The relationship of the slope of the heart rate graph regression with linear and nonlinear heart rate dynamics in stationary short-time series

The relationship of the slope of the heart rate graph regression curve (b ₁) with periodic (linear) and nonlinear heart rate dynamics has been studied in stationary short-time series (256 points). For estimating nonlinear dynamics, a parameter derived from correlation dimension has been used, which has made it possible to estimate chaotic processes in short-time series. According to the results of the study, the heart rate dynamics in short-time series may be represented as a sum of linear (periodic) and nonlinear (stochastic) processes. The relationships of b ₁ with both the linear and the nonlinear heart rate dynamics have been demonstrated. Equations for calculating the absolute and relative (to the periodic oscillation amplitude) noises in the heart rate dynamics in short-time series are proposed. Stochastic nonlinear dynamics in different physiological states of humans have been compared. It has been found that the increase in the relative noise intensity in the heart rate dynamics with an increase in respiration rate is determined not only by the decrease in the amplitude of respiratory waves, but also by an increase in the amplitude of the noise itself. The absolute noise intensity is decreased in the states of neurotic excitement, fatigue, and, especially, mental stress. In the state of rest, nonlinear (stochastic) processes dominate over linear (periodic) ones.     

Nonlinear dynamics of heart rate variability in cocaine-exposed neonates during sleep

The aim of this study was to determine the effects of prenatal cocaine exposure (PCE) on the dynamics of heart rate variability in full-term neonates during sleep. R-R interval (RRI) time series from 9 infants with PCE and 12 controls during periods of stable quiet sleep and active sleep were analyzed using autoregressive modeling and nonlinear dynamics. There were no differences between the two groups in spectral power distribution, approximate entropy, correlation dimension, and nonlinear predictability. However, application of surrogate data analysis to these measures revealed a significant degree of nonlinear RRI dynamics in all subjects. A parametric model, consisting of a nonlinear delayed-feedback system with stochastic noise as the perturbing input, was employed to estimate the relative contributions of linear and nonlinear deterministic dynamics in the data. Both infant groups showed similar proportional contributions in linear, nonlinear, and stochastic dynamics. However, approximate entropy, correlation dimension, and nonlinear prediction error were all decreased in active versus quiet sleep; in addition, the parametric model revealed a doubling of the linear component and a halving of the nonlinear contribution to overall heart rate variability. Spectral analysis indicated a shift in relative power toward lower frequencies. We conclude that 1) RRI dynamics in infants with PCE and normal controls are similar; and 2) in both groups, sympathetic dominance during active sleep produces primarily periodic low-frequency oscillations in RRI, whereas in quiet sleep vagal modulation leads to RRI fluctuations that are broadband and dynamically more complex.     

Mean and quasideterministic equivalence for linear stochastic dynamics

In linear, stochastic dynamics it is shown that the quasideterministic population size is equivalent to the mean population size. The quasideterministic dynamics are defined by the conditional infinitesimal mean of the process. The stochastic component of the dynamics includes both Gaussian and Poisson white noise, with amplitude coefficients proportional to the population size. Generalizations are given for nonautonomous coefficients and for distributed Poisson jump amplitudes. A counter example--an exactly integrable nonlinear jump model--shows that the equivalence result does not hold for nonlinear stochastic dynamics.     

Respiratory influences on non-linear dynamics of heart rate variability in humans

