A quantitative comparison of different methods to detect cardiorespiratory coordination during night-time sleep
© Cysarz et al; licensee BioMed Central Ltd. 2004
Received: 02 September 2004
Accepted: 25 November 2004
Published: 25 November 2004
The univariate approaches used to analyze heart rate variability have recently been extended by several bivariate approaches with respect to cardiorespiratory coordination. Some approaches are explicitly based on mathematical models which investigate the synchronization between weakly coupled complex systems. Others use an heuristic approach, i.e. characteristic features of both time series, to develop appropriate bivariate methods.
In this study six different methods used to analyze cardiorespiratory coordination have been quantitatively compared with respect to their performance (no. of sequences with cardiorespiratory coordination, no. of heart beats coordinated with respiration). Five of these approaches have been suggested in the recent literature whereas one method originates from older studies.
The methods were applied to the simultaneous recordings of an electrocardiogram and a respiratory trace of 20 healthy subjects during night-time sleep from 0:00 to 6:00. The best temporal resolution and the highest number of coordinated heart beats were obtained with the analysis of 'Phase Recurrences'. Apart from the oldest method, all methods showed similar qualitative results although the quantities varied between the different approaches. In contrast, the oldest method detected considerably fewer coordinated heart beats since it only used part of the maximum amount of information available in each recording.
The method of 'Phase Recurrences' should be the method of choice for the detection of cardiorespiratory coordination since it offers the best temporal resolution and the highest number of coordinated sequences and heart beats. Excluding the oldest method, the results of the heuristic approaches may also be interpreted in terms of the mathematical models.
The time intervals between successive heartbeats (e.g. the RR tachogram or, equivalently, the series of instantaneous heart rates) may be analyzed with different tools to obtain information about e.g. heart rate variability (HRV) , regularity of the time series [2–7], or large-scale correlations in the time series [8–10]. Each technique provides different information about the 'time-structure' contained in the series of successive heartbeats. Some of them, like information obtained from HRV, may be linked to the sympathetic and parasympathetic activity of the autonomic nervous system [1, 11]. Hence, they offer the possibility to interpret the derived quantities in terms of physiology. Furthermore, when applied to data obtained from patients suffering from heart diseases, these different information may be combined and used for risk stratification [12, 13]. A large body of work concentrates on these univariate time series analysis and standards have been established .
To gain information that cannot be extracted from a single time series, the interaction between two (or more) physiological (sub-)systems has been investigated. This is often done on the basis of simultaneously recorded time series of each system. Hence, new bivariate techniques have been developed that aim to quantify the imprint of the physiological interaction between the two systems [14–17]. Especially, techniques which analyze the cardiorespiratory interaction have been developed recently [18–24]. Among these techniques the coordination between the series of successive heartbeats and the series of successive respiratory cycles, further denoted as cardiorespiratory coordination, is one focus of attention.
In fact, physiologists had already investigated cardiorespiratory coordination in the human organism as early as the 1960ies . Calculating the distance between an inspiratory onset and its preceding R-peak they found intermittent coordination between heartbeat and respiration. In the 1970ies this interesting topic was no longer followed up (except in one study ), presumably because the physiological interpretation of the results was limited, although the last reviews of this era appeared in the late 1980ies [27–29]. The investigation of cardiorespiratory coordination has recently been revived mainly by physicists and mathematicians. They first studied the interaction of two weakly coupled chaotic oscillators and found synchronization of the phases whereas the amplitudes of the oscillators remained uncorrelated . This class of mathematical models may serve as an approximate qualitative model for the interaction of heartbeat and respiration . On the basis of these models new techniques for the analysis of bivariate time series data recorded from these physiological systems have been developed [22, 31, 32]. Other approaches have also been used to develop methods to analyze the cardiorespiratory interaction since the significance of this topic was deemed high [23, 33–41].
The similarities or differences between the definitions of the different techniques and the advantages of each technique are not known yet. Unfortunately, a complete review and a (qualitative and quantitative) comparison of the recent literature is too extensive and beyond the scope of this paper. This study is restricted to the comparison of the performance of six different techniques in the detection of cardiorespiratory coordination. Although the comparison is limited, the insight that is drawn from it provides valuable information with regard to techniques that are not considered in this comparison. A second aim of this study is to investigate the cardiorespiratory interaction during night-time sleep since previous studies only dealt with relatively short recordings (no more than one hour).
Data acquisition and pre-processing
The electrocardiogram (ECG, standard lead II) and the uncalibrated nasal airflow (derived by a thermistor-technique) were recorded simultaneously in 20 healthy subjects (7 female, median age: 34.9 years, interquartile range: 13.7 years) using an ambulatory device (Medikorder, Tom-Signal, Graz). In each subject the night-time sleeping period was recorded, in some subjects the complete evening before the sleep period was also recorded. Hence, the total recording time varied from 7–16 hours. The device's internal sampling rate of the ECG was 3000 Hz. Thus, the times of the automatically identified R-peaks had an accuracy <1 ms and were written into a file. To save memory, the ECG was recorded using a rate of 250 Hz. The data were transferred to a PC and analyzed using Matlab (The Mathworks, Natick, Mass, USA) and C routines. The times of the automatically identified R-peaks were visually controlled, i.e. the times of the R-waves were marked in the ECG, and edited if the times of the automatically identified R-peaks did not match the R-peak in the ECG (<0.1% of all R-peaks). The times of the edited R-peaks had an accuracy of 4 ms because the recorded ECG had a lower sampling rate. Ectopies and artefacts were marked and excluded from the analysis.
