictory with respect to the validity of the techniques for characterizing the corresponding physiological systems. Indeed, some authors have recently suggested that, for instance, the cardiac interbeat signals cannot be used for such purpose [68-70], whereas others have stated just the opposite [60, 65, 71]. There exists, however, a more or less general consensus about the interest of applying these techniques in the field of cardiovascular physiology (see, for example, the issue n. 3 of Cardiovascular Research, 1996). Besides, the results of several studies in different animals, such as rabbits [72], dogs [63], humans [57, 60] and more recently rats [34, 67], support the existence of non-linear components in the cardiovascular variability signals in the basal state. In the following section, we will review those aspects related to the origin of this non-linearity in the light of these results. 7.2. How Do They Become Non-Linear? There is a current discussion about what might be the factors that give rise to the existence of non-linear components in the cardiovascular variability signals [73, 74], although it seems clear that the degree of non-linearity is associated with both the degree of maturity of the ANS and its correct functioning [75, 76]. Some studies have reported the decrease of non-linearity in the HRV signal after

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parasympathetic blockade both in rabbits [72, 77] and piglets [58]. In general terms, it seems that the HRV signal becomes more regular and less sensitive to the initial conditions after the application of parasympathetic blockers such as atropine [33, 34]. In humans, the respiratory component of the HRV mainly mediated by the parasympathetic system- seems to also be involved in the non-linearity of this signal [76]. It is clear, however, that this cannot be the only source of the non-linear characteristics of this signal. In fact, this nonlinearity remained unchanged after the administration of either a sympathetic agonist (phenylephrine) or a nitric oxide synthesis inhibitor (NOSi) [34], both of which induce an increase in the respiratory component of the HRV signal [78]. Concerning the possible role of the beta-sympathetic system, the administration of beta-blockers to animals whose parasympathetic systems had been previously blocked led to a further decline in the non-linear characteristics of the HRV signal [34, 58, 72]. The non-specific beta-sympathetic blocker propranolol has an effect contrary to that of atropine [33, 34]. Thus, it seems that the non-linearity of the HRV signal in the basal state arises from the interaction of the beta-sympathetic and parasympathetic systems. This conclusion is also supported by the fact that NOSis, which induce an increase in reflex parasympathetic cardiac activity [78], do not give rise to any increase in the non-linear structure of this signal [34]. It is therefore, plausible that the two nervous systems interact in a typically non-linear way to modulate the duration of the heart period: while each of them has a specific action, their final effect depends on the degree of activity of the other. Furthermore, the results from the nonlinear analysis of the cardiac variability signals represent an improvement in regard with those obtained from the spectral analysis. Thus, although changes in the spectral power are characteristic of some pathological situations [2], it is often difficult to decide whether these changes are due to a parasympathetic or a sympathetic malfunction. However, parasympathetic alterations produce results in opposite direction than sympathetic ones as far as non-linear indices are concerned. A summary of the effects of the different drugs on the cardiac interbeat signal is shown in (Fig. 10), where we plot the results of a non-linear prediction algorithm applied to the RRV signal in the rat before and after the administration of the drugs. The situation is different in the case of the BPV signal. Results have shown a decrease in the complexity of the signal after baroreceptor denervation [63]. But recent studies have shown that only an alpha_sympathetic agonist and an antagonist have a significant effect on the non-linear properties of the signal [34, 35]. The BPV signal became more regular after the administration of such drugs, something similar to what happens after parasympathetic blocking and NOSi administration, as well as with the betasympathetic blockers. These results allowed a tentative guess to be made about the mechanisms that give rise to the nonlinear structure of the BPV signal [34]. Unlike the case of the cardiac signal, the role of the parasympathetic nervous system seems less important: neither the non-linear characteristics nor the mean value of the BPV signal change after it is blocked. However, it is clear that it has some effect, since different non-linear indices were modified after its

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Fig. (10). Normalized nonlinear prediction error for the RRV signal before (doted line) and after the drugs infusion (solid line) as a function of the prediction time (bars indicate the SE of the mean). PSB: atropine, a parasympathetic blocker; ßSNB: propranolol, a ß-sympathetic nonspecific blockers; ßSSB: atenolol, a ß-sympathetic specific blockers; αSB: prazosin, an α−sympathetic blocker; αSA: phenylephrine, an α−sympathetic agonist; NOI: N-monomethyl-L-arginine, a nitric oxide synthesis inhibitor.

blocking [33, 34], suggesting that the signal became less complex and more predictable. On the other hand, the sympathetic system (especially the alpha-sympathetic one), does seem to have a major role: both an alpha-sympathetic agonist and an antagonist modify the characteristics of the BPV signal, making it less complex and more predictable [34]. One explanation might be that one of the ANS variables that could be modified before the administration of these drugs (namely, the degree of alpha-sympathetic activity) is fixed at an extreme value after it. It is also noteworthy that a NOSi, which has no influence on the HRV signal (except to produce a significant bradycardia), also produced a similar effect to that of the alpha-sympathetic drugs. Indeed, an earlier spectral-analysis-based study suggested that NOSi in rats produces a decrease in alpha_sympathetic activity [78]. This indicates that the nonlinear structure of the BPV signal may be indirectly affected by other control mechanisms than the alpha-sympathetic system [34], which would also explain why the BPV signal is more complex than the HRV signal [34], since more mechanisms are involved in regulating the former as compared to the latter. As in the case of the cardiac signal, a summary of the effects of the different drugs on the BPV signal is shown in (Fig. 11) by applying the same non-linear prediction algorithm to this signal in rat before and after the administration of the drugs. 7.3. The Multivariate Approach The main application of the multivariate non-linear approach in cardiovascular physiology has been the

