On the validity of estimating EEG correlation dimension from a spatial embedding

We demonstrate by using simulations that spatial embedding of single-variable time series data does not reliably reconstruct state-space dynamics. Instead, correlation dimension estimated from spatially embedded data is largely a measure of linear cross-correlation in the data set. For actual electroencephalographic (EEG) data, we demonstrate a high negative correlation between spatial correlation dimension and the average amount of lag-zero cross-correlation between “nearest-neighbor” embedding channels (the greater the cross-correlation, the lower the dimension). We also show that the essential results obtained from spatially embedding EEG data are also obtained when one spatially embeds across a set of highly cross-correlated stochastic (second-order autoregressive) processes. Although, with appropriate surrogate data, correlation dimension estimated from spatially embedded data detects nonlinearity, its use is not recommended because correlation dimension estimated from temporally embedded data both reconstructs state-space dynamics and, with appropriate surrogate data, detects nonlinearity as well.     

Nonlinear analysis of the EEG of schizophrenics with optimal embedding dimension

We estimated the correlation dimensions of EEGs in patients with schizophrenia to investigate the dynamical properties underlying the EEG. We employed a new method, proposed by Kennel et al. (Kennel MB, Brown R, Abarbanel HDI. Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys Rev A 1992;45:3403-11), to calculate the correlation dimension D2. That method determined the proper minimum embedding dimension by looking at the behaviour of nearest neighbours under a change in the embedding dimension d from d to d + 1. We demonstrated that for limited noisy data, our algorithm was strikingly faster and more accurate than previous ones. We estimated the D2 of EEGs from 16 channels in patients with schizophrenia according to DSM-IV whereas previous studies, which estimated chaoticity of EEG in schizophrenia, recorded EEG only in a limited number of channels. Schizophrenic patients had a lower correlation dimension in the left inferior frontal and anterior temporal regions compared with controls. Our finding of decreased left frontal and temporal chaotic activity in schizophrenics is in line with the findings of a hypofrontality and hypotemporality reported in previous clinical studies such as EEG, blood flow, brain MRI and positron emission tomography studies in schizophrenia. This result suggests that chaos analysis may be a useful tool in analysing EEG data to explore the brain mechanism of schizophrenia.     

认知负载状态下的精神分裂症患者脑电图特异性研究

认知功能损害是精神分裂症的三大原发症状之一,在疾病早期发现和高危人群风险预警等方面具有重要价值。为了研究精神分裂症患者在认知负载状态下的脑电图特异性,本试验收集17例精神分裂症患者和19例健康受试者的脑电信号作为对照,基于小波变换提取各频段信号,计算非线性动力学及脑功能网络属性等特征,并利用机器学习算法将两类人群进行自动分类分析。试验结果表明,两组受试者在认知负载状态下,Fp1和Fp2导联在α、β、θ、γ这4个频带的关联维数和样本熵的差异均具有统计学意义,提示大脑额叶功能损伤是精神分裂症认知功能损害的重要原因。进一步基于机器学习的自动分类分析结果表明,将非线性动力学与脑功能网络属性相结合作为分类器的输入特征,所得分类效果最优,其结果显示准确率为76.77%、敏感度为72.09%、特异性为80.36%。本研究结果表明,脑电信号的非线性动力学和脑功能网络属性等特征,或可作为精神分裂症早期筛查和辅助诊断的潜在生物标记物。     

Why bother to spatially embed EEG? Comments on Pritchard et al., Psychophysiology, 33, 362–368, 1996

In a recent paper, Pritchard, Krieble, and Duke (Psychophysiology, 33, 362-368, 1996) studied the validity of spatial embedding of electroencephalographic (EEG) data and rejected this method in favor of time-delay embedding. The present paper describes the nonlinear characterization of brain dynamics using either spatial or time-delay embedding. We discuss the arguments published in Pritchard et al. (1996) and demonstrate that the spatial embedding cannot be rejected on this basis. We also point out the limitations of both spatial and time-delay embeddings related to the spatial extension and the high-dimensional dynamics of brain activity.     

多导信号的相空间重构及在脑电信号分析中的应用

本研究提出了一种基于时间延迟和序列相关性的多导信号相空间重构方法。根据信号的相关系数重组多导联信号序列，并利用信号的可确定性选取时间延迟参数重构相空间。对仿真数据的研究表明这种重构方法在信号的确定性和相关维数计算中具有良好的性能，受噪声、延迟量和嵌入维数等参数变化和单导重构相比影响较小，结果更稳定和可靠。对实验思维脑电数据的非线性分析得到了脑电和思维复杂性，关联性等性质的联系。该方法适用于短时多导含噪信号的非线性分析和脑电的在线研究等。     

On the validity of spatial embedding: a reply to Pezard et al.

