Fractals in the Neurosciences, Part I: General Principles and Basic Neurosciences

Abstract

The natural complexity of the brain, its hierarchical structure, and the sophisticated topological architecture of the neurons organized in micronetworks and macronetworks are all factors contributing to the limits of the application of Euclidean geometry and linear dynamics to the neurosciences. The introduction of fractal geometry for the quantitative analysis and description of the geometric complexity of natural systems has been a major paradigm shift in the last decades. Nowadays, modern neurosciences admit the prevalence of fractal properties such as self-similarity in the brain at various levels of observation, from the microscale to the macroscale, in molecular, anatomic, functional, and pathological perspectives. Fractal geometry is a mathematical model that offers a universal language for the quantitative description of neurons and glial cells as well as the brain as a whole, with its complex three-dimensional structure, in all its physiopathological spectrums. For a holistic view of fractal geometry of the brain, we review here the basic concepts of fractal analysis and its main applications to the basic neurosciences.

Quantification of thioacetamide-induced liver necrosis using fractal analysis

“…Through the use of fractal dimension, it is possible to quantify the irregularity and complexity of structures in biological systems (10). It is widely used in many different areas of biomedical sciences, such as neurosciences (11)(12)(13), tumor and liver pathology (14-16) and many other areas where the use of image analysis is necessary (17,18).…”

The living realm depicted by the fractal geometry

“…cancer, is probabilistically correlated with prognosis, up to cure" [47]. The brain is now accepted as one of nature's complete networks [48], while the hierarchical organization of the brain, seen at multiple scales from genes to molecular micronetworks and macronetworks organized in building neurons, has a fractal structure as well [49] with various modules that are interconnected in small-world topology [50]. The theoretical significance is that the fractality found in DNA and organisms, for a long time "apparently unrelated," was put into a "cause and effect" relationship by the principle of recursive genome function [47].…”

Fractal and gray level co-occurrence matrix texture analysis of senescent and non-senescent deciduous teeth stem cells: A pilot study

“…The same procedure was used to calculate D bin , but with an addition of the Fil command after the binarization process, which in this case, represented the last step in the process of image transformation ( Figures 2D and 2H) The fractal analysis was done using the box-counting method (pathway: ImageJ>Analyze>Tools>Fractal Box Count). By default, the box sizes for counting the fractal dimension using the ImageJ software are 2, 3, 4, 6,8,12,16,32,64. In this paper, the box sizes were 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, obtained as an increasing geometric progression 2 n where n = 0, 1, 2…10, as also used in our previous papers [9,19].…”

“…It is widely used in several different areas of biological and medical sciences, such as neuroscience [8][9][10][11], histological diagnosis and quantification of neoplasms [12][13][14][15], blood vessels changes [16,17], assessment of cell aging [18] and many others. New methods and modifications of fractal analysis are also being discovered [19], thus allowing this mathematical method to be applied in different areas of natural sciences.…”