Revisited measles and chickenpox dynamics through orthogonal transformation

The question addressed is whether or not childhood epidemics such as measles and chickenpox are characterized by low-dimensional chaos. We propose a new method for the detection and extraction of hidden periodic components embedded in an irregular cyclical series, and study the characterization of the epidemiological series in terms of the characteristic features or periodicity attributes of the extracted components. It is shown that the measles series possesses two periodic components each having a period of one year. Both the periodic components have time-varying pattern, and the process is nonlinear and deterministic; there is no evidence of strong chaoticity in the measles dynamics. The chickenpox series has one seasonal component with stable pattern, and the process is deterministic but linear, and hence non-chaotic. We also propose surrogate generators based on null hypotheses relating to the variability of the periodicity attributes to analyse the dynamics in the epidemic series. The process dynamics is also studied using seasonally forced SEIR epidemic model, and the characterization performance of the proposed schemes is assessed.     

Analysis of cedar pollen time series: no evidence of low-dimensional chaotic behavior

Much of the current interest in pollen time series analysis is motivated by the possibility that pollen series arise from low-dimensional chaotic systems. If this is the case, short-range prediction using nonlinear modeling is justified and would produce high-quality forecasts that could be useful in providing pollen alerts to allergy sufferers. To date, contradictory reports about the characterization of the dynamics of pollen series can be found in the literature. Pollen series have been alternatively described as featuring and not featuring deterministic chaotic behavior. We showed that the choice of test for detection of deterministic chaos in pollen series is difficult because pollen series exhibit power spectra. This is a characteristic that is also produced by colored noise series, which mimic deterministic chaos in most tests. We proposed to apply the Ikeguchi–Aihara test to properly detect the presence of deterministic chaos in pollen series. We examined the dynamics of cedar (Cryptomeria japonica) hourly pollen series by means of the Ikeguchi–Aihara test and concluded that these pollen series cannot be described as low-dimensional deterministic chaos. Therefore, the application of low-dimensional chaotic deterministic models to the prediction of short-range pollen concentration will not result in high-accuracy pollen forecasts even though these models may provide useful forecasts for certain applications. We believe that our conclusion can be generalized to pollen series from other wind-pollinated plant species, as wind speed, the forcing parameter of the pollen emission and transport, is best described as a nondeterministic series that originates in the high dimensionality of the atmosphere.     

Chaotic Signatures of Heart Rate Variability and Its Power Spectrum in Health,Aging and Heart Failure

A paradox regarding the classic power spectral analysis of heart rate variability (HRV) is whether the characteristic high- (HF) and low-frequency (LF) spectral peaks represent stochastic or chaotic phenomena. Resolution of this fundamental issue is key to unraveling the mechanisms of HRV, which is critical to its proper use as a noninvasive marker for cardiac mortality risk assessment and stratification in congestive heart failure (CHF) and other cardiac dysfunctions. However, conventional techniques of nonlinear time series analysis generally lack sufficient sensitivity, specificity and robustness to discriminate chaos from random noise, much less quantify the chaos level. Here, we apply a ‘litmus test’ for heartbeat chaos based on a novel noise titration assay which affords a robust, specific, time-resolved and quantitative measure of the relative chaos level. Noise titration of running short-segment Holter tachograms from healthy subjects revealed circadian-dependent (or sleep/wake-dependent) heartbeat chaos that was linked to the HF component (respiratory sinus arrhythmia). The relative ‘HF chaos’ levels were similar in young and elderly subjects despite proportional age-related decreases in HF and LF power. In contrast, the near-regular heartbeat in CHF patients was primarily nonchaotic except punctuated by undetected ectopic beats and other abnormal beats, causing transient chaos. Such profound circadian-, age- and CHF-dependent changes in the chaotic and spectral characteristics of HRV were accompanied by little changes in approximate entropy, a measure of signal irregularity. The salient chaotic signatures of HRV in these subject groups reveal distinct autonomic, cardiac, respiratory and circadian/sleep-wake mechanisms that distinguish health and aging from CHF.     

Do the oscillations of cardiovascular parameters persist during voluntary apnea in humans?

