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The Fractal Geometry of the Brain: AnOverview

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The Fractal Geometry of the Brain

Part of the book series: Advances in Neurobiology ((NEUROBIOL,volume 36))

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Abstract

The first chapter of this book introduces some history, philosophy, and basic concepts of fractal geometry and discusses how the neurosciences can benefit from applying computational fractal-based analysis. Further, it compares fractal with Euclidean approaches to analyzing and quantifying the brain in its entire physiopathological spectrum and presents an overview of the first section of this book as well.

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References

  1. Losa GA. The fractal geometry of life. Riv Biol. 2009;102:29–59.

    PubMed  Google Scholar 

  2. Barnsley MF. Fractals everywhere. New York: Academic Press; 1988.

    Google Scholar 

  3. Mandelbrot BB. The fractal geometry of nature. New York: W.H. Freeman; 1982.

    Google Scholar 

  4. Mandelbrot BB. Is nature fractal? Science. 1998;279:783–4.

    Article  CAS  Google Scholar 

  5. Cross SS. Fractals in pathology. J Pathol. 1997;182:1–8.

    Article  CAS  PubMed  Google Scholar 

  6. Cross SS. The application of fractal geometric analysis to microscopic images. Micron. 1994;25:101–13.

    Article  CAS  PubMed  Google Scholar 

  7. Rigaut JP. An empirical formulation relating boundary lengths to resolution in specimens showing “non-ideally fractal” dimensions. J Microsc. 1984;133:41–54.

    Article  Google Scholar 

  8. Di Ieva A, Grizzi F, Jelinek H, Pellionisz AJ, Losa GA. Fractals in the neurosciences, part I: general principles and basic neurosciences. Neuroscientist. 2013;20:403–17.

    Article  PubMed  Google Scholar 

  9. Di Ieva A, Esteban FJ, Grizzi F, Klonowski W, Martin-Landrove M. Fractals in the neurosciences, part II: clinical applications and future perspectives. Neuroscientist. 2015;21:30–43.

    Article  PubMed  Google Scholar 

  10. Prigogine I. Les lois du chaos. Paris: Flammarion; 1997.

    Google Scholar 

  11. Mandelbrot BB. Les fractales, les monstres et la beauté. In: Balaubre G, editor. L’irruption des géométries fractales dans les sciences. Paris: Editions de l’Académie Européenne Interdisciplinaire des Sciences (AEIS); 2006. p. 317–41.

    Google Scholar 

  12. Di Ieva A. Memetics in neurosurgery and neuroscience. NeuroQuantology. 2008;6:182–93.

    Google Scholar 

  13. Mandelbrot B. How long is the coast of britain? Statistical self-similarity and fractional dimension. Science. 1967;156:636–8.

    Article  CAS  PubMed  Google Scholar 

  14. Normant F, Tricot C. Method for evaluating the fractal dimension of curves using convex hulls. Phys Rev A. 1991;43:6518–25.

    Article  CAS  PubMed  Google Scholar 

  15. Sernetz M, Gelleri B, Hofmann J. The organism as bioreactor. Interpretation of the reduction law of metabolism in terms of heterogeneous catalysis and fractal structure. J Theor Biol. 1985;117:209–30.

    Article  CAS  PubMed  Google Scholar 

  16. Losa GA, Nonnenmacher TF. Self-similarity and fractal irregularity in pathologic tissues. Mod Pathol. 1996;9:174–82.

    CAS  PubMed  Google Scholar 

  17. Weibel ER. Design of biological organisms and fractal geometry. In: Nonnenmacher TF, Losa GA, Weibel ER, editors. Fractals in biology and medicine, vol. I. Basel: Birkhäuser Press; 1994.

    Google Scholar 

  18. Baish JW, Jain RK. Fractals and cancer. Cancer Res. 2000;60:3683–8.

    CAS  PubMed  Google Scholar 

  19. Losa GA. Fractal morphometry of cell complexity. Riv Biol. 2002;95:239–58.

    PubMed  Google Scholar 

  20. Mandelbrot BB. Stochastic models for the earth’s relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands. Proc Natl Acad Sci U S A. 1975;72:3825–8.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  21. Abry P, Baraniuk R, Flandrin P, Riedi R, Veitch R, Veitch D. Multiscale nature of network traffic. IEEE Signal Proc Mag. 2002;3:28–46.

    Article  Google Scholar 

  22. Mandelbrot BB. Fractals and scaling in finance: discontinuity, concentration, risk. New York: Springer; 1997.

    Book  Google Scholar 

  23. Peng CK, Buldyrev SV, Goldberger AL, Havlin S, Mantegna RN, Simons M, et al. Statistical properties of DNA sequences. Phys A. 1995;221:180–92.

    Article  CAS  Google Scholar 

  24. Lieberman-Aiden E, van Berkum NL, Williams L, Imakaev M, Ragoczy T, Telling A, et al. Comprehensive mapping of long-range interactions reveals folding principles of the human genome. Science. 2009;326:289–93.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  25. Pellionisz AJ. The principle of recursive genome function. Cerebellum. 2008;7:348–59.

    Article  CAS  PubMed  Google Scholar 

  26. Strogatz SH. Exploring complex networks. Nature. 2001;410:268–76.

    Article  CAS  PubMed  Google Scholar 

  27. Brown JH, Gupta VK, Li BL, Milne BT, Restrepo C, West GB. The fractal nature of nature: power laws, ecological complexity and biodiversity. Philos Trans R Soc Lond Ser B Biol Sci. 2002;357:619–26.

    Article  Google Scholar 

  28. West GB, Brown JH, Enquist BJ. The fourth dimension of life: fractal geometry and allometric scaling of organisms. Science. 1999;284:1677–9.

