Mathematical modeling and computational analysis of neuronal cell images: Application to dendritic arborization of Golgi-impregnated neurons in dorsal horns of the rat spinal cord

The Morphology of Brain Neurons: The Box-Counting Method in the Quantitative Analysis of 2D Images

This chapter begins by showing the difference between fractal geometry and fractal analysis. The text shows the difference between mathematical and natural fractals and how they are best defined by explaining the concept of fractal analysis. Furthermore, the text presents the most famous technique of fractal analysis: the box-counting method. Defining this method and showing the methodology that leads to the precise value of the fractal (i.e., the box) dimension is done by demonstrating the images of human dentate neurons. A more detailed explanation of the methodology was presented in the previous version of this chapter.This version promotes the notion of monofractal analysis and shows how three types of the same neuronal images can quantify four image properties. The results showed that monofractal parameters successfully quantified four image properties in three nuclei of the cerebellum. Finally, the author discusses the results of this chapter and previously published conclusions. The results show how the monofractal parameters discriminate images of neurons from the three nuclei of the human cerebrum. These outcomes are discussed along with the results of previous studies.

Cell image area as a tool for neuronal classification

The measurement of the area of a shapeless plane region is one of the basic problems in traditional calculus. In order to calculate the 'true' area of such a region, we have superimposed a net of identical squares on this region, counted the squares containing at least one point of the region, and calculated the sum of the areas of said squares. This sum represents an approximation of the region's area. By mathematical modelling and computational techniques we have investigated the law governing the decrease of these areas with the decrease of the length of the square's side. In theory, the prediction of the 'true' area could then be performed if the side of the net's squares tend to zero. Of course, the accuracy of the calculated area strongly depends on the computational potential and the statistical possibilities. Several morphometric parameters are currently in use for the quantitative analysis of the morphology of neuronal cell images. The cell image area has not yet been used and evaluated as a classification parameter - but it has the potential to be chosen over some other alternatives due to the high mathematical accuracy at which it is defined. By adopting mathematical modelling and computational techniques we show that this parameter can lead to successful distinction between 2 types of morphologically very similar cells (large boundary neuron and large asymmetrical neuron) in the dentate nucleus of the rhesus monkey (Macaca mulatta), while some other parameters failed to achieve positive results.

A confirmation of Rexed's laminar hypothesis using the Sholl linear method complemented by nonparametric statistics

Images of Golgi-impregnated neurons from laminae I to VI in the dorsal horn of the cat spinal cord were subjected to the linear Sholl analysis of concentric circles to support Rexed's hypothesis on the laminar organization of spinal gray matter in mammals. Since Rexed's determination of the laminae is based upon size, location, and grouping of cell bodies, neglecting one of the principal morphologic attributes of the neuron-the dendritic tree, the purpose of the present study was to evaluate Rexed's hypothesis testing the structure of dendritic arborization patterns of neurons. The differences in the complexity of dendritic trees between the groups of neurons from different laminae were evaluated by nonparametric statistics. Data obtained using Sholl's method is not always subjected to complete statistical analysis. The problem becomes particularly apparent in the quantitative examination of dendritic structures. Our aim was also to perform a careful analysis of our data for normality, in order to choose the appropriate statistical method for data processing. In the linear Sholl analysis, it is important to properly represent and interpret the frequency functions. The objective of this study was also to investigate the problems of determining the frequency functions, plotting the corresponding lines of regression, and measuring the degree of fluctuation of experimental data points around these lines. The main result of our testing is a confirmation of Rexed's laminar scheme: we have proved that there are 6 out of 10 possible pairs of samples where one member significantly differs from the other, i.e. one lamina is significantly distinguishable from the other.