Can the analytic techniques of nonlinear dynamics distinguish periodic, random and chaotic signals?

Recent advances in the mathematical discipline of nonlinear dynamics have led to its use in the analysis of many biologic processes. But the ability of the tools of nonlinear dynamic analysis to identify chaotic behavior has not been determined. We analyzed a series of signals--periodic, chaotic and random--with five tools of nonlinear dynamics. Periodic signals were sine, square, triangular, sawtooth, modulated sine waves and quasiperiodic, generated at multiple amplitudes and frequencies. Chaotic signals were generated by solving sets of nonlinear equations including the logistic map, Duffing's equation, Lorenz equations and the Silnikov attractor. Random signals were both discontinuous and continuous. Gaussian noise was added to some signals at magnitudes of 1, 2, 5, 10 and 20% of the signal's amplitude. Each signal was then subjected to tools of nonlinear dynamics (phase plane plot, return map, Poincaré section, correlation dimension and spectral analysis) to determine the relative ability of each to characterize the underlying system as periodic, chaotic or random. In the absence of noise, phase plane plots and return maps were the most sensitive detectors of chaotic and periodic processes. Spectral analysis could determine if a process was periodic or quasiperiodic, but could not distinguish between chaotic and random signals. Correlation dimension was useful to determine the overall complexity of a signal, but could not be used in isolation to identify a chaotic process. Noise at any level effaced the structure of the phase plane plot. Return maps were relatively immune to noise at levels of up to 5%. Spectral analysis and correlation dimension were insensitive to noise. Accordingly, we recommend that unknown signals be subjected to all of the techniques to increase the accuracy of identification of the underlying process. Based on these data, we conclude that no single test is sufficiently sensitive or specific to categorize an unknown signal as chaotic.     

Synaptic transmission of chaotic spike trains between primary afferent fiber and spinal dorsal horn neuron in the rat

Primary sensory neurons can generate irregular burst firings in which the existence of significant deterministic behaviors of chaotic dynamics has been proved with nonlinear time series analysis. But how well the deterministic characteristics and neural information of presynaptic chaotic spike trains were transmitted into postsynaptic spike trains is still an open question. Here we investigated the synaptic transmission of chaotic spike trains between primary Adelta afferent fiber and spinal dorsal horn neuron. Two kinds of basic stimulus unit, brief burst and single pulse, were employed by us to comprise chaotic stimulus trains. For time series analysis, we defined "events" as the longest sequences of spikes with all interspike intervals less than or equal to a certain threshold and extracted the interevent intervals (IEIs) from spike trains. Return map analysis of the IEI series showed that the main temporal structure of chaotic input trains could be detected in postsynaptic output trains, especially under brief-burst stimulation. Using correlation dimension and nonlinear prediction methods, we found that synaptic transmission could influence the nonlinear characteristics of chaotic trains, such as fractal dimension and short-term predictability, with greater influence made under single-pulse stimulation. By calculating the mutual information between input and output trains, we found the information carried by presynaptic spike trains could not be completely transmitted at primary afferent synapses, and that brief bursts could more reliably transmit the information carried by chaotic input trains across synapses. These results indicate that although unreliability exists during synaptic transmission, the main deterministic characteristics of chaotic burst trains can be transmitted across primary afferent synapses. Moreover, brief bursts that come from the periphery can more reliably transmit neural information between primary afferent fibers and spinal dorsal horn neurons.     

Assessment of Chaotic Parameters in Nonstationary Electrocardiograms by Use of Empirical Mode Decomposition

This study addressed the issue of assessing chaotic parameters from nonstationary electrocardiogram (ECG) signals. The empirical mode decomposition (EMD) was proposed as a method to extract intrinsic mode functions (IMFs) from ECG signals. Chaos analysis methods were then applied to the stationary IMFs without violating the underlying assumption of stationarity. Eight ECG data sets representing normal and various abnormal rhythms were obtained from the American Heart Associate Ventricular Arrhythmia database. The chaotic parameters including Lyapunov exponent, entropy, and correlation dimension were computed. The results consistently showed that the 10th IMF (IMF-10) was stationary and preserved sufficient nonlinearity of the ECG signals. Each IMF-10 from the data sets (n = 8) gave a positive dominate Lyapunov exponent (0.29-0.64, p < 0.0001), a positive entropy (0.039-0.061, p < 0.0001), and a noninteger correlation dimension (1.1-1.9). These were evidences of a chaotic dynamic system. We therefore concluded that the original ECG signals must also have chaotic properties. The chaotic parameters did not show significant differences among the eight data sets representing normal sinus rhythm and various abnormalities. This study has demonstrated an effective way to characterize nonlinearities in nonstationary ECG signals by combining the empirical mode decomposition and the chaos analysis methods.     

