A fractal analysis of cultured rat optic nerve glial growth and differentiation

Fractal Analysis in Neurodegenerative Diseases

Neurodegenerative diseases are defined by progressive nervous system dysfunction and death of neurons. The abnormal conformation and assembly of proteins is suggested to be the most probable cause for many of these neurodegenerative disorders, leading to the accumulation of abnormally aggregated proteins, for example, amyloid β (Aβ) (Alzheimer's disease and vascular dementia), tau protein (Alzheimer's disease and frontotemporal lobar degeneration), α-synuclein (Parkinson's disease and Lewy body dementia), polyglutamine expansion diseases (Huntington disease), or prion proteins (Creutzfeldt-Jakob disease). An aberrant gain-of-function mechanism toward excessive intraparenchymal accumulation thus represents a common pathogenic denominator in all these proteinopathies. Moreover, depending upon the predominant brain area involvement, these different neurodegenerative diseases lead to either movement disorders or dementia syndromes, although the underlying mechanism(s) can sometimes be very similar, and on other occasions, clinically similar syndromes can have quite distinct pathologies. Non-Euclidean image analysis approaches such as fractal dimension (FD) analysis have been applied extensively in quantifying highly variable morphopathological patterns, as well as many other connected biological processes; however, their application to understand and link abnormal proteinaceous depositions to other clinical and pathological features composing these syndromes is yet to be clarified. Thus, this short review aims to present the most important applications of FD in investigating the clinical-pathological spectrum of neurodegenerative diseases.

Fractal analysis of extra-embryonic vascularization in Japanese quail embryos exposed to extremely low frequency magnetic fields

Magnetic fields (MF) can alter the dynamic behavior of vascular tissue and may have a stimulatory or inhibitory effect on blood vessel growth. Fractal geometry has been used in several studies as a tool to describe the development of blood vascular networks. Due to its self-similarity, irregularity, fractional dimension, and dependence on the scale of vessel dimensions, vascular networks can be taken as fractal objects. In this work, we calculated the fractal dimension by the methods of box counting (D(bc)) and information dimension (D(inf)) to evaluate the development of blood vessels of the yolk sac membrane (YSM) from quail embryos exposed to MF with a magnetic flux density of 1 mT and a frequency of 60 Hz. The obtained results showed that when the MF was applied to embryos aged between 48 and 72 h, in sessions of 2 h (6 h/day) and 3 h (9 h/day) with exposure intervals between 6 and 5 h, respectively, blood vascular formation was inhibited. Exposure sessions shorter than 2 h or longer than 3 h had no observable change on the vascular process. In contrast, the magnetic field had no observable change on the YSM vascular network for embryos aged between 72 and 96 h, irrespective of the exposure time. In conclusion, these results show a "window effect" regarding exposure time.

Rho/Rho-kinase signaling pathway controls axon patterning of a specified subset of cranial motor neurons

Cranial motor neurons, which are divided into somatic motor (SM), branchiomotor (BM) and visceral motor (VM) neurons, form distinct axonal trajectories to innervate their synapse targets. Rho GTPase regulates various neuronal functions through one of the major effector proteins, Rho-kinase. Here, we addressed the in vivo role of the Rho/Rho-kinase signaling pathway in axon patterning of cranial motor neurons. We performed conditional expression of a dominant-negative mutant for RhoA or Rho-kinase in transgenic mice by using the Cre-loxP system to suppress the activity of these molecules in developing cranial motor neurons. Blockade of the Rho/Rho-kinase signaling pathway caused defects in the patterning of SM axons but not in that of BM/VM axons, in which defects were accompanied by reduced muscle innervation and reduced synapse formation by SM neurons. In addition, blockade of the signaling pathway shifted the trajectory of growing SM axons in explant cultures, whereas it did not appear to affect the rate of spontaneous axonal outgrowth. These results indicate that the Rho/Rho-kinase signaling pathway plays an essential role in the axon patterning of cranial SM neurons during development.

