Wavelet packet fractal analysis of neuronal morphology

An image analysis method called two-dimensional wavelet packet analysis (2D WPA) is introduced to quantify branching complexity of neurons. Both binary silhouettes and contour profiles of neurons were analyzed to determine accuracy and precision of the fractal dimension in cell classification tasks. Two-dimensional WPA plotted the slope of decay for a sorted list of discrete wavelet packet coefficients belonging to the adapted wavelet best basis to obtain the fractal dimension for test images and binary representations of neurons. Two-dimensional WPA was compared with box counting and mass-radius algorithms. The results for 2D WPA showed that it could differentiate between neural branching complexity in cells of different type in agreement with accepted methods. The importance of the 2D WPA method is that it performs multiresolution decomposition in the horizontal, vertical, and diagonal orientations.     

Use of the fractal dimension for the analysis of electroencephalographic time series

 Electroencephalogram (EEG) traces corresponding to different physiopathological conditions can be characterized by their fractal dimension, which is a measure of the signal complexity. Generally this dimension is evaluated in the phase space by means of the attractor dimension or other correlated parameters. Nevertheless, to obtain reliable values, long duration intervals are needed and consequently only long-term events can be analysed; also much calculation time is required. To analyse events of brief duration in real-time mode and to apply the results obtained directly in the time domain, thus providing an easier interpretation of fractal dimension behaviour, in this work we optimize and propose a new method for evaluating the fractal dimension. Moreover, we study the robustness of this evaluation in the presence of white or line noises and compare the results with those obtained with conventional spectral methods. The non-linear analysis carried out allows us to investigate relevant EEG events shorter than those detectable by means of other linear and non-linear techniques, thus achieving a better temporal resolution. An interesting link between the spectral distribution and the fractal dimension value is also pointed out. Received: 21 November 1996 / Accepted in revised form: 1 July 1997     

Scale dependency and fractal dimension of rarity

The rarity of species in a country is usually determined by counting the number of grid cells occupied by those species on a geographical observation grid. In this paper, we present a measure of rarity that is less sensitive to the shape and size of a country. We demonstrate that the distribution of species on a national grid is not monofractal. Consequently, rarity figures cannot be scaled down to a finer grid merely using scale-area plots. We propose a downscaling method that takes into account the non-monofractal distribution of species. Rarity figures have often been published on a scale comprising a limited number of rarity classes. This article finally provides an insight into the degree of accuracy of such classes.     

Local fractal dimension based ECG arrhythmia classification

We propose a local fractal dimension based nearest neighbor classifier for ECG based classification of arrhythmia. Local fractal dimension (LFD) at each sample point of the ECG waveform is taken as the feature. A nearest neighbor algorithm in the feature space is used to find the class of the test ECG beat. The nearest neighbor is found based on the RR-interval-information-biased Euclidean distance, proposed in the current work. Based on the two algorithms used for estimating the LFD, two classification algorithms are validated in the current work, viz. variance based fractal dimension estimation based nearest neighbor classifier and power spectral density based fractal dimension estimation based nearest neighbor classifier. Their performances are evaluated based on various figures of merit. MIT-BIH (Massachusetts Institute of Technology - Boston’s Beth Israel Hospital) Arrhythmia dataset has been used to validate the algorithms. Along with showing good performance against all the figures of merit, the proposed algorithms also proved to be patient independent in the sense that the performance is good even when the test ECG signal is from a patient whose ECG is not present in the training ECG dataset.     

Fractal dimension analysis and mathematical morphology of structural changes in actin filaments imaged by electron microscopy

In this work, we examined structural changes of actin filaments interacting with myosin visualized by quick freeze deep-etch replica electron microscopy (EM) by using a new method of image processing/analysis based on mathematical morphology.In order to quantify the degree of structural changes, two characteristic patterns were extracted from the EM images. One is the winding pattern of the filament shape (WP) reflecting flexibility of the filament, and the other is the surface pattern of the filament (SP) reflecting intra-molecular domain-mobility of actin monomers constituting the filament. EM images were processed by morphological filtering followed by box-counting to calculate the fractal dimensions for WP (D_WP) and SP (D_SP). The result indicates that D_WP was larger than D_SP irrespective of the state of the filament (myosin-free or bound) and that both parameters for myosin-bound filaments were significantly larger than those for myosin-free filaments. Overall, this work provides the first quantitative insight into how conformational disorder of actin monomers is correlated with the myosin-induced increase in flexibility of actin filaments along their length as suggested by earlier studies with different techniques. Our method is yet to be improved in details, but promising as a powerful tool for studying the structural change of protein molecules and their assemblies, which can potentially be applied to a wide range of biological and biomedical images.     