Introducing Statistical Persistence Decay: A Quantification of Stride-to-Stride Time Interval Dependency in Human Gait

Stride-to-stride time intervals during human walking are characterised by predictability and statistical persistence quantified by sample entropy (SaEn) and detrended fluctuation analysis (DFA) which indicates a time dependency in the gait pattern. However, neither analyses quantify time dependency in a physical or physiological interpretable time scale. Recently, entropic half-life (ENT½) has been introduced as a measure of the time dependency on an interpretable time scale. A novel measure of time dependency, based on DFA, statistical persistence decay (SPD), was introduced. The present study applied SaEn, DFA, ENT½, and SPD in known theoretical signals (periodic, chaotic, and random) and stride-to-stride time intervals during overground and treadmill walking in healthy subjects. The analyses confirmed known properties of the theoretical signals. There was a significant lower predictability (p = 0.033) and lower statistical persistence (p = 0.012) during treadmill walking compared to overground walking. No significant difference was observed for ENT½ and SPD between walking condition, and they exhibited a low correlation. ENT½ showed that predictability in stride time intervals was halved after 11–14 strides and SPD indicated that the statistical persistency was deteriorated to uncorrelated noise after ~50 strides. This indicated a substantial time memory, where information from previous strides affected the future strides.     

帕金森病患者加速度信号的嵌入维和关联维特征

帕金森病作为一种运动调节功能障碍性疾病,会显著地降低人体中枢神经系统对肢体的控制能力。本文将G-P算法直接应用于人体行走时产生的加速度信号的混沌动力学研究。通过对帕金森病患者和健康人群加速度信号的动力学特征进行比较后发现,相对于健康人,帕金森病患者加速度信号的嵌入维和关联维均显著减小。结果表明,帕金森疾病会使患者运动神经控制系统的混沌程度减弱,复杂性降低;而且可以通过对有限长一维加速度信号的重构捕捉患者运动神经控制系统动力学特征的改变。本研究的结论对于人体运动神经控制系统的生理建模以及帕金森病的临床诊断具有一定的理论和应用价值。     

Radial basis function neural networks for the characterization of heart rate variability dynamics

This study introduces new neural network based methods for the assessment of the dynamics of the heart rate variability (HRV) signal. The heart rate regulation is assessed as a dynamical system operating in chaotic regimes. Radial-basis function (RBF) networks are applied as a tool for learning and predicting the HRV dynamics. HRV signals are analyzed from normal subjects before and after pharmacological autonomic nervous system (ANS) blockade and from diabetic patients with dysfunctional ANS. The heart rate of normal subjects presents notable predictability. The prediction error is minimized, in fewer degrees of freedom, in the case of diabetic patients. However, for the case of pharmacological ANS blockade, although correlation dimension approaches indicate significant reduction in complexity, the RBF networks fail to reconstruct adequately the underlying dynamics. The transient attributes of the HRV dynamics under the pharmacological disturbance is elucidated as the explanation for the prediction inability.     

采用分块耦合广义分形维的语音端点检测技术研究

为了克服传统广义分形维算法存在的边缘效应,提高低信噪比环境中语音信号起止点的检测能力,我们提出了一种分块耦合(Boxes-coupling)的广义维数计算方法,并将其应用于语音信号的端点检测中.首先确定各分块之间的耦合边长,利用耦合的分块覆盖待检测的语音信号,求得容量维数、信息维数和关联维数,以构成三维特征向量;然后根据特征距离和相似度之间的关系,用一般距离度量进行特征提取;最后采用双门限模式分类进行决策分类.实验结果表明,该方法适合多种噪声环境下的语音端点检测.与传统广义维数(Original generalized dimension, OGD)以及谱熵(Spectral entropy, SE)算法相比,在低信噪比情形下,具有检测精度高、抗干扰能力强以及强系统稳定性(Robustness)等优点.     

