Non-linear analysis of epileptic seizures I. Correlation-dimension measurements for absence epilepsy and near-periodic signals

The study of six absence seizures from two patients confirmed the efficacy, in the search for low correlation dimensions, of using scaled-structure analysis, combined with the appropriate checking procedures. The analysis is directed towards characterizing an attractor not only by its correlation dimension, but also by its “quality” and by the probability for genuine identification. For near-periodic dynamics, we warn against: (1) artefacts that appear at high values of the correlation integral, in the form of apparent Grassberger-Procaccia scaling at very low values of the dimension (near-periodicity artefact); (2) erroneous interpretation of phase-randomization data, owing to destruction of the artefact by randomization rather than any evidence for low-dimensional dynamics. In single-channel analyses of two patients and six seizures altogether, high-quality attractors were found only for one seizure in two channels, at correlation dimensions 4.7 and 6, respectively. Furthermore, no attractor of measurable dimension was found from multichannel space reconstructions over durations approaching those of typical seizures. Both these results show that in an absence seizure, spatial extension of low-dimensional dynamics must be lost over such durations. Received: 27 April 1998 / Accepted in revised form: 10 November 1998     

On the correlation dimension of optokinetic nystagmus eye movements: computational parameters, filtering, nonstationarity, and surrogate data

 We discuss the estimation of the correlation dimension of optokinetic nystagmus (OKN), a type of reflexive eye movement. Parameters of the time-delay reconstruction of the attractor are investigated, including the number of data points, the time delay, the window duration, and the duration of the signal being analyzed. Adequate values are recommended. Digital low-pass filtering causes the dimension to increase as the filter cutoff frequency decreases, in accord with a previously published prediction. The stationarity of the correlation dimension is examined; the dimension appears to decrease over the course of 120 s of continuous stimulation. Implications for the reliable estimation of the dimension are considered. Several surrogate data sets are constructed, based on both early (0–30 s) and late (100–130 s) OKN segments. Most of the surrogate data sets randomize some aspect of the original OKN, while maintaining other aspects. Dimensions are found for all surrogates and for the original OKN. Evidence is found that is consistent with some amount of deterministic and nonlinear dynamics in OKN. When this structure is randomized in the surrogate, the dimension changes or the dimension algorithm ceases to converge to a finite value. Implications for further analysis and modeling of OKN are discussed. Received: 30 August 1996/Accepted in revised form: 13 November 1996     

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

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

     

Fractal geometry of root systems: Field observations of contrasting genotypes of common bean (Phaseolus vulgaris L.) grown under different phosphorus regimes

Root growth and architecture are important for phosphorus acquisition due to the relative immobility of P in the soil. Fractal geometry is a potential new approach to the analysis of root architecture. Substantial genetic variation in root growth and architecture has been observed in common bean. Common bean (Phaseolus vulgaris L.) genotypes with contrasting root architecture were grown under moderate and low P conditions in a field experiment. Linear and planar fractal dimension were measured by tracing root intercepts with vertical planes. Linear fractal dimension increased over time in efficient genotypes, but remained fairly constant over time in inefficient genotypes. Planar fractal dimension increased over time for all genotypes, but was higher in efficient than inefficient genotypes at the end of the experiment. Planar fractal dimension of medium P plants was found to correlate with shoot P content indicating fractal dimension to be a possible indicator for root P uptake. The increasing fractal dimension over time indicates that fractal analysis is a sensitive measure of root branching intensity. A less destructive method for acquisition of data that allows for continuous analysis of fractal geometry and thereby screening for more P efficient genotypes in the field is suggested. This method will allow the researcher to conduct fractal analysis and still complete field trials with final yield evaluation.     