Careful consideration of the issues raised by Pezard and colleagues (in this issue of Psychophysiology) allows for the conclusion that spatial embedding may be valid as a method of dynamical reconstruction. However, two problems with the technique cannot be ignored. First, spatial embedding of EEG invariably involves linear cross-correlation among channels, which distorts the dynamical reconstruction due to compression toward the main state-space diagonal. Further, before spatially embedding across a set of channels, one must first check for at least "similar" dynamics among them, using, for example, a measure such as estimated mutual dimension. Measuring a "whole cortex" state via spatial embedding may also be possible in principle, except for the nontrivial obstacle of separating local dynamics that are heterogeneous across the cortex from activity reflecting the "unified field" of the cortex as a whole.     

一种脑电信号相空间分析的新方法

提出一种新的脑电(EEG)信号相空间分析方法。通过计算相空间状态点间的欧氏距离,定义相对嵌入维数的态密度和态方差,并与反映非线性动力学系统混沌特征的关联维数作比较。对各种实测的脑电时间序列的计算结果表明,态密度和态方差不仅计算简单,计算结果一致可靠,而且和关联维数相比,更能有效地反映非线性动力学系统的某些特征。此外,计算了基于距离协方差的脑电信号的奇异谱,并对实验结果作了分析。     

A Comparison of Fractal Dimensions of EEG Time Series of Partial EpiIepsy

IntroductionTheDiagnosisofepilepsymainlydependsonclinicalcasehistoryandEEGexamination.Butalmostallthepatientsshownosignofepilepticattackwhenhavingaclinicaltest,duringtheinterparoxysmalpause,about5o%ofpatients'EEGhasnoepilepticeIectricalactivities.Therefore,lackofobjectiveevidencemakesitdifficuIttoformacorrectdiagnosisonadisease.Inrecentyears,thetheoryofnonIineardynamicshasbroughthopetotheresearchofbrainwave.Somecurrentresearchshowsthatwhenbrainsuffersepilepticaffection,someneurons'repetit…     

电刺足三里穴脑电信号的非线性动力学方法初探

为探讨电刺足三里穴位引起的大脑活动的变化，本文以脑电为手段，用非线性动力学方法对电刺前后的两种脑电的非线性特征进行分析。实验结果显示，非线性特征指数，如关联维数、Lyapunov指数和测度熵，在电刺前后均能很好的区分（P〈0．05）。结果表明，电刺后大脑的随机性和无序度降低，大脑相应脑区的活动更趋于有序，且这种有序性将随着电刺次数的增加而渐趋于稳定。     

Non-linear dynamical analysis of multichannel EEG: Clinical applications in dementia and Parkinson's disease

The irregular, aperiodic character of the EEG is usually explained by a stochastic model. In this view the EEG is linearly filtered noise. According to chaos theory such irregular signals can also result from low dimensional deterministic chaos. In this case the underlying dynamics is nonlinear, and has only few effective degrees of freedom. In contrast, stochastic models are less efficient, because they require in principle infinite degrees of freedom. Chaotic dynamics in the EEG can be studied by calculating the correlation dimension (D2). Although it has become clear that D2 calculations alone cannot prove chaos, the D2 has potential value as an EEG diagnostic. In this study we investigated whether D2 could be used to discriminate EEGs from normal controls, demented patients and Parkinson patients. We have analyzed epochs (20 channels; 2.5 s) from 52 EEGs (20 controls; 15 patients with dementia; 17 patients with Parkinson's disease). Controls had a mean D2 of 6.5 (0.9); demented patients of 4.4 (1.5), and Parkinson patients of 5.3 (0.9). Both groups were significantly different from controls (p < 0.001). There was a significant positive correlation between D2 and relative power in the beta band (r=0.81) and a significant negative correlation between D2 and power in the delta (r=–0.60) and theta band (r=–0.37). These results suggest the possible usefulness of multichannel D2 estimations in a clinical setting.     