The aim of this study was to ascertain the persistence of heart rate and blood pressure oscillations at the onset of voluntary apnea in humans and to assess the dependence of the fluctuations parameters on the chemoreceptor activity. In 24 young subjects (10 males, 14 females, mean age 20.4 years) heart rate (represented by its reciprocal value--RR-intervals), systolic blood pressure (SBP) and diastolic blood pressure (DBP) during controlled breathing (CB) of atmospheric air and oxygen followed by apnea were recorded continuously. The cosine functions were then fitted by nonlinear regression analysis to the heart rate, SBP and DBP oscillations during CB and at the onset of apnea. The parameters of oscillations were different during atmospheric air breathing compared to oxygen breathing. During oxygen breathing there was an increase of the RR-interval oscillations--relative bradycardia and enhanced magnitude of respiratory sinus arythmia. During apnea, the base level of the blood pressure oscillations was higher after breathing of atmospheric air compared to oxygen breathing. At least one cosine-like wave oscillation was present at the onset of apnea in the heart rate, SBP and DBP and the second wave was present in all assessed parameters in at least 70% of recordings. The oscillations in RR-intervals are, to some extent, independent of blood pressure oscillations. No significant gender differences were found either in the duration of breath holding or in the RR and SBP oscillations parameters.     

神经元的确定性与随机性整数倍放电

在大鼠损伤背根节神经元的自发放电中发现了整数倍放电, 为了阐明这种放电所产生的原因, 首先研究神经元模型中确定性混沌所引起的整数倍放电与噪声所诱发的整数倍放电的峰峰间期（ＩＳＩ） 序列,通过分析得到前者的ＩＳＩ序列是非线性可预报的,具有确定的非线性特性,但由噪声所诱发的整数倍放电的ＩＳＩ序列是不可预报的, 这表明这两种机制所产生的整数倍放电具有不同的特点,存在着定性的差别,并且混沌运动所产生的整数倍放电是由混沌中各阶不稳定周期轨道决定的。从这种差别出发,分析了实验中整数倍放电的ＩＳＩ 序列,得到该ＩＳＩ 序列是可非线性预报的,这表明大鼠损伤背根节神经元自发放电中的整数倍放电更可能是由确定性机制所产生的     

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。     

Cardiorespiratory interactions during resistive load breathing

The addition to the respiratory system of a resistive load results in breathing pattern changes and in negative intrathoracic pressure increases. The aim of this study was to use resistive load breathing as a stimulus to the cardiorespiratory interaction and to examine the extent of the changes in heart rate variability (HRV) and respiratory sinus arrhythmia (RSA) in relation to the breathing pattern changes. HRV and RSA were studied in seven healthy subjects where four resistive loads were applied in a random order during the breath and 8-min recording made in each condition. The HRV spectral power components were computed from the R-R interval sequences, and the RSA amplitude and phase were computed from the sinusoid fitting the instantaneous heart rate within each breath. Adding resistive loads resulted in 1) increasing respiratory period, 2) unchanging heart rate, and 3) increasing HRV and changing RSA characteristics. HRV and RSA characteristics are linearly correlated to the respiratory period. These modifications appear to be linked to load-induced changes in the respiratory period in each individual, because HRV and RSA characteristics are similar at a respiratory period obtained either by loading or by imposed frequency breathing. The present results are discussed with regard to the importance of the breathing cycle duration in these cardiorespiratory interactions, suggesting that these interactions may depend on the time necessary for activation and dissipation of neurotransmitters involved in RSA.     

Is there chaos in plankton dynamics?

A controversial issue in ecosystem modeling is whether the irregularfluctuations that one observes in nature are due solely to randomenvironmental factors or whether, at least partially, a deterministicmechanism is responsible for the unpredictable behavior. Thissecond alternative is called deterministic chaos and the issuein this paper is to decide if actual plankton time series canvindicate the hypothesis of chaotic dynamics. The near-neighborforecasting method is a recent technique for detecting determinismin a time series and we apply it to measurements of phytoplanktonand zooplankton biomass obtained at a single station in theMiddle Atlantic Bight. Although the results do not concludethe presence of chaos, they do give some support to the ideathat deterministic non-linear trophic dynamics may account forat least some of the variability that is seen in the data, particularlyin terms of inferring zooplankton oscillations from those ofphytoplankton.     