 The goal of our study was to determine whether evidence for chaos in heart rate variability (HRV) can be observed when the respiratory input to the autonomic controller of heart rate is forced by the deterministic pattern associated with periodic breathing. We simultaneously recorded, in supine healthy volunteers, RR intervals and breathing volumes for 20 to 30 min (1024 data point series) during three protocols: rest (control), fixed breathing (15 breath/min) and voluntary periodic breathing (3 breaths with 2 s inspiration and 2 s expiration followed by an 8 s breath hold). On both the RR interval and breathing volume series we applied the non-linear prediction method (Sugihara and May algorithm) to the original time series and to distribution-conserved isospectral surrogate data. Our results showed that, in contrast to the control test, during both fixed and voluntary periodic breathing the variability of breathing volumes was clearly deterministic non-chaotic. During all the three protocols, the RR-interval series’ non-linear predictability was consistent with one of a chaotic series. However, at rest, no clear difference was observed between the RR-interval series and their surrogates, which means that no clear chaos was observed. During fixed breathing a difference appeared, and this difference seemed clearer during voluntary periodic breathing. We concluded that HRV dynamics were chaotic when respiration was forced with a deterministic non-chaotic pattern and that normal spontaneous respiratory influences might mask the normally chaotic pattern in HRV. Received: 7 August 1995 / Accepted in revised form: 20 March 1997     

存在睡眠呼吸异常的慢病患者呼吸影响心率变异性的初步分析*

目的: 基于整体整合生理学医学理论提出的呼吸引起循环指标变异的假说,分析研究存在睡眠呼吸异常的慢病患者睡眠期间呼吸和心率变异之间的相关关系。方法: 纳入存在睡眠呼吸异常且呼吸暂停低通气指数（AHI）≥15次/小时的慢病患者11例,签署知情同意书后完成标准化症状限制性极限运动的心肺运动试验（CPET）和睡眠呼吸监测,计算分析病人睡眠期间波浪式呼吸（OB）期与正常平稳呼吸期的呼吸鼻气流、心电图R-R间期心率变异的规律。结果: 存在睡眠呼吸异常的慢病患者CPET峰值摄氧量（Peak VO₂）和无氧阈（AT）为（70.8±13.6）%pred和（71.2±6.1）%pred;CPET有5例存在运动诱发的波浪式呼吸（EIOB）,6例为呼吸不稳定,提示整体功能状态低于正常人。本组慢病患者AHI为每小时（28.8±10.0）次,睡眠呼吸异常总时间占睡眠总时间的比值为（0.38±0.25）;OB周期的平均时间长度为（51.1±14.4）s。本组慢病患者正常平稳呼吸期的呼吸周期数与心率变异周期数的比值（B-n/HRV-B-n）为1.00±0.04,每个呼吸周期节律的心率变异平均幅度（HRV-B-M）为（2.64±1.59） bpm,虽然低于正常人（P<0.05）,但却与无睡眠呼吸异常的慢病患者相似（P>0.05）;HRV-B-M的变异度CV（HRV-B-M的SD/x）为( 0.33±0.11）,期间血氧饱和度（SpO₂）虽略低,但并无明显规律性下降与上升。本组慢病患者的OB期间呼吸周期数与心率变异周期数（OB-B-n/OB-HRV-B-n）比值为（1.22±0.18）,OB期每个呼吸周期节律的心率变异平均幅度（OB-HRV-B-M）为（3.56±1.57）bpm及其变异度（OB-CV =OB-HRV-B-M的SD/x）为（0.59±0.28）,每个OB周期节律的心率变异平均幅度（OB-HRV-OB-M）为（13.75±4.25）bpm,OB期间低通气时SpO₂出现明显的下降,OB期间SpO₂平均变异幅度（OB-SpO₂-OB-M）为（4.79±1.39）%,OB期的OB-B-n/OB-HRV-B-n比值、OB-HRV-OB-M比其正常平稳呼吸期对应指标显著增大（P<0.01）。OB-HRV-B-M虽然与正常平稳呼吸期HRV-B-M相比差异无统计学意义（P>0.05）,但其变异度OB-CV却显著增大（P<0.01）。结论: 睡眠呼吸异常的慢病患者OB期的心率变异幅度大于其正常平稳呼吸期,当呼吸模式发生改变时心率变异也发生明显改变,其平稳呼吸期的呼吸周期数与心率变异周期数的比值与正常人以及无睡眠呼吸异常的慢病患者相同,证实心率变异为呼吸源性;而其OB期间心率变异周期数相对于呼吸周期减少直接源于此时的低通气或者呼吸暂停,心率变异也是呼吸源性。     

