Use and abuse of fractal theory in neuroscience

The Fractal Geometry of the Brain: AnOverview

The first chapter of this book introduces some history, philosophy, and basic concepts of fractal geometry and discusses how the neurosciences can benefit from applying computational fractal-based analysis. Further, it compares fractal with Euclidean approaches to analyzing and quantifying the brain in its entire physiopathological spectrum and presents an overview of the first section of this book as well.

Estimating the Fractal Dimensions of Vascular Networks and Other Branching Structures: Some Words of Caution

Branching patterns are ubiquitous in nature; consequently, over the years many researchers have tried to characterize the complexity of their structures. Due to their hierarchical nature and resemblance to fractal trees, they are often thought to have fractal properties; however, their non-homogeneity (i.e., lack of strict self-similarity) is often ignored. In this paper we review and examine the use of the box-counting and sandbox methods to estimate the fractal dimensions of branching structures. We highlight the fact that these methods rely on an assumption of self-similarity that is not present in branching structures due to their non-homogeneous nature. Looking at the local slopes of the log–log plots used by these methods reveals the problems caused by the non-homogeneity. Finally, we examine the role of the canopies (endpoints or limit points) of branching structures in the estimation of their fractal dimensions.

How neurons exploit fractal geometry to optimize their network connectivity

We investigate the degree to which neurons are fractal, the origin of this fractality, and its impact on functionality. By analyzing three-dimensional images of rat neurons, we show the way their dendrites fork and weave through space is unexpectedly important for generating fractal-like behavior well-described by an ‘effective’ fractal dimension D. This discovery motivated us to create distorted neuron models by modifying the dendritic patterns, so generating neurons across wide ranges of D extending beyond their natural values. By charting the D-dependent variations in inter-neuron connectivity along with the associated costs, we propose that their D values reflect a network cooperation that optimizes these constraints. We discuss the implications for healthy and pathological neurons, and for connecting neurons to medical implants. Our automated approach also facilitates insights relating form and function, applicable to individual neurons and their networks, providing a crucial tool for addressing massive data collection projects (e.g. connectomes).

The Sholl analysis of neuronal cell images: Semi-log or log–log method?

Although the Sholl analysis is a quantitative method for morphometric neuronal studies and its application provides many benefits to neurobiology since it is obvious, common and meaningful, there are many unresolved theoretical issues that need to be addressed. Nevertheless, it can be practiced without much background or sophistication. The two different methods of the Sholl analysis--log-log and semi-log--have been applied previously without a clear basis as to what to use. To make an adequate choice of the method, one should try and accept that one which gives a better result. We consider that some of the underlying principles, assumptions and limitations for the Sholl analysis can be found in basic mathematics. In order to compare the results of applications of the semi-log and log-log methods to the same neuron, we introduce the concept of determination ratio as the ratio of the coefficient of determination for the semi-log method and that for the log-log method. If the semi-log method is better as related to the log-log method, the value of this parameter is larger than 1. Having in mind that dendrites exhibit enormously diverse forms, we point out that the semi-log Sholl method is more frequently utilizable in practice. Only the neurons, whose dendritic trees have one or a few sparsely ramified dendrites being much longer than the rest ones, could be successfully and exactly analysed using the log-log method. We also compare the Sholl analysis with fractal analysis for the characterization of neuronal arborization patterns and found that between the Sholl and fractal analysis exist various and important analogies.

Morphometric analysis of neurons from the marginal and substantia gelatinosa layers of human spinal cord: Classification according to laminar organization

The main goal of morphometric analysis of neuronal images, except for getting information about their geometry and dendritic branching patterns, is their classification based on laminar organization. The majority of contemporary techniques for image analysis are based on the application of fractal theory, which has some limitations on results analysis. For that reasons, the new, mostly nonfractal techniques for image analysis had been designed in the past few years. This study shows the analysis of morphometry of the human spinal cord neurons from the marginal (lamina I) and substantia gelatinosa (laminae I-II). For the analysis of neuron images two techniques of morphometric analysis were used: box-counting method as a mainly used technique for fractal analysis, and circle-counting method as a newly designed technique for measuring the length of dendrites. The use of these methods for neurons of the mentioned regions of human spinal cord showed that circlecounting method had given more accurate results than fractal analysis method. When the proposed method was used for the analysis of neuronal images, it was possible to classify neurons according to their laminar position.

