Formation of functional associations across time scales in the fetal autonomic control system--a multifractal analysis.

During fetal development, different control systems mediated by the autonomic nervous system form functional connections over a wide range of time scales. Using multiscale multifractal analysis (MMA) of fetal heart rate variability (HRV), we describe fundamental relationships in the developing scale-wide adjustments within fetal behavior states as well as across state changes. MMA yields the local Hurst exponent surface h(q,s) with q as the multifractal parameter and s as the scale. In 30-minute recordings of healthy fetuses between 24 and 36weeks of gestation (n=25 in quiet sleep, n=29 in active sleep, n=30 changing sleep state) we investigated the dependency of h(q,s) on gestation age. In univariate models, we found a decreasing persistence for short scales and small amplitudes in the quiet (s1=39, q1=-0.7, R(2)=0.52) and in the active (s1=69, q1=-1.4, R(2)=0.23) sleep in contrast to an increasing persistency for long scales and large amplitudes (s1=147, q1=2.4, R(2)=0.29) in the mixed state. Bivariate models (additional scales considered) presented increased coefficients of determination R(2)=0.56, 0.4, and 0.43, respectively. Persistency increasing with age in connection with the sleep state changes (independent of the age related short range dependencies within the separate homogeneous sleep states) is reported here for the first time. The MMA indices obtained for the fetal HRV represent characteristics of the maturating scale-wide cardiovascular control in the context of the evolving sleep state dynamics, which have so far not been considered. They should be incorporated in the search for HRV indices for prenatal diagnosis of developmental disorders and risk assessment.

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.

On the risk of aortic valve replacement surgery assessed by heart rate variability parameters.

In recent years the number of arterial stenosis (AS) patients has grown rapidly and valvular disease is expected to be the next great epidemic. We studied a group of 385 arterial valve replacement (AVR) surgery patients, of whom 16 had died in the postoperational period (up to 30 d after the operation). Each patient had a heart rate variability (HRV) recording made prior to the operation in addition to a full set of medical diagnostics including echocardiography. We formed 16 age, sex, New York Heart Association (NYHA) class, and BMI adjusted control pairs for each person who died in the perioperative period. Our aim was to find indications of the risk from AVR surgery based on the medical data and HRV properties. Besides standard, linear HRV methods, we used indexes of time irreversibility introduced by Guzik (G%), Porta (P%), Ehlers (index E) and Hou (index D). In addition, we analyzed the multiscale multifractal properties of HRV calculating the Hurst surface. The nonlinear analysis methods show statistically significant indications of the risk of AVR surgery in an increase of multifractality and an increase of time irreversibility of the HRV measured prior to the operation.

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.

Multiscale multifractal analysis of heart rate variability recordings with a large number of occurrences of arrhythmia.

Human heart rate variability, in the form of time series of intervals between heart beats, shows complex, fractal properties. Recently, it was demonstrated many times that the fractal properties vary from point to point along the series, leading to multifractality. In this paper, we concentrate not only on the fact that the human heart rate has multifractal properties but also that these properties depend on the time scale in which the multifractality is measured. This time scale is related to the frequency band of the signal. We find that human heart rate variability appears to be far more complex than hitherto reported in the studies using a fixed time scale. We introduce a method called multiscale multifractal analysis (MMA), which allows us to extend the description of heart rate variability to include the dependence on the magnitude of the variability and time scale (or frequency band). MMA is relatively immune to additive noise and nonstationarity, including the nonstationarity due to inclusions into the time series of events of a different dynamics (e.g., arrhythmic events in sinus rhythm). The MMA method may provide new ways of measuring the nonlinearity of a signal, and it may help to develop new methods of medical diagnostics.

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.

Spiral wave breakup in excitable media with an inhomogeneity of conduction anisotropy.