Classification of electroencephalographic seizure recordings into ictal and interictal files using correlation sum

This study provides a performance evaluation of the correlation sum in terms of accuracy, sensitivity, and specificity in its ability to classify seizure files from non-seizure files. The main thrust of the study is whether computable properties (“metrics”) of EEG tracings over time allow a seizure to be detected. This study evaluates raw intracranial EEG (iEEG) recordings with the intent to detect a seizure and classify different EEG epoch files. One hundred twenty-six iEEG files from eleven sequential patients are processed and the correlation sum is extracted from non-overlapping scrolling windows of 1-s duration.The novelty of this research is in defining a generalized nonlinear approach to classify EEG seizure segments by introducing nonlinear decision functions with the flexibility in choosing any degree of complexity and with any number of dimensions, lending resiliency to data overlap and opportunity for multidimensional data analysis. A singular contribution of this work is in determining a 2-D decision plane, in this case, where duration is one dimension and window-based minima of the correlation sum is the second dimension. Also, experimental observations clearly indicate that a significant drop in the magnitude of the correlation sum signal actually coincides with the clinical seizure onset more so than the electrographic seizure onset as provided by the medical experts. The method with k-fold cross validation performed with an accuracy of 91.84%, sensitivity of 92.31%, and specificity of 91.67%, which makes this classification method most suitable for offline seizure detection applications.     

脑电混沌维数复杂度连续检测方法的研究

一些研究表明,脑电信号具有低维的混沌动力学特性,脑电的维数则反映脑信息处理过程中的神经元集群的活动状态,是研究脑电信号的重要非线性参数.在Grassberger-Procaccia计算关联维数算法的基础上,通过改进计算关联积分的过程,提出一种适合于移动重叠窗口的维数复杂度连续检测方法.另外,还对关联积分线性区间选择上做了改进,使得所求得的参数在严格意义上为维数复杂度.数值仿真验证了该方法的正确性,并在此基础上对视觉输入变化的EEG信号进行动态数据分析,结果显示睁眼时脑电的复杂度明显高于闭眼时脑电的复杂度.所提出的维数复杂度计算方法减少了数据分析的负荷,适合于连续数据分析,能够准确地反映脑电维数复杂度的连续变化过程.     

A spatio-temporal wavelet-chaos methodology for EEG-based diagnosis of Alzheimer's disease

A spatio-temporal wavelet-chaos methodology is presented for analysis of EEGs and their delta, theta, alpha, and beta sub-bands for discovering potential markers of abnormality in Alzheimer's disease (AD). The non-linear dynamics of the EEG and EEG sub-bands are quantified in the form of the correlation dimension (CD), representing system complexity, and the largest Lyapunov exponent (LLE), representing system chaoticity. The methodology is applied to two groups of EEGs: healthy subjects and AD patients. The eyes open and eyes closed conditions are investigated to evaluate the effect of visual input and attention. EEGs from different loci in the brain are investigated to discover areas of the brain responsible for or affected by changes in CD and LLE. It is found that the wavelet-chaos methodology and the sub-band analysis developed in this research accurately characterizes the non-linear dynamics of non-stationary EEG-like signals with respect to the EEG complexity and chaoticity. It is concluded that changes in the brain dynamics are not spread out equally across the spectrum of the EEG and over the entire brain, but are localized to certain frequency bands and electrode loci. New potential markers of abnormality were discovered in this research for both eyes open and closed conditions.     

Analysis of wavelet-filtered tonic-clonic electroencephalogram recordings

EEG signals obtained during tonic-clonic epileptic seizures can be severely contaminated by muscle and physiological noise. Heavily contaminated EEG signals are hard to analyse quantitatively and also are usually rejected for visual inspection by physicians, resulting in a considerable loss of collected information. The aim of this work was to develop a computer-based method of time series analysis for such EEGs. A method is presented for filtering those frequencies associated with muscle activity using a wavelet transform. One of the advantages of this method over traditional filtering is that wavelet filtering of some frequency bands does not modify the pattern of the remaining ones. In consequence, the dynamics associated with them do not change. After generation of a ‘noise free’ signal by removal of the muscle artifacts using wavelets, a dynamic analysis was performed using non-linear dynamics metric tools. The characteristic parameters evaluated (correlation dimension D₂ and largest Lyapunov exponent λ₁) were compatible with those obtained in previous works. The average values obtained were: D₂=4.25 and λ₁=3.27 for the pre-ictal stage; D₂=4.03 and λ₁=2.68 for the tonic seizure stage; D₂=4.11 and λ₁=2.46 for the clonic seizure stage.     

