Central pattern generator based on self-sustained oscillator coupled to a chain of oscillatory circuits.

We have proposed and studied both numerically and experimentally a multistable system based on a self-sustained Van der Pol oscillator coupled to passive oscillatory circuits. The number of passive oscillators determines the number of multistable oscillatory regimes coexisting in the proposed system. It is shown that our system can be used in robotics applications as a simple model for a central pattern generator (CPG). In this case, the amplitude and phase relations between the active and passive oscillators control a gait, which can be adjusted by changing the system control parameters. Variation of the active oscillator's natural frequency leads to hard switching between the regimes characterized by different phase shifts between the oscillators. In contrast, the external forcing can change the frequency and amplitudes of oscillations, preserving the phase shifts. Therefore, the frequency of the external signal can serve as a control parameter of the model regime and realize a feedback in the proposed CPG depending on the environmental conditions. In particular, it allows one to switch the regime and change the velocity of the robot's gate and tune the gait to the environment. We have also shown that the studied oscillatory regimes in the proposed system are robust and not affected by external noise or fluctuations of the system parameters. Moreover, using the proposed scheme, we simulated the type of bipedal locomotion, including walking and running.

Directional couplings between the respiration and parasympathetic control of the heart rate during sleep and wakefulness in healthy subjects at different ages.

Cardiorespiratory interactions are important, both for understanding the fundamental processes of functioning of the human body and for development of methods for diagnostics of various pathologies. The properties of cardiorespiratory interaction are determined by the processes of autonomic control of blood circulation, which are modulated by the higher nervous activity. We study the directional couplings between the respiration and the process of parasympathetic control of the heart rate in the awake state and different stages of sleep in 96 healthy subjects from different age groups. The detection of directional couplings is carried out using the method of phase dynamics modeling applied to experimental RR-intervals and the signal of respiration. We reveal the presence of bidirectional couplings between the studied processes in all age groups. Our results show that the coupling from respiration to the process of parasympathetic control of the heart rate is stronger than the coupling in the opposite direction. The difference in the strength of bidirectional couplings between the considered processes is most pronounced in deep sleep.

Directional Couplings Between Electroencephalogram and Interbeat Intervals Signals in Awake State and Different Stages of Sleep.

Purpose of the work is to identify the directional coupling between the structures of the brain and the autonomic control of the heart rate variability, to analyze the changes in these coupling in sleep and in wakefulness. Infra-slow oscillations of the electroencephalograms potential and low-frequency components (0.04-0.15 Hz) of the interbeat intervals signal where analyzed using a sensitive method for identifying the directional coupling. The technique, based on modeling the dynamics of instantaneous phases of oscillations, made it possible to reveal the presence and quantify the directional couplings between the structures of the brain and the autonomic control of the heart rate variability. It was shown that the coupling coefficients in the frequency band of 0.04-0.15 Hz (associated mainly with sympathetic control of blood circulation), on average, decrease with falling asleep. We have also shown the asymmetry of coupling. At the same time, stronger connections were revealed in the direction from the autonomic control of the heart rate variability to the brain structures than in the opposite direction. It has been shown that the strength of such couplings decreases with increasing of sleep depth.

Low-frequency component of photoplethysmogram reflects the autonomic control of blood pressure.

The question of how much information the photoplethysmogram (PPG) signal contains on the autonomic regulation of blood pressure (BP) remains unsolved. This study aims to compare the low-frequency (LF) and high-frequency components of PPG and BP and assess their correlation with oscillations in interbeat (RR) intervals at similar frequencies. The PPG signal from the distal phalanx of the right index finger recorded using a reflective PPG sensor at green light, the BP signal from the left hand recorded using a Finometer, and RR intervals were analyzed. These signals were simultaneously recorded within 15 min in a supine resting condition in 17 healthy subjects (12 males and 5 females) aged 33 ± 9 years (mean ± SD). The study revealed the high coherence of LF components of PPG and BP with the LF component of RR intervals. The high-frequency components of these signals had low coherence. The analysis of the signal instantaneous phases revealed the presence of high-phase coherence between the LF components of PPG and BP. It is shown that the LF component of PPG is determined not only by local myogenic activity but also reflects the processes of autonomic control of BP.

Synchronization of the Processes of Autonomic Control of Blood Circulation in Humans Is Different in the Awake State and in Sleep Stages.