The impact of stochasticity on the behaviour of nonlinear population models: synchrony and the Moran effect

Environmental variation is ubiquitous, but its effects on nonlinear population dynamics are poorly understood. Using simple (unstructured) nonlinear models we investigate the effects of correlated noise on the dynamics of two otherwise independent populations (the Moran effect), i.e. we focus on noise rather than dispersion or trophic interaction as the cause of population synchrony. We find that below the bifurcation threshold for periodic behaviour (1) synchrony between populations is strongly dependent on the shape of the noise distribution but largely insensitive to which model is studied, (2) there is, in general, a loss of synchrony as the noise is filtered by the model, (3) for specially structured noise distributions this loss can be effectively eliminated over a restricted range of distribution parameter values even though the model might be nonlinear, (4) for unstructured models there is no evidence of correlation enhancement, a mechanism suggested by Moran, but above the bifurcation threshold enhancement is possible for weak noise through phase-locking, (5) rapid desynchronisation occurs as the chaotic regime is approached. To carry out the investigation the stochastic models are (a) reformulated in terms of their joint asymptotic probability distributions and (b) simulated to analyse temporal patterns.     

A stochastic limit cycle oscillator model of the EEG

We present an empirical model of the electroencephalogram (EEG) signal based on the construction of a stochastic limit cycle oscillator using Itô calculus. This formulation, where the noise influences actually interact with the dynamics, is substantially different from the usual definition of measurement noise. Analysis of model data is compared with actual EEG data using both traditional methods and modern techniques from nonlinear time series analysis. The model demonstrates visually displayed patterns and statistics that are similar to actual EEG data. In addition, the nonlinear mechanisms underlying the dynamics of the model do not manifest themselves in nonlinear time series analysis, paralleling the situation with real, non-pathological EEG data. This modeling exercise suggests that the EEG is optimally described by stochastic limit cycle behavior.     

Vagal-dependent nonlinear variability in the respiratory pattern of anesthetized, spontaneously breathing rats

Physiological rhythms, including respiration, exhibit endogenous variability associated with health, and deviations from this are associated with disease. Specific changes in the linear and nonlinear sources of breathing variability have not been investigated. In this study, we used information theory-based techniques, combined with surrogate data testing, to quantify and characterize the vagal-dependent nonlinear pattern variability in urethane-anesthetized, spontaneously breathing adult rats. Surrogate data sets preserved the amplitude distribution and linear correlations of the original data set, but nonlinear correlation structure in the data was removed. Differences in mutual information and sample entropy between original and surrogate data sets indicated the presence of deterministic nonlinear or stochastic non-Gaussian variability. With vagi intact (n = 11), the respiratory cycle exhibited significant nonlinear behavior in templates of points separated by time delays ranging from one sample to one cycle length. After vagotomy (n = 6), even though nonlinear variability was reduced significantly, nonlinear properties were still evident at various time delays. Nonlinear deterministic variability did not change further after subsequent bilateral microinjection of MK-801, an N-methyl-D-aspartate receptor antagonist, in the K?lliker-Fuse nuclei. Reversing the sequence (n = 5), blocking N-methyl-D-aspartate receptors bilaterally in the dorsolateral pons significantly decreased nonlinear variability in the respiratory pattern, even with the vagi intact, and subsequent vagotomy did not change nonlinear variability. Thus both vagal and dorsolateral pontine influences contribute to nonlinear respiratory pattern variability. Furthermore, breathing dynamics of the intact system are mutually dependent on vagal and pontine sources of nonlinear complexity. Understanding the structure and modulation of variability provides insight into disease effects on respiratory patterning.     