Effects of fractal surface on C6 glioma cell morphogenesis and differentiation in vitro

Neurons and glial cells in the brain are surrounded by a fractal environment. A fractal alkylketene dimmer (AKD) surface was shown to provide such a biomimetic environment for glial cell culture. However, little is known about the effects of fractal surface on the complexity of cell morphology. In particular, whether fractal surface induces glial cell differentiation remains to be elucidated. The present work, thus determined the fractal dimension (FD) of cell complexity with a geometrically calculational parameter, the expressions of GFAP gene and protein in C6 glioma cells on fractal AKD, non-fractal AKD and PLL-coated surfaces. Fractal surface suppressed the proliferation of glioma cell, and significantly increased the length and number of cell process. Furthermore, the enhanced values of FD were accompanied with the expressions of GFAP gene and protein, especially that of gene. However, cells on non-fractal and PLL surface proliferated gradually along with the culture time, showing the fibroblast-like morphology, and accompanied with the consistent expressions of GFAP gene and protein. These results suggested that C6 glioma cell differentiation can be induced by fractal AKD surface.

Anatomy and flow in normal and ischemic microvasculature based on a novel temporal fractal dimension analysis algorithm using contrast enhanced ultrasound

Strategies for improvement of blood flow by promoting new vessel growth in ischemic tissue are being developed. Recently, contrast-enhanced ultrasound (CEU) imaging has been used to assess tissue perfusion in models of ischemia-related angiogenesis, growth-factor mediated angiogenesis, and tumor angiogenesis. In these studies, microvascular flow is measured in order to assess the total impact of adaptations at different vascular levels. High-resolution methods for imaging larger vessels have been developed in order to derive "angiograms" of arteries, veins, and medium to large microvessels. We describe a novel method of vascular bed (microvessel and arterial) characterization of vessel anatomy and flow simultaneously, using serial measurement of the fractal dimension (FD) of a temporal sequence of CEU images. This method is proposed as an experimental methodology to distinguish ischemic from nonischemic tissue. Moreover, an improved approach for extracting the FD unique to this application is introduced.

Quantitative morphological study of microglial cells in the ischemic rat brain using principal component analysis

Pathogenic stimuli induce alterations in the morphology of microglial cells. We analysed changes in lectin-stained cells on the 1st, 3rd, 7th or 14th day after transient global ischemia. Three areas differing in the degree of microglial reaction were selected for analysis: the upper cerebral cortex, the hippocampal CA1 area, and the hilus of the dentate gyrus. Nine morphological parameters, including fractal dimension, lacunarity, self-similarity range, solidity, convexity and form factor were determined. Then the resultant data were processed using principal component analysis (PCA). We found that the two first principal components together explained more than 73% of the observed variability, and may be sufficient both to describe the morphological diversity of the cells, and to determine the dynamics and direction of the changes. In both hippocampal areas, the transformation to hypertrophied and phagocytic cells was observed, but changes in the hilus were faster than in the CA1. In contrast, in the cortex, a microglial reaction was characterised by an increase in the complexity of processes. The results presented show that the quantitative morphological analysis can be an effective tool in research on the reactive behaviour of microglia and, particularly, in the detection of small and early changes in the cells.

Fractal Analysis of the Vascular Tree in the Human Retina

The retinal circulation of the normal human retinal vasculature is statistically self-similar and fractal. Studies from several groups present strong evidence that the fractal dimension of the blood vessels in the normal human retina is approximately 1.7. This is the same fractal dimension that is found for a diffusion-limited growth process, and it may have implications for the embryological development of the retinal vascular system. The methods of determining the fractal dimension for branching trees are reviewed together with proposed models for the optimal formation (Murray Principle) of the branching vascular tree in the human retina and the branching pattern of the human bronchial tree. The limitations of fractal analysis of branching biological structures are evaluated. Understanding the design principles of branching vascular systems and the human bronchial tree may find applications in tissue and organ engineering, i.e., bioartificial organs for both liver and kidney.

