Mathematical model of heart rate variability

The study presents a mathematical model of non-linear dynamics of the heart rate variability (HRV). The model is based on quantitative characteristics of pulse conduction in the heart conducting system: the delays of sinoatrial (SA) and atrioventricular (AV) pulse conduction and refractors periods of the SA and AV nodes. The model predicts heart rate disturbances in fast electric activity of the atria, increase in the delay of the AV conduction, the critical value of atrial period where transition to non-linear dynamics of the heart rate variability starts. The correlation between indexes of HRV and period of stimulation of atria for 1-contour cardiac control model has been demonstrated.     

Analysis of changes in heart rate variability during fast extracardial impulsation in the sinoatrial node

It is known that fast extracardial impulsation in the sinoatrial node modifies the degree of heart rare variability. The present study presents theoretical and experimental investigation of this effect. Theoretical investigation is based on the mathematical modelling of impulse conduction in the cardiac conduction system. Experimental investigation on dogs revealed quantitative correlations between the frequency of extracardial impulsation in the sinoatrial node and changes of heart rate variability. Computer simulation shows that the mathematical model can account for most principal properties of the heart rate disturbances during fast extracardial impulsation in the sinoatrial node: transitions between different regimens of heart rate dynamics; increase in the delay of atrio-ventricular conduction; Wenkebach's periodicity.     

Analysis of stability of different regimes of heart rate dynamics by computer modeling

Computer modeling revealed the following three regimes of heart rate dynamics: linear dynamics, “1st degree chaos,” and “2nd degree chaos.” This study investigated a stability of these regimes with respect to changes in initial conditions. The results show that the greatest stability is notable for the linear regime. For this regime small errors in values of initial conditions can not sharply change the initial dynamics of RR intervals. Both nonlinear regimes of heart rate dynamics are unstable, and a degree of instability of regime “2nd degree chaos” is higher in comparison with regime “1st degree chaos.” The results of computer modeling are in agreement with experimental data pointing to the existence of a relationship between the degree of heart rate irregularity and cardiac electrical stability.     

Effect of emotional stress on the cardiac rhythm variability in rats

A degree of irregularities of the heart rhythm was studied by two methods: chaos-analysis and the HRV (heart rate variability) analysis. Our study shows an individual response in 3 groups of animals: 1--animals with low initial level of chaos (correlation dimension (PD2 < 2); 2--animals with high level of chaos (PD2 > 4); and animals with middle level of chaos (2 < PD2 < 4). The first two groups proved to be more sensitive to stress than the third group. Moreover we found that the electrical stability of the heart as measured by the fibrillation threshold, was higher for the chaos third group. The animals of the first two groups had low cardiac stability and high risk of stress-induced cardiac disturbances.     

Theoretical and clinical assessment of the sensitivity of heart rate variability indices

The application of modern methods of mathematical processing of non-stationary quasi-periodic data to the analysis of heart-rate variability is considered. Methods for the assessment of new parameters in non-linear variability analysis are described in detail. Mathematical models of heart rhythm are developed with the presence of various noise processes taken into account. A model of the state of the cardiovascular system based on the analysis of heart-rate variability has been developed. A theoretical estimate of the sensitivity of heart-rate variability indices to changes in the state of the cardiovascular system has been obtained for model data. Clinical studies of the parameters of heart-rate variability included in the analysis have been performed within the framework of cardiological screening for coronary heart disease.     

基于复杂度的心率变异性分析研究进展

心率变异性（HRV）是当前心电图分析的一个前沿热点,它反映了交感神经和副交感神经对心血管系统的综合调节作用,是评价心血管系统功能的重要指标。随着非线性动力学和复杂性科学的发展,HRV信号被普遍认为是混沌或含有混沌成分的信号。复杂度是用来表征一个心率非线性动力学系统复杂程度的量度,以其简单快速的优点引起了众多研究者的兴趣,并广泛应用于心率变异性分析。本文综述了国内外复杂度算法的研究进展及基于复杂度的心率变异性分析的临床应用及前景。     

Parameters of atrioventricular conduction and stability of different regimes of heart rate dynamics

By method of computer modeling we investigated a stability of different regimes of heart rate dynamics in relation to the change in the atrioventricular conduction parameters: refractory period, minimum atrioventricular delay and curvatures of delay functions in sinoatrial and atrioventricular nodes. It is shown that curvatures of delay functions in sinoatrial and atrioventricular nodes have the most significant influence on the stability of different regimes of heart rate dynamics. The minimum delays in sinoatrial and atrioventricular nodes have smaller impact. The parameters determining the refractory periods of sinoatrial and atrioventricular nodes are least significant in terms of stability of heart rate dynamics.     

