Fractal probability measures of learning

Herein we introduce a technique for determining the fractal dimension of time series obtained from complex systems, in particular brain-wave data in which the fractal dimension is arguably a measure of awareness and learning. The technique is based on determining the probability distribution for the degree of irregularity in random time series and has been shown to be superior in terms of efficiency and reliability to more commonly used methods that rely on the correlation function. We speculate that the scaling behavior of the probability measure is an indicator of an allometric relation between learning and brain-wave activity.     

Use of fractal theory in neuroscience: methods, advantages, and potential problems

Fractal analysis has already found widespread application in the field of neuroscience and is being used in many other areas. Applications are many and include ion channel kinetics of biological membranes and classification of neurons according to their branching characteristics. In this article we review some practical methods that are now available to allow the determination of the complexity and scaling relationships of anatomical and physiological patterns. The problems of describing fractal dimensions are discussed and the concept of fractal dimensionality is introduced. Several related methodological considerations, such as preparation of the image and estimation of the fractal dimensions from the data points, as well as the advantages and problems of fractal geometric analysis, are discussed.     

Fat fractal and multifractals for protein and enzyme surfaces.

The roughness and irregularity of the surfaces in the protein and enzyme are fractal features that may be characterized by fractal dimensions and mass exponents. The surface fractal dimensions calculated by the variation method are different from those obtained by other methods, since the former is applicable to the self-affine system. Thus the results reported here are reliable for the surfaces. However, the fat fractal and multifractal features of proteins and enzymes are studied by simulation. The surface mass exponents are regarded as another kind of scaling exponent, and the spectrum f(alpha) provides further detailed information about the surfaces of enzyme and protein. The applications of the spectrum f(alpha) to the enzymatic reactions is also discussed.     

Microbial growth patterns described by fractal geometry.

Fractal geometry has made important contributions to understanding the growth of inorganic systems in such processes as aggregation, cluster formation, and dendritic growth. In biology, fractal geometry was previously applied to describe, for instance, the branching system in the lung airways and the backbone structure of proteins as well as their surface irregularity. This investigation applies the fractal concept to the growth patterns of two microbial species, Streptomyces griseus and Ashbya gossypii. It is a first example showing fractal aggregates in biological systems, with a cell as the smallest aggregating unit and the colony as an aggregate. We find that the global structure of sufficiently branched mycelia can be described by a fractal dimension, D, which increases during growth up to 1.5. D is therefore a new growth parameter. Two different box-counting methods (one applied to the whole mass of the mycelium and the other applied to the surface of the system) enable us to evaluate fractal dimensions for the aggregates in this analysis in the region of D = 1.3 to 2. Comparison of both box-counting methods shows that the mycelial structure changes during growth from a mass fractal to a surface fractal.     

遗传算法支持下土地利用空间分形特征尺度域的识别

针对土地利用空间分形特征只存在于有限尺度域的现象,采用无标度区内离散点拟合的离差平方和平均值最小作为目标函数,提出了一种基于遗传算法的土地利用空间分形特征尺度域的识别方法,用于准确计算分形维数的有效区间范围。以武汉市武昌区水域空间分形特征为例,利用Quickbird多光谱遥感影像提取土地利用空间信息,重点讨论了基于遗传算法识别土地利用空间分形特征尺度域范围的总体思路、适应度函数和遗传算子等环节;然后分别从测定系数、标准差和无标度区间3个角度,将其同人工判断法、相关系数法以及强化系数法进行对比讨论;并结合研究区域实际的水域空间分布格局,分析不同方法计算所得半径维数的合理性。结果表明,土地利用分形特征尺度域的范围对分形维数计算结果有较大影响;相对于传统计算方法来说,遗传算法在尺度无标度区间识别上具有更高的精度,可以为土地利用空间格局分形特征的研究提供客观指导意见。     

Fractal properties of forest spatial structure

The definition of fractal dimension of natural objects, which enables to deal with scale dependence of fractal dimension is discussed. Abrupt changes of fractal dimension of spatial structure of terrestrial ecosystems are considered in the context of hierarchical paradigm. On this ground the procedure is proposed for segmentation of a territory, which takes into account the scale dependence of spatial variability of ecological parameters. Using remotely sensed data — normalized difference vegetation index (NDVI) and thermal radiation in the infrared band — fractal dimensions and critical scales are evaluated for different forest types with the help of software, developed for this purpose. The results obtained corroborate the potentialities of fractal approach in ecology. These methods and results can be used for discrimination of remotely sensed data; but further investigations, including detailed comparison of fractal characteristics of remotely sensed forest images with results of on-site field studies are necessary to validate them.     