Chaotic dynamics of resting ventilatory flow in humans assessed through noise titration

The mammalian ventilatory behaviour exhibits nonlinear dynamics as reflected by certain nonlinearity or complexity indicators (e.g. correlation dimension, approximate entropy, Lyapunov exponents, etc.) but this is not sufficient to determine its possible chaotic nature. To address this, we applied the noise titration technique, previously shown to discern and quantify chaos in short and noisy time series, to ventilatory flow recordings obtained in quietly breathing normal humans. Nine subjects (8 men and 1 woman, 24-42 years) were studied during 15-min epochs of ventilatory steady-state (10.1+/-3.0 breaths/min, tidal volume 0.63+/-0.2 L). Noise titration applied to the unfiltered signals subsampled at 5 Hz detected nonlinearity in all cases (noise limit 20.2+/-12.5%). Noise limit values were weakly correlated to the correlation dimension and the largest Lyapunov exponent of the signals. This study shows that the noise titration approach evidences a chaotic dimension to the behavior of ventilatory flow over time in normal humans during tidal breathing.     

Linguistic analyses of speech disorder in psychosis

Much theory and research has been devoted to characterizing speech disorders in psychotic individuals and to understanding the connections between language productions and underlying cognitive processes. Studies in the literature have used primarily traditional clinical assessments and experimental task methods to study speech phenomena in psychotic patients. More recently, however, the range of study methods has expanded to include models based on normal language use, borrowed from the field of psycholinguistics. This article reviews linguistic studies of the discourse of psychotic patients with a focus on understanding speech disorders in the context of schizophrenia and mania. The studies are grouped according to the method of analysis used, including frequency counts, type-token ratios, cloze analyses, contextual constraint, pause and hesitation patterns, and discourse level analyses. Methodological problems and theoretical issues are summarised, followed by a statement of current status. The use of multiple measures applied to whole discourse speech samples offers particular promise for clarifying diagnostic issues and testing hypotheses based on cognitive models.     

Examining the timing of miniature endplate potential releases at the frog and mouse neuromuscular junctions for deviations from Poisson expectations

The intervals between miniature endplate potentials (MEPPs) were measured at frog and mouse neuromuscular junctions in several different solutions. Data sets with monotonic trends in MEPP frequency were discarded. The remaining data sets had between 283 and 5024 MEPPs. The fit to the exponential distribution, which describes a Poisson process, was tested with Sherman's statistic. If the intervals are not distributed exponentially, this statistic indicates whether they deviate because they are more concentrated or more diffuse. In 6 of 20 data sets from the frog and in 3 of 7 data sets from the mouse the intervals between MEPPs were not distributed exponentially. Some of these were clustered, while others were diffuse. In one frog data set release was periodic. In all data sets releases appeared to be independent, judging from the integrated power spectra. One data set may show fractal behavior. Calculations of the approximate entropy gave no indication of an underlying regularity, so there is no evidence for a chaotic process. The lack of fit to the exponential distribution due to either concentrated or diffuse interval distributions is mimicked by a model in which release is Poisson, but with a mean rate that randomly shifts to a different level.     

Nonlinear analysis of the performance and reliability of wavelet singularity detection based denoising for Doppler ultrasound fetal heart rate signals

Many studies on the physiology of the cardiovascular system revealed that nonlinear chaotic dynamics govern the generation of the heart rate signal. This is also valid for the fetal heart rate (FHR) variability, where however the variability is affected by many more factors and is significantly more complicated than for the adult case. Recently an adaptive wavelet denoising method for the Doppler ultrasound FHR recordings has been introduced. In this paper the performance and reliability of that method is confirmed by the observation that for the wavelet denoised FHR signal, a deterministic nonlinear structure, which was concealed by the noise, becomes apparent. It provides strong evidence that the denoising process removes actual noise components and can therefore be utilized for the improvement of the signal quality. Hence by observing after denoising a significant improvement of the 'chaoticity' of the FHR signal we obtain strong evidence for the reliability and efficiency of the wavelet based denoising method. The estimation of the chaoticity of the FHR signal before and after the denoising is approached with three nonlinear analysis methods. First, the rescaled scale analysis (RSA) technique reveals that the denoising process increases the Hurst exponent parameter as happens when additive noise is removed from a chaotic signal. Second, the nonlinear prediction error evaluated with radial basis function (RBF) prediction networks is significantly lower at the denoised signal. The significant gain in predictability can be attributed to the drastic reduction of the additive noise from the signal by the denoising algorithm. Moreover, the evaluation of the correlation coefficient between actual and neural network predicted values as a function of the prediction time displays characteristics of chaos only for the denoised signal. Third, a chaotic attractor, reconstructed with the embedding dimension technique, becomes evident for the denoised signal, while it is completely obscured for the original signals. The correlation dimension of the reconstructed attractor for the denoised signal tends to reach a value independent of the embedding dimension, a sign of deterministic chaotic signal. In contrast for the original signal the correlation dimension increases steadily with the embedding dimension, a fact that indicates strong contribution of noise.     

