A dynamical system analysis of the development of spontaneous lower extremity movements in newborn and young infants

This study's aim was to evaluate the characteristics of newborn and young infants' spontaneous lower extremity movements by using dynamical systems analysis. Participants were 8 healthy full-term newborn infants (3 boys, 5 girls, mean birth weight and gestational age were 3070.6 g and 39 weeks). A tri-axial accelerometer measured limb movement acceleration in 3-dimensional space. Movement acceleration signals were recorded during 200 s from just below the ankle when the infant was in an active alert state and lying supine (sampling rate 200 Hz). Data were analyzed linearly and nonlinearly. As a result, the optimal embedding dimension showed more than 5 at all times. Time dependent changes started at 6 or 7, and over the next four months decreased to 5 and from 6 months old, increased. The maximal Lyapnov exponent was positive for all segments. The mutual information is at its greatest range at 0 months. Between 3 and 4 months the range in results is narrowest and lowest in value. The mean coefficient of correlation for the x-axis component was negative and y-axis component changed to a positive value between 1 month old and 4 months old. Nonlinear time series analysis suggested that newborn and young infants' spontaneous lower extremity movements are characterized by a nonlinear chaotic dynamics with 5 to 7 embedding dimensions. Developmental changes of an optimal embedding dimension showed a U-shaped phenomenon. In addition, the maximal Lyapnov exponents were positive for all segments (0.79-2.99). Infants' spontaneous movement involves chaotic dynamic systems that are capable of generating voluntary skill movements.     

基于混沌理论的蛋白质序列特性的研究

蛋白质序列特性的研究对于蛋白质的结构及功能具有重要意义。该文为了研究蛋白质序列是否具有混沌行为,先将蛋白质序列通过氨基酸电子离子相互作用势(electron interaction potential,EIIP)转化为时间序列,再根据混沌理论对其进行相空间重构,利用去偏自相关系数,经典G-P算法确定系统的时间延迟t和嵌入维数m,系统的最大Lyapunov指数则用改进的最大Lyapunov指数计算方法计算,其结果绝大多数为正,从而确认了蛋白质时间序列的混沌行为,并对特例进行了说明。     

Ecological implications of a latitudinal gradient in inter-annual climatic variability: a test using fractal and chaos theories

Although measures of seasonality are common, ecologists have seldom measured inter-annual variation or tested for the presence of a latitudinal gradient. Yet, species diversity and range size may be influenced by latitudinal gradients in climatic variability. Fractal and chaos theories offer new techniques to measure climatic unpredictability by analyzing the dynamics of nonlinear time-series data. We analyzed data on the number of freezing degree-days and date of first permanent ice from 12 weather stations (43–83°N) in Canada to describe latitudinal gradients in the timing of seasonal events. Time-series (ca 30 yr) of climatic variables were found to be nonlinear dynamic systems that were neither stochastic nor cyclic. The analyses reveal a latitudinal gradient in system memory length, with high latitude locations displaying shorter memory length (R/S fractal analysis and Lyapunov exponent). This latitudinal gradient is also characterized by more random and irregular fractal patterns (Hurst exponent) associated with higher latitudes. Both of these patterns may be due to the simpler climatic systems in high latitudes as indicated by fewer dynamic variables (lower correlation dimension). Application of fractal and chaos theories show that the timing of seasonal events (i.e. inter-annual variation) comprises a chaotic-dynamic system of comparatively low dimension.     

胃电的非线性动力学分析

我们用非线性动力学的方法分析了两例健康人的胃电图（Ｅｌｅｃｔｒｏｇａｓｔｒｏｇｒａｍ,简称ＥＧＧ）,得出它的功率谱、计算了它的关联维数和最大Ｌｙａｐｕｎｏｖ指数。结果显示：它的功率谱为一有尖峰的连续谱;其关联维数介于６和７之间;它的最大Ｌｖａａｕｎｏｖ指数大于０。由此,我们可以初步推知ＥＧＧ不是随机噪声,它服从确定性的动力学规律。     

Sensitivity of Chaos Measures in Detecting Stress in the Focusing Control Mechanism of the Short-Sighted Eye