Fractal dimension in human cerebellum measured by magnetic resonance imaging

Fractal dimension has been used to quantify the structures of a wide range of objects in biology and medicine. We measured fractal dimension of human cerebellum (CB) in magnetic resonance images of 24 healthy young subjects (12 men and 12 women). CB images were resampled to a series of image sets with different 3D resolutions. At each resolution, the skeleton of the CB white matter was obtained and the number of pixels belonging to the skeleton was determined. Fractal dimension of the CB skeleton was calculated using the box-counting method. The results indicated that the CB skeleton is a highly fractal structure, with a fractal dimension of 2.57 +/- 0.01. No significant difference in the CB fractal dimension was observed between men and women.

Characterization of Melanophore Morphology by Fractal Dimension Analysis

Fractal or focal dimension (FD) analysis is a valuable tool to identify physiologic stimuli at the cellular and tissue levels that allows for quantification of cell perimeter complexity. The FD analysis was determined on fluorescence images of caffeine- or epinephrine-treated (or untreated control) killifish Fundulus heteroclitus (Linneaus) melanophores in culture. Cell perimeters were indicated by rhodamine-phalloidin labeling of cortical microfilaments using box-counting FD analysis. Caffeine-treated melanophores displayed dispersed melanosomes in cells with less serrated edges and reduced FD and complexity. Complexity in epinephrine-treated cells was significantly higher than the caffeine-treated cells or in the control. Cytoarchitectural variability of the cell perimeter is expected because cells change shape when cued with agents. Epinephrine-treated melanophores demonstrated aggregated melanosomes in cells with more serrated edges, significantly higher FD and thus complexity. Melanophores not treated with caffeine or epinephrine produced variable distributions of melanosomes and resulted in cells with variably serrated edges and intermediate FD with a larger SE of the regression and greater range of complexity. Dispersion of melanosomes occurs with rearrangements of the cytoskeleton to accommodate centrifugal distribution of melanosomes throughout the cell and to the periphery. The loading of melanosomes onto cortical microfilaments may provide a less complex cell contour, with the even distribution of the cytoskeleton and melanosomes. Aggregation of melanosomes occurs with rearrangements of the cytoskeleton to accommodate centripetal distribution of melanosomes. The aggregation of melanosomes may contribute to centripetal retraction of the cytoskeleton and plasma membrane. The FD analysis is, therefore, a convenient method to measure contrasting morphologic changes within stimulated cells.

Morphological development of microglia in the postnatal rat brain

Morphological transformation of lectin-positive microglia/macrophages in the developing rat cerebral hemisphere was analysed using quantitative methods. During the first postnatal month, the cells showed increases in their size and fractal dimension accompanied by a simultaneous decrease in their solidity. Regional variations in dynamics of the process indicated the existence of spatio-temporal developmental gradients within the cerebral hemisphere wall which might correspond with regional patterns of neuronal differentiation. Results of the present study prove that the quantitative methods can be the source of reliable data replacing subjective cell typologies.

Morphological transformations of cells immunopositive for GFAP, TrkA or p75 in the CA1 hippocampal area following transient global ischemia in the rat. A quantitative study

Transient global ischemia induces intensive neuronal degeneration in the hippocampal CA1 pyramidal layer, accompanied by reactive transformation of glial cells. Previously, we have shown using the double immunostaining method that the NGF receptors (NGFR) p75 and TrkA are expressed mainly on subpopulations of GFAP+ astrocytes, and this expression increases progressively after ischemia. In the presented study, we analyzed quantitatively the morphological transformations of cells immunopositive for GFAP or NGF receptors in the stratum radiatum of the CA1 hippocampal area in different survival periods after ischemia, evoked by 10-min cardiac arrest in adult rats. In control brains, NGF receptors were expressed only on small cells with poorly ramified processes. After ischemia, the NGFR+ cells increased in size and morphological complexity (measured using fractal analysis). However, even 2 weeks after ischemia these cells did not reach the size and value of the fractal dimension typical of the largest GFAP+ astrocytes. Moreover, the reaction of NGFR+ cells was significantly delayed in comparison with the total astrocyte population. The obtained results suggest that NGF receptors are expressed mainly by immature astrocytes and ischemia induces the maturation of these cells.