Testing a model for the dynamics of actin structures with biological parameter values

A simple mathematical model for the dynamics of network-bundle transitions in actin filaments has been previously proposed and some of its mathematical properties have been described. Other models in this class have since been considered and investigated mathematically. In this paper, we have made the first steps in connecting parameters in the model with biologically measurable quantities such as published values of rate constants for filament-crosslinker association. We describe how this connection was made, and give some preliminary numerical simulation results for the behavior of the model under biologically realistic parameter regimes. A key result is that filament length influences the bundle-network transition.     

Routes to chaos in a model of a bursting neuron.

Chaotic regimens have been observed experimentally in neurons as well as in deterministic neuronal models. The R15 bursting cell in the abdominal ganglion of Aplysia has been the subject of extensive mathematical modeling. Previously, the model of Plant and Kim has been shown to exhibit both bursting and beating modes of electrical activity. In this report, we demonstrate (a) that a chaotic regime exists between the bursting and beating modes of the model, and (b) that the model approaches chaos from both modes by a period doubling cascade. The bifurcation parameter employed is the external stimulus current. In addition to the period doubling observed in the model-generated trajectories, a period three "window" was observed, power spectra that demonstrate the approaches to chaos were generated, and the Lyaponov exponents and the fractal dimension of the chaotic attractors were calculated. Chaotic regimes have been observed in several similar models, which suggests that they are a general characteristic of cells that exhibit both bursting and beating modes.     

Intrinsic variability of latency to first-spike

The assessment of the variability of neuronal spike timing is fundamental to gain understanding of latency coding. Based on recent mathematical results, we investigate theoretically the impact of channel noise on latency variability. For large numbers of ion channels, we derive the asymptotic distribution of latency, together with an explicit expression for its variance. Consequences in terms of information processing are studied with Fisher information in the Morris–Lecar model. A competition between sensitivity to input and precision is responsible for favoring two distinct regimes of latencies.     

A model of heart ventricles for electrocardiographic evaluation of the state of the myocardium

A mathematical model of heart excitation processes has been developed for describing an electrocardiogram. A verified archive of model electrocardiograms has been created with the use of the model. The model has been used to study how electrocardiograms are affected by individual variability in ventricle shape and heart position in the norm, in myocardial infarction of different localizations, and in ventricular hypertrophy. Correspondence of the specific features of real and model electrocardiograms is discussed.     

Effect of the low-frequency impulse magnetic field on the autonomic nervous system in animals

The effect of weak (up to 3.5 mT) low-frequency (up to 100 Hz) impulse magnetic field on the state of the vegetative nervous system of animals has been studied by analyzing the variability of the heart rate. The effect of the magnetic field was estimated by a specially designed complex for recording cardiac signals of animals. Several specially selected regimes of impulse magnetic fields were studied. It was shown that the impulse magnetic field possesses a high biological activity at all regimes used, and the indices of the vegetative nervous system after the exposure to the impulse magnetic field approach the values typical for normotonic animals. This makes it possible to use magnetic fields at these regimes in magnetotherapy.     

健康人心率变异性中的不稳定周期轨道

为刻划心脏节律存在的确定性动力学特征,运用不稳定周期轨道分析方法对健康青年人的RR间期时间序列数据进行分析。研究结果揭示健康人心脏节律中存在显著的不稳定周期轨道及不稳定周期轨道分级（周期1、周期2,周期3,周期4）现象,表明健康青年人心脏节律的动力学特性中包含着显著的确定性行为。通过跟踪不周期轨道随时间的演变,迹表明心脏节律的变化中存在着因有的非平稳性。     

Diversity of temporal self-organized behaviors in a biochemical system.

The numerical study of a glycolytic model formed by a system of three delay-differential equations revealed a notable richness of temporal structures which included the three main routes to chaos, as well as a multiplicity of stable coexisting states. The Feigenbaum, intermitency and quasiperiodicity routes to chaos can emerge in the biochemical oscillator. Moreover, different types of birhythmicity, trirhythmicity and hard excitation emerge in the phase space. For a single range of the control parameter it can be observed the coexistence of two quasiperiodicity routes to chaos, the coexistence of a stable steady state with a stable torus, and the coexistence of a strange attractor with different stable regimes such as chaos with different periodic regimes, chaos with bursting behavior, and chaos with torus. In most of the numerical studies, the biochemical oscillator has been considered under periodic input flux being the mean input flux rate 6 mM/h. On the other hand, several investigators have observed quasiperiodic time patterns and chaotic oscillations by monitoring the fluorescence of NADH in glycolyzing yeast under sinusoidal glucose input flux. Our numerical results match well with these experimental studies.     

Ventricular fibrillation: the current methods for analysing the degree of irregularity of the process

Ventricular fibrillation has traditionally been described as "chaotic" and in recent years there has been discussions that fibrillation may be an instance of deterministic chaos in the context of nonlinear dynamical systems theory. The current paper summarizes modern methods of mathematical analysis of the degree of electrical irregularities of the heart during VF. The traditional methods of Fourier analysis of electrocardiographic data as well as concepts of chaos theory--fractal dimension, entropy, reconstruction of attractors and some new methods such as spatial coherence have been considered. The results are discussed in context of mathematical models and hypothesis of mechanisms of VF.     

