Fractal Analysis: Pitfalls and Revelations in Neuroscience

Fractal analysis has become a popular method in all branches of scientific investigation including ecology, physics and medicine. The method is often used to determine effects such as impact of cattle grazing, the distribution of stars within a galaxy or whether tissue is pathological. However several aspects of fractal analysis are not often considered when interpreting results communicated in the literature. These include the concept that no presentation of any pattern on a computer, even for an ideal fractal, is truly fractal. Pre-processing that is also required, such as scanning of images and resizing play a role in the variation of the final fractal dimension. In addition D is also a function of the fractal analysis method used and how the final fractal dimension is determined. To obtain a better overview of the effects of the steps involved in fractal analysis and the utility of this method, this chapter describes, using biological material from neuroscience, a non fractal based method, Sholl analysis and continues by discussing various processing options and the results obtained using fractal analysis. The effect of different fractal analysis methods, different computer applications of the same method, scale and resolution as well as regression analysis, which is for most methods the final step in determining D are discussed. This provides a platform for a better understanding of fractal analysis in research fields other than physics and mathematics and a more meaningful interpretation of results.

[1]  S. Cross,et al.  The Applications of Fractal Geometry in Pathology , 1995 .

[2]  Sholl Da.  Dendritic organization in the neurons of the visual and motor cortices of the cat. , 1953 .

[3]  Herbert F Jelinek,et al.  Neurons and fractals: how reliable and useful are calculations of fractal dimensions? , 1998, Journal of Neuroscience Methods.

[4]  D. V. van Essen,et al.  The pattern of interhemispheric connections and its relationship to extrastriate visual areas in the macaque monkey , 1982, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[5]  H. E. Stanley,et al.  Determination of fractal dimension of physiologically characterized neurons in two and three dimensions , 1995, Journal of Neuroscience Methods.

[6]  W. B. Marks,et al.  A fractal analysis of cell images , 1989, Journal of Neuroscience Methods.

[7]  G. Elston,et al.  The occipitoparietal pathway of the macaque monkey: comparison of pyramidal cell morphology in layer III of functionally related cortical visual areas. , 1997, Cerebral cortex.

[8]  E Louis,et al.  Are neurons multifractals? , 1999, Journal of Neuroscience Methods.

[9]  Herbert F. Jelinek,et al.  Is there meaning in fractal analyses , 1998 .

[10]  G. Elston.  Cortex, cognition and the cell: new insights into the pyramidal neuron and prefrontal function. , 2003, Cerebral cortex.

[11]  G. Elston,et al.  Dendritic branching patterns of pyramidal cells in the visual cortex of the new world marmoset monkey, with comparative notes on the old world macaque monkey , 2001 .

[12]  Joseph J. Capowski,et al.  Computer Techniques in Neuroanatomy , 1989, Springer US.

[13]  H F Jelinek,et al.  Use of fractal theory in neuroscience: methods, advantages, and potential problems. , 2001, Methods.

[14]  R. Porter,et al.  A fractal analysis of pyramidal neurons in mammalian motor cortex , 1991, Neuroscience Letters.

[15]  M. J. Friedlander,et al.  Morphogenesis and territorial coverage by isolated mammalian retinal ganglion cells , 1991, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[16]  H. Uylings,et al.  Topological Analysis of Individual Neurons , 1989 .

[17]  H. Kolb,et al.  Complexity and scaling properties of amacrine, ganglion, horizontal, and bipolar cells in the turtle retina , 1994, The Journal of comparative neurology.

[18]  Herbert F. Jelinek,et al.  Segmentation of retinal fundus vasculature in nonmydriatic camera images using wavelets: Advanced Segmentation Techniques , 2003 .

[19]  P Sterling,et al.  Retinal neurons and vessels are not fractal but space‐filling , 1995, The Journal of comparative neurology.

[20]  M. Khokha,et al.  Fractal Geometry in Biological Systems: An Analytical Approach , 1996 .

[21]  E. Weibel,et al.  Fractals in Biology and Medicine , 1994 .

[22]  Jaap van Pelt,et al.  A shape analysis framework for neuromorphometry , 2002, Network.

[23]  G. Elston,et al.  PYRAMIDAL NEURONES IN MACAQUE VISUAL CORTEX: INTERAREAL PHENOTYPIC VARIATION OF DENDRITIC BRANCHING PATTERNS , 2001 .

[24]  A. Reichenbach,et al.  Comparative morphometry of Bergmann glial (Golgi epithelial) cells , 1991, Anatomy and Embryology.

[25]  Benoit B. Mandelbrot,et al.  A Fractal’s Lacunarity, and how it can be Tuned and Measured , 1994 .