Note that in the following the term 'cardiorespiratory coordination' is used descriptively in the sense that the times of the R-waves and the times of the inspiratory onsets show some kind of temporal incidence. The temporal incidence does not imply a physiological coupling between heartbeat and respiration, i.e. 'cardiorespiratory synchronization'.
Temporal distance between inspiratory onsets and successive R-peaks:
absolute distance: t i = R i - I j (1)
temporal distance between an inspiratory onset and the preceding R-peak:
absolute distance: t j = I j - R i-1 (3)
Note that for the calculation of the absolute distances t i and n > 1 the appropriate settings of the indices i and j have to be considered. Furthermore, the relative distance φ i corresponds to a definition of a phase angle of an oscillator on the interval [0, n]. This phase (multiplied by 2π) is equivalent to the cyclic phase calculated via the Hilbert-transformation [32, 42]. Similar phases may also be calculated using e.g. the Fourier-transformation . Hence, with respect to the variable φ i , the terms 'relative distance' and 'phase' are used synonymously throughout this paper. Compared to φ i and t i the calculation of t j and β j only needs the timings of the R-peaks before and after the inspiratory onset. Thus, these variables contain less information. They have been mainly used in early studies on cardiorespiratory coordination and are not common at present [44, 45]. In order to keep this comparison practicable, the calculation of the absolute distances t j and the subsequent analysis of these quantities is left out in this study.
The fact that cardiorespiratory coordination is present if structures with horizontal lines occur leads to the task of detecting these structures in the different diagrams. This task can be carried out by different techniques based on mathematical models or heuristic approaches. Some relevant approaches are described in the following.
Detection of horizontal structures in Synchrograms or Post Event Time Series
1. Detection via 'Synchronization-λ'
Along with the development of the Synchrogram a quantification on the basis of a mathematical model has been proposed [32, 47, 48]. Consider two coupled oscillators with appropriately defined phases φ 1 and φ 2 that may intermittently show a 1:1-synchronization (for simplicity both phases are defined on the interval [0, 2π]). In the synchronized case the difference φ 1 - φ 2 is constant. In real world data the phases φ 1 and φ 2 are contaminated by noise. Thus, even during synchronization the difference φ 1 - φ 2 is not constant. Rather it fluctuates around a constant. To take these influences into account a 'stroboscopic technique' has proven to be useful. Each time the phase φ 1 exceeds a pre-defined value θ the phase φ 2 is registered. During a synchronization the different values of φ 2 are almost constant whereas during complete de-synchronization φ 2 spreads equally over the interval [0, 2π]. This approach may easily be generalized for m:n-synchronization if the phases φ 1 and φ 2 are replaced by φ 1/m and φ 2/n. The distribution of φ 2 may be quantified by the first Fourier mode.
The formalization of this idea is as follows: Observe the phase φ 2 at the times t the phase φ 1 is φ 1= θ :
Next, calculate the first Fourier mode of the distribution of ξ i which is known as circular variance  (M values of φ 2 have been observed):
If both oscillators are synchronized λ = 1, the completely de-synchronized case yields λ = 0. To get a more reliable result, λ may be calculated for different values of θ and subsequently averaged.
This method is easily applicable to the detection of cardiorespiratory coordination. Define the phases φ 1,2 as follows: φ 1 increases linearly on the interval [0, 2π] between two successive R-peaks and φ 2 between two successive inspiratory onsets, respectively. To detect coordination for different n:m-ratios this method has to be applied for each desired m:n-ratio. In practice, coordination is present if λ exceeds a pre-defined threshold λ ≥ thres λ . In this case the respective R-peaks are marked as coordinated.
Note that although this method deals with synchronization of two coupled oscillators it is not able to detect cardiorespiratory synchronization in a physical sense. A detection of synchronization on the basis of two simultaneously recorded time series would require supplementary information, e.g. the variation of the strength of coupling. Such supplementary information is not available for the cardiorespiratory interaction. Thus, the term cardiorespiratory coordination denotes a temporal incidence of events in both time series without claiming that this incidence is based on a coupling between both systems.
In this study, the window length M is set to M = 20. To obtain a better time resolution of the λ-values the window is forwarded one R-peak or accompanying phase, respectively. To obtain more stable values of λ the average of 10 different values of θ, equally distributed over the interval [0, 2π], is used. The threshold is set to thres λ = 0.85. This procedure is successively carried out for the following m:n-coordinations: for n = 1: m = 2,...,8 and for n = 2: m = 5,7,9,11,13,15.
2. Detection via 'Phase Recurrences'
This method is based on a heuristic approach. Consider a synchrogram containing a sequence with m parallel horizontal lines indicating a m:n-coordination. In this sequence the relative distance φ i of each m-th R-peak has to be approximately the same. Otherwise parallel horizontal lines would not appear. This 'recurrence of phases' may be used to identify coordinated sequences  (a slightly different approach is presented in ).