Genetic analysis of melanophore development in zebrafish embryos

Vertebrate pigment cells are derived from neural crest, a tissue that also forms most of the peripheral nervous system and a variety of ectomesenchymal cell types. Formation of pigment cells from multipotential neural crest cells involves a number of common developmental processes. Pigment cells must be specified; their migration, proliferation, and survival must be controlled and they must differentiate to the final pigment cell type. We previously reported a large set of embryonic mutations that affect pigment cell development from neural crest (R. N. Kelsh et al., 1996, Development 123, 369-389). Based on distinctions in pigment cell appearance between mutants, we proposed hypotheses as to the process of pigment cell development affected by each mutation. Here we describe the cloning and expression of an early zebrafish melanoblast marker, dopachrome tautomerase. We used this marker to test predictions about melanoblast number and pattern in mutant embryos, including embryos homozygous for mutations in the colourless, sparse, touchdown, sunbleached, punkt, blurred, fade out, weiss, sandy, and albino genes. We showed that in homozygous mutants for all loci except colourless and sparse, melanoblast number and pattern are normal. colourless mutants have a pronounced decrease in melanoblast cell number from the earliest stages and also show poor melanoblast differentiation and migration. Although sparse mutants show normal numbers of melanoblasts initially, their number is reduced later. Furthermore, their distribution indicates a defect in melanoblast dispersal. These observations permit us to refine our model of the genetic control of melanophore development in zebrafish embryos.     

Characterization of pellet morphology during submerged growth of Streptomyces tendae by image analysis

Many filamentous bacteria and fungi tend to form pellets, or mixtures of dispersed mycelium and pellets in liquid fermentation broths. In some cases, a specific kind of morphology is required for optimum product yield. When quantitative analysis and characterization of the pellet morphology are needed, an image processing system can be used. It allows a fast and reproducible analysis of the frequency distribution of pellet size, mean pellet size, contents of pellets, or their shape. The use of such a system allows for an on-line analysis. For a demonstration of the method, results of two fermentations of Streptomyces tendae are shown.     

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.     

The arterial pattern and fractal dimension of the dog kidney.

A method has been developed by which it is possible to measure the fractal dimension of the arterial tree of the kidney. The objective of this work is to determine a method which permits us to discriminate between the architectures of specific organs by reference to a unique number, namely the fractal dimension of the arterial tree of that organ. This method opens the possibility of a new taxonomy for normal organs and for the pathological injiries related to the vascular morphology of those organs. The method that we have devised uses as its input the volume which is taken up by the arterial tree of the kidney. In order to calculate this volume we first obtained a plastic cast (the arteries were filled with Araldite CY233 plastic resin after which the organic tissues were corroded); thereafter we constructed a theoretical arterial tree having the same volume as the renal one. From this simplified tree, we were able to calculate its fractal dimension. The complete process of constructing the theoretical arterial tree and the subsequent calculation of its fractal dimension was carried out automatically by way of a computer programme to which we have given the name fractal program.     

Linear correlation between fractal dimension of EEG signal and handgrip force

Fractal dimension (FD) has been proved useful in quantifying the complexity of dynamical signals in biology and medicine. In this study, we measured FDs of human electroencephalographic (EEG) signals at different levels of handgrip forces. EEG signals were recorded from five major motor-related cortical areas in eight normal healthy subjects. FDs were calculated using three different methods. The three physiological periods of handgrip (command preparation, movement and holding periods) were analyzed and compared. The results showed that FDs of the EEG signals during the movement and holding periods increased linearly with handgrip force, whereas FD during the preparation period had no correlation with force. The results also demonstrated that one method (Katz’s) gave greater changes in FD, and thus, had more power in capturing the dynamic changes in the signal. The linear increase of FD, together with results from other EEG and neuroimaging studies, suggest that under normal conditions the brain recruits motor neurons at a linear progress when increasing the force.     

Fungal fractal morphology of pellet formation in Aspergillus niger

Mycelial fractal values were compared to the conventional fungal morphological parameters: average total mycelial length, average number of tips and average growth unit. The fractal values were between 1.47 to 1.3 for the various submerged culture conditions of Aspergillus niger. The average pellet diameter was 1.4 mm at the fractal value of 1.47. The mycelia with fractal values close to 1 were less branched and slim.     

