Teodor Buchner

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.

Symbolic dynamics and complexity in a physiological time series

Abstract A general approach to non-stationary data from a non-linear dynamical time series is presented. As an application, the RR intervals extracted from the 24 h electrocardiograms of 60 healthy individuals 16–64 yr of age are analyzed with the use of a sliding time window of 100 intervals. This procedure maps the original time series into a time series of the given complexity measure. The state of the system is then given by the properties of the distribution of the complexity measure. The relation of the complexity measures to the level of the catecholamine hormones in the plasma, their dependence on the age of the subject, their mutual correlation and the results of surrogate data tests are discussed. Two different approaches to analyzing complexity are used: pattern entropy as a measure of statistical order and algorithmic complexity as a measure sequential order in heart rate variability. These two complexity measures are found to reflect different aspects of the neuroregulation of the heart. Finally, in some subjects (usually younger persons) the two complexity measures depend on their age while in others (mostly older subjects) they do not – in which case the correlation between is lost.

On the nature of heart rate variability in a breathing normal subject: A stochastic process analysis.

Human heart rate is moderated by the autonomous nervous system acting predominantly through the sinus node (the main cardiac physiological pacemaker). One of the dominant factors that determine the heart rate in physiological conditions is its coupling with the respiratory rhythm. Using the language of stochastic processes, we analyzed both rhythms simultaneously taking the data from polysomnographic recordings of two healthy individuals. Each rhythm was treated as a sum of a deterministic drift term and a diffusion term (Kramers-Moyal expansion). We found that normal heart rate variability may be considered as the result of a bidirectional coupling of two nonlinear oscillators: the heart itself and the respiratory system. On average, the diffusion (noise) component measured is comparable in magnitude to the oscillatory (deterministic) term for both signals investigated. The application of the Kramers-Moyal expansion may be useful for medical diagnostics providing information on the relation between respiration and heart rate variability. This interaction is mediated by the autonomous nervous system, including the baroreflex, and results in a commonly observed phenomenon--respiratory sinus arrhythmia which is typical for normal subjects and often impaired by pathology.

Local time properties of 24-hour Holter recordings of RR intervals: a nonlinear quantitative method through pattern entropy of 3-D return maps

An easy to implement, consistent new measure of the complexity of heart rate variability has been developed. It is well suited for nonstationary data such as that of Holter ECG recordings and allows to assess the risk of cardiac arrest.

HRV strongly depends on breathing. Are we questioning the right suspect

The fact that the heart rate variability (HRV) depends on breathing is well known. The HRV is an important phenomenon which reflects the functional state of the autonomous nervous system (ANS), although there are some doubts concerning the actual interpretation of spectral components of HRV and their postulated balance. The assessment of the functional state of the ANS is the task of paramount importance in risk stratification of cardiological patients. HRV is considered to depend mainly on the properties of the sinus node (SN), which achieves neurohumoral input from the ANS. Interestingly, there is growing evidence that the relation between the heart rate (HR) and breathing rate (BR) is really strong. The variety of breathing-related effects that are present in HRV is very rich, including respiratory sinus arrhythmia (RSA), cardiorespiratory synchronization and vivid heart rate response to breathing disorders. If the mean frequency of any of rhythms is changed, the other rhythm adjusts itself. This provokes the question on the actual source of the dynamics observed in the HRV. Is it possible that we observe mainly the dynamics of the respiratory rhythm which is just transduced by the heart effector? What might be the role of the intrinsic dynamics of this effector? Is the RSA a product of neural regulation or rather a by-product: what is its teleological role? In consequence: if we concentrate on the sinus node and its properties in order to understand the nature of the HRV — are we questioning the right suspect? The reasoning is supplied by suitable choice of literature and by the analysis of the computational model. Various consequences are discussed.

Day-to-day reproducibility of Holter beat-by-beat analysis of repolarisation

Reproducibility of Holter QT analysis is not well established and has been assessed only in one study. Study design — We evaluated the day-to-day reproducibility of different QT parameters – mean and max (four beats basis) 24h QT and QTc (Bazett formula), QT for heart rate 55-60 [QT60], 75-80 [QT80] and 95-100 [QT100] beats/min and QT/RR slope (calculated in moving window of 3000 beats in 50 beat steps). QT intervals were measured from 48h digital ECG (sampled at 256 Hz) recordings using Del Mar Medical’s QT software in beat-to-beat fashion. The analysed group consisted of 6 women and 24 men – 13 patients with hypertrophic cardiomyopathy, 5 healthy family members of the patients with hypertrophic cardiomyopathy, 7 patients with CAD and 5 with other diseases (hypertension, arrhythmia, aborted sudden death without organic heart disease). Reproducibility was analysed with the methods proposed by Bland and Altman. Results — The overall reproducibility of repolarisation parameters was acceptable. Coefficient of reproducibility for mean 24h QT was 24ms, mean QTc 12ms, max QT 22ms, max QTc 24ms. The best reproducibility was observed for QT60, QT80 and QT100 – 12ms, respectively. The poorest day-to-day reproducibility was recorded for the QT/RR slope parameters, which was related to lower heart rate reproducibility. Conclusions — We can conclude that day-to-day reproducibility of Holter repolarisation analysis is acceptable. QT measurement in narrow heart rate windows has the best reproducibility. Accurate QT analysis requires good quality recording, T wave amplitude above 0.2mV and an interactive QT measurement tool which includes verification, editing abilities.