When fixating on a stationary object, the power of the eye’s lens fluctuates. Studies have suggested that changes in these so-called microfluctuations in accommodation may be a factor in the onset and progression of short-sightedness. Like many physiological signals, the fluctuations in the power of the lens exhibit chaotic behaviour. A breakdown or reduction in chaos in physiological systems indicates stress to the system or pathology. The purpose of this study was to determine whether the chaos in fluctuations of the power of the lens changes with refractive error, i.e. how short-sighted a subject is, and/or accommodative demand, i.e. the effective distance of the object that is being viewed. Six emmetropes (EMMs, non-short-sighted), six early-onset myopes (EOMs, onset of short-sightedness before the age of 15), and six late-onset myopes (LOMs, onset of short-sightedness after the age of 15) took part in the study. Accommodative microfluctuations were measured at 22 Hz using an SRW-5000 autorefractor at accommodative demands of 1 D (dioptres), 2 D, and 3 D. Chaos theory analysis was used to determine the embedding lag, embedding dimension, limit of predictability, and Lyapunov exponent. Topological transitivity was also tested for. For comparison, the power spectrum and standard deviation were calculated for each time record. The EMMs had a statistically significant higher Lyapunov exponent than the LOMs (\(0.64\pm 0.33\) vs. \(0.39\pm 0.20~\hbox {D}/\hbox {s}\)) and a lower embedding dimension than the LOMs (\(3.28\pm 0.46\) vs. \(3.67\pm 0.49\)). There was insufficient evidence (non-significant p value) of a difference between EOMs and EMMs or EOMs and LOMs. The majority of time records were topologically transitive. There was insufficient evidence of accommodative demand having an effect. Power spectrum analysis and assessment of the standard deviation of the fluctuations failed to discern differences based on refractive error. Chaos differences in accommodation microfluctuations indicate that the control system for LOMs is under stress in comparison to EMMs. Chaos theory analysis is a more sensitive marker of changes in accommodation microfluctuations than traditional analysis methods.     

Nonlinear analysis of epileptic activity in rabbit neocortex

We report on the nonlinear analysis of electroencephalogram (EEG) recordings in the rabbit visual cortex. Epileptic seizures were induced by local penicillin application and triggered by visual stimulation. The analysis procedures for nonlinear signals have been developed over the past few years and applied primarily to physical systems. This is an early application to biological systems and the first to EEG data. We find that during epileptic activity, both global and local embedding dimensions are reduced with respect to nonepileptic activity. Interestingly, these values are very low () and do not change between preictal and tonic stages of epileptic activity, also the Lyapunov dimension remains constant. However, between these two stages the manifestations of the local dynamics change quite drastically, as can be seen, e.g., from the shape of the attractors. Furthermore, the largest Lyapunov exponent is reduced by a factor of about two in the second stage and characterizes the difference in dynamics. Thus, the occurrence of clinical symptoms associated with the tonic seizure activity seems to be mainly related to the local dynamics of the nonlinear system. These results thus seem to give a strong indication that the dynamics remains much the same in these stages of behavior, and changes are due to alterations in model parameters and consequent bifurcations of the observed orbits. Received: 5 February 1997 / Accepted in revised form: 18 September 1997     

Biological variation of glucose and insulin includes a deterministic chaotic component

Serial data of glucose and insulin values of individual patients vary over short periods of time; this phenomenon has been called biological variation. The classic homeostatic control model assumes that the physiological mechanisms maintaining the concentrations of glucose and insulin are linear. The only deviations over a short period of time one should observe are in relation to a glucose load or major hormonal disturbance. Otherwise, the values of these analytes should be constant and any variations seen are due to random disturbances. We investigated previously published serial data (three for glucose and one for insulin) with nonlinear analytical methods, such as embedding space, correlation dimension, Lyapunov exponents, singular value decomposition and phase portraits, as well as linear methods, such as power spectra and autocorrelation functions. The power spectra failed to show dominant frequencies, but the autocorrelation functions showed significant correlation, consistent with a deterministic process. The correlation dimension was finite, around 4.0, the first Lyapunov exponent was positive, indicative of a deterministic chaotic process. Furthermore, the phase portraits showed directional flow. Therefore, the short-term biological variation observed for analytes arises from nonlinear, deterministic chaotic behavior instead of random variation.     