    Article  CAS  PubMed  Google Scholar 

  29. Najafi E, Darooneh AH. The fractal patterns of words in a text: a method for automatic keyword extraction. PLoS One. 2015;10:e0130617.

    Article  PubMed  PubMed Central  Google Scholar 

  30. West BJ, Scafetta N. Nonlinear dynamical model of human gait. Phys Rev E Stat Nonlinear Soft Matter Phys. 2003;67:051917.

    Article  Google Scholar 

  31. Liu S, Meng T, Russo C, Di Ieva A, Berkovsky S, Peng L, Dou W, Qian L. Brain volumetric and fractal analysis of synthetic MRI: a comparative study with conventional 3D T1-weighted images. Eur J Radiol. 2021;141:109782. https://doi.org/10.1016/j.ejrad.2021.109782.

    Article  PubMed  Google Scholar 

  32. Newport RA, Russo C, Liu S, Suman AA, Di Ieva A. SoftMatch: comparing scanpaths using combinatorial spatio-temporal sequences with fractal curves. Sensors (Basel). 2022;22(19):7438. https://doi.org/10.3390/s22197438.

    Article  PubMed  Google Scholar 

  33. Andronache I, Marin M, Fischer R, Ahammer H, Radulovic M, Ciobotaru AM, Jelinek HF, Di Ieva A, Pintilii RD, Drăghici CC, Herman GV, Nicula AS, Simion AG, Loghin IV, Diaconu DC, Peptenatu D. Dynamics of forest fragmentation and connectivity using particle and fractal analysis. Sci Rep. 2019;9(1):12228. https://doi.org/10.1038/s41598-019-48277-z.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  34. Peptenatu D, Andronache I, Ahammer H, Radulovic M, Costanza JK, Jelinek HF, Di Ieva A, et al. A new fractal index to classify forest fragmentation and disorder. Landsc Ecol. 2023; https://doi.org/10.1007/s10980-023-01640-y.

  35. Robles KE, Roberts M, Viengkham C, Smith JH, Rowland C, Moslehi S, Stadlober S, Lesjak A, Lesjak M, Taylor RP, Spehar B, Sereno ME. Aesthetics and psychological effects of fractal based design. Front Psychol. 2021;12:699962. https://doi.org/10.3389/fpsyg.2021.699962.

    Article  PubMed  PubMed Central  Google Scholar 

  36. Goldberger AL. Fractals and the birth of gothic: reflections on the biologic basis of creativity. Mol Psychiatry. 1996;1:99–104.

    CAS  PubMed  Google Scholar 

  37. Grizzi F, Chiriva-Internati M. The complexity of anatomical systems. Theor Biol Med Model. 2005;2:26.

    Article  PubMed  PubMed Central  Google Scholar 

  38. Hermansen TD, Ventegodt S, Rald E, Clausen B, Nielsen ML, Merrick J. Human development I: twenty fundamental problems of biology, medicine, and neuro-psychology related to biological information. Sci World J. 2006;6:747–59.

    Article  Google Scholar 

  39. Ventegodt S, Hermansen TD, Nielsen ML, Clausen B, Merrick J. Human development II: we need an integrated theory for matter, life and consciousness to understand life and healing. Sci World J. 2006;6:760–6.

    Article  Google Scholar 

  40. Losa GA, Di Ieva A, Grizzi F, De Vico G. On the fractal nature of nervous cell system. Front Neuroanat. 2011;5:45. https://doi.org/10.3389/fnana.2011.00045.

    Article  PubMed  PubMed Central  Google Scholar 

  41. Hofman MA. The fractal geometry of convoluted brains. J Hirnforsch. 1991;32:103–11.

    CAS  PubMed  Google Scholar 

  42. Keipes M, Ries F, Dicato M. Of the british coastline and the interest of fractals in medicine. Biomed Pharmacother. 1993;47:409–15.

    Article  CAS  PubMed  Google Scholar 

  43. Peitgen HO, Richter PH. The beauty of fractals: images of complex dynamical systems. Berlin: Springer; 1986.

    Book  Google Scholar 

  44. Murray JD. Use and abuse of fractal theory in neuroscience. J Comp Neurol. 1995;361:369–71.

    Article  CAS  PubMed  Google Scholar 

  45. Gazit Y, Berk DA, Leunig M, Baxter LT, Jain RK. Scale-invariant behavior and vascular network formation in normal and tumor tissue. Phys Rev Lett. 1995;75:2428–31.

    Article  CAS  PubMed  Google Scholar 

  46. Tolle CR, McJunkin TR, Rohrbaugh DT, LaViolette RA. Lacunarity definition for ramified data sets based on optimal cover. Phys A. 2003;179:129–52.

    Google Scholar 

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Authors and Affiliations

  1. Computational NeuroSurgery (CNS) Lab & Macquarie Neurosurgery, Macquarie Medical School, Faculty of Medicine, Human and Health Sciences, Macquarie University, Sydney, NSW, Australia

    Antonio Di Ieva

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  1. Antonio Di Ieva
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Correspondence to Antonio Di Ieva .

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Editors and Affiliations

  1. Computational NeuroSurgery (CNS) Lab & Macquarie Neurosurgery Macquarie Medical School, Faculty of Medicine, Human and Health Sciences, Macquarie University, Sydney, NSW, Australia

    Antonio Di Ieva

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Di Ieva, A. (2024). The Fractal Geometry of the Brain: AnOverview. In: Di Ieva, A. (eds) The Fractal Geometry of the Brain. Advances in Neurobiology, vol 36. Springer, Cham. https://doi.org/10.1007/978-3-031-47606-8_1

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