Correlation Dimension Maps of EEG from Epileptic Absences

The understanding of brain activity, and in particular events such as epileptic seizures, lies on the characterisation of the dynamics of the neural networks. The theory of non-linear dynamics provides signal analysis techniques which may give new information on the behaviour of such networks. Methods: We calculated correlation dimension maps for 19-channel EEG data from 3 patients with a total of 7 absence seizures. The signals were analysed before, during and after the seizures. Phase randomised surrogate data was used to test chaos. Results: In the seizures of two patients we could distinguish two dynamical regions on the cerebral cortex, one that seemed to exhibit chaos whereas the other seemed to exhibit noise. The pattern shown is essentially the same for seizures triggered by hyperventilation, but differ for seizures triggered by light flashes. The chaotic dynamics that one seems to observe is determined by a small number of variables and has low complexity. On the other hand, in the seizures of another patient no chaotic region was found. Before and during the seizures no chaos was found either, in all cases. Conclusions: The application of non-linear signal analysis revealed the existence of differences in the spatial dynamics associated to absence seizures. This may contribute to the understanding of those seizures and be of assistance in clinical diagnosis.     

局限性癫痫脑电时间序列的分形维数计算比较

为探索应用非线性动力学理论诊断癫痫病的新方法，对局限性癫痫病患者脑电时间序列进行了相关维数（Ｄｃ）和波形维数（Ｄｗ）的计算比较。观察到，痫性导联脑电的分形维数多低于对侧导联的值；相关维数Ｄｃ的相对变化量较波形维数Ｄｗ大；而波形维数Ｄｗ对痫性棘波敏感；结果提示，脑电时间序列的分形维数有可能成为诊断癫痫的特征参数，值得进一步深入研究。     

A nonlinear approach to brain function: deterministic chaos and sleep EEG.

In order to perform a nonlinear dimensional analysis of the sleep electroencephalogram (EEG), we applied an algorithm proposed by Grassberger and Procaccia to calculate the correlation dimension D2 of different sleep stages under Lorazepam medication versus placebo. This correlation dimension characterizes the dynamics of the sleep EEG and it estimates the degrees of freedom of the signal under study. We demonstrate that slow-wave sleep depicts a much smaller dimensionality than light or rapid eye movement (REM) sleep, and that Lorazepam does not alter the EEG's dimensionality except in stage II and REM.     

基于非线性时空动力学的意识障碍患者脑电图研究

目前,意识障碍患者的意识恢复过程仍然不是很清楚.大多数相关研究采用组间比较方法,而意识恢复不仅是一个动态过程,而且会涉及不同脑区间的相互作用.因此,阐明意识恢复机制需要从时间和空间两个维度对大脑活动进行跟踪.利用脑电图的时空分辨率优势,跟踪41例意识障碍患者,共采集161例脑电信号.之后,比较不同意识恢复阶段患者脑电信...     

EEG/EP: New techniques

The topographic analysis of electrical brain activity consists of the extraction of quantitative features which adequately describe the scalp recorded electrical fields of the brain. In the beginning of brain electrical activity mapping most methods centered mainly around the graphical display of multichannel EEG and evoked potential data. Meanwhile quantitative analysis strategies have been developed, and such methods are applied to topographic EEG and evoked potential data enabling the statistical evaluation of the effects of different experimental conditions as well as the comparison of various clinical populations. Major new analysis techniques comprise the computation of global field power and global dissimilarity for determination of components of evoked potential fields, the segmentation of map series by topographical features, time range analysis, FFT approximation for the spatial analysis of EEG frequency bands as well as correlation analysis and spatial principal components analysis (Spatial PCA). Data from experiments dealing with evoked brain activity will illustrate the application of these quantitative methods that also can be used for the analysis of the spontaneous EEG.     

Hemispheric laterality and dimensional complexity in schizophrenia under sound and light stimulation.

The aim of the present study is to investigate the relationship between linear and non-linear activities in human electroencephalograms (EEGs) by examining the linear lateral asymmetry index and the correlation dimension as a non-linear measure of complexity and to typify the characteristics of EEGs between schizophrenic patients and normal controls. We recorded the EEG from 16 electrodes in 10 schizophrenics (6 males and 4 females) and 10 age-matched normal controls (10 males), and calculated their asymmetry indices. The asymmetry index shows which hemispheric activity is dominant through examination of interhemispheric pairs in the frequency domain with EEGs between two regions. We also estimated correlation dimension. Remarkably, lower dimensional complexities appeared on the brain regions, which had significantly lower brain activity, as determined by a lateral asymmetry analysis, in schizophrenics before sound and light (SL) stimulation. We may suggest the possibility of co-varying of both linear and non-linear properties. This co-varying phenomenon maintained after the SL stimulation. Furthermore, schizophrenic patients revealed opposite asymmetric patterns compared to normal controls, as well as reversal phenomena and abnormalities in the left frontal region when SL stimuli were applied.