Familial aggregation of heart rate variability based on short recordings – the kibbutzim family study

The objective of this study was to assess the familial aggregation of heart rate variability (HRV), a readily measurable noninvasive reflection of cardiac autonomic function. Familial correlations were analyzed in 451 kibbutz members aged 15–97 years belonging to 80 kindreds. Five-minute duration Holter recordings made during silent supine spontaneous breathing and metronomic breathing were analyzed in the time and frequency domains. The present analysis considers the familial correlations and the heritability estimates of two time-domain indices, the standard deviation (SD) of the R-R interval (RR), reflecting total variability, and the root mean square of successive differences in RR intervals (RMSSD), reflecting vagal (parasympathetic) tone. During free breathing, age- and sex-adjusted correlations between parents and their children (r=0.24 for both indices) and between adult siblings above 30 years of age (r=0.24 and r=0.34 for SD and RMSSD, respectively) were statistically significant, whereas spouse correlations (r=–0.04, r=–0.02 for SD and RMSSD, respectively) and correlations in younger siblings (r=–0.22 and r=0.01, respectively) were not. Significant heritability estimates were demonstrated for the two indices (h ²=0.41 for SD and h ²=0.39 for RMSSD). These findings suggest that familial aggregation of HRV characteristics is determined mostly by genetic factors and less so by environmental factors and provide a basis for continuing the investigation into the underlying genetic influences on HRV. Received: 26 August 1997 / Accepted: 15 March 1998     

Biological Experimental Observations of an Unnoticed Chaos as Simulated by the Hindmarsh-Rose Model

An unnoticed chaotic firing pattern, lying between period-1 and period-2 firing patterns, has received little attention over the past 20 years since it was first simulated in the Hindmarsh-Rose (HR) model. In the present study, the rat sciatic nerve model of chronic constriction injury (CCI) was used as an experimental neural pacemaker to investigate the transition regularities of spontaneous firing patterns. Chaotic firing lying between period-1 and period-2 firings was observed located in four bifurcation scenarios in different, isolated neural pacemakers. These bifurcation scenarios were induced by decreasing extracellular calcium concentrations. The behaviors after period-2 firing pattern in the four scenarios were period-doubling bifurcation not to chaos, period-doubling bifurcation to chaos, period-adding sequences with chaotic firings, and period-adding sequences with stochastic firings. The deterministic structure of the chaotic firing pattern was identified by the first return map of interspike intervals and a short-term prediction using nonlinear prediction. The experimental observations closely match those simulated in a two-dimensional parameter space using the HR model, providing strong evidences of the existence of chaotic firing lying between period-1 and period-2 firing patterns in the actual nervous system. The results also present relationships in the parameter space between this chaotic firing and other firing patterns, such as the chaotic firings that appear after period-2 firing pattern located within the well-known comb-shaped region, periodic firing patterns and stochastic firing patterns, as predicted by the HR model. We hope that this study can focus attention on and help to further the understanding of the unnoticed chaotic neural firing pattern.     

Exciting chaos with noise: unexpected dynamics in epidemic outbreaks

 In this paper, we identify a mechanism for chaos in the presence of noise. In a study of the SEIR model, which predicts epidemic outbreaks in childhood diseases, we show how chaotic dynamics can be attained by adding stochastic perturbations at parameters where chaos does not exist apriori. Data recordings of epidemics in childhood diseases are still argued as deterministic chaos. There also exists noise due to uncertainties in the contact parameters between those who are susceptible and those who are infected, as well as random fluctuations in the population. Although chaos has been found in deterministic models, it only occurs in parameter regions that require a very large population base or other large seasonal forcing. Our work identifies the mechanism whereby chaos can be induced by noise for realistic parameter regions of the deterministic model where it does not naturally occur. Received: 13 October 2000 / Revised version: 15 May 2001 / Published online: 7 December 2001     

Biomechanics and regulation of external respiration under conditions of five-day dry immersion