The influence of higher nervous activity on the processes of autonomic control of the cardiovascular system and baroreflex regulation is of considerable interest, both for understanding the fundamental laws of the functioning of the human body and for developing methods for diagnostics and treatment of pathologies. The complexity of the analyzed systems limits the possibilities of research in this area and requires the development of new tools. Earlier we propose a method for studying the collective dynamics of the processes of autonomic control of blood circulation in the awake state and in different stages of sleep. The method is based on estimating a quantitative measure representing the total percentage of phase synchronization between the low-frequency oscillations in heart rate and blood pressure. Analysis of electrocardiogram and invasive blood pressure signals in apnea patients in the awake state and in different sleep stages showed a high sensitivity of the proposed measure. It is shown that in slow-wave sleep the degree of synchronization of the studied rhythms is higher than in the awake state and lower than in sleep with rapid eye movement. The results reflect the modulation of the processes of autonomic control of blood circulation by higher nervous activity and can be used for the quantitative assessment of this modulation.

Mathematical modeling of the cardiovascular autonomic control in healthy subjects during a passive head-up tilt test.

A mathematical model is proposed for the autonomic control of cardiovascular system, which takes into account two separated self-exciting sympathetic control loops of heart rate and peripheral vascular tone. The control loops are represented by self-exciting time-delay systems and their tone depends on activity of the aortic, carotid, and lower-body baroreceptors. The model is used to study the dynamics of the adaptive processes that manifest in a healthy cardiovascular system during the passive head-up tilt test. Computer simulation provides continuous observation of the dynamics of the indexes and variables that cannot be measured in the direct experiment, including the noradrenaline concentration in vessel wall and heart muscle, tone of the sympathetic and parasympathetic control, peripheral vascular resistance, and blood pressure. In the supine and upright positions, we estimated the spectral characteristics of the model variables, especially in the low-frequency band, and the original index of total percent of phase synchronization between the low-frequency oscillations in heart rate and blood pressure signals. The model demonstrates good quantitative agreement with the dynamics of the experimentally observed indexes of cardiovascular system that were averaged for 50 healthy subjects.

Simulating Dynamics of Circulation in the Awake State and Different Stages of Sleep Using Non-autonomous Mathematical Model With Time Delay.

We propose a mathematical model of the human cardiovascular system. The model allows one to simulate the main heart rate, its variability under the influence of the autonomic nervous system, breathing process, and oscillations of blood pressure. For the first time, the model takes into account the activity of the cerebral cortex structures that modulate the autonomic control loops of blood circulation in the awake state and in various stages of sleep. The adequacy of the model is demonstrated by comparing its time series with experimental records of healthy subjects in the SIESTA database. The proposed model can become a useful tool for studying the characteristics of the cardiovascular system dynamics during sleep.

Model of human cardiovascular system with a loop of autonomic regulation of the mean arterial pressure.

A model of human cardiovascular system is proposed which describes the main heart rhythm, the regulation of heart function and blood vessels by the autonomic nervous system, baroreflex, and the formation of arterial blood pressure. The model takes into account the impact of respiration on these processes. It is shown that taking into account nonlinearity and introducing the autonomous loop of mean arterial blood pressure in the form of self-oscillating time-delay system allow to obtain the model signals whose statistical and spectral characteristics are qualitatively and quantitatively similar to those for experimental signals. The proposed model demonstrates the phenomenon of synchronization of mean arterial pressure regulatory system by the signal of respiration with the basic period close to 10 seconds, which is observed in the physiological experiments.

Autonomic control of cardiovascular system in pre- and postmenopausal women: a cross-sectional study.

OBJECTIVE: The aim of this cross-sectional study was to assess the features of autonomic control of the cardiovascular system in pre- and postmenopausal women. MATERIAL AND METHODS: We studied 185 postmenopausal women aged 59.3±8.5 years (mean±SD) and 104 premenopausal women aged 45.1±5.8 years. Standard indices of heart rate variability (HRV) (mean heart rate, coefficient of variation, standard deviation of the NN interval (the time elapsing between two consecutive R waves in the electrocardiogram with normal sinus rhythm) (SDNN), square root of the mean squared differences of successive NN intervals (RMSSD), proportion derived by dividing RR50, the number of interval differences of successive NN intervals greater than 50 ms, by the total number of NN intervals (PNN50), and power of low frequency (LF) and high frequency (HF) bands in absolute values and percentages of total spectral power) and index S of synchronization between the 0.1-Hz rhythms in heart rate and photoplethysmogram were compared between these two groups at rest. We assessed the following sex hormones: estradiol, follicle stimulating hormone, dehydroepiandrosterone sulfate, and testosterone. RESULTS: Mean heart rate and power of LF and HF bands were significantly different (p<0.05) in pre- and postmenopausal women. The autonomic indices were similar in women with natural and surgical menopause. Some indices (coefficient of variation, SDNN, RMSSD, PNN50, and power of LF and HF bands) showed weak correlation with menopause time in women with natural menopause. In women with surgical menopause, a moderate statistically significant correlation was observed only between menopause time and S index (r=-0.41, p=0.039). In premenopausal women, only testosterone correlated weakly with coefficient of variation, SDNN, PNN50, RMSSD, and power of HF band. In postmenopausal women, no correlations were found. We did not find any significant relationship between autonomic indices and hot flashes, assessed by hot flash diary. CONCLUSION: We did not find a clinically important relationship between cardiovascular autonomic control and menopausal status in women.