Nonlinear analysis of gait kinematics to track changes in oxygen consumption in prolonged load carriage walking: A pilot study

Linking human mechanical work to physiological work for the purpose of developing a model of physical fatigue is a complex problem that cannot be solved easily by conventional biomechanical analysis. The purpose of the study was to determine if two nonlinear analysis methods can address the fundamental issue of utilizing kinematic data to track oxygen consumption from a prolonged walking trial: we evaluated the effectiveness of dynamical systems and fractal analysis in this study. Further, we selected, oxygen consumption as a measure to represent the underlying physiological measure of fatigue. Three male US Army Soldier volunteers (means: 23.3 yr; 1.80 m; 77.3 kg) walked for 120 min at 1.34 m/s with a 40-kg load on a level treadmill. Gait kinematic data and oxygen consumption (VO₂) data were collected over the 120-min period. For the fractal analysis, utilizing stride interval data, we calculated fractal dimension. For the dynamical systems analysis, kinematic angle time series were used to estimate phase space warping based features at uniform time intervals: smooth orthogonal decomposition (SOD) was used to extract slowly time-varying trends from these features. Estimated fractal dimensions showed no apparent trend or correlation with independently measured VO₂. While inter-individual difference did exist in the VO₂ data, dominant SOD time trends tracked and correlated with the VO₂ for all volunteers. Thus, dynamical systems analysis using gait kinematics may be suitable to develop a model to predict physiologic fatigue based on biomechanical work.     

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.     

Analysis of fractal dimension of O2A glial cells differentiating in vitro

Fractal dimension is a quantitative measure of morphological complexity. Glial cells of the oligodendrocyte-type 2 astrocyte (O2A) lineage exhibit increasing morphological complexity as they differentiate in vitro. Enriched populations of O2A progenitor cells isolated from neonatal rat cerebral hemispheres or optic nerves were allowed to differentiate in vitro, and their fractal dimensions were measured over time. The fractal dimensions of the maturing cells correlated with perceived complexity; cells with elaborate process branching had larger fractal dimensions than cells with a simpler morphology. An analysis of changes in fractal dimension revealed distinct rates of growth for both oligodendrocytes and type 2 astrocytes. The fractal dimension remained constant over a 10-fold range in optical magnification, demonstrating that cultured O2A glial cells exhibit self-similarity, a defining characteristic of fractal objects. These results illustrate that fractal dimension analysis of maturing cell populations is a useful method for quantitatively describing the process of cell differentiation.     

Perinatal model of human transition from hypogravity to Earth’s gravity based on nonlinear electromyogram characteristics

Interferential electromyogram (iEMG) was analyzed in healthy newborns (n = 29) during the first 24 h of life as a model of transition from hypogravity (intrauterine immersion) to Earth’s gravity (the postnatal period). Nonlinear methods of iEMG analysis (correlation dimension, entropy, and fractal dimension) reflecting the complexity, chaotic character, and predictability of signals from the leg and arm antagonistic muscles were used. The iEMG fractal dimension was shown to grow in all the muscles as the postnatal period extended, except the m. gastrocnemius during the first 24 h of life. Lower fractal and correlation dimensions and entropy were found to be characteristic of the flexors, especially at a low iEMG amplitude suggesting better congenital programming of the flexors compared to the extensors. It was concluded that the early ontogenesis model can be potentially useful for studying the evolution and the states of antigravity functions.     

A critique of field methodology for determining fish feeding periodicity

Synopsis Two major criticisms are directed at studies which attempt to determine changes in the feeding intensity of fish in the natural environment over a 24 hour period. A confusing vocabulary in the literature has lead to misinterpretation of data and erroneous conclusions about the diel nature of fish feeding. Workers also tend to conclude that there is variation in stomach content (and by implication feeding rate) over time, without testing observed differences statistically. Large variations in data render such conclusions suspect.This paper attempts to clarify the existing vocabulary by documenting the more commonly used terms and by providing precise definitions of keywords and phrases. A methodology using the analysis of covariance is proposed which establishes statistical criteria upon which more valid inferences about the temporal continuity of fish feeding can be made.Covariant analysis of stomach content data of the longhorn sculpin, Myoxocephalus octodecemspinosus, is used to refute conclusions based on apparent daily feeding activity patterns and exemplifies the need for applying rigorous inferential analysis.     

