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.