人体通气状态对动脉血气波浪式幅度的影响*

目的: 本研究旨在发现不同通气模式下动脉血气的变化特点。方法: 选择心功能正常,需要连续监测动脉血流动力学变化的患者6 例,4男2女,年龄（59.00±16.64）岁,体质量（71.67±10.37）kg,左心射血分数（LVEF）（61.33±2.16）%。患者签署知情同意书后,分别于正常呼吸、憋气20 s以及高潮气量过度通气状态下连续15~16次心跳桡动脉、颈静脉逐搏取血,测定PO₂,用于分析三种呼吸状态下动、静脉血气的变化特点。分别比较患者相邻最高和最低值,以验证三种呼吸状态下动、静脉血气是否都存在周期性波浪式信号变化;此外,将患者动、静脉血气周期性波浪式信号的变化幅度进行统计学t检验分析,比较有无差异。结果: 共6例ICU 住院监护患者, 抽取动、静脉血液充满肝素化细长塑化管需要15~16次心跳,即取血需要15~16次心跳,全部覆盖超过2个呼吸周期。患者正常呼吸、憋气20 s以及高潮气量通气状态下动脉血气中PaO₂呈现波浪式变化,幅度分别是（9.96±5.18）mmHg,（5.33±1.55）mmHg和（13.13±7.55）mmHg,分别是各自均值的（8.09±2.43）%,（5.29±2.19）%,（10.40±2.68）%,高通气量呼吸模式波浪式变化幅度大于正常呼吸模式（P<0.05）,正常呼吸模式波浪式变化幅度大于憋气状态（P<0.05）。正常呼吸、憋气20 s以及高潮气量通气状态下静脉血气中PO₂未呈现波浪式变化,幅度分别是（1.63±0.41）mmHg,（1.13±0.41）mmHg和（1.31±0.67）mmHg,分别是各自均值的（3.91±1.22）%,（2.92±1.12）%,（3.33±1.81）%,都显著低于同状态下动脉血气,但组间差异不明显。结论: 分别于三种通气状态下采用连续逐搏动脉取血血气分析法证实,患者高通气状态呼吸时动脉血气的周期性波浪式变化信号增强,憋气时波浪式呼吸变化信号变弱,而静脉血氧分压波浪式变化幅度于三种呼吸状态下都不明显。说明肺通气导致肺换气是影响动脉血液波浪式信号幅度的直接决定性因素。     

周期激励下FHN模型的两种多峰分布放电节律

文章揭示了外界周期脉冲激励下神经元系统产生的随机整数倍和混沌多峰放电节律的关系.随机节律统计直方图呈多峰分布、峰值指数衰减、不可预报且复杂度接近1;混沌节律统计直方图呈不同的多峰分布,峰值非指数衰减、有一定的可预报性且复杂度小于1.混沌节律在激励脉冲周期小于系统内在周期且刺激强度较大时产生,参数范围较小;而随机节律在激励脉冲周期大于系统内在周期且脉冲刺激强度小时,可与随机因素共同作用而产生,产生的参数范围较大.上述结果揭示了两类节律的动力学特性,为区分两类节律提供了实用指标.     

Quasicycles revisited: apparent sensitivity to initial conditions

Environmental noise is known to sustain cycles by perturbing a deterministic approach to equilibrium that is itself oscillatory. Quasicycles produced in this way display a regular period but varied amplitude. They were proposed by Nisbet and Gurney (Nature 263 (1976) 319) as one possible explanation for population fluctuations in nature. Here, we revisit quasicyclic dynamics from the perspective of nonlinear time series analysis. Time series are generated with a predator-prey model whose prey's growth rate is driven by environmental noise. A method for the analysis of short and noisy data provides evidence for sensitivity to initial conditions, with a global Lyapunov exponent often close to zero characteristic of populations 'at the edge of chaos'. Results with methods restricted to long time series are consistent with a finite-dimensional attractor on which dynamics are sensitive to initial conditions. These results are compared with those previously obtained for quasicycles in an individual-based model with heterogeneous spatial distributions. Patterns of sensitivity to initial conditions are shown to differentiate phase-forgetting from phase-remembering quasicycles involving a periodic driver. The previously reported mode at zero of Lyapunov exponents in field and laboratory populations may reflect, in part, quasicyclic dynamics.     