The formalization of this concept is straightforward. For a m:n-coordination check whether the phase difference between the phase of R-peak i + m and the phase of R-peak i is within a pre-defined tolerance ε. This condition has to be fulfilled for at least k successive R-peaks:
N r is the total number of R-peaks. In principle, the parameter k is not pre-defined. But, to be compatible with the description of 'parallel horizontal lines' during coordination, k ≥ m needs to be fulfilled. This procedure allows a detection of a structure of parallel horizontal lines already with a length of 2m successive relative distances φ i . E.g. a 4:1-coordination may be identified already in a minimal sequence of 8 R-peaks. In the case coordination is detected, the respective R-peaks are marked as coordinated. To detect horizontal structures for different m:n-ratios this method has to be applied for each ratio. It is easy to implement and it can be used for absolute distances t i analogously.
In this paper the Phase Recurrence is analyzed with respect to absolute distances t i and relative distances φ i for the following m:n-coordinations: n = 1: m = 2,...,8 and n = 2: m = 5,7,9,11,13,15. In the case of absolute distances t i the tolerance ε is set to ε = 0.075, i.e. 75 ms, and in the case of relative distances φ i the tolerance is set to ε = 0.025.
3. Detection via 'Quantification of Histograms'
Another technique to detect cardiorespiratory coordination makes use of a sliding window comprising n F successive phases φ i that is moved over the entire series of phases [34, 46]. First, a distribution of the phases φ i is calculated for each window. If cardiorespiratory coordination, i.e. a structure with parallel horizontal lines, is present the distribution of phases φ i shows some distinct equidistant local maxima (e.g. four local maxima in the case of 4:1 coordination). In the un-coordinated case the phases φ i are randomly distributed and local maxima do not appear. Next, each distribution is quantified by means of a Fourier Transformation. In the case of coordination, the distribution of the phases φ i contains several local maxima resulting in a huge maximum in the power spectrum. In the un-coordinated case the power spectrum does not contain any pronounced maximum since the phases φ i are equally distributed. Thus, the appearance of pronounced local maxima in the power spectrum is used to detect cardiorespiratory coordination.
This idea may be formalized as follows . First, choose the length n F of the sliding window and the number of bins k to calculate the distribution. In the completely un-coordinated case the value of each bin x l is n F /k. If a perfect m:n-coordination is present the following distribution is obtained:
Here, l 0 is the index of the bin that contains the first local maximum of the distribution. This distribution is quantified by means of a Fourier transformation:
The average of all bins n F /k is subtracted from each bin to avoid a dc-component in the power spectrum. In the un-coordinated case P x (f) = 0 for all frequencies f. In the case of a m:n-coordination m local maxima at the frequencies f = a m, a ≤ k/(2m) appear. These local maxima are conserved even if the parallel horizontal lines are not exactly mapped into m bins but also some surrounding bins are filled. Thus, the distinction between coordinated and un-coordinated cardiorespiratory interaction may be carried out using the difference between the maximum and the minimum of the power spectrum.
diff(P x ) = max(P x ) - min(P x ) (10)
In the coordinated case this difference is large whereas in the un-coordinated case the difference is small. In practice, the R-peaks in the analyzed window are being marked as coordinated if diff (P x ) exceeds a pre-defined threshold, i.e. if diff (P x ) ≥ thres F . This procedure may analogously be applied to absolute distances t i .
Practically, as a compromise between the temporal resolution and a practicable estimation of the distribution, the window length is set n F = 20 and the window is forwarded 1 R-peak each time. The distribution has a bin-width of 0.1 sec for absolute distances t i and 0.025 for phases φ i , the number of bins k is adjusted accordingly. The threshold for identifying coordination is set to thres F = 12 for both, absolute distances t i and phases φ i . Notice that this implementation does not require a pre-selection of the integers m and n of a m:n-ratio.
4. Detection via 'Quantification of the distribution of inspiratory onsets in RR-intervals'
In this study, the relative distances β j are also analyzed. They were used in the early studies of the 1960ies and in the subsequent studies of these research groups [25, 27–29]. Although this technique does not use all the information available in the recording it is successfully used at present by one research group [44, 51–53]. It has to be kept in mind that this technique yields coordinated inspiratory onsets instead of coordinated R-peaks. Thus, in order to get results which are comparable with the previously described techniques, the coordinated inspiratory onsets are replaced by R-peaks with the following rule: all R-peaks in a respiratory cycle that follow a coordinated inspiratory onset are marked as coordinated.
The quantification of the distribution of β j is carried out as follows. Since the values of β j are in the interval [0,1] they may be mapped onto the interval [0,2π] by multiplication with 2π. In this study, the distribution of β j in a data window of length n F is quantified by the calculation of the first Fourier mode of the distribution (the circular variance):
Analogously to the 'Synchronization-λ' defined above, the range of γ is 0 ≤ γ ≤ 1. If γ = 0 β j is equally distributed indicating the completely de-coordinated case. If γ = 1 the β j is constant in the data window indicating the coordinated case. Practically, the inspiratory onsets in a data window are marked as being coordinated if γ ≥ thres γ . Subsequently, the R-peaks in the respiratory cycle that follow a coordinated inspiratory onset are marked as coordinated. The window length n F is set to 10 and thres γ = 0.5. The data window is forwarded one inspiratory onset to achieve the maximal temporal resolution.