Contribution of Ca2+ calmodulin-dependent protein kinase II and mitogen-activated protein kinase kinase to neural activity-induced neurite outgrowth and survival of cerebellar granule cells

In this report we describe our studies on intracellular signals that mediate neurite outgrowth and long-term survival of cerebellar granule cells. The effect of voltage-gated calcium channel activation on neurite complexity was evaluated in cultured cerebellar granule cells grown for 48 h at low density; the parameter measured was the fractal dimension of the cell. We explored the contribution of two intracellular pathways, Ca2+ calmodulin-dependent protein kinase II and mitogen-activated protein kinase kinase (MEK1), to the effects of high [K+ ]e under serum-free conditions. We found that 25 mm KCl (25K) induced an increase in calcium influx through L subtype channels. In neurones grown for 24-48 h under low-density conditions, the activation of these channels induced neurite outgrowth through the activation of Ca2+ calmodulin-dependent protein kinase II. This also produced an increase in long-term neuronal survival with a partial contribution from the MEK1 pathway. We also found that the addition of 25K increased the levels of the phosphorylated forms of Ca2+ calmodulin-dependent protein kinase II and of the extracellular signal-regulated kinases 1 and 2. Neuronal survival under resting conditions is supported by the MEK1 pathway. We conclude that intracellular calcium oscillations can triggered different biological effects depending on the stage of maturation of the neuronal phenotype. Ca2+ calmodulin-dependent protein kinase II activation determines the growth of neurites and the development of neuronal complexity.

Identification of living oligodendrocyte developmental stages by fractal analysis of cell morphology

The Mandelbrot's fractal dimension (D), a measure of shape complexity, has been used to quantify the complex morphology of living cells. Previous studies on glial cells have shown that as cells increase in morphological complexity, their "D" value increases, suggesting that "D" could be used to estimate their stage of differentiation. In the present study the box-counting method was used to calculate the "D" values of rat cerebellar oligodendrocytes during their differentiation in primary culture. These values were correlated with the immunoreactivity of cells to antigenic markers commonly used for assessing their stages of differentiation: A2B5, O4 and anti-galactocerebroside (Gal-C). Our results show that changes of the fractal dimension during differentiation follow the well known pattern of markers expression by these cells. These results demonstrate that A2B5-, O4-, and Gal-C-expressing oligodendrocytes can be confidently estimated from their respective fractal dimension values. Based on this immunocytochemical calibration, the calculation of "D" allows an easy and fast determination of the developmental stage of living (unstained) oligodendrocytes before the study of their physiological characteristics. Using this method we precisely identified living oligodendrocyte progenitors and early pro-oligodendrocytes expressing voltage-activated sodium currents that is a common characteristic of these two immature developmental stages (Sontheimer et al. [1989b] Neuron 2:1135-1145).

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.

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.

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.

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.

Fractal methods and results in cellular morphology--dimensions, lacunarity and multifractals

This paper discusses the concepts of fractal geometry in a cellular biological context. It defines the concept of the fractal dimension. D, as a measure of complexity and illustrates the two different general ways of quantitatively measuring D by length-related and mass-related methods. Then, these several Ds are compared and contrasted. A goal of the paper is to find methods other than length-related measures that can distinguish between two objects that have the same D but are structurally different. The mass-related D is shown potentially to be such a measure. The concept of lacunarity, L, is defined and methods of measuring L are illustrated. L is also shown to be a potentially distinguishing measure. Finally, the notion of multifracticality is defined and illustrated to exist in certain individual nerve and glial cells.