A fractal model for the characterization of mycelial morphology

A new technique based on a fractal model has been developed for the quantification of the macroscopic morophology of mycelia. The morphological structuring is treated as a fractal object, and the fractal dimension, determined by an ultrasonic scattering procedure developed for the purpose, serves as a quantitative morphological index. Experimental observations reported earlier and simulations of mycelial growth, carried out using a probabilistic-geometric growth model developed for the purpose, both validate the applicability of the fractal model. In experiments with three different species, the fractal dimensions of pelletous structures were found to be in the range 1.45-2.0 and those of filamentous structures were in the range 1.9-2.7, with values around 2.0 representing mixed morphologies. Fractal dimensions calculated from simulated mycelia are in rough agreement with these ranges. The fractal dimension is also found to be relatively insensitive to the biomass concentration, as seen by dilution of the original broths. The relation between morphology and filtration properties of the broths has also been studied. The fractal dimension shows a strong correlation with the index of cake compressibility and with the Kozeny constant, two filtration parameters that are known to be morphology dependent. This technique could thus be used to develop correlations between the morphology, represented by the fractal dimension, and important morphology-dependent process variables. (c) 1993 John Wiley & Sons, Inc.     

On the relationship between fractal dimension and encounters in three-dimensional trajectories

The encounter of individuals-prey, predators and mates-living in the surrounding environment is a fundamental process in the life of an organism. Along with the sensory abilities, this process will be regulated by the movement rules adopted by the individual. In this work we discuss the encounter-enhancement effect due to different natatorial modes by calculating the number of encounters realised by differently convoluted trajectories in two homogeneous distributions of particles. Using numerically generated trajectories representative of specific swimming behaviour, we demonstrate that high values of three-dimensional fractal dimension D(3D)(>1.9) are beneficial only at high concentration, whereas at low concentration less tortuous tracks (D(3D) approximately 1.5) are almost equally efficient. In the light of our results it is possible to better understand the behavioural adaptations evolved by individuals to thrive in their environment.     

Protein surface representation and analysis by dimension reduction

Background

Protein structures are better conserved than protein sequences, and consequently more functional information is available in structures than in sequences. However, proteins generally interact with other proteins and molecules via their surface regions and a backbone-only analysis of protein structures may miss many of the functional and evolutionary features. Surface information can help better elucidate proteins' functions and their interactions with other proteins. Computational analysis and comparison of protein surfaces is an important challenge to overcome to enable efficient and accurate functional characterization of proteins.

Methods

In this study we present a new method for representation and comparison of protein surface features. Our method is based on mapping the 3-D protein surfaces onto 2-D maps using various dimension reduction methods. We have proposed area and neighbor based metrics in order to evaluate the accuracy of this surface representation. In order to capture functionally relevant information, we encode geometric and biochemical features of the protein, such as hydrophobicity, electrostatic potential, and curvature, into separate color channels in the 2-D map. The resulting images can then be compared using efficient 2-D image registration methods to identify surface regions and features shared by proteins.

Results

We demonstrate the utility of our method and characterize its performance using both synthetic and real data. Among the dimension reduction methods investigated, SNE, LandmarkIsomap, Isomap, and Sammon's mapping provide the best performance in preserving the area and neighborhood properties of the original 3-D surface. The enriched 2-D representation is shown to be useful in characterizing the functional site of chymotrypsin and able to detect structural similarities in heat shock proteins. A texture mapping using the 2-D representation is also proposed as an interesting application to structure visualization.

     

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.     

Burrow fractal dimension and foraging success in subterranean rodents: a simulation

For animals that forage underground, the success with whichfood items are located may be closely related to burrow architecture.Fractal dimension, which describes how a burrow explores thesurrounding area in a way that is independent of burrow length,is an obvious choice for a single metric describing burrow shape.Although it is often assumed that burrows of high fractal dimensionwill be associated with greater foraging success, this has notpreviously been demonstrated. In this study, we use computersimulations to study the success with which burrows of differentfractal dimensions locate randomly distributed food items. Inaddition, we examine the effect of different patterns of fooddistribution (in particular the patchiness with which food itemsare distributed) and consider how using different criteria forlocating food items affects the relationship between fractaldimension and foraging success. We conclude that, under a widerange of plausible assumptions about the ways in which subterraneanrodents forage, burrows of high fractal dimension are more successfulat locating food items than burrows of lower fractal dimension.