Local entropies as a measure of ordering in discrete maps

Abstract Local order measures are applied to a simple dynamical system: the logistic map. The results on pattern entropy are compared with those obtained using other, well-known order measures such as Renyi entropy, Shannon entropy and Lyapunov exponent (an analytical expression). Two modes of calculation are used: the global one and the local one (i.e. calculated in a sliding time window). The properties of local measures are discussed. It is concluded that the local measures defined here may be a natural tool for analysing non-stationary signals.

Complex activity patterns in arterial wall

Using a dynamical model of smooth muscle cells in an arterial wall, defined as a system of coupled five-dimensional nonlinear oscillators, on a grid with cylindrical symmetry, we compare the admissible activity patterns with those known from the heart tissue. We postulate on numerical basis the possibility to induce a stable spiral wave in the arterial wall. Such a spiral wave can inhibit the propagation of the axial calcium wave and effectively stop the vasomotion. We also discuss the dynamics of the circumferential calcium wave in comparison to rotors in venous ostia that are a common source of supraventricular ectopy. We show that the velocity and in consequence the frequency range of the circumferential calcium wave is by orders of magnitude too small compared to that of the rotors. The mechanism of the rotor is not likely to involve the calcium-related dynamics of the smooth muscle cells. The calcium-related dynamics which is voltage-independent and hard to be reset seems to actually protect the blood vessels against the electric activity of the atria. We also discuss the microreentry phenomenon, which was found in numerical experiments in the studied model.

Cardiorespiratory coupling in young healthy subjects

OBJECTIVE To quantify the presence of cardiorespiratory interaction in a group of 41 healthy subjects performing a subset of the Ewing test battery. APPROACH We measure the empirical distribution of the cardiorespiratory coupling time (RI), defined as the time from inspiration onset to R peaks in the ECG. The study protocol is a subset of the Ewing test battery. The respiratory function was measured with a thoracic belt and heart rate was obtained from a two channel ECG measurement. Both series of fiducial points were determined using custom software. Additionally, we determine the presence of cardiorespiratory coupling (CRC) and cardiorespiratory interaction (CRI) using Shannon entropy, synchrograms and coordigrams. MAIN RESULTS We observe that the RI distribution is asymmetric and nonuniform. These features are a manifestation of the causal relation between heart action and respiration. The preceding R peak strongly affects a position of inspiration onset. From the asymmetry of the RI distribution we conclude that this relation is stronger than the relation between inspiration onset and the following R peak. We use a suitable choice of surrogate data to prove that the result cannot be falsified. We observe a dual structure of the RI histograms, which may be related to the respiratory rhythmogenesis. We compare the sensitivity of RI histograms with other measures of CRI and CRC. In 46% of subjects, CRC appears in at least one stage of the examination, most often in resting states. In states of increased stress-orthostasis or physical (exercise)-the strength of coupling is visibly diminished. The nonuniform structure of the RI histogram is more sensitive to the presence of CRI than synchrograms or coordigrams are, as is well visible in the group averages. We also refer to the question of the most proper mathematical description of cardiorespiratory dynamics (phase domain or time domain). Finally, we formulate the hypothesis that the arterial blood pressure is a common driver of cardiac and respiratory rhythms. SIGNIFICANCE Analysis of the asymmetry of RI histograms is an interesting and sensitive method to study cardiorespiratory interaction and autonomic balance, in order to assess physical and mental health. The dual structure of the RI histograms which we have observed suggests the possible presence of a twofold mechanism for respiratory rhythmogenesis, as proposed by Galletly and Larsen.

Concealed conduction effects in the atrium

Here, we present new results obtained from a 1-D model of the atrium, with both the sinoatrial (SA) and atrioventricular (AV) nodes included. This model is able to reproduce forward and backward propagation between the SA and AV node. These nodes were approximated by a 1-D chain of diffusively coupled, modified relaxation oscillators. The atrial muscle was modeled using a chain of modified FitzHugh-Nagumo (FHN) equations. The FHN model captures the key features of excitable media and is widely used as a simple model of cardiac muscle electrical activity. The complete model consists of three segments: the SA node (15 elements), the atrial muscle (90 elements), and the AV node (15 elements) coupled diffusively at the interfaces and was solved numerically using the Euler method. The model is dimensionless, but the parameters were set in such a way that the period of the oscillations was numerically of the order of the length of RR intervals in human heart rate variability recordings. The model is able to reproduce the low-pass filtering properties of the SA node. The interspike intervals (ISIs) of the calculated action potentials of the AV node of our model were compared, with RR intervals obtained from two selected 24-h Holter recordings of patients of the Institute of Cardiology at Warszawa: one recorded in a 11-year-old girl with hypertrophic cardiomyopathy and a second one recorded in a patient with a possible SA block and no ventricular arrhythmia.