Complexity analysis of oligodendroglial processes expressing myelin-associated glycoprotein

Oligodendroglia synthesize myelin in the mammalian central nervous system. Mature oligodendroglia have been identified in culture by two criteria; the expression of molecules characteristic of myelin, such as galactocerebroside (galC) and myelin-associated glycoprotein (MAG), and the elaboration of complex processes. Myelin gene expression can be documented by the binding of specific antibodies and antisera to the myelin-specific molecules; process complexity can be described by the fractal dimension, D. In this study, anti-MAG antisera was used to document MAG expression in the processes of oligodendroglia. Eighty percent of the galC+ oligodendroglia bound anti-MAG antiserum. With time in culture, MAG immunoreactivity seemed to extend from the cell soma into the oligodendroglial processes. To quantify this observation, fractal dimensions were calculated using either galC or MAG immunoreactivity to visualize oligodendroglial processes. A fractal dimension of 1.5 was calculated for O1+ processes by day 4 of culture; this value for D remained constant over the course of 1 month in culture. The fractal dimension calculated for MAG+ processes increased from 1.2 to 1.5 over the course of 28 days in culture. This change in fractal dimension confirms our visual impression that galC-containing processes acquire MAG slowly over the course of several weeks in culture.

Fractal analysis of spinal dorsal horn neuron discharges by means of sequential fractal dimension D

The present study describes a new method for converting a typical point process, such as a train of neuronal action potentials (spikes), into a planar curve which is then processed by means of a fast algorithm to calculate and display the fractal dimension D values of each of a sequence of blocks having an equal and preselectable number of interspike intervals, hence the term sequential fractal dimension D (SFD). This method is fast, does not require special computing facilities, and provides a continuous, high temporal resolution display of the neuronal discharge complexity along the course of spontaneous activity or event relating changes. The method affords insight into short duration changes in neuronal behaviour in a way independent of its discharge rate. SFD analysis of spike trains from spinal dorsal horn neurons suggests that the neuronal response to a given stimulus can be expressed as changes in the discharge pattern complexity, thus revealing a novel sensory coding strategy.

Determination of fractal dimension of physiologically characterized neurons in two and three dimensions

Although there is a growing interest in the application of fractal analysis in neurobiology, questions about the methodology have restricted its wider application. In this report we discuss some of the underlying principles for fractal analysis, we propose the cumulative-mass method as a standard method and we extend the applicability of fractal analysis to both 2 and 3 dimensions. We have examined the relationship between the method of log-log Sholl analysis and fractal analysis and have found that they correlate well. Measurements of physiologically characterized retinal ganglion cells indicate that different cell types can have significantly different fractal dimensions. Such differences may allow the correlation of the physiological type of a neuron with its morphological fractal dimension.

Quantification of tight junction complexity by means of fractal analysis

The concept of fractal geometry provides an elegant tool for the quantitative and objective structural description of various objects, the fractal analysis. Fractal analysis quantifies the structural complexity of objects by a characteristic singular value, the fractal dimension (FD). It can be estimated, e.g. by the box-counting method and provides a highly integrated measure in the range 1 < FD < 2 for curves extending within a plane. In this study, fractal analysis is used for the first time to evaluate the complexity of the tight junction network between adjoining cells. Bovine brain endothelial cells were cultured under various experimental conditions and the tight junctions were drawn to scale as visualized by the freeze fracture technique. These drawings were analyzed by fractal analysis, and by two other methods commonly used in this field, viz. the strand counting (SC) and complexity index (CI) methods. In contrast to the latter methods, the FD shows no directional preference and therefore no assumptions on the dynamic properties of the network's complexity are required. Thus, FD is demonstrated to provide the most sensitive, reliable and complete measure of tight junction complexity. In combination with SC and CI, additional information can be achieved concerning the directionality of the altered arrangement of tight junctional strands. Our analysis allows for the following conclusions. (1) Defined experimental influences can modify the complexity of tight junctions that are formed between endothelial cells in vitro, and (2) these structural modifications of the tight junctions are mainly due to an altered strand branching pattern.