Ontogenetic changes in the fractal geometry of the bronchial tree in Rattus norvegicus

Respiration and metabolism change dramatically over the course of the development of vertebrates. In mammals these changes may be ascribed to organogenesis and differentiation of structures involved in gas exchange and transport and the increase in size. Since young as well as mature individuals must be well-designed if the species is to survive, the physiological changes during the development should be matched with geometrical or structural adjustments of the respiratory system. The aim of this study was to evaluate changes in the fractal geometry of the bronchial tree during the postnatal development of the rat. The average fractal dimension of the bronchial tree of the rats was 1.587, but that of juveniles was larger than that of the adults. We found a significant negative correlation between age and fractal dimension. This correlation could be considered be misleading because of the difficulty of separating age/body size effects. Nevertheless, because fractal dimensions of the bronchial tree of rabbits and humans are known to be similar, 1.58 and 1.57 respectively, the body size effect may be nil. To our knowledge, this is the first report of ontogenetic changes in the fractal dimension of the bronchial tree in mammals.     

Incomplete quality of life data in lung transplant research: comparing cross sectional,repeated measures ANOVA,and multi-level analysis

Background

In longitudinal studies on Health Related Quality of Life (HRQL) it frequently occurs that patients have one or more missing forms, which may cause bias, and reduce the sample size. Aims of the present study were to address the problem of missing data in the field of lung transplantation (LgTX) and HRQL, to compare results obtained with different methods of analysis, and to show the value of each type of statistical method used to summarize data.

Methods

Results from cross-sectional analysis, repeated measures on complete cases (ANOVA), and a multi-level analysis were compared. The scores on the dimension ''energy'' of the Nottingham Health Profile (NHP) after transplantation were used to illustrate the differences between methods.

Results

Compared to repeated measures ANOVA, the cross-sectional and multi-level analysis included more patients, and allowed for a longer period of follow-up. In contrast to the cross sectional analyses, in the complete case analysis, and the multi-level analysis, the correlation between different time points was taken into account. Patterns over time of the three methods were comparable. In general, results from repeated measures ANOVA showed the most favorable energy scores, and results from the multi-level analysis the least favorable. Due to the separate subgroups per time point in the cross-sectional analysis, and the relatively small number of patients in the repeated measures ANOVA, inclusion of predictors was only possible in the multi-level analysis.

Conclusion

Results obtained with the various methods of analysis differed, indicating some reduction of bias took place. Multi-level analysis is a useful approach to study changes over time in a data set where missing data, to reduce bias, make efficient use of available data, and to include predictors, in studies concerning the effects of LgTX on HRQL.     

Development and application of a T-RFLP data analysis method using correlation coefficient matrices

Environmental microbiology studies commonly use terminal restriction fragment length polymorphism (T-RFLP) of 16S rRNA genes, for example, to analyze changes in community structure in relation to changing physicochemical and biological conditions over space and time. Although T-RFLP is most useful for comparing samples from different environments, a large number of samples makes effective analysis difficult using the Web-based tools that are currently available. To resolve this dilemma, we used a new approach for calculating data from multiple T-RFLP samples by estimating terminal fragment combinations, then applying a correlation analysis using two different fluorescent dyes generated simultaneously from all samples. This calculation was based on the expectation that the proportions of two terminal fragments from one full-length polymerase chain reaction fragment would be nearly the same in each analysis. Using this program, the oral microflora in 73 human saliva samples were analyzed, and 24 bacterial groups, with peak areas of at least 0.5% and correlation coefficients of 0.55 or greater, were identified from the T-RFs within 40 s.     

Comparative variation of morphological and molecular evolution through geologic time: 28S ribosomal RNA versus morphology in echinoids.

