Study of the cardiorespiratory interaction: a non-linear approach.
ERNESTO PEREDA1, JULIÁN GONZÁLEZ2, DULCE M. DE LA CRUZ2, LUIS DE VERA2
1Department of Basic Physics, University of La Laguna
Avda. Astrofísico Fco. Sánchez s/n, La Laguna, 38205, Tenerife, SPAIN
2Department of Physiology, University of La Laguna, Ctra. La Cuesta-Taco s/n, 38320, Tenerife, SPAIN
Abstract: - Multivariate non-linear analysis methods are used to study the interdependence between heart rate, blood pressure and respiratory activity in rats both in basal conditions and after the application of two different drugs affecting the cardiovascular system. The results showed that this approach can be used to explore changes in the interdependence between signals of different complexity, which might be very helpful in exploring the cardiovascular control system.
Key-Words: - Multivariate analysis, Non-linear time series methods, Phase synchronization, cardiovascular system, short term variability.
1 Introduction
The non-linear analysis of time series has been widely applied during the last decade to study several physiological systems [1]. In particular, the EEG and different signals from the cardiovascular control system (such as hear rate and blood pressure variability) are among the most commonly analyzed. The idea behind applying this new methodology is assessing not only the main features of the signal itself, but only to gather information about the system that produces it.
Historically, the first branch developed was the characterization of individual signals (henceforth, univariate analysis), in which only one signal was studied to quantify its main non-linear features. The use of tests for nonlinearity allowed solving the problem of complex but otherwise stochastic signals that fooled the non-linear algorithms, thus producing spurious results [2]. The univariate analysis proved useful, for instance, in searching for differences among groups of subjects and among different experimental conditions. However, the idea of jointly analyzing two or more signals coming from interacting subsystems is more closely related to the concept of dynamical systems. In this approach (henceforth, multivariate analysis), it is the very interaction among the different variables what is investigated, to understand how changes in this interaction might affect the performance of the system.
The multivariate analysis developed later than the univariate one, but the current trend seems to favor the former rather than the later. The finding that even complex non-linear system undergoing chaotic regime might synchronize has boosted the study of the interdependence between natural systems by using non-linear analysis methods in the state space. In this context, different synchronization indexes have been proposed to measure the degree of interdependence between physiological signals. These nonlinear indexes excelled the linear ones because the former are sensitive to any kind of interdependences and asymmetric (i.e., they give information about the directionality of the interdependence). They have been mainly applied to multivariate EEGs [3], but there have been no applications to the cardiovascular system, which might be due to the fact that the indexes present some problems when the signals under study are of different complexity [3,4]. However, recent refinements of the methodology have allowed overcoming this drawback [3], thus paving the way for applications in cardiovascular physiology.
On the other hand, the concept of phase synchronization has also been applied successfully in searching for synchronous regimes between nearly periodic physiological signals [5]. This idea has been used to explore the cardiorespiratory interaction mainly in humans. Although the early indexes proposed for this aim have the drawback of being symmetric, the latest ones give information about the directionality of the interdependence as well as its strength [5].
In this paper we apply both methods for the analysis of the interdependence between the cardiac interval and the respiratory signal in rats in different experimental situations. By making use of the newest indexes from both approaches we aiming at getting further insight not only on the cardiovascular system itself but also on the applicability of these methodologies to the studied signals.
2 Material and methods
The EKG and the simultaneous respiratory signal respiratory signal were recorded from Sprague Dawley male rats during basal condition as well as after the administration of atropine (parasympathetic blockade) and propranolol (unspecific b-sympathetic blockade). The complete recording procedure is described elsewhere [6]. From the EKG, the values of the RR intervals were obtained by using appropriated software. Artifact-free stationary segments of 1024 RR intervals and the simultaneous respiratory amplitude (RP signal) where firstly chosen for a total 10 rat in each experimental situation. Because the synchronization index is intended to study the interdependence between nearly periodic signals, instead of carrying out the analysis on the raw data we first filtered the data using a high pass filter (cut-off frequency 0.9 Hz) to obtain narrow band signals in the high frequency band (0.9-2.5 Hz), where the main respiratory component of the rat is situated.
2.1 The index N
As commented in the Introduction, the concept of generalized synchronization can be used to quantify the degree of interdependence between two signals in their state spaces. Briefly, if a set of state vectors are close in the state space of signal X, then the simultaneous state vectors of Y must be also close in its state space due to the synchronization. Therefore, by assessing the average closeness of all the state vectors of X to their mutual neighbors (those vectors in X bearing the same time indexes of the nearest neighbors of Y) it is possible to quantify statistically whether there is any interdependence between the signals. Recent results demonstrate that the index N(X|Y) is the most robust tool for this purpose [4]. N ranges from 1 (identical signals) to 0 (independent signals), and has the advantage of being relatively independent of the complexity of signal, so that N(X|Y)¹ N(Y|X) might be interpreted as the existence of directionality in the interdependence (with N(X|Y)> N(Y|X) suggesting that X is the source of activity and therefore influences more on Y than vice versa [3,4]).