Dynamic responses of presynaptic terminal membrane pools to electrical stimulation

The anatomical tenets of the quantal-vesicular hypothesis of neurotransmission are a 1:1 ratio between numbers of releasable quanta and vesicles, a reciprocal response between vesicle and terminal membrane pools and constancy of the total membrane pool. We have used electrical stimulation and morphometry to study these relationships in the cholinergic presynaptic terminals of Torpedo electric organ. Our results show that during neurotransmission changes in vesicle numbers do not correlate with quantal release, vesicle and terminal membranes do not change in reciprocal fashion and total nerve terminal membrane does not remain constant. We conclude that these vesicular tenets of quantal release are not verifiable at the Torpedo electric organ junction.

Complexity and scaling properties of amacrine, ganglion, horizontal, and bipolar cells in the turtle retina

In the present study we have evaluated the complexity and scaling properties of the morphology of retinal neurons using fractal dimension as a quantitative parameter. We examined a large number of cells from Pseudemys scripta and Mauremys caspica turtles that had been labeled using Golgi-impregnation techniques, intracellular injection of Lucifer Yellow followed by photooxidation, intracellular injection of rhodamine conjugated horseradish peroxidase, or intracellular injection of Lucifer Yellow or horseradish peroxidase alone. The fractal dimensions of two-dimensional projections of the cells were calculated using a box counting method. Discriminant analysis revealed fractal dimension to be a significant classification parameter among several other parameters typically used for placing turtle retinal neurons in different cell classes. The fractal dimension of amacrine cells was significantly correlated with dendritic field diameters, while the fractal dimensions of ganglion cells did not vary with dendritic field span. There were no significant differences between the same cell types in two different turtle species, or between the same types of neurons in the same species after labeling with different techniques. The application of fractal dimension, as a quantitative measure of complexity and scaling properties and as a classification criterion of neuronal types, appears to be useful and may have wide applicability to other parts of the central nervous system.

Comparative fractal analysis of cultured glia derived from optic nerve and brain demonstrate different rates of morphological differentiation

O-2A progenitor cells derived from neonatal rat cerebral hemispheres or optic nerves, were induced to differentiate in culture into either oligodendrocytes or type 2 astrocytes. The fractal dimensions, a measure of morphological complexity, of the differentiating glial cells were measured over time. Analysis of the changes in fractal dimension (D) with respect to time revealed specific rates of growth for each glial phenotype and a specific final D. The time course of these changes is well fit by a simple mathematical model. While brain-derived oligodendrocytes matured faster than the astrocytes, they ultimately attained comparable levels of complexity, with similar maximum fractal dimensions. Oligodendrocytes from nerve also matured faster than nerve derived astrocytes, in contrast, however, they attained a greater morphological complexity than nerve astrocytes. While the brain-derived oligodendrocytes showed a faster rate of maturation than their optic nerve counterparts, astrocytes from both regions had similar rates of morphological differentiation. Self-similarity, a defining property of fractal objects was investigated, by determining the fractal dimension of cells over a range of magnifications. The calculated fractal dimension remained constant over a 10-fold range in optical magnification, illustrating that cultured glial cells exhibit this important characteristic of fractal objects. In addition, we analyzed the branching patterns of glial processes by the Sholl method and found that the results were not as interpretable or meaningful as those of fractal analysis.

The application of fractal geometric analysis to microscopic images

Fractal geometry is a relatively new tool for the quantitative microscopist that is a more valid way of measuring dimensions of complex irregular objects than the integer-dimensional geometries (such as Euclidean geometry). This review discusses the theory of fractal geometry using the classic examples of the Von Koch curve, the Cantor set and the Sierpinski gasket. The problems of describing the dimensions of these objects are discussed and the concept of fractal dimensionality is introduced. Methods for measuring fractal dimensions are discussed, including their implementation on microcomputer-based image analysis systems . The advantages and problems of fractal geometric analysis are discussed and current applications in the field of microscopy are reviewed.