Quantitative comparison of the methods, Statistics
The total number of coordinated sequences and the total number of coordinated R-peaks during night-time sleep, i.e. from 0:00 to 6:00 assuming that all subjects were continuously asleep, served as quantitative measures of the performance of each method. They were calculated for each recording. Furthermore, the matrix of Pearson's correlation coefficients r between the coordination diagrams of the six methods was obtained as follows. Each correlation is calculated comparing the percentages of coordinated R-peaks with a m:n-ratio of two different methods for each data window. Furthermore, to enhance the validity of this comparison, only those pairs of percentages are used in which at least one percentage is greater than zero. Otherwise the large number of pairs with (0%,0%), i.e. the white area that both coordination diagrams have in common, would lead to a strong correlation even if the other parts of the coordination diagrams show apparent divergent m:n-coordination ratios.
An example of a 'coordination diagram' is shown in Figure 4. This example was analyzed with the method of 'Phase Recurrences' of relative distances φ i . Since the heart rate decreases during night-time sleep whereas the respiratory rate remains almost constant the m:n-ratios of cardiorespiratory coordination also decrease during this period. Thus the sleeping period may be estimated from the 'coordination diagram' by looking at the low m:n-ratios. In the example, the subject slept from 23:30 to 6:30 and had a predominant 4:1-coordination during night-time sleep. At midnight the maximum of approximately 68% is reached, i.e., 68% of all R-peaks in a window of 500 R-peaks showed a 4:1-coordination with respect to respiration. A second local maximum of 4:1-coordination is observable at 1:30. The rest of the night shows sequences with 4:1-, 7:2- (shown in between 4:1- and 3:1-coordination) and also some 3:1-coordination. Although not shown, it has to be added that qualitatively similar results were obtained with the other techniques. Furthermore, the analysis of the other subjects showed that the predominant m:n-coordination ratio during night-time sleep of each subject varies inter-individually and is one of the following ratios: 3:1, 4:1, 5:1, 7:2 or 9:2.
The additional line drawn in the coordination diagram represents the 'total coordination', i.e., the percentage of coordinated R-peaks in a window of 500 consecutive R-peaks regardless of the m:n-ratio. As indicated by the time course of the separate m:n-coordinations the total coordination is not constant or a simple function of time. Instead, the total coordination seems to oscillate. In the course of the night-time sleep the frequency of the oscillation increases whereas the amplitude decreases. Similar qualitative results were obtained using the other techniques.
Number of detected sequences with cardiorespiratory coordination. Median and inter-quartile range during night-time sleep of the following parameters: N R total number of coordinated R-peaks, N seq total number of coordinated sequences, % N R relative amount of coordinated R-peaks with respect to the total number of R-peaks.
PhRec t i
PhRec φ i
Hist t i
Hist φ i
Hist β j
% N R
Correlations between the different methods to detect cardiorespiratory coordination. Median and inter-quartile range of the correlations coefficients between the different methods to detect cardiorespiratory coordination.
PhRec t i
PhRec φ i
Hist t i
Hist φ i
Hist β j
PhRec t i
PhRec φ i
Hist t i
Hist φ i
Hist β j
In recent years a variety of different methods have been proposed to detect cardiorespiratory interaction on the basis of simultaneously recorded time series of heart beat and respiration. In this study, six different methods to identify cardiorespiratory coordination based on mathematical models or heuristic approaches have been quantitatively compared. The main issue of the different methods is the recognition of coordination in synchrograms and post event time series, i.e. the detection of structures with horizontal lines in these representations. Thus, the procedures permit the quantification of qualitative information that is contained in these representations and may provide new information that is not extractable from the isolated univariate time series.
The quantitative comparison of the six different methods to detect cardiorespiratory coordination showed that the group's median total number of coordinated R-peaks and the median total number of coordinated sequences varied depending on the detection method used. The largest number of coordinated R-peaks (30.4% of all R-peaks during night-time sleep) and coordinated sequences was detected by the 'Phase Recurrences' of phases φ i . Since the coordinated R-peaks are distributed in the largest number of coordinated sequences this method has the best temporal resolution compared to all other methods used in this study. Furthermore, it is able to detect even very short and intermittent cardiorespiratory coordination because the implementation used in this study only required at least 2m R-peaks to detect a m:n-coordination (typically, m = 2,3,...,8). But, if the minimal number of required coordinated R-peaks is increased to e.g. 2(m + 8), this method shows approximately the same number of coordinated epochs and R-peaks as e.g. the 'Quantification of Histograms'. Hence, if the 'window length' of this method is increased, the results are similar to those of the methods with a window length of 20 R-peaks. Since the 'Phase Recurrences' of absolute distances t i also showed a large number of coordinated R-peaks and coordinated sequences, respectively, and the correlation to the results of the 'Phase Recurrences' of phases φ i is strong, these procedures showed approximately the same characteristic features. All other methods were less sensitive in the detection of cardiorespiratory coordination because the identification of cardiorespiratory coordination required a minimal window length of 20 R-peaks. The method of 'Quantification of inspiratory onsets in RR-intervals' had an even longer window length, provided that the window length of 10 inspiratory onsets is equivalent to approximately 40 R-peaks since the average ratio of heart rate and respiratory rate, irregardless of any cardiorespiratory coordination, is 4 . Thus, these methods are not as suitable for the detection of cardiorespiratory coordination as the method of 'Phase Recurrences'. Nevertheless, although all methods had different definitions and different implementation details, they revealed similar qualitative features of cardiorespiratory coordination during night-time sleep, i.e., the coordination diagrams showed approximately the same structure and the 'total coordination' (percentage of all coordinated R-peaks regardless of the m:n- ratio) was not constant but oscillated during night-time sleep. Since this oscillation correlates with heart rate variability (HRV)  and different sleep stages also correlate with HRV [54, 55], this oscillation may have its origin in the sleep architecture.