The comparatively good fossil record of post-Palaeozoic echinoids allows rates of morphological change to be estimated over the past 260 million years and compared with rates of molecular evolution. Parsimony analysis of morphological data, based predominantly on skeletal characteristics, and parsimony, distance and maximum likelihood analyses of molecular data, from the first 380 bases from the 5' end of the 28S rRNA molecule, for 10 species of echinoid produce congruent phylogenies. The molecular sequence chosen is demonstrably far from saturation and sister groups have divergence times ranging from about 15 to 260 Ma. Parsimony analysis allows the great majority of molecular and morphological apomorphies to be placed in one of 18 independent geological time intervals, providing a direct measure of rates of evolution for periods in the geological past. Because most molecular fixed point mutations in our sequences cannot be polarized unambiguously by outgroup comparison (making the outgroup states effectively random), distance and parsimony analyses both tend spuriously to root the echinoid tree on the longest internal branch. A topology identical to that derived from morphological data is, however, obtained using Maximum Likelihood and also parsimony analysis where outgroup rooting is restricted to more conserved regions. This is taken as the correct topology for assessing rates of evolution. Overall, both morphological and molecular changes show a moderately strong correlation with time elapsed, but a weaker correlation with one another. Statistically significant differences in evolutionary rate are found between some, but not all, pair-wise comparisons of sister lineages for both molecular and morphological data. The molecular clock rate for echinaceans is three times faster than that for cidaroids and irregular echinoids. Spearman's rank correlation test, which requires only relative magnitude of changes to be known, suggests that morphological change has a slightly better correlation with time than does molecular change, averaged over all ten species. However, when just echinaceans are considered an extremely good correlation is found between the number of molecular changes and time elapsed, whereas morphological change remains poorly correlated. Thus, molecular rates approximate to a clocklike model within restricted echinoid clades, but vary significantly between clades. Averaging results over all echinoids produces a correlation that is no better than the correlation between morphological change and time elapsed.     

Application of the correlation integral to respiratory data of infants during REM sleep

 Non-linear time sequence analysis has been performed on infant sleep measurement data in order to obtain more information about the respiratory processes. As a first step, respiration data during REM sleep were analysed with methods from non-linear dynamics, especially, the correlation integral and the slope of its log-log plot, representing the correlation dimension. Before calculation of the correlation integral, a special kind of filtering has to be applied to the data. This filtering algorithm is a state space and singular value decomposition-based noise reduction method, and it is used to separate the noise and signal subspaces. The dynamics of a signal (in our case data from the respiratory process) and its degrees of freedom can be characterised by the correlation integral and by the correlation dimension, respectively. The main result of this study is that the highly irregular-looking breathing patterns during REM sleep could be described by a deterministic system, and finally the physiological significance of this finding is discussed. Received: 17 June 1994/Accepted in revised form: 18 November 1994     

A cDNA microarray assessment of gene expression in the liver of rainbow trout (Oncorhynchus mykiss) in response to a handling and confinement stressor