Structural complexity in mussel beds: the fractal geometry of surface topography

Seafloor topographic complexity is ecologically important because it provides habitat structure and alters boundary-layer flow over the bottom. Despite its importance, there is little agreement on how to define and measure surface complexity. The purpose of this investigation was to utilize fractal geometry of vertical cross-section profiles to characterize the surface topography of the soft-bottom mussel bed (Mytilus edulis L.) at Bob's Cove, ME, USA. Mussels there have been shown previously to have spatially ordered fractal characteristics in the horizontal plane. Two hypotheses were tested. The first was that the bed surface is fractal over the spatial scale of 1.44-200 mm, with fractal dimension less than or equal to 1.26, the value for the Koch curve, our model for bed profiles. The second was that bed surface topography (i.e., in vertical profile) is less complex than the mussel bed spatial pattern (i.e., aerial view in the horizontal plane). Both hypotheses were supported. Cross-sections of plaster casts of the bed produced 88 surface profiles, all of which were fractal over the entire spatial scale of more two orders of magnitude employed in the analysis. Fractal dimension values (D) for individual profiles ranged from 1.031 to 1.310. Fractal dimensions of entire casts ranged up to mean (1.242+/-0.046) and median (1.251) values similar to 1.26, the theoretical value of the Koch curve. The bed surface was less complex than the bed spatial pattern because every profile had D<1.36, the smallest value previously obtained from aerial views of the bed. The investigation demonstrated for the first time that surface topography of a soft-bottom mussel bed was fractal at a spatial scale relevant to hydrodynamic processes and habitat structure important for benthic organisms. The technique of using cross-section profiles from casts of the bed surface avoided possible underestimates of fractal dimension that can result from other profiling methods reported in the literature. The results demonstrate that fractal dimension can be useful in the analysis of habitat space and water flow over any irregular seafloor surface because it incorporates the size, shape, and scale of roughness elements into a simple, numerical metric.     

基于样带的唐古特白刺灌丛沙包空间格局尺度研究

黄河河岸与库布齐沙漠之间生态过渡带上的唐古特白刺(Nitraria Tangutorum)灌丛沙包由于遭受到不同程度的破坏而表现出一定程度的退化,空间格局尺度从一定程度上表征了其破碎化程度。我们分别应用样方方差法,小波分析,分维,谱分析以及空隙度分析等方法对鄂尔多斯高原北缘唐古特白刺灌丛沙包的空间格局尺度进行了比较研究,旨在选出适合分析干旱半干旱区白刺灌丛沙包空间格局尺度的方法,并确定研究区内唐古特白刺灌丛沙包的空间格局尺度,为保护唐古特白刺灌丛沙包提供科学依据。结果表明:(1)TTLQV法得到的格局尺度过大,而小波分析得到的格局尺度过小,分维分析得到的格局尺度除了相对较小之外,趋势也不明显,因此都不适合分析干旱半干旱区白刺灌丛沙包的空间格局尺度;(2)而3TLQV,tQV,谱分析和空隙度分析所得到的结果则相对容易接受,其大小顺序是:3TLQV>tQV>空隙度分析>谱分析;(3)独贵塔拉、呼和木独和巴拉贡唐古特白刺灌丛沙包的空间格局尺度分别为16.7～27.1m,10.2～25.0m和8.8～17.0m,这一结果将有助于在今后的研究中合理地确定研究范围和尺度。     

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.     

Fractal geometry: a tool for describing spatial patterns of plant communities

Vegetation is a fractal because it exhibits variation over a continuum of scales. The spatial structure of sandrim, bryophyte, pocosin, suburban lawn, forest tree, and forest understory communities was analyzed with a combination of ordination and geostatistical methods. The results either suggest appropriate quadrat sizes and spacings for vegetation research, or they reveal that a sampling design compatible with classical statistics is impossible. The fractal dimensions obtained from these analyses are generally close to 2, implying weak spatial dependence. The fractal dimension is not a constant function of scale, implying that patterns of spatial variation at one scale cannot be extrapolated to other scales.     