Another quantitative comparison was carried out by intra-individually correlating the 'coordination diagrams' of each method. The correlation coefficients served as a measure of similarity between the different diagrams. The strongest correlation was observed between the coordination diagrams derived from the 'Synchronization-λ' and the 'Quantification of Histograms' of relative distances φ i . Furthermore, the 'Phase Recurrences' of relative distances φ i and of absolute distances t i correlated strongly. These results indicate that the amount of information contained in the relative distances φ i , i.e. the relative timings of the R-peaks in a full respiratory cycle, and in the absolute distances t i , i.e. the timings of R-peaks with regard to the preceding inspiratory onset, was approximately the same. Furthermore, the results of the 'Synchronization-λ' showed strong correlations to the 'Phase Recurrences' and the 'Quantification of Histograms'. Thus, the latter methods, which have heuristic origins, may also be explained mathematically by the mathematical models used to develop the method of 'Synchronization-λ'. However, the correlation coefficients concerning the 'Quantification of Histograms' of relative distances β j , i.e. the relative timing of an inspiratory onset in the respective RR-interval, were lower. Thus, these distances seem to contain different information. This difference is attributed to the fact that the calculation of the relative distances β j is not based on the full amount of information available in the recording.
Taken these findings together, it is apparent that the different methods may lead to similar results if the maximum amount of information is used. It is important to keep in mind that all quantifications are based on the 'Post Event Time Series' or the 'Synchrogram'. Thus, although some methods have heuristic origins and some have a mathematical model as its origin and although the different approaches differ considerably with respect to their calculations, the relevant information of the 'Post Event Time Series' and the 'Synchrogram' is captured by the different methods.
A debate exits whether the cardiorespiratory coordination is due to physiological interaction between the involved systems or other mechanisms might be responsible, like e.g. a reduced variability in at least one of the involved systems [36, 56]. Surrogate data have been used to give at least a quantitative answer to this debate [34, 38, 57]. The analysis of surrogate data suggests that cardiorespiratory coordination is sometimes due to some kind of physiological interaction. Another approach that may be of importance is the stochastic phase synchronization, i.e. the synchronization between coupled systems due to stochastic stimuli [58, 59]. However, this kind of mechanism has not been explored for the cardiorespiratory system and the implications are not clear yet. Irrespective of this debate, it is an intriguing phenomenon that heart rate and respiratory frequency show a 4:1-ratio on average during night-time sleep [35, 60]. The present results also indicate that there seem to be some constraints with respect to variability and other features of the involved systems to achieve certain m:n-ratios. Further methods and models have to be developed to explore these features of the cardiorespiratory system.
In conclusion, the different methods to detect cardiorespiratory coordination show similar qualitative results. The method 'Phase Recurrences' has the ability to detect most sequences of heart beats coordinated with respiration. It is able to maximize the temporal resolution because even very short sequences may be detected. Thus, this method should be preferred. The method of 'Quantification of Histograms' of relative distances β j contains different quantitative information since only parts of the full amount of information available is used. Hence, this method is not recommended because a comparison with the other methods is limited. Generally, using appropriate methods, cardiorespiratory coordination is detectable in approximately 20–25% of all heart beats during night-time sleep. Future work should focus on the prerequisites of the appearance of cardiorespiratory coordination during night-time sleep and whether this appearance is linked to sleep stages. Furthermore, a preliminary study has shown the loss of cardiorespiratory coordination in patients after acute myocardial infarction [61, 62]. Hence, the gain of prognostic information of this bivariate analysis should be explored further.
DC and HB acknowledge financial support from the Weleda AG, Schwäbisch Gmünd, Germany. The authors would like to thank Roland Zerm and Matthias Kröz for providing data.