2.2 The index of phase synchronization
The assessment of the phase synchronization between experimental signals is a two-step process. First, it is necessary to obtain the phase associated to the signals, and then to quantify the degree of synchronization between the phases. Different strategies can be followed to obtain the phases of a signal, among which the Hilbert transform is one of the most used. With this strategy, a real signal is turned into a complex one, so that we obtain a value of the phase for each sample. The assessment of the degree of synchronization is then carried out by following the procedure detailed in a recent paper [5], where the dependence of the phase increments of the signal on the phase of the other is supposed to be weak and mathematically modeled by means of a function, which is periodic in both phases. By integrating this function is possible to obtain a directionality index d1,2 that ranges from 1 (unidirectional coupling 1®2) to -1 (unidirectional coupling 1¬2).
2.3 Statistical comparisons
The values of the index d1,2 before and after the administration of the drugs were compared by means of a t test for dependent samples. The values of the indexes N(X|Y) and N(Y|X) were compared by using an ANOVA test for repeated measures and a Scheffe post-hoc test when necessary. Differences were considered significant if P<0.05.
3 Results
The results for the d1,2 index between the RR signal and the respiratory are shown in figure 1. The values in basal condition are slightly negative, indicating that in this situation the high frequency component of the RR variability is modulated by the respiratory signal. The administration of the parasympathetic blockade changes this scenario, with the index becoming positive (left panel). The unspecific b-sympathetic blockade, however, does not manage to produce any significant change (right panel).
As for the N index, results are shown in figures 2 and 3. In basal condition, we found N(RR|RP)<N(RP|RR), suggesting that the RR signal depends more on the respiratory signal that vice versa. The parasympathetic blockade changed this, drastically decreasing the values of the index (figure 2). Further, the asymmetry in the interdependence is also destroyed after atropine. The b-sympathetic blockade had a less pronounced effect, where only the asymmetry disappeared, but the value of the indexes prior and after propranolol where similar (figure 3).
4 Conclusions
It is a well-known fact that the respiration modulates the heart rate in basal condition (the so-called sinus arrhythmia, SA). This modulation is mediated by the sympathetic and parasympathetic nervous system in a balanced way. The SA has been used in the context of non-linear analysis to study the degree of maturation in healthy newborns along time [5]. Our purpose was to use two recently developed indexes for the assessment of synchronization between experimental signals in order to check whether they were able to detect and quantify the SA in the basal state. Further, we aimed at determining how the SA changed in response to the administration of two different blockades affecting separately the two systems that mediate it.
In effect, both indexes show that the cardiorespiratory interaction in basal condition is markedly unidirectional, with the respiratory signal modulating the RR cardiac interval. Additionally, they also agree that it is the parasympathetic blockade the one affecting more the interdependence between the signals, with the unspecific b-blockade playing a less significant role. This is an expected result, in agreement with the current knowledge in physiology, where the cardiorespiratory interaction in mammals is supposed to be mediated mainly by the sympathetic nervous systems. The main difference between the indexes is that the value of the phase synchronization one (d12) summarizes the information about the directionality, whereas the state space index N must be calculated in both signals to determine whether there is asymmetric coupling. In any case, they seem to perform very well in identifying both the direction and the degree of cardiorespiratory interaction and might be used to study the interdependence between cardiovascular signals in other situations.
References:
[1] Elbert T, Ray WJ, Kowalik ZJ, Skinner JE, Graf KE, Birbaumer N, Chaos and physiology: deterministic chaos in excitable cell assemblies, Physiol. Rev., 74, 1994, pp. 1-47.
[2] Theiler J, Rapp P, Re-examination of the evidence for low-dimensional, nonlinear structure in the human electroencephalogram, Electroenceph. clin. Neurophysiol., 98, 1996, pp. 213-222.
[3] Quiroga RQ, Kraskov A, Kreuz T, Grassberger P, Performance of different synchronization measures in real data: A case study on electroencephalographic signals, Phys. Rev. E 65, 2002, 041903.
[4] Quiroga RQ, Arnhold J, Grassberger P, Learning drive-response relationships from synchronization patterns, Phys. Rev. E 61, 2000, pp.5142-5148.
[5] Rosenblum MG, Cimponeriu L, Bezerianos A, Patzak A, Mrowka R, Identification of coupling direction: Application to cardiorespiratory interaction, Phys. Rev. E 65, 2002, 041909.
[6] González J, Cordero JJ, Feria M, Pereda E, Detection and sources of non-linearity in the variability of cardiac R-R intervals and blood pressure in rats, Am. J. of Physiol., 279, 2000, pp. H3040-3046.