Nonlinear analysis of EEG signals: Surrogate data analysis

ObjectivesThe electroencephalogram (EEG) signal contains information about the state and condition of the brain. The aim of the study is to conduct a nonlinear analysis of the EEG signals and to compare the differences in the nonlinear characteristics of the EEG during normal state and the epileptic state.DataThe EEG data used for this study – which consisted of epileptic EEG and normal EEG – were obtained from the EEG database available with the Bonn University, Germany.ResultsThe attractors seen in normal and epileptic human brain dynamics were studied and compared. Surrogate data analyses were conducted on two nonlinear measures, namely the largest Lyapunov exponent and the correlation dimension, to test the hypothesis whether EEG signals were in accordance with linear stochastic models.DiscussionsThe existence of deterministic chaos in brain activity is confirmed by the existence of a chaotic attractor; also, saturation of the correlation dimension towards a definite value is the manifestation of a deterministic dynamics. Also a reduction is observed between the dimensionalities of the brain attractors from normal state to the epileptic state. The evaluation of the largest Lyapunov exponent also confirms the lowering of complexity during an episode of seizure.ConclusionIn case of Lyapunov exponent of EEG data, the change due to surrogating is small suggesting that it is not representing the system complexity properly but there is a marked change in the case of correlation dimension value due to surrogating.     

Ventilatory chaos is impaired in carotid atherosclerosis

Ventilatory chaos is strongly linked to the activity of central pattern generators, alone or influenced by respiratory or cardiovascular afferents. We hypothesized that carotid atherosclerosis should alter ventilatory chaos through baroreflex and autonomic nervous system dysfunctions. Chaotic dynamics of inspiratory flow was prospectively evaluated in 75 subjects undergoing carotid ultrasonography: 27 with severe carotid stenosis (>70%), 23 with moderate stenosis (<70%), and 25 controls. Chaos was characterized by the noise titration method, the correlation dimension and the largest Lyapunov exponent. Baroreflex sensitivity was estimated in the frequency domain. In the control group, 92% of the time series exhibit nonlinear deterministic chaos with positive noise limit, whereas only 68% had a positive noise limit value in the stenoses groups. Ventilatory chaos was impaired in the groups with carotid stenoses, with significant parallel decrease in the noise limit value, correlation dimension and largest Lyapunov exponent, as compared to controls. In multiple regression models, the percentage of carotid stenosis was the best in predicting the correlation dimension (p<0.001, adjusted R²: 0.35) and largest Lyapunov exponent (p<0.001, adjusted R²: 0.6). Baroreflex sensitivity also predicted the correlation dimension values (p = 0.05), and the LLE (p = 0.08). Plaque removal after carotid surgery reversed the loss of ventilatory complexity. To conclude, ventilatory chaos is impaired in carotid atherosclerosis. These findings depend on the severity of the stenosis, its localization, plaque surface and morphology features, and is independently associated with baroreflex sensitivity reduction. These findings should help to understand the determinants of ventilatory complexity and breathing control in pathological conditions.     

Pseudo-Lyapunov exponents and predictability of Hodgkin-Huxley neuronal network dynamics

We present a numerical analysis of the dynamics of all-to-all coupled Hodgkin-Huxley (HH) neuronal networks with Poisson spike inputs. It is important to point out that, since the dynamical vector of the system contains discontinuous variables, we propose a so-called pseudo-Lyapunov exponent adapted from the classical definition using only continuous dynamical variables, and apply it in our numerical investigation. The numerical results of the largest Lyapunov exponent using this new definition are consistent with the dynamical regimes of the network. Three typical dynamical regimes—asynchronous, chaotic and synchronous, are found as the synaptic coupling strength increases from weak to strong. We use the pseudo-Lyapunov exponent and the power spectrum analysis of voltage traces to characterize the types of the network behavior. In the nonchaotic (asynchronous or synchronous) dynamical regimes, i.e., the weak or strong coupling limits, the pseudo-Lyapunov exponent is negative and there is a good numerical convergence of the solution in the trajectory-wise sense by using our numerical methods. Consequently, in these regimes the evolution of neuronal networks is reliable. For the chaotic dynamical regime with an intermediate strong coupling, the pseudo-Lyapunov exponent is positive, and there is no numerical convergence of the solution and only statistical quantifications of the numerical results are reliable. Finally, we present numerical evidence that the value of pseudo-Lyapunov exponent coincides with that of the standard Lyapunov exponent for systems we have been able to examine.     