Effects of antihypertensive treatment on cardiovascular autonomic control: a prospective study.

OBJECTIVE: The aim of study was to propose an approach to the control of dynamics of autonomic dysfunction in cardiovascular system (CVS) under antihypertensive treatment (AT) in patients with arterial hypertension (AH), based on individual features of synchronization of 0.1-Hz rhythms in heart rate (HR) and photoplethysmogram (PPG) and spectral indices of heart rate variability (HRV). METHODS: We designed prospective cohort diagnostic accuracy and studied 105 AH patients (66 females) aged 47±8 years during 8 weeks. The HRV spectral indices and the index S of synchronization between the 0.1-Hz rhythms in HR and PPG during a tilt test are compared in their ability to control the AT with angiotensin-converting enzyme inhibitors (ACE-Is) (fosinopril or enalapril) and ß-blockers (atenolol or metoprolol). We apply Shapiro-Wilk, Mann-Whitney U and Wilcoxon tests. RESULTS: It is shown that the power of low-frequency (LF) band in HRV spectrum and index S can be used as criteria for initial assessment of the status of autonomic regulation in AH patients. The patients with S<25% in vertical body's position and LF>250 ms2 in horizontal body's position require ACE-Is treatment. The AH patients with LF<350 ms2 and S<30% in vertical body's position require ß-blocker treatment. The AH patients with S>25% and LF>250 ms2 in horizontal body's position do not require any ACE-Is or ß-blocker treatment. Both drug groups can be used in patients with low values of index S and low power of LF band in HRV spectrum. CONCLUSION: The control of AT can be carried out in AH patients taking into account the individual features of autonomic dysfunction in CVS. Sensitivity and specificity of our approach were 65% and 73%, respectively.

Intermittency of intermittencies.

A phenomenon of intermittency of intermittencies is discovered in the temporal behavior of two coupled complex systems. We observe for the first time the coexistence of two types of intermittent behavior taking place simultaneously near the boundary of the synchronization regime of coupled chaotic oscillators. This phenomenon is found both in the numerical and physiological experiments. The laws for both the distribution and mean length of laminar phases versus the control parameter values are analytically deduced. A very good agreement between the theoretical results and simulation is shown.

Autonomous and forced dynamics of oscillator ensembles with global nonlinear coupling: an experimental study.

We perform experiments with 72 electronic limit-cycle oscillators, globally coupled via a linear or nonlinear feedback loop. While in the linear case we observe a standard Kuramoto-like synchronization transition, in the nonlinear case, with increase of the coupling strength, we first observe a transition to full synchrony and then a desynchronization transition to a quasiperiodic state. However, in this state the ensemble remains coherent so that the amplitude of the mean field is nonzero, but the frequency of the mean field is larger than frequencies of all oscillators. Next, we analyze effects of common periodic forcing of the linearly or nonlinearly coupled ensemble and demonstrate regimes when the mean field is entrained by the force whereas the oscillators are not.

Evaluation of 5-year risk of cardiovascular events in patients after acute myocardial infarction using synchronization of 0.1-Hz rhythms in cardiovascular system.

