Noise robust approach to reconstruction of van der Pol-like oscillators and its application to Granger causality.

Van der Pol oscillators and their generalizations are known to be a fundamental model in the theory of oscillations and their applications. Many objects of a different nature can be described using van der Pol-like equations under some circumstances; therefore, methods of reconstruction of such equations from experimental data can be of significant importance for tasks of model verification, indirect parameter estimation, coupling analysis, system classification, etc. The previously reported techniques were not applicable to time series with large measurement noise, which is usual in biological, climatological, and many other experiments. Here, we present a new approach based on the use of numerical integration instead of the differentiation and implicit approximation of a nonlinear dissipation function. We show that this new technique can work for noise levels up to 30% by standard deviation from the signal for different types of autonomous van der Pol-like systems and for ensembles of such systems, providing a new approach to the realization of the Granger-causality idea.

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.

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.

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.

Application of adaptive nonlinear Granger causality: disclosing network changes before and after absence seizure onset in a genetic rat model.

BACKGROUND: Advanced methods of signal analysis of the preictal and ictal activity dynamics characterizing absence epilepsy in humans with absences and in genetic animal models have revealed new and unknown electroencephalographic characteristics, that has led to new insights and theories. NEW METHOD: Taking into account that some network associations can be considered as nonlinear, an adaptive nonlinear Granger causality approach was developed and applied to analyze cortico-cortical, cortico-thalamic and intrathalamic network interactions from local field potentials (LFPs). The outcomes of adaptive nonlinear models, constructed based on the properties of electroencephalographic signal and on statistical criteria to optimize the number of coefficients in the models, were compared with the outcomes of linear Granger causality. RESULTS: The nonlinear adaptive method showed statistically significant preictal changes in Granger causality in almost all pairs of channels, as well as ictal changes in cortico-cortical, cortico-thalamic and intrathalamic networks. Current results suggest rearrangement of interactions in the thalamo-cortical network accompanied the transition from preictal to ictal phase. COMPARISON WITH EXISTING METHOD(S): The linear method revealed no preictal and less ictal changes in causality. CONCLUSIONS: Achieved results suggest that this proposed adaptive nonlinear method is more sensitive than the linear one to dynamics of network properties. Since changes in coupling were found before the seizure-related increase of LFP signal amplitude and also based on some additional tests it seems likely that they were not spurious and could not result from signal to noise ratio change.

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.

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 couplings in ensembles of stochastic oscillators.

The problem of detection and quantitative characterization of directional couplings in an ensemble of noisy oscillators from a time series is addressed. We suggest estimators for the strengths of couplings which are based on modeling the observed oscillations with a set of stochastic phase oscillators and easily interpreted from a physical viewpoint. Moreover, we present an analytic formula for a statistical significance level allowing to reveal an architecture of couplings reliably from a relatively short time series. The technique applies to weakly coupled nonsynchronized oscillators. It is introduced for oscillators with close basic frequencies but can be readily generalized to the case of arbitrary frequencies. Efficiency of the technique is demonstrated in numerical experiments.

Estimation of interaction strength and direction from short and noisy time series.

A technique for determination of character and intensity of interaction between the elements of complex systems based on reconstruction of model equations for phase dynamics is extended to the case of short and noisy time series. Corrections, which eliminate systematic errors of the estimates, and expressions for confidence intervals are derived. Analytic results are presented for a particular case of linear uncoupled systems, and their validity for a much wider range of situations is demonstrated with numerical examples. The technique should be useful for the analysis of nonstationary processes in real time, including the situations of significant noise and restrictions on the observation time.

Choice of dynamical variables for global reconstruction of model equations from time series.

The success of modeling from an experimental time series is determined to a significant extent by the choice of dynamical variables. We propose a method for preliminary investigation of a time series whose purpose is to find out whether a global dynamical model with smooth functions can be constructed for the chosen variables. The method consists in the estimation of single valuedness and continuity of relations between dynamical variables and variables to enter left-hand sides of model equations. The method is explained with numerical examples. Its efficiency is demonstrated by modeling a real nonlinear electric circuit.