- Task Force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology: Heart rate variability: standards of measurement, physiological interpretation, and clinical use. Circulation 1996, 93: 1043–1065.View ArticleGoogle Scholar
- Pincus S, Cummins TR, Haddad GG: Heart rate control in normal and aborted-SIDS infants. Am J Physiol 1993, 264: R638-R646.Google Scholar
- Pincus SM: Approximate entropy in cardiology. Herzschr Elektrophys 2000, 11: 139–150. 10.1007/s003990070033View ArticleGoogle Scholar
- Pincus S: Approximate entropy (ApEn) as a complexity measure. Chaos 1995, 5: 110–117. 10.1063/1.166092View ArticleGoogle Scholar
- Wessel N, Schirdewan A, Malik M, Voss A: Symbolic dynamics – an independent method for detecting nonlinear phenomena of heart rate regulation. Biomed Tech (Berl) 1998,43(Suppl):510–511.Google Scholar
- Wessel N, Ziehmann C, Kurths J, Meyerfeldt U, Schirdewan A, Voss A: Short-term forecasting of life-threatening cardiac arrhythmias based on symbolic dynamics and finite-time growth rates. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 2000, 61: 733–739. 10.1103/PhysRevE.61.733Google Scholar
- Wessel N, Voss A, Kurths J, Schirdewan A, Hnatkova K, Malik M: Evaluation of renormalised entropy for risk stratification using heart rate variability data. Med Biol Eng Comput 2000, 38: 680–685.View ArticleGoogle Scholar
- Ashkenazy Y, Ivanov PC, Havlin S, Peng CK, Goldberger AL, Stanley HE: Magnitude and sign correlations in heartbeat fluctuations. Phys Rev Lett 2001, 86: 1900–1903. 10.1103/PhysRevLett.86.1900View ArticleGoogle Scholar
- Peng CK, Havlin S, Hausdorff JM, Mietus J, Stanley HE, Goldberger AL: Fractal mechanisms and heart rate dynamics: Long-range correlations and their breakdown with disease. J Electrocardiol 1995, 28 Suppl: 59–65.View ArticleGoogle Scholar
- Bunde A, Havlin S, Kantelhardt JW, Penzel T, Peter HJ, Voigt K: Correlated and uncorrelated regions in heart-rate fluctuations during sleep. Phys Rev Lett 2000, 85: 3736–3739. 10.1103/PhysRevLett.85.3736View ArticleGoogle Scholar
- Akselrod S, Gordon D, Ubel FA, Shannon DC, Berger AC, Cohen RJ: Power spectrum analysis of heart rate fluctuation: a quantitative probe of beat-to-beat cardiovascular control. Science 1981, 213: 220–222.View ArticleGoogle Scholar
- Makikallio TH, Huikuri HV, Hintze U, Videbaek J, Mitrani RD, Castellanos A, Myerburg RJ, Moller M: Fractal analysis and time- and frequency-domain measures of heart rate variability as predictors of mortality in patients with heart failure. Am J Cardiol 2001, 87: 178–182. 10.1016/S0002-9149(00)01312-6View ArticleGoogle Scholar
- Voss A, Hnatkova K, Wessel N, Kurths J, Sander A, Schirdewan A, Camm AJ, Malik M: Multiparametric analysis of heart rate variability used for risk stratification among survivors of acute myocardial infarction. Pacing Clin Electrophysiol 1998, 21: 186–192.View ArticleGoogle Scholar
- Porta A, Baselli G, Lombardi F, Montano N, Malliani A, Cerutti S: Conditional entropy approach for the evaluation of the coupling strength. Biol Cybern 1999, 81: 119–129. 10.1007/s004220050549View ArticleGoogle Scholar
- Terry JR, Breakspear M: An improved algorithm for the detection of dynamical interdependence in bivariate time-series. Biol Cybern 2003, 88: 129–136. 10.1007/s00422-002-0368-4View ArticleGoogle Scholar
- Baselli G, Cerutti S, Civardi S, Liberati D, Lombardi F, Malliani A, Pagani M: Spectral and cross-spectral analysis of heart rate and artrial blood pressure variability signals. Comput Biomed Res 1986, 19: 520–534. 10.1016/0010-4809(86)90026-1View ArticleGoogle Scholar
- Palus M, Komarek V, Hrncir Z, Sterbova K: Synchronization as adjustment of information rates: detection from bivariate time series. Phys Rev E Stat Nonlin Soft Matter Phys 2001, 63: 046211. 10.1103/PhysRevE.63.046211View ArticleGoogle Scholar
- Schäfer C, Rosenblum MG, Kurths J, Abel HH: Heartbeat synchronized with ventilation. Nature 1998, 392: 239–240. 10.1038/32567View ArticleGoogle Scholar
- Schäfer C, Rosenblum MG, Abel HH, Kurths J: Synchronization in the human cardiorespiratory system. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 1999, 60: 857–870. 10.1103/PhysRevE.60.857Google Scholar
- Hoyer D, Hoyer O, Zwiener U: A new approach to uncover dynamic phase coordination and synchronization. IEEE Trans Biomed Eng 2000, 47: 68–74. 10.1109/10.817621View ArticleGoogle Scholar