Chaotic behavior in the locomotion ofamoeba proteus

Summary The locomotion ofAmoeba proteus has been investigated by algorithms evaluating correlation dimension and Lyapunov spectrum developed in the field of nonlinear science. It is presumed by these parameters whether the random behavior of the system is stochastic or deterministic. For the analysis of the nonlinear parameters, n-dimensional time-delayed vectors have been reconstructed from a time series of periphery and area ofA. proteus images captured with a charge-coupled-device camera, which characterize its random motion. The correlation dimension analyzed has shown the random motion ofA. proteus is subjected only to 3–4 macrovariables, though the system is a complex system composed of many degrees of freedom. Furthermore, the analysis of the Lyapunov spectrum has shown its largest exponent takes positive values. These results indicate the random behavior ofA. proteus is chaotic and deterministic motion on an attractor with low dimension. It may be important for the elucidation of the cell locomotion to take account of nonlinear interactions among a small number of dynamics such as the sol-gel transformation, the cytoplasmic streaming, and the relating chemical reaction occurring in the cell.     

Research on the relation of EEG signal chaos characteristics with high-level intelligence activity of human brain

Using phase space reconstruct technique from one-dimensional and multi-dimensional time series and the quantitative criterion rule of system chaos, and combining the neural network; analyses, computations and sort are conducted on electroencephalogram (EEG) signals of five kinds of human consciousness activities (relaxation, mental arithmetic of multiplication, mental composition of a letter, visualizing a 3-dimensional object being revolved about an axis, and visualizing numbers being written or erased on a blackboard). Through comparative studies on the determinacy, the phase graph, the power spectra, the approximate entropy, the correlation dimension and the Lyapunov exponent of EEG signals of 5 kinds of consciousness activities, the following conclusions are shown: (1) The statistic results of the deterministic computation indicate that chaos characteristic may lie in human consciousness activities, and central tendency measure (CTM) is consistent with phase graph, so it can be used as a division way of EEG attractor. (2) The analyses of power spectra show that ideology of single subject is almost identical but the frequency channels of different consciousness activities have slight difference. (3) The approximate entropy between different subjects exist discrepancy. Under the same conditions, the larger the approximate entropy of subject is, the better the subject's innovation is. (4) The results of the correlation dimension and the Lyapunov exponent indicate that activities of human brain exist in attractors with fractional dimensions. (5) Nonlinear quantitative criterion rule, which unites the neural network, can classify different kinds of consciousness activities well. In this paper, the results of classification indicate that the consciousness activity of arithmetic has better differentiation degree than that of abstract.     

Effect of naloxone on body temperature in postmenopausal women with Parkinson's disease

The role exerted by the endogenous opioid system on thermoregulation has been studied in six postmenopausal women affected by Parkinson's disease and in 6 age-matched, normal postmenopausal women, as controls. The women randomly received an infusion of the opioid antagonist naloxone (1.6 mg/h for 4 h) or of saline on two consecutive days. Body temperature, as evaluated by rectal temperature, was significantly lower (p less than 0.05) in Parkinsonian than in normal women, and it did not vary during saline infusion, in either groups. Naloxone infusion significantly reduced (p less than 0.01) body temperature in normal postmenopausal women, but it was unable to modify body temperature in women affected by Parkinson's disease. The low basal body temperature values and the inability of naloxone to exert a hypothermic effect in women suffering from Parkinson's disease seem to constitute further evidence for an impaired regulation of body temperature and impaired activity of the endogenous opioid system in this pathology.     

Nonlinear dynamics of the blood flow studied by Lyapunov exponents

In order to gain an insight into the dynamics of the cardiovascular system throughout which the blood circulates, the signals measured from peripheral blood flow in humans were analyzed by calculating the Lyapunov exponents. Over a wide range of algorithm parameters, paired values of both the global and the local Lyapunov exponents were obtained, and at least one exponent equaled zero within the calculation error. This may be an indication of the deterministic nature and finite number of degrees of freedom of the cardiovascular system governing the blood-flow dynamics on a time scale of minutes. A difference was observed in the Lyapunov dimension of controls and athletes.     

Low-dimensional chaotic attractors in the rat brain

The existence of chaotic attractors for discrete time series, derived from the occurrences of spikes during electrophysiological recordings, was investigated. The time series included between 800 and 5200 points per analyzed record. The spike trains were recorded in the substantia nigra pars reticulata (n=13) and in the auditory thalamus (n=14). The experiments were performed on anesthetized rats during spontaneous activity and during auditory stimulation. According to standard methods of dynamical systems theory, an embedding space was constructed using delay coordinates. The embedding and correlation dimensions were computed by means of the correlation integrals. For 7 of 27 samples, a deterministic structure with a low embedding dimension (ranging between 2 and 6) and a correlation dimension between 0.14 and 3.3 could be determined. Evidence was found that the sensory stimulation may affect the chaotic behavior. Single units recorded simultaneously from the same electrode tip may display different chaotic dynamics, even with a similar time-locked response to the stimulus onset.     