The functional state of external respiration and the features of its regulation in healthy persons were studied under conditions of microgravity simulated using dry immersion. The lung volume, the ratio of thoracic and abdominal components during quiet breathing and performing various respiratory maneuvers, as well as the parameters that characterize the regulation of breathing (the duration of breath holding and the ability to voluntarily control respiratory movements), were recorded during the baseline period, on days 2 and 4 of dry immersion, and after the end of the dry immersion. It has been shown that the breathing pattern did not significantly change under conditions of dry immersion compared to the baseline period; however, the inspiratory reserve volume increased (p < 0.05), while the expiratory reserve volume decreased (p < 0.01). Dry immersion did not alter pulmonary ventilation, yet most of the subjects trended toward an increase in the contribution of the abdominal component of breathing movements during quiet breathing and demonstrated a statistically significant increase in this parameter during the lung vital capacity maneuver. The durations of the inspiratory and expiratory maximal breath holding under conditions of immersion did not differ from the background values. During the immersion, the accuracy of voluntary control of breathing increased. We believe that immersion, similar to microgravity, leads to changes in the reserve lung volume, which are partly because of changes in the body position; changes in relative contributions of the thoracic and abdominal components in the breathing movements; and changes in voluntary breath regulation.     

Chaotic attractor in two-prey one-predator system originates from interplay of limit cycles

We investigate the appearance of chaos in a microbial 3-species model motivated by a potentially chaotic real world system (as characterized by positive Lyapunov exponents (Becks et al., Nature 435, 2005). This is the first quantitative model that simulates characteristic population dynamics in the system. A striking feature of the experiment was three consecutive regimes of limit cycles, chaotic dynamics and a fixed point. Our model reproduces this pattern. Numerical simulations of the system reveal the presence of a chaotic attractor in the intermediate parameter window between two regimes of periodic coexistence (stable limit cycles). In particular, this intermediate structure can be explained by competition between the two distinct periodic dynamics. It provides the basis for stable coexistence of all three species: environmental perturbations may result in huge fluctuations in species abundances, however, the system at large tolerates those perturbations in the sense that the population abundances quickly fall back onto the chaotic attractor manifold and the system remains. This mechanism explains how chaos helps the system to persist and stabilize against migration. In discrete populations, fluctuations can push the system towards extinction of one or more species. The chaotic attractor protects the system and extinction times scale exponentially with system size in the same way as with limit cycles or in a stable situation.     

Comparative Evaluation of Heart Rate Variability Based on the Data of ECG and Blood Pressure Measurements

The development of new approaches to the assessment of heart rate variability (HRV) is an important problem, since HRV reflects the functioning of cardiovascular control and is affected by various diseases. The purpose of this study was to evaluate the informative value of statistical and spectral HRV parameters calculated from pulse interval (PI) data of blood pressure as compared with those calculated from RR-interval data of electrocardiograms (ECG). We recorded ECG in conscious rats using skin adhesive electrodes simultaneously with blood pressure signal obtained through a catheter in the femoral artery. It has been found that the PI sequence can be used to calculate the statistical HRV indices that describe the HRV at time intervals about 1 min or longer, but statistical indices of the PI and RR intervals may differ in the analysis of beat-tobeat variations. The power spectra of the RR intervals and PI coincide in the low-frequency region, including the band of baroreflex cardiac rhythm oscillation. However, they can differ in the high-frequency region (at respiration frequency and above).     

Model of fringing reef development in response to progressive sea level fall over the last 7000 years – (Kikai-jima, Ryukyu Islands, Japan)

 Kikai-jima in the central Ryukyu Islands of Japan is fringed by exposed terraces of Holocene reefs, which formed as a result of periodic local tectonic uplift associated with subduction/collision. The terraces form four topographically distinct features (TI-IV) around the island and represent reefs that grew to sea level at 9000–6065 y BP, 6065–3390 y BP, 3790–2630 y BP, and 2870 to 1550 y BP. The modern reef terrace has been growing since approximately 1550 y BP. The reef terraces were uplifted sequentially around 6050 y BP (4 m), 3390–3790 y BP (2.5 m), 2630–2870 y BP (1 m) and 1550 y BP (2.5 m). Five sites were studied to define reef development in response to periodic relative sea level fall and different stillstand recovery periods. Thirty coral genera and 70 species were recorded from four distinct shallow reef flat to upper reef slope and one deeper reef slope coral assemblage. Significant lateral variations in total coral abundance, genera number, diversity, and the coverage density of Acropora spp. and Faviids occur both within and between the terraces. Stratigraphically, drill core and outcrop data recorded shallowing upward sequences characterised by tabulate Acropora spp. overlying massive Porites sp. and Faviids. The biological variations may represent growth strategies responding to initial colonisation, episodic perturbation (relative sea level fall) and differing recovery times during stillstands, and indicate a reef ecosystem stable and strong enough to recover after substantial perturbations. However, this study suggests that relatively small geological changes have had substantial biological effects, and modelling indicates that such changes would have been more profound had a third factor, such as substrate angle, varied more dramatically. In such a case, the drowning growth strategy exhibited in the drill core transect may have been more prevalent, and reefs would be struggling to grow around Kikai-jima today. Accepted: 27 May 1998     

Can noise induce chaos?