- Kotani K, Takamasu K, Ashkenazy Y, Stanley HE, Yamamoto Y: Model for cardiorespiratory synchronization in humans. Phys Rev E Stat Nonlin Soft Matter Phys 2002, 65: 051923. 10.1103/PhysRevE.65.051923View ArticleGoogle Scholar
- Rosenblum MG, Cimponeriu L, Bezerianos A, Patzak A, Mrowka R: Identification of coupling direction: application to cardiorespiratory interaction. Phys Rev E Stat Nonlin Soft Matter Phys 2002, 65: 041909.View ArticleGoogle Scholar
- Rzeczinski S, Janson NB, Balanov AG, McClintock PV: Regions of cardiorespiratory synchronization in humans under paced respiration. Phys Rev E Stat Nonlin Soft Matter Phys 2002, 66: 051909. 10.1103/PhysRevE.66.051909View ArticleGoogle Scholar
- Stefanovska A, Haken H, McClintock PVE, Hozic M, Bajrovic F, Ribaric S: Reversible transitions between synchronization states of the cardiorespiratory system. Phys Rev Lett 2000, 85: 4831–4834. 10.1103/PhysRevLett.85.4831View ArticleGoogle Scholar
- Stutte KH, Hildebrandt G: Untersuchungen über die Koordination von Herzschlag und Atmung beim Menschen. Pflugers Arch 1966, 289: R47-R48.Google Scholar
- Pessenhofer H, Kenner T: Method for the continuous measurement of the phase relation between heart beat and respiration. Pflugers Arch 1975, 355: 77–83. 10.1007/BF00584801View ArticleGoogle Scholar
- Raschke F: Coordination in the circulatory and respiratory systems. In Temporal disorder in human oscillatory systems. Edited by: Rensing L, an der Heiden U, Mackey MC. Berlin: Springer; 1987:152–158.View ArticleGoogle Scholar
- Hildebrandt G: The autonomous time structure and its reactive modifications in the human organism. In Temporal disorder in human oscillatory systems. Edited by: Rensing L, an der Heiden U, Mackey MC. Berlin: Springer; 1987:160–175.View ArticleGoogle Scholar
- Raschke F: The respiratory system – features of modulation and coordination. In Rhythms in physiological systems. Edited by: Haken H, Koepchen HP. Berlin: Springer; 1991:155–164.View ArticleGoogle Scholar
- Rosenblum MG, Pikovsky AS, Kurths J: Phase synchronization of chaotic oscillators. Phys Rev Lett 1996, 76: 1804–1807. 10.1103/PhysRevLett.76.1804View ArticleGoogle Scholar
- Rosenblum MG, Kurths J: Analysing synchronization phenomena from bivariate data by means of the Hilbert transform. In Nonlinear analysis of physiological data. Edited by: Kantz H, Kurths J, Mayer-Kress G. Berlin: Springer; 1998:91–99.View ArticleGoogle Scholar
- Rosenblum MG, Pikovsky AS, Schäfer C, Tass P, Kurths J: Phase synchronization: from theory to data analysis. In Neuro-Informatics and Neural Modeling. Edited by: Moss F, Gielen S. Amsterdam: Elsevier Science; 2001:279–321.View ArticleGoogle Scholar
- Hoyer D, Bauer R, Walter B, Zwiener U: Estimation of nonlinear couplings on the basis of complexity and predictability – A new method applied to cardiorespiratory coordination. IEEE Trans Biomed Eng 1998, 45: 545–552. 10.1109/10.668739View ArticleGoogle Scholar
- Seidel H, Herzel H: Analyzing entrainment of heartbeat and respiration with surrogates. IEEE Eng Med Biol Mag 1998, 17: 54–57. 10.1109/51.731321View ArticleGoogle Scholar
- Bettermann H, Cysarz D, Van Leeuwen P: Detecting cardiorespiratory synchronization by respiratory pattern analysis of heart period dynamics: the musical rhythm approach. Int J Bifurcat Chaos 2000, 10: 2349–2360. 10.1142/S021812740000150XGoogle Scholar
- Bettermann H, Cysarz D, Van Leeuwen P: Comparison of two different approaches in the detection of intermittent cardiorespiratory coordination during night sleep. BMC Physiol 2002, 2: 18. 10.1186/1472-6793-2-18View ArticleGoogle Scholar
- Palus M: Detecting phase synchronization in noisy systems. Phys Lett A 1997, 235: 341–351. 10.1016/S0375-9601(97)00635-XMathSciNetView ArticleGoogle Scholar
- Palus M, Hoyer D: Detecting nonlinearity and phase synchronisation with surrogate data. IEEE Eng Med Biol Mag 1998, 17: 40–45. 10.1109/51.731319View ArticleGoogle Scholar
- Janson NB, Balanov AG, Anishchenko VS, McClintock PV: Phase synchronization between several interacting processes from univariate data. Phys Rev Lett 2001, 86: 1749–1752. 10.1103/PhysRevLett.86.1749View ArticleGoogle Scholar
- Janson NB, Balanov AG, Anishchenko VS, McClintock PVE: Phase relationships between two or more interacting processes from one-dimensional time series. II. Application to heart-rate-variability data. Phys Rev E Stat Nonlin Soft Matter Phys 2002, 65: 036212. 10.1103/PhysRevE.65.036212View ArticleGoogle Scholar