Structure-function correlation in transient amacrine cells of goldfish retina: basic and multifractal analyses of dendritic trees in distinct synaptic layers

Amacrine cells generating light-evoked transient ON-OFF responses were stained by intracellular injection of horseradish peroxidase after determining their input-output (voltage response vs. light intensity) profiles. Ten cells specifically having bistratified dendritic trees were analyzed. The cross-sectional area of the dendrites in each sublamina (a and b) of the inner plexiform layer was initially measured. Although some variability was observed, there was no statistically significant overall difference in the cross-sectional areas of the dendritic trees in sublaminae a and b. Also, the amplitudes of the ON and OFF responses, generated by a midrange criterion stimulus, could not be correlated with the cross-sectional areas or the number of branches of the dendrites in sublaminae b and a, respectively. On the other hand, determination of the generalized fractal spectra revealed that the negative (up to -3) and zero-order fractal dimensions of the dendritic trees in sublamina a were consistently higher than those for sublamina b. Furthermore, there was a positive correlation between response amplitude and some part of the generalized fractal dimension in the respective parts of the dendritic trees. It is concluded that dendritic tree characteristics differ in the two halves of the inner plexiform layer and that these can be related to the cells' light-evoked response amplitudes. Furthermore, generalized fractal analysis appears to be a useful method for correlating structure and function in retinal amacrine cells with complex dendritic trees.

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.

Morphology of reactive microglia in the injured cerebral cortex. Fractal analysis and complementary quantitative methods

The present study focuses on application of quantitative methods measuring differences between particular morphological types of microglial cells as well as between their proliferating and non-proliferating examples. On the basis of subjective classification, microglial cells of three morphological types (ramified, hypertrophied and bushy) were selected from the neocortex of injured rat brain. Thereafter, the morphological complexity of each cell was assessed by calculation its fractal dimension as well as its form factor, convexity, ramification factor and solidity. The fractal dimension seemed a good parameter for detecting small changes in the space-filing capacity of cells, for example, it shows differences between ramified cells from control and injured brains. This measure seemed insensitive to some aspects of cell morphology. To obtain precise quantification of observed changes other morphological parameters had to be applied. Proliferating and non-proliferating microglial cells displayed significant differences in their solidity and ramification factors, but not in fractal dimension and convexity. The results indicated that proliferating microglia were more massive and less-ramified but they did not reduce their spatial complexity.

Physiology of angiogenesis

Angiogenesis is a key prerequisite for growth in all vertebrate embryos and in many tumors. Rapid growth requires efficient transport of oxygen and metabolites. Hence, for a better understanding of tissue growth, biophysical properties of vascular systems, in addition to their molecular mechanisms, need to be investigated. The purpose of this article is twofold: (1) to discuss the biophysics of growing and perfused vascular systems in general, emphasizing non-sprouting angiogenesis and remodeling of vascular plexuses; and (2) to report on cellular details of sprouting angiogenesis in the initially non-perfused embryonic brain and spinal cord. It is concluded that (1) evolutionary optimization of the circulatory system corresponds to highly conserved vascular patterns and angiogenetic mechanisms; (2) deterministic and random processes contribute to both extraembryonic and central nervous system vascularization; (3) endothelial cells interact with a variety of periendothelial cells during angiogenesis and remodeling; and that (4) mathematical models integrating molecular, morphological and biophysical expertise improve our understanding of normal and pathological angiogenesis and account for allometric relations.

Generic modeling of chemotactic based self-wiring of neural networks

The proper functioning of the nervous system depends critically on the intricate network of synaptic connections that are generated during the system development. During the network formation, the growth cones migrate through the embryonic environment to their targets using chemical communication. A major obstacle in the elucidation of fundamental principles underlying this self-wiring is the complexity of the system being analyzed. Hence much effort is devoted to in vitro experiments of simpler (two-dimensional) 2D model systems. In these experiments neurons are placed on Poly-L-Lysine (PLL) surfaces, so it is easier to monitor their self-wiring. We developed a model to reproduce the salient features of the 2D systems, inspired by the study of the growth of bacterial colonies and the aggregation of amoebae. We represent the neurons (each composed of cell's soma, neurites and growth cones) by active elements that capture the generic features of the real neurons. The model also incorporates stationary units representing the cells' soma and communicating walkers representing the growth cones. The stationary units send neurites one at a time, and respond to chemical signaling. The walkers migrate in response to chemotaxis substances emitted by the soma and communicate with each other and with the soma by means of chemotactic "feedback". The interplay between the chemo-repulsive and chemo-attractive responses is determined by the dynamics of the walker's internal energy which is controlled by the soma. These features enable the neurons to perform the complex task of self-wiring. We present numerical experiments of the model to demonstrate its ability to form fine structures in simple networks of few neurons. Our results raise two fundamental issues: (1) one needs to develop characterization methods (beyond number of connections per neuron) to distinguish the various possible networks; (2) what are the relations between the network organization and its computational properties and efficiency?