Uses and abuses of fractal methodology in ecology

Fractals have found widespread application in a range of scientific fields, including ecology. This rapid growth has produced substantial new insights, but has also spawned confusion and a host of methodological problems. In this paper, we review the value of fractal methods, in particular for applications to spatial ecology, and outline potential pitfalls. Methods for measuring fractals in nature and generating fractal patterns for use in modelling are surveyed. We stress the limitations and the strengths of fractal models. Strictly speaking, no ecological pattern can be truly fractal, but fractal methods may nonetheless provide the most efficient tool available for describing and predicting ecological patterns at multiple scales.     

Fractal analysis of the uterine contractions.

The fractal dimension D may be calculated in many ways, since its strict definition, the Hausdorff definition is too complicated for practical estimation. In this paper we perform a comparative study often methods of fractal analysis of time series. In Benoit, a commercial program for fractal analysis, five methods of computing fractal dimension of time series (rescaled range analysis, power spectral analysis, roughness-length, variogram methods and wavelet method) are available. We have implemented some other algorithms for calculating D: Higuchi's fractal dimension, relative dispersion analysis, running fractal dimension, method based on mathematical morphology and method based on intensity differences. For biomedical signals results obtained by means of different algorithms are different, but consistent.     

Fractal methods to analyze ion channel kinetics

We describe the traditional nonfractal and the new fractal methods used to analyze the currents through ion channels in the cell membrane. We discuss the hidden assumptions used in these methods and how those assumptions lead to different interpretations of the same experimental data. The nonfractal methods assumed that channel proteins have a small number of discrete states separated by fixed energy barriers. The goal was to determine the parameters of the kinetic diagram, which are the number of states, the pathways between them, and the kinetic rate constants of those pathways. The discovery that these data have fractal characteristics suggested that fractal approaches might provide more appropriate tools to analyze and interpret these data. The fractal methods determine the characteristics of the data over a broad range of time scales and how those characteristics depend on the time scale at which they are measured. This is done by using a multiscale method to accurately determine the probability density function over many time scales and by determining how the effective kinetic rate constant, the probability of switching states, depends on the effective time scale at which it is measured. These fractal methods have led to new information about the physical properties of channel proteins in terms of the number of conformational substates, the distribution of energy barriers between those states, and how those energy barriers change with time. The new methods developed from the fractal paradigm shifted the analysis of channel data from determining the parameters of a kinetic diagram to determining the physical properties of channel proteins in terms of the distribution of energy barriers and/or their time dependence.     

Fractal and spectral dimension analysis of liver fibrosis in needle biopsy specimens

OBJECTIVE: To evaluate the usefulness of a reliable and reproducible mathematical scoring system based on fractal geometry for quantifying the irregular pattern in fibrosis commonly seen in liver biopsy specimens from chronic liver diseases. STUDY DESIGN: The study used 26 standard liver biopsy specimens obtained from patients with chronic hepatitis C virus-related liver disease. The degree of fibrosis in each specimen was estimated using a quantitative scoring system based on the computer-assisted evaluation of both the fractal and spectral dimensions of deposited collagen. The fractal dimension was then compared with the percent area of collagen measured using an image analysis system. RESULTS: The fractional dimension of its irregular shape defines fibrosis as a natural fractal structure. The complex distribution of its collagenous components (unmeasurable by means of the usual morphometric parameters) can be optimally quantified using a single numerical score that seems to be a better alternative to the semiquantitative methods adopted so far. The proposed method is reproducible, rapid and inexpensive; furthermore, supported by specific software, its mathematical approach excludes subjectivity and eliminates the external factors capable of influencing staging and classification. CONCLUSION: This study demonstrated that it is possible to quantify the irregularity of the structures of the liver in an objective manner and that the box-counting fractal dimension does not depend on the amount of collagen deposited on the slide. Furthermore, as has been found in other fields of investigation, study of the fractal properties of the liver is likely to reveal more about its structure and the pathogenesis of liver diseases.     

