Coupling of local muscle deoxygenation and autonomic control depends on exercise intensity-insights from transfer entropy analysis.

OBJECTIVE: We analyzed the driving component between the periods of adjacent heartbeats (R-R intervals) and vastus lateralis-deoxygenation (%HHb) during incremental cycling. Considering a tight matching of local metabolism with systemic and local perfusion, a coupling between indices of cardiovascular control (R-R variability) and %HHb is suggested. Further, an intensity-dependent coupling between R-R variability and %HHb was hypothesized, because a multitude of feedback and feedforward mechanisms to autonomic cardiovascular control as well as local vasodilating mechanisms are associated with muscle metabolism and thus exercise intensity. APPROACH: Ten male triathletes (age: 34 ± 8 years) completed a test, including baseline (BAS, 50 W), a 25 W * min-1 ramp incremental phase until exhaustion and a recovery period (REC, 50 W). R-R intervals, %HHb and respiratory responses were simultaneously recorded. Five corresponding data segments were selected: BAS, before the first ventilatory threshold (preGET), between GET and the respiratory compensation point (preRCP), above RCP (postRCP), and REC. Bivariate transfer entropy (BTE) was applied to determine the signal coupling between R-R and %HHb. MAIN RESULTS: During preGET and preRCP, the analysis yielded the dominating direction from %HHb to R-R intervals, while for postRCP the direction was reversed. No significant signal coupling was detectable for the BAS and REC segments. SIGNIFICANCE: Assuming that %HHb is related to the metabolic state of the working muscle, BTE results support the role of metaboreceptors in the systemic blood flow regulation at lower exercise intensities, while other mechanisms (e.g. baroreceptor and mechanoreceptor feedback, central command) that modulate cardiovascular control may override this coupling at higher intensities.

A fixed mass method for the Kramers-Moyal expansion--application to time series with outliers.

Extraction of stochastic and deterministic components from empirical data-necessary for the reconstruction of the dynamics of the system-is discussed. We determine both components using the Kramers-Moyal expansion. In our earlier papers, we obtained large fluctuations in the magnitude of both terms for rare or extreme valued events in the data. Calculations for such events are burdened by an unsatisfactory quality of the statistics. In general, the method is sensitive to the binning procedure applied for the construction of histograms. Instead of the commonly used constant width of bins, we use here a constant number of counts for each bin. This approach-the fixed mass method-allows to include in the calculation events, which do not yield satisfactory statistics in the fixed bin width method. The method developed is general. To demonstrate its properties, here, we present the modified Kramers-Moyal expansion method and discuss its properties by the application of the fixed mass method to four representative heart rate variability recordings with different numbers of ectopic beats. These beats may be rare events as well as outlying, i.e., very small or very large heart cycle lengths. The properties of ectopic beats are important not only for medical diagnostic purposes but the occurrence of ectopic beats is a general example of the kind of variability that occurs in a signal with outliers. To show that the method is general, we also present results for two examples of data from very different areas of science: daily temperatures at a large European city and recordings of traffics on a highway. Using the fixed mass method, to assess the dynamics leading to the outlying events we studied the occurrence of higher order terms of the Kramers-Moyal expansion in the recordings. We found that the higher order terms of the Kramers-Moyal expansion are negligible for heart rate variability. This finding opens the possibility of the application of the Langevin equation to the whole range of empirical signals containing rare or outlying events. Note, however, that the higher order terms are non-negligible for the other data studied here and for it the Langevin equation is not applicable as a model.

Extraction of stochastic dynamics from time series.

We present a method for the reconstruction of the dynamics of processes with discrete time. The time series from such a system is described by a stochastic recurrence equation, the continuous form of which is known as the Langevin equation. The deterministic f and stochastic g components of the stochastic equation are directly extracted from the measurement data with the assumption that the noise has finite moments and has a zero mean and a unit variance. No other information about the noise distribution is needed. This is contrary to the usual Langevin description, in which the additional assumption that the noise is Gaussian (δ-correlated) distributed as necessary. We test the method using one dimensional deterministic systems (the tent and logistic maps) with Gaussian and with Gumbel noise. In addition, results for human heart rate variability are presented as an example of the application of our method to real data. The differences between cardiological cases can be observed in the properties of the deterministic part f and of the reconstructed noise distribution.

Stochastic analysis of heart rate variability and its relation to echocardiography parameters in hypertrophic cardiomyopathy patients.

The heart rate variability of 10 healthy males (age 26 - 4/+ 3 y) and 49 patients with hypertrophic cardiomyopathy (HCM) (25 males, 24 females, age 29.5 - 11.5/+ 10.5 y) was studied. We applied Kramers-Moyal expansion to extract the drift and diffusion terms of the Langevin equation for the RR interval time series. These terms may be used for a stochastic reconstruction of the time series and for description of the properties of heart rate variability. New parameters characterizing the diffusion term are proposed: the coefficients of the linear fit to the left (LCF) and right (RCF) branch of the dependence of the diffusion term on the rescaled heart rate. Relations of the new parameters to classical echocardiography parameters were studied. Using the relation between the difference LCF-RCF and the left ventricular systolic diameter, the HCM patients studied were divided into three groups. In addition, comparison of the properties of the heart rate variability in the HCM group with that obtained for the healthy young men showed that the parameter LCF-RCF may be treated as a measure of the effect of HCM on heart rate variability and may have diagnostic value.

Kramers-Moyal coefficients in the analysis and modeling of heart rate variability.

Modeling of recorded time series may be used as a method of analysis for heart rate variability studies. In particular, the extraction of the first two Kramers-Moyal coefficients has been used in this context. Recently, the method was applied to a wide range of signal analysis: from financial data to physiological and biological time series. Modeling of the signal is important for the prediction and interpretation of the dynamics underlying the process. The method requires the determination of the Markov time. Obtaining the drift and diffusion term of the Kramers-Moyal expansion is crucial for the modeling of the original time series with the Langevin equation. Both Tabar [Comput. Sci. Eng. 8, 54 (2006)] and T. Kuusela [Phys. Rev. E 69, 031916 (2004)] suggested that these terms may be used to distinguish healthy subjects from those with heart failure. The research groups applied a somewhat different methodology and obtained substantially different ranges of the Markov time. We show that the two studies may be considered consistent with each other as Kuusela analyzed 24 h recordings while Tabar analyzed daytime and nighttime recordings, separately. However, both groups suggested using the Langevin equation for modeling of time series which requires the fluctuation force to be a Gaussian. We analyzed heart rate variability recordings for ten young male (age 26-4+3 y ) healthy subjects. 24 h recordings were analyzed and 6-h-long daytime and nighttime fragments were selected. Similar properties of the data were observed in all recordings but all the nighttime data and seven of the ten 24 h series exhibited higher-order, non-negligible Kramers-Moyal coefficients. In such a case, the reconstruction of the time series using the Langevin equation is impossible. The non-negligible higher-order coefficients are due to autocorrelation in the data. This effect may be interpreted as a result of a physiological phenomenon (especially occurring for nighttime data): respiratory sinus arrhythmia (RSA). We detrended the nighttime recordings for the healthy subjects and obtained an asymmetry in the dependence of the diffusion term on the rescaled heart rate. This asymmetry seems to be an effect of different time scales during the inspiration and the expiration phase of breathing. The asymmetry was significantly decreased in the diffusion term found for detrended nighttime recordings obtained from five hypertrophic cardiomyopathy (HCM) patients. We conclude that the effect of RSA is decreased in the heart rate variability of HCM patients-a result which may contribute to a better medical diagnosis by supplying a new quantitative measure of RSA.