Comparison of mathematical indices of fetal heart rate variability with visual assessment in the human and sheep.

Mathematical indices for quantitation of fetal heart rate variability have been proposed by numerous authors, but there have only been infrequent attempts to determine which such indices correspond to the semi-subjective evaluation of variability observed by clinicians. We have previously examined most of the published indices by using them for calculation of the variability of sets of computer-generated numbers, and seeing if they fulfill certain criteria of validity. Two sets of indices (each measuring short-term and long-term variability) were selected as acceptable. Segments of fetal heart rate records from both humans and sheep, with a wide range of subjective variability, were used to compare the mathematically derived indices with the semi-subjective evaluation of three observers. The results show that the mathematical indices of short-term variability compare closely to its subjective evaluation of being present or absent. The long-term variability of indices also increase progressively with the observers' evaluations of increasing variability. The agreement among observers, measured by Cohen's kappa test, is generally "substantial", although for some indices the agreement was "moderate" to "almost perfect". We conclude that the two sets of indices examined do quantitate what is clinically regarded as fetal heart rate variability.     

A mathematical model of human heart including the effects of heart contractility varying with heart rate changes

Incorporating the intrinsic variability of heart contractility varying with heart rate into the mathematical model of human heart would be useful for addressing the dynamical behaviors of human cardiovascular system, but models with such features were rarely reported. This study focused on the development and evaluation of a mathematical model of the whole heart, including the effects of heart contractility varying with heart rate changes. This model was developed based on a paradigm and model presented by Ottesen and Densielsen, which was used to model ventricular contraction. A piece-wise function together with expressions for time-related parameters were constructed for modeling atrial contraction. Atrial and ventricular parts of the whole heart model were evaluated by comparing with models from literature, and then the whole heart model were assessed through coupling with a simple model of the systemic circulation system and the pulmonary circulation system. The results indicated that both atrial and ventricular parts of the whole heart model could reasonably reflect their contractility varying with heart rate changes, and the whole heart model could exhibit major features of human heart. Results of the parameters variation studies revealed the correlations between the parameters in the whole heart model and performances (including the maximum pressure and the stroke volume) of every chamber. These results would be useful for helping users to adjust parameters in special applications.     

Analysis of Heart Rate Variability Based on the Graph Method

A method of analysis of heart rate variability based on the graph theory principle was suggested. The main parameters of the heart rate graph structure were determined and analyzed using models of harmonic oscillations, white noise, and various functional tests (including controllable respiration and mental load). The efficiency of the use of parameters of the heart rate graph for diagnosing some functional states was considered. A correlation of the parameters of the heart rate graph structure with the frequency characteristics of heart rate variability was studied. A general model of changes in the heart rate graph structure parameters at different levels of mental activity was constructed in terms of entropy changes.     

An Examination of the Relationship Between Resting Heart Rate Variability and Heart Rate Reactivity to a Mental Arithmetic Stressor

Resting heart rate variability can be an index of sympathetic or parasympathetic dominance, according to the frequency of the variability studied. Sympathetic dominance of this system has been linked to increased risk of cardiovascular disease (CVD). Similarly, rapid and dramatic increases in heart rate reactivity to a stressor task have also been suggested as indicating increased risk of CVD via atherogenesis. Although both of these variables have been related to the development of cardiovascular disease, and both may be related to increased sympathetic activity or parasympathetic withdrawal, most research studies have tended to focus on either variable independently of the other. In order to investigate whether these two indices of stressor reactivity were related in relatively young and healthy subjects, resting heart rate variability data were collected from 80 volunteers for 20 minutes. In addition, heart rate reactivity data were collected during a 2-minute mental arithmetic stressor, which has been previously shown to induce significant increases in heart rate. After classifying subjects according to whether their heart rate variability data were above or below the mean for their gender, heart rate reactivity data were examined via MANOVA to detect significant differences between subject groups. Females showed significant effects, and males showed nonsignificant trends, but these two sets of data were in different directions, suggesting that gender may be a confounding factor in the relationship between heart rate reactivity and heart rate variability.     

Effect of low-frequency pulsatile AC magnetic field on the state of the autonomic nervous system in rats

The effect of weak (1.4–3.15 mT) pulsatile 100-Hz AC magnetic field (PAMF) on the state of the autonomic nervous system in rats has been studied by analyzing the heart rate variability with a specially designed complex. PAMF evoked appreciable effects in single 30-min exposures at all intensity regimes tested; the initially broadly ranging indices of autonomic regulation converged after the exposure upon values typical of normotonic animals. These data support the potentialities of such fields in magnetotherapy.