Many conditions remodel the heart muscle such that it results in a perturbation of cells coupling. The effect of this perturbation on the stability of the spiral waves of electrochemical activity is not clear. We used the FitzHugh-Nagumo model of an excitable medium to model the conduction of the activation waves in a two-dimensional system with inhomogeneous anisotropy level. Inhomogeneity of the anisotropy level was modeled by adding Gaussian noise to diffusion coefficients corresponding with lateral coupling of the cells. Low noise levels resulted in a stable propagation of the spiral wave. For large noise level conduction was not possible due to insufficient coupling in direction perpendicular to fibers. For intermediate noise intensities, the initial wave broke up into several independent spiral waves or waves circulating around conduction obstacles. At an optimal noise intensity, the number of wavelets was maximized-a form of anti-coherent resonance was obtained. Our results suggest that the inhomogeneity of conduction anisotropy may promote wave breakup and hence play an important role in the initiation and perpetuation of the cardiac arrhythmias.

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.

Integration of the data from electroanatomical mapping system and CT imaging modality.

Heart mapping systems allow approximate reconstruction of the heart chamber geometry which is used as a base for the representation of the spatial distribution of electrophysiological parameters. Main limitation lies in the difficulty of the reconstruction of the geometry of more complicated areas of the heart. Here, we propose a new method of representation of the spatial distribution of the electrophysiological parameters-an integration of the data points collected by a classical mapping system with the geometry reconstructed from a computed tomography (CT) image. CARTO maps of activation and bipolar viability of seven patients undergoing atrial fibrillation ablation were integrated with the geometry of the left atria reconstructed from the CT image. In all cases, integration was successful with the registration error measured as the distance between objects equal to 2.52 +/- 0.25 mm. Bipolar viability and activation maps were reconstructed on the CT geometry. Our method allowed us to create maps of electrophysiological parameters of anatomically complex structures without the need for their detailed mapping.

Nonlinear oscillator model reproducing various phenomena in the dynamics of the conduction system of the heart.

A dedicated nonlinear oscillator model able to reproduce the pulse shape, refractory time, and phase sensitivity of the action potential of a natural pacemaker of the heart is developed. The phase space of the oscillator contains a stable node, a hyperbolic saddle, and an unstable focus. The model reproduces several phenomena well known in cardiology, such as certain properties of the sinus rhythm and heart block. In particular, the model reproduces the decrease of heart rate variability with an increase in sympathetic activity. A sinus pause occurs in the model due to a single, well-timed, external pulse just as it occurs in the heart, for example due to a single supraventricular ectopy. Several ways by which the oscillations cease in the system are obtained (models of the asystole). The model simulates properly the way vagal activity modulates the heart rate and reproduces the vagal paradox. Two such oscillators, coupled unidirectionally and asymmetrically, allow us to reproduce the properties of heart rate variability obtained from patients with different kinds of heart block including sino-atrial blocks of different degree and a complete AV block (third degree). Finally, we demonstrate the possibility of introducing into the model a spatial dimension that creates exciting possibilities of simulating in the future the SA the AV nodes and the atrium including their true anatomical structure.

Association between short term and long term communication in pathological autonomic control.

Autonomic Information Flow (AIF) reflects the time scale dependence of autonomic communications such as vagal, sympathetic, and slower rhythms and their complex interplay. We investigated the hypothesis that pathologically disturbed short term control is associated with simplified complex long term control. This particular characteristic of altered autonomic communication was evaluated in different medical patient groups. Holter recordings were assessed in patients with multiple organ dysfunction (MODS) (26 survivors, 10 non-survivors); with heart failure (14 low risk-without history of aborted cardiac arrest (CA), 13 high risk--with history of CA); with idiopathic dilated cardiomyopathy (IDC) (26 low risk, 11 high risk of CA), after myocardial infarction (MI) (1221 low risk--survivors, 55 high risk--non-survivors); after abdominal aorta surgery (AAS, 32 length of stay in hospital LOS>7 days, 62 LOS < or =7 days). AIF of short and long time scales was investigated. We found a fundamental association of increased short term randomness and decreased long term randomness due to pathology. Concerning risk, high risk patients were characterized by increased short term complexity and decreased long term complexity in all patients groups with the exception of the IDC patients. We conclude that different time scales of AIF represent specific pathophysiological aspects of altered autonomic communication and control. The association of altered short term control with simplified long term behavior might be a pathophysiologically relevant compensation mechanism in the case of a disturbed fastest actuator. This knowledge might be useful for the development of comprehensive therapeutic strategies besides the predictive implications.