Non Linear Analysis of Magnetoencephalographic Signals as a Tool for Assessing Malignant Lesions of the Brain: First Results

Summary Background and purpose: Non linear signal analysis is a powerful technique that reveals qualitative and quantitative differentiations between different dynamical systems (biological or otherwise). Presented here are the first results of a work in progress to investigate the Magnetoencephalograms (MEG) from patients with malignant CNS lesions and from healthy volunteers. Methods: We present MEG recordings of 10 patients diagnosed with malignant CNS lesions and the corresponding ones from 10 healthy volunteers. A 122-channel SQUID biomagnetometer in an electromagnetically shielded room was used to record the MEG signals and the Grassberger-Procaccia method for the estimation of the correlation dimension was applied on the phase space reconstruction of the recorded signal from each patient. Results: Evidence linking MEG signal characteristics (existence of low dimensionality chaotic dynamics) with the existence of the tumour was found from this analysis. Conclusion: The obtained results substantiate our hypothesis of a relation between tumours of the brain and the mathematically chaotic nature of the neural dynamics derived from their MEG recordings.     

HAI实验中EEG信号的非线性动力学研究

采用一维时间序列相空间重构技术和混沌的定量判据，对缺氧窒息而引起的中枢神经损伤（Hypoxic-Asphyxic Injury，HAl）实验中仔猪的脑电（Electroencephalogram，EEG）信号进行了分析与计算。通过对生理和损伤状态下仔猪EEG信号的相图、功率谱、关联维数和Lyapunov指数的对比研究，得出如下结论：（1）EEG的相图、功率谱、关联维数和Lyapunov指数反映了大脑的总体动态特征，它们可作为一种定量研究EEG的新方法进行脑损伤的早期诊断；（2）在正常的生理状态下EEG是混沌的，而在损伤状态下则趋于有序。     

大鼠癫痫模型脑电信息的不稳定周期轨道分析

为了进一步探索应用非线性动力学理论对癫痫脑电信息进行分析。在采用非线性动力学指标:近似熵和相关维对大鼠癫痫发作过程的整个脑电信号进行分析研究的基础上,运用新的用于神经元系统的复杂性行为研究的非线性动力学方法——不稳定周期轨道,分析研究癫痫不同发作时期脑电信号的变化规律。结果显示癫痫发作时,脑电信号中存在具有统计显著性的周期1和周期2轨道,而在癫痫发作前期仅存在具有统计显著性的周期1轨道。从而进一步验证了癫痫发作整个过程,脑电信号复杂度的变化规律。     

Time series analyses of breathing patterns of lung cancer patients using nonlinear dynamical system theory

The underlying requirements for successful implementation of any efficient tumour motion management strategy are regularity and reproducibility of a patient's breathing pattern. The physiological act of breathing is controlled by multiple nonlinear feedback and feed-forward couplings. It would therefore be appropriate to analyse the breathing pattern of lung cancer patients in the light of nonlinear dynamical system theory. The purpose of this paper is to analyse the one-dimensional respiratory time series of lung cancer patients based on nonlinear dynamics and delay coordinate state space embedding. It is very important to select a suitable pair of embedding dimension 'm' and time delay 'τ' when performing a state space reconstruction. Appropriate time delay and embedding dimension were obtained using well-established methods, namely mutual information and the false nearest neighbour method, respectively. Establishing stationarity and determinism in a given scalar time series is a prerequisite to demonstrating that the nonlinear dynamical system that gave rise to the scalar time series exhibits a sensitive dependence on initial conditions, i.e. is chaotic. Hence, once an appropriate state space embedding of the dynamical system has been reconstructed, we show that the time series of the nonlinear dynamical systems under study are both stationary and deterministic in nature. Once both criteria are established, we proceed to calculate the largest Lyapunov exponent (LLE), which is an invariant quantity under time delay embedding. The LLE for all 16 patients is positive, which along with stationarity and determinism establishes the fact that the time series of a lung cancer patient's breathing pattern is not random or irregular, but rather it is deterministic in nature albeit chaotic. These results indicate that chaotic characteristics exist in the respiratory waveform and techniques based on state space dynamics should be employed for tumour motion management.     