Computer-vision-based extraction of neural dendrograms

This work introduces a new approach to the characterization of neural cells by means of semi-automated generation of dendrograms; data structures which describe the inherently hierarchical nature of neuronal arborizations. Dendrograms describe the branched structure of neurons in terms of the length, average thickness and bending energy of each of the dendritic segments and allow in a straightforward manner, the inclusion of additional measures. The bending energy quantifies the complexity of the shape and can be used to characterize the spatial coverage of the arborizations (the bending energy is an alternative for other complexity measures such as the fractal dimension). The new approach is based on the partitioning of the cell's outer contour as a function of the high curvature points followed by a syntactical analysis of the segmented contours. The semi-automated method is robust and is an improvement on the time consuming manual generation of the dendrograms. Several experimental results are included in this paper which illustrate and corroborate the effectiveness of the approach. The technique presented in this paper is limited to planar neurons but could be extended to a 3D approach.

Stochastic fractal behavior in concentration fluctuation and fluorescence correlation spectroscopy

Fluctuations in the concentration of Brownian particles in one and two dimensions, or any reasonable measurement of the concentration such as in fluorescence correlation spectroscopy, is shown to be a stochastic fractal with a long tail. Being singular at omega = 0, the power spectrum of the fluctuation S(omega) approximately omega-1/2 for diffusion in one dimension, approximately log omega in two dimensions, but non-singular in three dimensions. This discovery provides one simple physical mechanism for possible long-memory fractal behavior, and its implications to various biological processes are discussed.

Neurons and fractals: how reliable and useful are calculations of fractal dimensions?

In the past 15 years it has become possible to determine the fractal dimension (Df) of complex objects, including neurons, by automated image analysis methods. However, there are many unresolved issues that need to be addressed. In this paper we discuss how the Df calculated by different methods may vary and how fractal analysis may be of use for retinal ganglion cell characterization. The goal of this work was to acknowledge inherent sources of variation during measurement and evaluate current fractal analysis methods for describing structure. Our results show that different algorithms and even the same algorithm performed by different computer programs and/or experimenters may give different but consistent numerical values. All described methods demonstrated their suitability for classifying cat retinal ganglion cells into distinct groups. Our results reinforce the idea that comparison of measurements of different profiles using the same measurement method may be useful and valid even if an exact numeric value of the dimension is not realised in practice.

Automated evaluation of angiogenic effects mediated by VEGF and PlGF homo- and heterodimers

The effects of growth factors on the blood vessel pattern of chick chorioallantoic membrane (CAM) were assessed with a fast and automated method (extended counting method, XCM; Sandau, 1996) that measures complexity, without assumptions about a fractal structure. XCM is a reliable measure of complexity not only in theory but also in practice: (1) it is robust with respect to thresholding; (2) it shows reduced variance due to pattern translation and rotation; (3) its properties come close to requirements of fractal geometry. It hence is superior to established fractal methods for distinguishing effects induced by various isoforms of vascular endothelial growth factor (VEGF121 and VEGF165), placenta growth factor (PlGF) isoforms, and control treatment. We here show that VEGF homo- and heterodimers and VEGF121/PlGF1 heterodimers increase vascular complexity, whereas PlGF1 and PlGF2 are not effective. PlGF1 and VEGF121 did not mutually influence each other when applied in adjacent fields on the same CAM. Since blood vessels in the CAM originate via nonfractal growth processes, their growth should be analyzed accordingly.

Fractals in pathology

Many natural objects, including most objects studied in pathology, have complex structural characteristics and the complexity of their structures, for example the degree of branching of vessels or the irregularity of a tumour boundary, remains at a constant level over a wide range of magnifications. These structures also have patterns that repeat themselves at different magnifications, a property known as scaling self-similarity. This has important implications for measurement of parameters such as length and area, since Euclidean measurements of these may be invalid. The fractal system of geometry overcomes the limitations of the Euclidean geometry for such objects and measurement of the fractal dimension gives an index of their space-filling properties. The fractal dimension may be measured using image analysis systems and the box-counting, divider (perimeter-stepping) and pixel dilation methods have all been described in the published literature. Fractal analysis has found applications in the detection of coding of coding regions in DNA and measurement of the space-filling properties of tumours, blood vessels and neurones. Fractal concepts have also been usefully incorporated into models of biological processes, including epithelial cell growth, blood vessel growth, periodontal disease and viral infections.