Analysis of neuronal networks in the visual system of the cat using statistical signals — Simple and complex cells

Superimposing additively a two-dimensional noise process to deterministic input signals (bars) the neurons of area 17 show a class-specific reaction for the task of signal extraction. Moving both parts of the signals simultaneously and varying the signal to noise ratio (S/N) the simple cells achieve the same performance as resulted from the psychophysical experiment. Type I complex cells extract moving deterministic signals (i.e. bars) from the stationary noise, whereas in the answers of Type II complex cells the statistical parts of the signals predominate. Considering the different cell types each as a series of a linear and a nonlinear system one obtains the cell specific space-time frequency and the amplitude characteristics.This work was supported by DFG Grant Ho 450/6 and Grant Se 251/9     

A general analysis of resource allocation by competing individuals.

A linear model for population dynamics in a stationary stochastic environment is introduced based on linearizing the N-species Lotka-Volterra competition equations in discrete time. Iteration of the linear model shows the sequence of population sizes to be formed from a simple linear operation on the sequence of carrying capacities. The transfer function for this operation is calculated and the spectral properties of time series data on population size follow directly.The above approach is illustrated with a symmetrical two-species competition system assuming white noise variation in the carrying capacities. The results are interpreted in detail with the following ideas. (1) The intrinsic rate of increase governs the “responsiveness” of the population to changes in the carrying capacity; (2) one effect of competition is to reduce the “effective rate of increase” of the population. Increasing competition can produce effects identical to that of lowering the intrinsic rate of increase; (3) the other effect of competition is to communicate the stochastic variation in one species' carrying capacity to its competitors. The end result of this communication depends critically on the cross-correlation scheme among the carrying capacities of the competing species.     

Detecting determinism in short time series, with an application to the analysis of a stationary EEG recording

We have developed a new method for detecting determinism in a short time series and used this method to examine whether a stationary EEG is deterministic or stochastic. The method is based on the observation that the trajectory of a time series generated from a differentiable dynamical system behaves smoothly in an embedded phase space. The angles between two successive directional vectors in the trajectory reconstructed from a time series at a minimum embedding dimension were calculated as a function of time. We measured the irregularity of the angle variations obtained from the time series using second-order difference plots and central tendency measures, and compared these values with those from surrogate data. The ability of the proposed method to distinguish between chaotic and stochastic dynamics is demonstrated through a number of simulated time series, including data from Lorenz, R?ssler, and Van der Pol attractors, high-dimensional equations, and 1/f noise. We then applied this method to the analysis of stationary segments of EEG recordings consisting of 750 data points (6-s segments) from five normal subjects. The stationary EEG segments were not found to exhibit deterministic components. This method can be used to analyze determinism in short time series, such as those from physiological recordings, that can be modeled using differentiable dynamical processes.     

Periodicities in respiration and heart rate in newborns

Periodic breathing is common in normal infants, but may be associated with prolonged apnea leading to crib death. The mechanisms of periodic breathing and its relation to normal breathing patterns are unclear. We recorded respiratory and heart rate (HR) patterns of 11 healthy newborn infants during quiet sleep, in both normal and periodic breathing. Spectral analysis of the respiratory pattern revealed a low-frequency (LF) periodicity in normal breathing approximately equal to the frequency of periodic breathing when this occurs. Periodic breathing thus appears to be an exaggeration of an underlying slow amplitude variation which is present in regular breathing. LF periodicity also appeared in the HR pattern in both normal and periodic breathing, suggesting an LF modulation of cardiovascular control as well. The lack of a definite phase relation between HR and ventilation at LF may indicate dominant peripheral, rather than central, interactions between HR and respiration at these frequencies.     

Anti-control of periodic firing in HR model in the aspects of position,amplitude and frequency

This paper proposes a novel controller to control position, amplitude and frequency of periodic firing activity in Hindmarsh–Rose model based on Hopf bifurcation theory which is composed of linear control gain and nonlinear control gain. First, we select the activation of the fast ion channel as control parameter. Based on explicit criterion of Hopf bifurcation, a series of conditions are obtained to derive the linear gains of controller responsible for control of the location where the periodic firing activity occurs. Then, based on the control parameter, a series of conditions are obtained to derive the nonlinear gains of controller responsible for controlling the amplitude and frequency of periodic firing activity by using center manifold and normal form. Finally, the numerical experiments show that our controller can make the periodic firing activity occur at designed value and control the amplitude and frequency of periodic firing activity by adjusting nonlinear control gain of controller.     