北京东灵山地区辽东栎林种群空间分布分形分析

通过植物个体的坐标点与模拟冠幅的分形维数的分析,对辽东栎林内不同种群沿海拔梯度变化的空间分布格局和种群动态进行了比较和讨论。结果表明:模拟冠幅分形分析方法更适于分析具有大小不等、复杂多样的冠幅的植物个体空间分布格局。随着海拔的升高,不同种群的空间占据能力也随之变化。辽东栎种群的空间分布占绝对优势地位,分形维数可达1.9811;五角枫种群则逐步下降,分形维数最低为0.1170。低海拔山坡六道木种群空间占据能力较强,高海拔山坡照山白种群有较大的空间占据能力。在一定的环境条件下,不同的种群可能具有相近的空间占据能力,但对乔木层与灌木层来说,相同的分形维数的内涵是不同的。冠幅的分形维数作为表征植物种群空间占据能力的工具,是种群动态分析和种群分布格局研究的的重要指标之一。     

Branching tree model with fractal vascular resistance explains fractal perfusion heterogeneity

Perfusion heterogeneities in organs such as the heart obey a power law as a function of scale, a behavior termed "fractal." An explanation of why vascular systems produce such a specific perfusion pattern is still lacking. An intuitive branching tree model is presented that reveals how this behavior can be generated as a consequence of scale-independent branching asymmetry and fractal vessel resistance. Comparison of computer simulations to experimental data from the sheep heart shows that the values of the two free model parameters are realistic. Branching asymmetry within the model is defined by the relative tissue volume being fed by each branch. Vessel ordering for fractal analysis of morphology based on fed or drained tissue volumes is preferable to the commonly used Strahler system, which is shown to depend on branching asymmetry. Recently, noninvasive imaging techniques such as PET and MRI have been used to measure perfusion heterogeneity. The model allows a physiological interpretation of the measured fractal parameters, which could in turn be used to characterize vascular morphology and function.     

Statistical discrimination of fractal and Markov models of single-channel gating.

A statistical comparison is presented of Markov and fractal models of ion channel gating. The analysis is based on single-channel data from two types of ion channels: open times from a 90 pS Ca-activated K channel from GH3 pituitary cells, and closed times from a nonselective channel from rabbit corneal endothelium (Liebovitch et al., 1987a). Maximum likelihood methods were used to fit the data. For both data sets the best Markov model had three exponential components. The best Markov model had a higher likelihood than the fractal model, and the Asymptotic Information Criterion favored the Markov model for each data set. A more detailed analysis, using the Monte Carlo methods described in Horn (1987), showed that the Markov model was not significantly better than the fractal model for the corneal endothelium channels. The inability to discriminate the models definitively in this case was shown to be due in part to the small size of the data set.     

黄土区农业景观空间格局分析

景观空间格局分析是景观生态学研究的核心问题之一。本文用地理信息系统,分维分析和统计分析相结合,以１∶１万土地利用现状图为基础,选取斑块大小、分维数、斑块伸长指数、多样性、优势度、相对丰富度、破碎度等指标,陕北米脂县泉家沟流域农业景观的空间格局。结果显示,乔木林地、坡耕地、果园、草地和梯田农地的分维数较低,坝地、水库、灌溉农田、居信用地和灌木林地的人维数较高,斑块形状较为复杂。随着斑块面积的增加,农     

毛乌素沙地沙生半灌木群落的空间异质性

在使用 5m× 50 0 0 m样带进行调查获得的植被盖度数据的基础上 ,通过半方差分析和分形分析揭示了毛乌素沙化草地沙生半灌木群落植被盖度的空间变异特点。半方差分析揭示出植被盖度的多尺度变异和等级斑块结构 ,在取样范围内至少存在着 3个等级。分形分析得出毛乌素沙地沙生半木群落植被盖度的分维 D=1 .90 ,接近于 2 ,指示着大部分空间变异发生在较小的尺度上 ;分段分形分析得出不同尺度范围内的分维 ,反映了空间异质性的多尺度变异性。这一结果揭示了荒漠化过程引起的生境破碎化