BACKGROUND: Synchronization between 0.1-Hz rhythms in cardiovascular system is deteriorated at acute myocardial infarction (AMI) leading to a disruption of natural functional couplings within the system of autonomic regulation. OBJECTIVE: This study evaluates the prognostic value of autonomic regulation indices for the 5-year risk of fatal and nonfatal cardiovascular events in patients after AMI. METHODS AND RESULTS: We studied 125 patients (53 [42%] female) after AMI aged between 30 and 83 years. The period of observation was 5 years with checkpoints at the first week after AMI and after each year after AMI. We compared the prognostic value of established clinical characteristics and degree S of synchronization between 0.1-Hz rhythms in heart rate and microcirculation for evaluation of the 5-year risk of mortality and recurrent myocardial infarction (MI) in patients after AMI. Acute heart failure Killip 2-4 at AMI and S < 20% at the first week after AMI were identified as the most important factors for evaluation of the risk of 5-year mortality in patients after AMI (χ(2) = 14.2, P = 0.003). Sensitivity and specificity of low S (<20%) at the first week after AMI were 76% and 43%, respectively. For evaluation of the 5-year risk of recurrent MI index S had no advantage over established clinical characteristics. CONCLUSION: The value of S below 20% in patients with AMI is a sensitive marker of high risk of mortality during the subsequent five years. It is characterized by better prognostic value than most of established clinical characteristics.

Experiments on oscillator ensembles with global nonlinear coupling.

We experimentally analyze collective dynamics of a population of 20 electronic Wien-bridge limit-cycle oscillators with a nonlinear phase-shifting unit in the global feedback loop. With an increase in the coupling strength we first observe formation and then destruction of a synchronous cluster, so that the dependence of the order parameter on the coupling strength is not monotonic. After destruction of the cluster the ensemble remains nevertheless coherent, i.e., it exhibits an oscillatory collective mode (mean field). We show that the system is now in a self-organized quasiperiodic state, predicted in Rosenblum and Pikovsky [Phys. Rev. Lett. 98, 064101 (2007)]. In this state, frequencies of all oscillators are smaller than the frequency of the mean field, so that the oscillators are not locked to the mean field they create and their dynamics is quasiperiodic. Without a nonlinear phase-shifting unit, the system exhibits a standard Kuramoto-like transition to a fully synchronous state. We demonstrate a good correspondence between the experiment and previously developed theory. We also propose a simple measure which characterizes the macroscopic incoherence-coherence transition in a finite-size ensemble.

Selection of optimal dose of beta-blocker treatment in myocardial infarction patients based on changes in synchronization between 0.1 Hz oscillations in heart rate and peripheral microcirculation.

BACKGROUND: Selection of the optimal dose of beta-blocker treatment in myocardial infarction (MI) patients is problematic because of a lack of well-established guidelines. METHODS: We evaluated changes in synchronization between 0.1 Hz oscillations in heart rate (HR) and plethysmographic peripheral microcirculation in response to a tilt-table test and to 3-month treatment with the highest tolerated beta-blocker (metoprolol) dose in 43 patients aged between 41 and 77 years with acute MI 6 months prior to the start of the study. Before the study the patients were treated with small doses of beta-blocker. Phase differences between HR and peripheral microcirculation oscillations were used to measure the degree of synchronization (S), and relative change in S from horizontal position was used to characterize the response to vertical tilt. RESULTS: Two groups of MI patients matched for clinical characteristics were identified on the basis of the results. The first group was composed of patients with decreased S as a response to vertical tilt at the beginning of the study. The patients with increased S during vertical tilt before treatment with the highest tolerated beta-blocker dose were attributed to the second group. The response to vertical tilt in the first group of patients was postulated to indicate the need to increase beta-blocker dose, and in turn, the response in the second group to indicate an already adequate beta-blocker dose. CONCLUSION: Assessment of synchronization of 0.1  Hz HR and peripheral microcirculation oscillations as a response to a tilt test can possibly be used as a guideline for selecting beta-blocker dose in post-MI patients.

Detection of synchronization from univariate data using wavelet transform.

A method is proposed for detecting from univariate data the presence of synchronization of a self-sustained oscillator by external driving with varying frequency. The method is based on the analysis of difference between the oscillator instantaneous phases calculated using continuous wavelet transform at time moments shifted by a certain constant value relative to each other. We apply our method to a driven asymmetric van der Pol oscillator, experimental data from a driven electronic oscillator with delayed feedback and human heartbeat time series. In the latest case, the analysis of the heart rate variability data reveals synchronous regimes between the respiration and slow oscillations in blood pressure.

Detecting synchronization of self-sustained oscillators by external driving with varying frequency.

We propose a method for detecting the presence of a synchronization of a self-sustained oscillator by external driving with linearly varying frequency. The method is based on a continuous wavelet transform of the signals of the self-sustained oscillator and external force and allows one to distinguish the case of true synchronization from the case of spurious synchronization caused by linear mixing of the signals. We apply the method to a driven van der Pol oscillator and to experimental data of human heart rate variability and respiration.