- Janson NB, Balanov AG, Anishchenko VS, McClintock PVE: Phase relationships between two or more interacting processes from one-dimensional time series. I. Basic theory. Phys Rev E Stat Nonlin Soft Matter Phys 2002, 65: 036211. 10.1103/PhysRevE.65.036211View ArticleGoogle Scholar
- Bracic-Lotric M, Stefanovska A: Synchronization and modulation in the human cardiorespiratory system. Physica A 2000, 283: 451–461. 10.1016/S378-437(00)00204-1View ArticleGoogle Scholar
- Bruns A: Fourier-, Hilbert- and wavelet-based signal analysis: are they really different approaches? J Neurosci Methods 2004, 137: 321–332. 10.1016/j.jneumeth.2004.03.002View ArticleGoogle Scholar
- Galletly DC, Larsen PD: The determination of cardioventilatory coupling from heart rate and ventilatory time series. Res Exp Med (Berl) 1999, 199: 95–99. 10.1007/s004330050136View ArticleGoogle Scholar
- Galletly DC, Larsen PD: Inspiratory timing during anaesthesia: a model of cardioventilatory coupling. Br J Anaesth 2001, 86: 777–788. 10.1093/bja/86.6.777View ArticleGoogle Scholar
- Seidel H: Nonlinear dynamics of physiological rhythms: the baroreflex. Berlin: Logos; 1998.Google Scholar
- Mrowka R, Patzak A, Rosenblum M: Quantitative analysis of cardiorespiratory synchronization in infants. Int J Bifurcat Chaos 2000, 10: 2479–2488. 10.1142/S0218127400001754Google Scholar
- Tass P, Rosenblum MG, Weule J, Kurths J, Pikovsky AS, Volkmann J, Schnitzler A, Freund HJ: Detection of n:m phase locking from noisy data: application to magnetoencephalography. Phys Rev Lett 1998, 81: 3291–3294. 10.1103/PhysRevLett.81.3291View ArticleGoogle Scholar
- Fisher NI: Statistical analysis of circular data. Cambridge: Cambridge University Press; 1995.Google Scholar
- Toledo E, Rosenblum MG, Schäfer C, Kurths J, Akselrod S: Quantification of cardiorespiratory synchronization in normal and heart transplant subjects. In Proc of the Int Symposium on Nonlinear Theory and Its Applications. Volume 1. Lausanne: Presses Polytechniques et Universitaires Romandes; 1998:171–174.Google Scholar
- Galletly DC, Larsen PD: Cardioventilatory coupling during anaesthesia. Br J Anaesth 1997, 79: 35–40.View ArticleGoogle Scholar
- Galletly DC, Larsen PD: Relationship between cardioventilatory coupling and respiratory sinus arrhythmia. Br J Anaesth 1998, 80: 164–168.View ArticleGoogle Scholar
- Galletly DC, Larsen PD: Cardioventilatory coupling in heart rate variability: methods for qualitative and quantitative determination. Br J Anaesth 2001, 87: 827–833. 10.1093/bja/87.6.827View ArticleGoogle Scholar
- Versace F, Mozzato M, De Min Tona G, Cavallero C, Stegagno L: Heart rate variability during sleep as a function of the sleep cycle. Biol Psychol 2003, 63: 149–162. 10.1016/S0301-0511(03)00052-8View ArticleGoogle Scholar
- Elsenbruch S, Harnish MJ, Orr WC: Heart rate variability during waking and sleep in healthy males and females. Sleep 1999, 22: 1067–1071.Google Scholar
- Geue D, Van Leeuwen P, Lange S, Grönemeyer D: Simulation des Kopplungsverhaltens von Herzrhythmen zur Untersuchung der Phasensynchronisation. Biomed Tech (Berl) 2002, 47: 229–232.View ArticleGoogle Scholar
- Toledo E, Akselrod S, Pinhas I, Aravot D: Does synchronization reflect a true interaction in the cardiorespiratory system? Med Eng Phys 2002, 24: 45–52. 10.1016/S1350-4533(01)00114-XView ArticleGoogle Scholar
- Bahar S, Moss F: Stochastic phase synchronization in the crayfish mechanoreceptor'photoreceptor system. Chaos 2003, 13: 138–144. 10.1063/1.1501899MathSciNetView ArticleGoogle Scholar
- Neiman AB, Russel DF: Synchronization of noise-induced bursts in noncoupled sensory neurons. Phys Rev Lett 2002, 88: 138103. 10.1103/PhysRevLett.88.138103View ArticleGoogle Scholar
- Hildebrandt G: Reactive modifications of the autonomous time structure in the human organism. J Physiol Pharmacol 1991, 42: 5–27.Google Scholar
- Leder U, Hoyer D, Sommer M, Baier V, Haueisen J, Zwiener U, Figulla HR: Cardiorespiratory desynchronisation after acute myocardial infarction. Z Kardiol 2000, 89: 630–637. 10.1007/s003920070214View ArticleGoogle Scholar
- Hoyer D, Leder U, Hoyer H, Pompe B, Sommer M, Zwiener U: Mutual information and phase dependencies: measures of reduced nonlinear cardiorespiratory interactions after myocardial infarction. Med Eng Phys 2002, 24: 33–43. 10.1016/S1350-4533(01)00120-5View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.