Is breathing in infants chaotic? Dimension estimates for respiratory patterns during quiet sleep

Wedescribe an analysis of dynamic behavior apparent in times-seriesrecordings of infant breathing during sleep. Three principal techniqueswere used: estimation of correlation dimension, surrogate dataanalysis, and reduced linear (autoregressive) modeling (RARM). Correlation dimension can be used to quantify the complexity of timeseries and has been applied to a variety of physiological andbiological measurements. However, the methods most commonly used toestimate correlation dimension suffer from some technical problems thatcan produce misleading results if not correctly applied. We used a newtechnique of estimating correlation dimension that has fewer problems.We tested the significance of dimension estimates by comparingestimates with artificial data sets (surrogate data). On the basis ofthe analysis, we conclude that the dynamics of infant breathing duringquiet sleep can best be described as a nonlinear dynamic system withlarge-scale, low-dimensional and small-scale, high-dimensionalbehavior; more specifically, a noise-driven nonlinear system with atwo-dimensional periodic orbit. Using our RARM technique, we identifiedthe second period as cyclic amplitude modulation of the same period asperiodic breathing. We conclude that our data are consistent withrespiration being chaotic.

     

Formulas for intrinsic noise evaluation in oscillatory genetic networks

The linear noise approximation is a useful method for stochastic noise evaluations in genetic regulatory networks, where the covariance equation described as a Lyapunov equation plays a central role. We discuss the linear noise approximation method for evaluations of an intrinsic noise in autonomously oscillatory genetic networks; in such oscillatory networks, the covariance equation becomes a periodic differential equation that provides generally an unbounded covariance matrix, so that the standard method of noise evaluation based on the covariance matrix cannot be adopted directly. In this paper, we develop a new method of noise evaluation in oscillatory genetic networks; first, we investigate structural properties, e.g., orbital stability and periodicity, of the solutions to the covariance equation given as a periodic Lyapunov differential equation by using the Floquet-Lyapunov theory, and propose a global measure for evaluating stochastic amplitude fluctuations on the periodic trajectory; we also derive an evaluation formula for the period fluctuation. Finally, we apply our method to a model of circadian oscillations based on negative auto-regulation of gene expression, and show validity of our method by comparing the evaluation results with stochastic simulations.     

Persistence in fluctuating environments

Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations’ invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka–Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions.

One day is fine, the next is black.—The Clash     

Modelling of the control of heart rate by breathing using a kernel method

The process of the breathing (input) to the heart rate (output) of man is considered for system identification by the input-output relationship, using a mathematical model expressed as integral equations. The integral equation is considered and fixed so that the identification method reduces to the determination of the values within the integral, called kernels, resulting in an integral equation whose input-output behaviour is nearly identical to that of the system. This paper uses an algorithm of kernel identification of the Volterra series which greatly reduces the computational burden and eliminates the restriction of using white Gaussian input as a test signal. A second-order model is the most appropriate for a good estimate of the system dynamics. The model contains the linear part (first-order kernel) and quadratic part (second-order kernel) in parallel, and so allows for the possibility of separation between the linear and non-linear elements of the process. The response of the linear term exhibits the oscillatory input and underdamped nature of the system. The application of breathing as input to the system produces an oscillatory term which may be attributed to the nature of sinus node of the heart being sensitive to the modulating signal the breathing wave. The negative-on diagonal seems to cause the dynamic asymmetry of the total response of the system which opposes the oscillatory nature of the first kernel related to the restraining force present in the respiratory heart rate system. The presence of the positive-off diagonal of the second-order kernel of respiratory control of heart rate is an indication of an escape-like phenomenon in the system.