Fractal character of the electrocardiogram: Distinguishing heart-failure and normal patients

Statistical analysis of the sequence of heartbeats can provide information about the state of health of the heart. We used a variety of statistical measures to identify the form of the point process that describes the human heartbeat. These measures are based on both intervent intervals and counts, and include the intervent-interval histogram, interval-based periodogram, rescaled range analysis, the event-number histogram, Fano-factor, Allan Factor, and generalized-rate-based periodogram. All of these measures have been applied to data from both normal and heart-failure patients, and various surrogate versions thereof. The results show that almost all of the interevent-interval and the long-term counting statistics differ in statistically significant ways for the two classes of data. Several measures reveal 1/f-type fluctuations (long-duration power-law correlation). The analysis that we have conducted suggests the use of a conveniently calculated, quantitative index, based on the Allan factor, that indicates whether a particular patient does or does not suffer from heart failure. The Allan factor turns out to be particularly useful because it is easily calculated and is jointly responsive to both short-term and long-term characteristics of the heartbeat time series. A phase-space reconstruction based on the generalized heart rate is used to obtain a putative attractor's capacity dimension. Though the dependence of this dimension on the embedding dimension is consistent with that of a low-dimensional dynamical system (with a larger apparent dimension for normal subjects), surrogate-data analysis shows that identical behavior emerges from temporal correlation in a stochastic process. We present simulated results for a purely stochastic integrate-and-fire model, comprising a fractal-Gaussian-noise kernel, in which the sequence of heartbeats is determined by level crossings of fractional Brownian motion. This model characterizes the statistical behavior of the human electrocardiogram remarkably well, properly accounting for the behavior of all of the measures studied, over all time scales.     

Nonlinear Analysis of Brain Activity in Magnetic Influenced Parkinson Patients

Magnetoencephalogram (MEG) recordings were obtained from the brain of patients suffering from Parkinson's disease (PD) using the Superconductive Quantum Interference Device (SQUID). For each patient the magnetic activity was recorded from a total of 64 points of the skull (32 points from each temporal lobe) as defined by a recording reference system, which is based on the 10-20 Electrode Placement System. Some of the recorded points were observed to exhibit abnormal rhythmic activity, characterized by high amplitudes and low frequencies. External magnetic stimulation (EMS) with intensity 1-7.5pT, and frequency the a-rhythm of the patient (8-13 Hz) was applied in the left-right temporal, frontal-occipital and vertex (2 minutes over each of the above regions) and the brain magnetic activity was recorded again. The application of the EMS resulted in rapid attenuation of the MEG activity of PD patients. Furthermore, chaotic dynamic methods were used, in order to estimate the correlation dimension D of the reconstructed phase spaces. The estimated values of D, in conjunction with the results derived from the other data analysis methods, strongly support the existence of low dimension chaotic structures in the dynamics of cortical activity of PD patients. In addition, the increased values of D of the MEG after the application of EMS when compared with the corresponding ones obtained from the MEGs prior to the EMS, suggest that the neural dynamics are strongly influenced by the application of EMS.     

Neural correlates of sensory and decision processes in auditory object identification

Physiological studies of auditory perception have not yet clearly distinguished sensory from decision processes. In this experiment, human participants identified speech sounds masked by varying levels of noise while blood oxygenation signals in the brain were recorded with functional magnetic resonance imaging (fMRI). Accuracy and response time were used to characterize the behavior of sensory and decision components of this perceptual system. Oxygenation signals in a cortical subregion just anterior and lateral to primary auditory cortex predicted accuracy of sound identification, whereas signals in an inferior frontal region predicted response time. Our findings provide neurophysiological evidence for a functional distinction between sensory and decision mechanisms underlying auditory object identification. The present results also indicate a link between inferior frontal lobe activation and response-selection processes during auditory perception tasks.     

Heart rate variability analysis using correlation dimension and detrended fluctuation analysis

The ECG is a representative signal containing information about the condition of the heart. The shape and size of the P-QRS-T wave, the time intervals between its various peaks, etc., may contain useful information about the nature of disease afflicting the heart. However, the human observer can not directly monitor these subtle details. Besides, since biosignals are highly subjective, the symptoms may appear at random in the time scale. Therefore, the heart rate variability signal parameters, extracted and analysed using computers, are highly useful in diagnostics. This paper deals with the classification of certain diseases using correlation dimension (CD) and detrended fluctuation analysis (DFA).