An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long-term average over the deterministic attractor while the SLE is the long-term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that "chaos" should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.     

Nonlinear heart rate variability measures under electromagnetic fields produced by GSM cellular phones

This study was designed to assess the nonlinear dynamics of heart rate variability (HRV) during exposure to low-intensity EMFs. Twenty-six healthy young volunteers were subjected to a rest-to-stand protocol to evaluate autonomic nervous system in quiet condition (rest, vagal prevalence) and after a sympathetic activation (stand). The procedure was conducted twice in a double-blind design: once with a genuine EMFs exposure (GSM cellular phone at 900 MHz, 2 W) and once with a sham exposure (at least 24 h apart). During each session, three-lead electrocardiograms were recorded and RR series extracted off-line. The RR series were analyzed by nonlinear deterministic techniques in every phase of the protocol and during the different exposures. The analysis of the data shows there was no statistically significant effect due to GSM exposure on the nonlinear dynamics of HRV.     

Effect of inspiratory muscle fatigue on breathing pattern

Our aim was to determine whether inspiratory muscle fatigue changes breathing pattern and whether any changes seen occur before mechanical fatigue develops. Nine normal subjects breathed through a variable inspiratory resistance with a predetermined mouth pressure (Pm) during inspiration and a fixed ratio of inspiratory time to total breath duration. Breathing pattern after resistive breathing (recovery breathing pattern) was compared with breathing pattern at rest and during CO2 rebreathing (control breathing pattern) for each subject. Relative rapid shallow breathing was seen after mechanical fatigue and also in experiments with electromyogram evidence of diaphragmatic fatigue where Pm was maintained at the predetermined level during the period of resistive breathing. In contrast there was no significant difference between recovery and control breathing patterns when neither mechanical nor electromyogram fatigue was seen. It is suggested that breathing pattern after inspiratory muscle fatigue changes in order to minimize respiratory sensation.     

Only noise can induce chaos in discrete populations

Motivated by the papers from Ellner and Turchin 2005 and Dennis et al. 2003 we investigate the possibility to detect chaos in noisy ecological systems. One message of our paper is that if a dynamic model is available and if this model predicts chaotic behaviour, one should consider its discrete-state, noisy version when fitting numerical predictions to observations. We emphasize that deterministic discrete-state models behave periodically, thus only the interaction of these deterministic skeletons with random noise can produce non-regular dynamics. We detect and describe a relatively sharply defined range of the noise (the grey zone) where the gradual transition from periodic to chaotic behaviour happens. This zone, the upper border of which can be predicted analytically, is identified in experimental data as well as in numerical simulations. In the grey zone the global, statistical behaviour will approach the statistics produced by the chaotic, continuous model, and in this sense we claim that noise can produce chaos.     

Time series prediction of lung cancer patients' breathing pattern based on nonlinear dynamics

This study focuses on predicting breathing pattern, which is crucial to deal with system latency in the treatments of moving lung tumors. Predicting respiratory motion in real-time is challenging, due to the inherent chaotic nature of breathing patterns, i.e. sensitive dependence on initial conditions. In this work, nonlinear prediction methods are used to predict the short-term evolution of the respiratory system for 62 patients, whose breathing time series was acquired using respiratory position management (RPM) system. Single step and N-point multi step prediction are performed for sampling rates of 5 Hz and 10 Hz. We compare the employed non-linear prediction methods with respect to prediction accuracy to Adaptive Infinite Impulse Response (IIR) prediction filters. A Local Average Model (LAM) and local linear models (LLMs) combined with a set of linear regularization techniques to solve ill-posed regression problems are implemented. For all sampling frequencies both single step and N-point multi step prediction results obtained using LAM and LLM with regularization methods perform better than IIR prediction filters for the selected sample patients. Moreover, since the simple LAM model performs as well as the more complicated LLM models in our patient sample, its use for non-linear prediction is recommended.