低维非线性呼吸系统的复杂性计算

首先应用混沌计算方法：关联维数D2和最大李亚普努夫指数LLE,以及替代数据分析法,分析了男性受试者在平静状态下记录的呼吸系统时间序列的混沌特征。同时,在呼吸变量的非线性分析中首次引进了被称为C0复杂度的新技术．它的应用将有助于更好地理解自主神经系统中潜在的生理过程。LLE计算的替代数据法分析结果显示,没有明确的证据可以证实受试者在平静状态下的呼吸时序的模态是混沌的。然而,C0复杂度的计算结果却表明大部分呼吸系统的时间序列表现为某种程度的复杂性,这为呼吸模态的非随机变化的属性提供了部分的实验和计算证据。更进一步,C0复杂度有可能以一种新的、确定的方式给出呼吸系统在激励状态下的量化改变。     

Dimensional analysis revisited

The applicability of dimensional analysis (DA) is discussed in relation to the metabolic scaling laws. The evolution of different theories of biological similarity has shown that the calculated reduced exponents (b) of Huxley's allometric equation are closely correlated with the numerical values obtained from the statistical analysis of empirical data. Body mass and body weight are not equivalent as biological reference systems, since in accordance to Newton's second law, the former has a dimension of a mass, while the latter should be dimensionally considered as a force (W = MLT-2). This distinction affects the coefficients of the mass exponent (alpha). This difference is of paramount importance in microgravity conditions (spaceflight) and of buoyancy during the fetal life in mammals. Furthermore, the coefficients (beta) of the length dimension, and (gamma) of the time dimension do not vary when mass or weight are utilized as reference systems. Consequently, the "specific metabolic time," that results from the ratio of basal oxygen consumption and body mass or body weight yields the "biological meaning" of the time dimension, which is of fractal nature.     

Singing of <Emphasis Type="Italic">Neoconocephalus robustus</Emphasis> as an example of deterministic chaos in insects

We use nonlinear time series analysis methods to analyse the dynamics of the sound-producing apparatus of the katydid Neoconocephalus robustus. We capture the dynamics by analysing a recording of the singing activity. First, we reconstruct the phase space from the sound recording and test it against determinism and stationarity. After confirming determinism and stationarity, we show that the maximal Lyapunov exponent of the series is positive, which is a strong indicator for the chaotic behaviour of the system. We discuss that methods of nonlinear time series analysis can yield instructive insights and foster the understanding of acoustic communication among insects.     

Nonlinearities in mating sounds of American crocodiles

We use nonlinear time series analysis methods to analyze the dynamics of the sound-producing apparatus of the American crocodile (Crocodylus acutus). We capture its dynamics by analyzing a recording of the singing activity during mating time. First, we reconstruct the phase space from the sound recording and thereby reveal that the attractor needs no less than five degrees of freedom to fully evolve in the embedding space, which suggests that a rather complex nonlinear dynamics underlies its existence. Prior to investigating the dynamics more precisely, we test whether the reconstructed attractor satisfies the notions of determinism and stationarity, as a lack of either of these properties would preclude a meaningful further analysis. After positively establishing determinism and stationarity, we proceed by showing that the maximal Lyapunov exponent of the recording is positive, which is a strong indicator for the chaotic behavior of the system, confirming that dynamical nonlinearities are an integral part of the examined sound-producing apparatus. At the end, we discuss that methods of nonlinear time series analysis could yield instructive insights and foster the understanding of vocal communication among certain reptile species.     

Lyapunov exponents computation for hybrid neurons

Lyapunov exponents are a basic and powerful tool to characterise the long-term behaviour of dynamical systems. The computation of Lyapunov exponents for continuous time dynamical systems is straightforward whenever they are ruled by vector fields that are sufficiently smooth to admit a variational model. Hybrid neurons do not belong to this wide class of systems since they are intrinsically non-smooth owing to the impact and sometimes switching model used to describe the integrate-and-fire (I&F) mechanism. In this paper we show how a variational model can be defined also for this class of neurons by resorting to saltation matrices. This extension allows the computation of Lyapunov exponent spectrum of hybrid neurons and of networks made up of them through a standard numerical approach even in the case of neurons firing synchronously.