Agnieszka Kaczkowska

Entropic Measures of Complexity of Short-Term Dynamics of Nocturnal Heartbeats in an Aging Population

Two entropy-based approaches are investigated to study patterns describing differences in time intervals between consecutive heartbeats. The first method explores matrices arising from networks of transitions constructed following events represented by a time series. The second method considers distributions of ordinal patterns of length three, whereby patterns with repeated values are counted as different patterns. Both methods provide estimators of dynamical aspects of short-term heartbeat signals obtained from nocturnal Holter electrocardiogram (ECG) recordings of healthy people of different ages and genders. The deceleration capacity, arising from the adjacency matrix of the network, and the entropy rate, resulting from the transition matrix of the network, are also calculated, and both significantly decay with aging. As people age, the permutation entropy grows, due to the increase in patterns with repeated values. All of these estimators describe in a consistent way changes in the beat-to-beat heart period dynamics caused by aging. An overall slowing down of heart period changes is observed, and an increase of permutation entropy results from the progressive increase of patterns with repeated values. This result points to the sympathetic drive becoming dominant in cardiac regulation of nocturnal heart rate with age.

Chronographic Imprint of Age-Induced Alterations in Heart Rate Dynamical Organization

Beat-to-beat changes in the heart period are transformed into a network of increments between subsequent RR-intervals, which enables graphical descriptions of short-term heart period variability. Three types of such descriptions are considered: (1) network graphs arising from a set of vertices and directed edges, (2) contour plots of adjacency matrices A, representing the networks and transition matrices T, resulting from A, and (3) vector plots of gradients of the matrices A and T. Two indices are considered which summarize properties of A and T: the approximate deceleration capacity and the entropy rate. The method, applied to time series of nocturnal RR-intervals recorded from healthy subjects of different ages, reveals important aspect of changes in the autonomic activity caused by biological aging. Independent of the subject’s age, following accelerations, a pendulum-like dynamics appears. With decelerations, this dynamics develops in line with the subject’s age. This aging transition can be graphically visualized by vectors connecting the maxima of the transition probabilities of T, which, metaphorically, resemble a chronometer or the hands of a clock.

Entropy Measures in the Assessment of Heart Rate Variability in Patients with Cardiodepressive Vasovagal Syncope

Sample entropy (SampEn) was reported to be useful in the assessment of the complexity of heart rate dynamics. Permutation entropy (PermEn) is a new measure based on the concept of order and was previously shown to be accurate for short, non-stationary datasets. The aim of the present study is to assess if SampEn and PermEn obtained from baseline recordings might differentiate patients with various outcomes of the head-up tilt test (HUTT). Time-domain heart rate variability (HRV) indices and several nonlinear parameters were calculated using 500 RR interval-long ECG recordings done before tilting in patients with a history suggesting vasovagal syncope. Groups of patients with so-called cardiodepressive vasovagal syncope (VVS_2) during HUTT and patients who did not faint during the test were compared. Two types of HUT tests were analyzed: with spontaneous (SB) or controlled breathing (CB). In our study, SampEn was higher in VVS_2 patients during SB, and PermEn was higher in VVS_2 patients during CB. Irrespective of the type of breathing during the test, SampEn and PermEn were similar in patients with the same type of reaction during HUTT. The use of several entropy-based parameters seems to be useful in HRV assessment in patients with vasovagal fainting.

Generalised heart rate statistics reveal neurally mediated homeostasis transients

Distributions of accelerations and decelerations, obtained from increments of heart rate recorded during a head-up tilt table (HUTT) test provide short-term characterization of the complex cardiovascular response to a rapid controlled dysregulation of homeostasis. A generalised statistic is proposed for evaluating the neural reflexes responsible for restoring the homeostatic dynamics. An evaluation of the effects on heart rate of the neural regulation involved in achieving homeostasis indicates a distinction between vasovagal patients and healthy subjects who are not susceptible to spontaneous fainting. A healthy cardiovascular response to the HUTT test is identified in the sympathetic tone appropriately punctuated by vagal activity.

Ordinal pattern statistics for RR intervals during head-up tilt test in patients with the history of vasovagal syncope

We apply ordinal pattern analysis to quantify differences in distribution of patterns of length 3 and 4 in basal state and during head-up tilt test (HUTT) in patients with the history of syncope and positive (HUTT(+)) or negative (HUTT(-)) responses to the test. We identify the patterns related to prevalence of sympathetic or parasympathetic cardiac modulation as well as describe the relations between the response to the test and distribution of the patterns.

Network approach to increments of RR-intervals for visualization of dynamics of cardiac regulation

The transition network for RR-increments is presented in a directed and weighted graph, with vertices representing RR-increments and edges corresponding to the order in a sequence of increments. The adjacency matrix and the transition matrix of this network provide a graphical tool which could be useful in the assessment of cardiac regulation. As an example, the method is applied in detecting differences between diurnal activity and nocturnal rest.

Temporal changes in complexity of cardiovascular regulation during head-up tilt test by entropic measures of fluctuations of heart period intervals and systolic blood pressure

Dynamical aspects of cardiovascular regulation in the head-up tilt test are revealed through the properties of the approximate entropy and sample entropy of increments between subsequent heart period intervals and increments between subsequent systolic blood pressure values. Marked differences are observed in the dynamical response to the tilt test of healthy subjects prone to spontaneous fainting and those who are less susceptible to fainting.

Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f.In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH2(M; ℚ) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.