The reconstruction, from a set of points, and analysis of the interior surface of the heart chamber.

Adequate description of heart muscle electrical activity is essential for the proper treatment of cardiac arrhythmias. Contemporary mapping and ablating systems allow a physician to introduce an electrode (catheter) into the human heart, to measure the position of the electrode in space and, simultaneously, the electrical activity timing and the bipolar and unipolar signal amplitudes--which correspond to the electrical viability of the heart muscle. If enough data points are collected, an approximate reconstruction of the heart chamber geometry (anatomy) is possible using also surface data such as the viability and local activity isochrones. Myocardial viability in patients after myocardial infarction is crucial for understanding and treating life threatening arrhythmias. Although there are commercial tools for heart chamber reconstruction, they lack the ability to quantitatively analyse the reconstructed data. Here, we show a method of reconstruction of the left ventricle of the heart from a measured set of data points and perform an interpolation of the measured voltages over the reconstructed surface. Next, we detect regions with voltage in a specified range and compute their areas and circumferences. Our methods allowed us to quantitatively describe the 'normal' muscle, the damaged or scar areas and the border zones between healthy muscle and the scars. In particular, we are able to find geometries of the damaged muscle areas that may be dangerous, e.g. when two such areas lie close to each other creating an isthmus--a macroreentry arrhythmia substrate. This work was inspired by a clinical hypothesis that the size of the border zone corresponds to the rate of occurrence of ventricular arrhythmia in patients after myocardial infarction.

Direct observation of homoclinic orbits in human heart rate variability.

Homoclinic trajectories of the interbeat intervals between contractions of ventricles of the human heart are identified. The interbeat intervals are extracted from 24-h Holter ECG recordings. Three such recordings are discussed in detail. Mappings of the measured consecutive interbeat intervals are constructed. In the second and in some cases in the fourth iterate of the map of interbeat intervals homoclinic trajectories associated with a hyperbolic saddle are found. The homoclinic trajectories are often persistent for many interbeat intervals, sometimes spanning many thousands of heartbeats. Several features typical for homoclinic trajectories found in other systems were identified, including a signature of the gluing bifurcation. The homoclinic trajectories are present both in recordings of heart rate variability obtained from patients with an increased number of arrhythmias and in cases in which the sinus rhythm is dominant. The results presented are a strong indication of the importance of deterministic nonlinear instabilities in human heart rate variability.

Logistic map with a delayed feedback: Stability of a discrete time-delay control of chaos.

The logistic map with a delayed feedback is studied as a generic model. The stability of the model and its bifurcation scheme is analyzed as a function of the feedback amplitude and of the delay. Stability analysis is performed semianalytically. A relation between the delay and the periodicity of the orbit, which explains why some terms used in chaos control are ineffective, was found. The consequences for chaos control are discussed. The structure of bifurcations is found to depend strongly on the parity and on the length of the delay. Boundary crisis, the tangent, the Neimark, as well as the period-doubling bifurcations occur in this system. The effective dimension of the model is also discussed.

Symbolic dynamics analysis of nonstationary data from a model of a magnetic system with solitons.

The probability distribution of a complexity measure is used to characterize chaotic states: an estimator of the algorithmic complexity of a time series of symbolic words is calculated within a fixed length time window, which sweeps through the time series analyzed. The words are derived through a symbolic dynamics scheme applied in an m-dimensional delay coordinate space. Time intervals instead of the variables of the system are used. The chaotic states of a model of a magnetic domain wall are characterized better by the methods presented than with the use of fractal dimensions and new intermittent states of the system were easily identified. Using an artificial nonstationary signal composed of different chaotic states of the Bloch wall as a test for chaos-chaos intermittency we demonstrate that the method developed is suitable for the detection and characterization of intermittency. It is also shown that nonstationarity in the form of a slow monotonic drift in the control parameter may extend the stability range of periodic states of the spatially extended system studied-a trackinglike phenomenon.