A purpose-designed microarray platform (Stressgenes, Phase 1) was utilised to investigate the changes in gene expression within the liver of rainbow trout during exposure to a prolonged period of confinement. Tissue and blood samples were collected from trout at intervals up to 648 h after transfer to a standardised confinement stressor, together with matched samples from undisturbed control fish. Plasma ACTH, cortisol, glucose and lactate were analysed to confirm that the neuroendocrine response to confinement was consistent with previous findings and to provide a phenotypic context to assist interpretation of gene expression data. Liver samples for suppression subtractive hybridisation (SSH) library construction were selected from within the experimental groups comprising “early” stress (2–48 h) and “late” stress (96–504 h). In order to reduce redundancy within the four SSH libraries and yield a higher number of unique clones an additional subtraction was carried out. After printing of the arrays a series of 55 hybridisations were executed to cover 6 time points. At 2 h, 6 h, 24 h, 168 h and 504 h 5 individual confined fish and 5 individual control fish were used with control fish only at 0 h. A preliminary list of 314 clones considered differentially regulated over the complete time course was generated by a combination of data analysis approaches and the most significant gene expression changes were found to occur during the 24 h to 168 h time period with a general approach to control levels by 504 h. Few changes in expression were apparent over the first 6 h. The list of genes whose expression was significantly altered comprised predominantly genes belonging to the biological process category (response to stimulus) and one cellular component category (extracellular region) and were dominated by so-called acute phase proteins. Analysis of the gene expression profile in liver tissue during confinement revealed a number of significant clusters. The major patterns comprised genes that were up-regulated at 24 h and beyond, the primary examples being haptoglobin, β-fibrinogen and EST10729. Two representative genes from each of the six k-means clusters were validated by qPCR. Correlations between microarray and qPCR expression patterns were significant for most of the genes tested. qPCR analysis revealed that haptoglobin expression was up-regulated approximately 8-fold at 24 h and over 13-fold by 168 h.     

山西太岳山兴唐寺红柄白鹃梅群落优势种的空间格局分析

使用点格局分析和分形分析(计盒维数、信息维数、关联维数)方法, 对山西太岳山兴唐寺红柄白鹃梅(Exochorda giraldii)群落进行了空间格局分析。结果表明: 1)红柄白鹃梅群落优势种在较小的尺度上呈现聚集分布的特点, 其中, 杠柳(Periploca sepium)和绣线菊(Spiraea salicifolia)在所有尺度上呈现聚集分布, 红柄白鹃梅和连翘(Forsythia suspensa)随着尺度的增加, 呈现聚集分布—随机分布—均匀分布的规律; 2)红柄白鹃梅群落优势种在不同尺度上表现出聚集分布的特点; 3)红柄白鹃梅群落优势种占据空间能力的大小为: 红柄白鹃梅>连翘>绣线菊>杠柳; 4)红柄白鹃梅群落优势种格局强度的尺度变化程度为: 杠柳>绣线菊>红柄白鹃梅>连翘; 5)红柄白鹃梅群落优势种个体空间相关程度为: 杠柳>红柄白鹃梅>连翘>绣线菊。点格局分析与分形分析结果一致, 揭示了暖温带落叶阔叶林遭到破坏后形成的次生灌丛的空间分布格局。     

A single-cell model to study changes in neuronal fractal dimension

Many methods have been developed to quantify neuronal morphology: measurement of neurite length, neurite number, etc. However, none of these approaches provides a comprehensive view of the complexity of neuronal morphology. In this work we have analyzed the evaluation of fractal dimension (D) as a tool to represent and quantify changes in complexity of the dendritic arbor, in in vitro cultures grown under low-density conditions. Neurons grown in isolation developed a bipolar morphology corresponding to a fractal dimension close to the unit. The analysis showed that neuronal complexity increased when cells were incubated with a depolarizing potassium concentration and there was a correlation with an increase in fractal dimension (D5 mM KCl = 1.08 +/- 0.01, D25 mM KCl =1.25 +/- 0.01). We conclude that fractal dimension is a suitable parameter to quantify changes in neuronal morphological complexity.     