assessment of the interdependence between the heart rate and respiration [21,37,41]. The concept of phase synchronization has turned out to be very useful in this regard, because it also allows the study of the statistical relationship between stochastic signals. Further, one can obtain different versions of the phase of a signal for this purpose, even by using the Wavelet Transform approach [22], thus making it possible to work with cardiovascular variability signals and equally sampled signals -such as the raw respiratory signal- at the same time [21, 41]. Lately, the concept of the joint state space has been applied for the same purpose [37]. However, the idea can also be used to assess the relationship between the RRV and the BPV. Recently, a thorough study in rats investigated the role of the different parts of the ANS in this relationship [34]. The values of the coupling indices, indicated a high degree of non-linear coordination influenced by the respiratory component, because this coupling was significantly reduced after the low-pass filtering of both signals. This was further supported by the decrease of such coupling after parasympathetic blockade with atropine, which always modified the respiratory component of the RRV [78]. The decrease in the RRV-SPV non-linear coupling after atropine and its increase after propranolol -as shown by the results presented in the study- stresses the relevance of the parasympathetic system in the heart rate baroreflex operation. In addition, the lower complexity of the BPV after atropine could also be considered a consequence of the reduction of baroreflex sensibility due to RRV-BPV decoupling.

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Fig. (11). Normalized nonlinear prediction error for the BPV signal before (doted line) and after the drugs infusion (solid line) as a function of the prediction time (bars indicate the SE of the mean). Abbreviations for the drugs as in Figure 10.

8. CONCLUSIONS AND FUTURE WORK We summarized the main results concerning the application of fractal and non-linear analysis to the study of the short-term cardiovascular variability signals. Although both approaches are similar in the sense that they provide an alternative explanation to the complexity of the signals, they come from different conceptual points of view. Fractal analysis sacrifices some descriptive power, so to say, and limits itself to the univariate study of the signals in return for using a more conservative hypothesis, in which the signals are regarded as stochastic [5, 6, 15]. The non-linear approach, on the other hand, allows a more thorough characterization of the signals and their interdependencies, but formally speaking requires the existence of some nonlinear, deterministic structure in them [7, 8, 20]. Despite the difficulties inherent to the analysis of short, noisy and often non-stationary signals, such as those typical of the cardiovascular system, the careful application of either methodology represents a convenient way of describing the signals and detecting possible changes in them that would remain undisclosed with traditional statistical and spectral analysis methods. The ability of detecting changes in the signals is the most important feature that would be useful for the application of these methodologies in clinical practice. As for the meaning of the different indices in basic cardiovascular physiology, we outlined the main insights provided for both approaches about the short-term cardiovascular control system. Overall, it seems clear that the cardiovascular signals from healthy subjects possess a greater complexity and lower predictability than pathological ones [15, 56, 57, 63, 70]. This greater complexity, associated

with the deviations of the time series from its basic homeostatic level, may be a sign of the ability of the system to respond to changes in the input signals, which allows a more effective response of the system to new physiological situations. In our opinion, there are two major open lines of research for future work. Concerning the univariate analysis, it is necessary to search for new, more efficient methods that are specially indicated to deal with non-stationary signals and can consider non-stationarity as a part of the signal’s intrinsic characteristics. Some steps have already been given in this direction in the past, for instance in order to improve the estimation of the correlation dimension in physiology [79]. Concerning the multivariate approach, the effective detection of driver-response relationships between the cardiovascular signals is yet to be carried out. Although the newest methodological advances in the field of phase synchronization analysis represent an important step in this direction [41], the use of other multivariate non-linear methods, such as those based on the concept of mutual neighbours would be very helpful for understanding the short-term cardiovascular control system of healthy and pathological subjects. This requires, however, the modification of these methods to deal with signals of different nature and complexity, a field in which they have shown to be little awkward [22]. This might be the explanation for the lack of applications on cardiovascular physiology of indices that are currently very popular in neuroscience [22, 80, 81]. Recent advances in the study of the joint state space of two subsystems also point to the same goal [38]. Additionally, the use of advanced non-linear

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statistical methods based on the transfer of information between the signals [82, 83] is another path yet to be fully explored in physiological applications.

[13]

In any case, the increase in conceptual complexity of the methodology involved strongly suggests the need of multidisciplinary research groups where physics, mathematics, physiologists and clinicians manage to efficiently deal with the different aspects of the problem under study.

[15]

9. APPENDIX: FIND IT IN THE WEB

[19]

Apart from the already mentioned Internet site www.physionet.org, which is the most important effort carried out, there are some other sites where the interested reader may find good research resources to begin the fractal and non-linear analysis of cardiovascular signals. Of course, the following list is neither comprehensive nor is guaranteed to remain valid. Instead, we have only included good quality sites, which are expected to last.

[14]

[16] [17] [18]

[20] [21] [22] [23]

A very good software package for the comprehensive univariate analysis of experimental signals is the TISEAN package [84], which can be downloaded for different platforms, at www.mpipks-dresden.mpg.de/~tisean/. As an additional advantage, the site contains an extensive online manual and examples.

[24] [25] [26] [27] [28] [29] [30]

The users of Matlab® can find a good toolbox for the analysis of univariate time series at http://www.physik3. gwdg.de/tstool/. If real time non-linear analysis is what you are up to, you may have a look at http://www.physik.tudarmstadt.de/NLyzer/, where some good software has been developed for such purpose.

[31] [32] [33]

And finally, instead of making this list endless, we recommend to visit both www.physionet.org and a useful site devoted to chaos theory in life sciences www.societyforchaostheory.org. In both of them there are a good links section, often updated and checked.

[36]

ACKNOWLEDGEMENTS

[40]

This work has been supported by the Grant PI020194 of the Spanish “Fondo de Investigaciones Sanitarias” (FIS).

[41]

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