Quantitative phylogenetic constancy of cerebellar Purkinje cell morphological complexity

Golgi-stained material of cerebellar cortices from 17 species was examined by measuring the fractal dimensions of the borders of Purkinje cells, which is a quantitative, objective measure of morphological complexity. Nine species (from birds to man) were chosen for a comparison with ANOVA and no statistically significant differences were found in their fractal dimensions. In contrast, a wide range of differences was found in the membrane areas across species lines. The Sholl coefficient, a measure of branch formation and termination away from the soma, showed no consistent pattern for each cell. We interpret our results as indicating a constancy in morphological cellular complexity of Purkinje cells during late evolutionary time.

Distinct effects of bFGF and PDGF on oligodendrocyte progenitor cells

We have compared the effects of platelet-derived (PDGF) and basic fibroblast (bFGF) growth factors on the shape, migration, and differentiation of oligodendrocyte progenitor cells, the precursors of myelin-forming cells in the CNS. In the presence of bFGF, oligodendrocyte progenitors purified from rat neonatal brain cultures were stellate, non-motile, and had a morphological complexity of 1.26 +/- 0.03 as measured by fractal dimension (D). These cells expressed transcripts encoding the POU-homeodomain transcription factor Oct-6, but not myelin genes. Upon addition of PDGF, bFGF-treated cells became motile and twofold less complex in shape (D = 1.19 +/- 0.03). These changes occurred within 6 +/- 4 h and were dependent on de novo transcription and translation, but not DNA synthesis. Upon removal of PDGF the cells reverted to their stellate shape (D = 1.26). Removal of both bFGF and PDGF resulted in oligodendrocyte differentiation after 3 days, with a fourfold increase in complexity of shape (D = 1.55 +/- 0.08), loss of Oct-6 transcripts, and gain of myelin transcripts. Thus PDGF is both necessary and sufficient to induce a motile state in progenitor cells growing in the presence of bFGF. Together with our previous data (McKinnon et al.: Neuron 5:603, 1990), our results suggest that bFGF and PDGF may control distinct phases of proliferation and migration of oligodendrocyte progenitor cells in vivo.

Early dendrite development in spinal cord cell cultures: a quantitative study

Neurons in dissociated cell culture provide a favorable system for the quantitative analysis of structural changes and the examination of structure-function relationships during development. Fragment C of tetanus toxin was used to label neurons in murine spinal cord cell cultures and dendrite outgrowth was monitored by a number of measures. The dissociated neurons increased in morphologic complexity from approximate spheres to highly branched structures during the first week in culture. Much of the structural complexity of the dendrite arbor, as quantified by fractal dimension, was established within 48 hr after plating, i.e., prior to the development of interneuronal contacts. During the first few days in culture, dendrite branching complexity increased more rapidly than dendrite size, whereas after 4 days, fractal dimension remained relatively constant while dendrites continued to grow. Fractal analysis has provided data which suggest that the early development of dendrite branching complexity is determined intrinsically. Fractal dimension, as an effective index of morphologic complexity, should be a useful tool for the further study of extrinsic signals which might modify the generation or stabilization of dendrite form.

A comparative fractal analysis of various mammalian astroglial cell types

Camera-lucida drawings of Golgi-impregnated astroglial cells and their processes are described by the fractal dimension of their borders, which is an objective, quantitative measure of morphological complexity. Protoplasmic astrocytes from human neocortex have fractal dimensions (D) that are larger than those of fibrous astrocytes from the cat optic nerve. Marginal astrocytes from monkey cerebropontile angle have two kinds of processes: (1) short, thick processes with endfeet abutting the pial surface, with relatively high D's, and (2) very long, thin processes extending into the neuronal tissue, with very low D's. These data indicate that short astrocytic processes may have a complex surface (and have a high D), whereas long processes are rather smooth (and have a low D). A comparison between transmission electron microscopy morphometry and measures of D at the light microscopic level, performed on different parts of rabbit retinal Müller glial cells, suggests that D is strongly correlated to the surface-to-volume ratio which, in part, determines the length constant of a cable for core-conductance of currents. We provide data supporting the hypothesis that astroglial cell geometry is adjusted to allow for sufficient spatial buffering K+ currents, even through very long processes.