Assessment of reproductive status in male echidnas

This study reports the development and application of techniques to assess the reproductive status of male echidnas. The pattern of testosterone secretion over a 24-h period in five echidnas was documented. Testosterone secretion after injection i.m. of either 1000 IU hCG (n=4) or 4 microg GnRH agonist (n=6) was determined to establish whether this could be used as a practical index of the prevailing steroidogenic capacity of the testes. hCG (1000 IU) was also used to assess seasonal changes in testosterone secretion in six echidnas over a 13-month period. Seasonal changes in testicular volume were examined by transabdominal ultrasonography. Electroejaculation was attempted to monitor seasonal changes in sperm production, which was also determined by spermatorrhea. There was no apparent diurnal pattern of testosterone secretion in echidnas and circulating concentrations of testosterone remained relatively low (maximum 1.2 ng/mL) and stable over 24h. Injection of hCG resulted in an increase (P<0.01; n=4) in testosterone concentration with a peak (2.9+/-0.3 ng/mL) approximately 4h after injection. GnRH also induced an increase (P<0.01; n=6) in circulating testosterone that was apparent after 1h (2.6+/-0.3 ng/mL) and concentrations remained elevated (3.4+/-0.3 ng/mL) for up to 8h after injection. Seasonal changes in testosterone secretion determined after injection of hCG, increased (P=0.03; n=6) from late-autumn, peaked in late-winter, and decreased by early-spring. Testicular volume followed a similar seasonal pattern (P<0.01; n=6) with an increase from late-autumn, peak in winter and a decline in mid-spring. There was no seasonal change in live weight. Electroejaculation was attempted throughout two breeding seasons but no semen was obtained. Spermatorrhoea in the echidna was described for the first time and was subsequently used to assess seasonal sperm production. Spermatozoa were found in the urine from June to September. This study has demonstrated that exogenous hormones can be used to obtain an index of the prevailing steroidogenic capacity of the testes in echidnas, which is not apparent with repetitive non-stimulated samples over 24 h. The assessment of testosterone secretion after injection of trophic hormones provides a valuable and practical procedure for the assessment of reproductive status. Testicular ultrasonography and spermatorrhea are useful in assessing reproductive status and in this study were successfully used to determine seasonal reproduction in captive echidnas.     

Nonlinear behavior of human respiratory movement during different sleep stages

To investigate the nonlinear properties of respiratory movement during different sleep stages, we applied an algorithm proposed by Grassberger and Procaccia to calculate the correlation dimension in rapid eye movement and non-rapid eye movement sleep. We also tested for nonlinearity in respiratory movement by comparing the correlation dimension for the original data with that for surrogate data. The study population included eight healthy volunteers. We recorded respiratory movement and the sleep electroencephalogram for 8 h. The correlation dimension for respiratory movement was 3.28+/-0.19 (mean +/- SD) during rapid eye movement sleep, 2.31+/-0.21 during light sleep (stage I) and 1.64+/-0.25 during deep slow-wave sleep (stage IV). Thus, the correlation dimension differed significantly by sleep stage (p < 0.001): it was least during stage IV sleep and greatest during REM. The correlation dimension for the original data also differed from that for surrogate data, confirming nonlinearity in original data. The results suggest that the nonlinear dynamics of respiratory movement in sleep changes with sleep stage, presumably due to the information processing by the cerebral cortex. The increased correlation dimension for respiratory movement in REM sleep may be related to increased cortical information processing associated with dreaming.     

Correlation between Antioxidative Activities and Metabolite Changes during Cheonggukjang Fermentation

Liquid chromatography mass spectrometry and multivariate analysis were employed to investigate the correlation between fermentation time-dependent metabolite changes in cheonggukjang, a traditional fermented soybean product, and changes in its antioxidant activity over 72 h. The metabolite patterns were clearly distinguished not by strains but by fermentation time, into patterns I (0–12 h), II (12–24 h), and III (24–72 h), which appeared as distinct clusters on principal component analysis. The compounds that significantly contributed to patterns I, II, and III were soyasaponins, isoflavonoid derivatives, and isoflavonoid aglycons respectively. Partial least square analysis for metabolite to antioxidant effects showed correlations between the ferric reducing/antioxidant power (FRAP) and 2,2-diphenyl-1-picrylhydrazyl (DPPH) assay during 24–36 h, and 2,2′-azinobis (3-ethylbenzothiazoline-6-sulfonic acid) (ABTS) test and total phenol content (TPC) during 36–72 h. Compared with the strong negative correlations of glucosylated-isoflavonoids with DPPH, ABTS and TPC during fermentation, the isoflavonoid aglycon displayed strong positive correlations with these compounds during fermentation.