The origin and development of retinal astrocytes in the mouse

Astrocytes, a class of glia which appear in the mammalian retina late in development, have been postulated either to originate in situ from Müller cells or extra-retinally from the optic stalk epithelium, only subsequently invading the eye. The site of origin and the developmental characteristics of retinal astrocytes were examined in the mouse, a species not previously studied for this purpose. Sections of normal eyes and stalks at different ages were examined. Cells positive for glial fibrillary acidic protein (GFAP) were first observed at post-conceptional day 17 at the optic disc end of the stalk. From this site, the GFAP-positive cells migrated into and across the retina at a rate of approximately 290 microns per day, reaching its edge by post-conceptional day 28. While migrating across the retina, the astrocytes progressively increased in size and morphological complexity, observations confirmed by measurement of their fractal dimension. Over the same period, a wave of differentiation swept along the stalk in the cranial direction. Further evidence that retinal astrocytes are born outside the retina emerged when foetal hemiretinae with or without optic stalks were explanted to the chorioallantoic membrane of the chick. When examined one to twelve days later, no explant cultured without the optic stalk contained GFAP-positive astrocytes, while explants with the stalk left attached contained relatively normal numbers of astrocytes. We observed, using fluorescence confocal microscopy, that retinal astrocytes in the mouse as in the rat, associate predominantly with blood vessels, not axonal bundles. It was of interest to determine whether this class of glia is essential to the normal cytoarchitectural development of the neural retina. Morphological analysis of the explants revealed no observable differences in cytoarchitecture or in the timing of developmental events between retinae maturing with or without astrocytes. It was therefore concluded that astrocytes may not be essential to the normal structural development of the murine retina.

A fractal analysis of pyramidal neurons in mammalian motor cortex

Pyramidal neurons in the mammalian cerebral cortex can be described by a fractal dimension (Mandelbrot, 1982), which is an objective, quantitative measure of the complexity of their soma/dendritic borders. In the cat, the fractal dimensions of lamina V cells, which include pyramidal tract neurons (PTN), indicate that these cells are more complex than other pyramidal neurons (PN) in the same region of motor cortex. The lamina V cells of the cat are also more complex than those in motor cortex of the monkey. Moreover, lamina III neurons in the monkey are more complex than monkey lamina V neurons. The fractal dimension of the intracortical axon collateral arborizations of the same pyramidal neurons indicated, in all cases, that the branching of these terminals is less complex than the branching of the dendrites of the same cells. In line with the observation that the fractal dimensions of some homologous cellular populations are different in different species, it is suggested that the fractal dimension and the degree of morphological complexity may relate to the requirement for the number of separable functions to be accommodated within one neuron. For example, as the size of the cortex and the number of neurons in a region increase, the opportunity exists within a given cortical zone, for individual functions to be segregated and for functional specialization to be accommodated with less morphological complexity of the individual neurons performing each of these functions.

Comparative morphometry of Bergmann glial (Golgi epithelial) cells. A Golgi study

Bergmann glial (Golgi epithelial) cells were Golgi-impregnated in the cerebella of species with great differences in the thickness of the molecular layer, in small African native mouse, rat, rhesus monkey, and man. The thickness of the molecular layer determines the length of the radial Bergmann cell processes. Whereas the overall morphology of the cells was found to be strikingly similar in all species studied, there were great quantitative differences in length and diameter of the stem processes. Species with thick molecular layers (man, monkey) have thicker stem processes than species with short distances between Bergmann glial cell soma and pial surface (rat, mouse). This could mean that larger animals with longer gestation periods allow for prolonged growth of cell volumes. On the other hand, an increase in the diameter of long processes should reduce the cytoplasmic resistance against ionic currents; this would be important when Bergmann glial cells--like retinal Müller cells--would act as "cables" for spatial buffering of potassium ions released by electrically active neurons. By contrast, the fractal dimension--i.e., a quantitative measure of the complexity of the cell's border--of the cell processes was lower in species with long processes. In an age series of rat cells, the fractal dimension is shown to increase slightly up to a very old age.