On the Fractal Distribution of Brain Synapses

  • Richard Crandall
Conference paper
First Online:
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 50)

Abstract

Herein we present mathematical ideas for assessing the fractal character of distributions of brain synapses. Remarkably, laboratory data are now available in the form of actual three-dimensional coordinates for millions of mouse-brain synapses (courtesy of Smithlab at Stanford Medical School). We analyze synapse datasets in regard to statistical moments and fractal measures. It is found that moments do not behave as if the distributions are uniformly random, and this observation can be quantified. Accordingly, we also find that the measured fractal dimension of each of two synapse datasets is 2.8 ± 0.05. Moreover, we are able to detect actual neural layers by generating what we call probagrams, paramegrams, and fractagrams—these are surfaces one of whose support axes is the y-depth (into the brain sample). Even the measured fractal dimension is evidently neural-layer dependent.

Key words

Brain Fractals Neural science Synapses 

Mathematics Subject Classifications (2010)

Primary 28Axx 28A80 28A25 Secondary 65Cxx 65C20 
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Notes

Acknowledgments

The author is grateful to S. Arch of Reed College, as well as N. Weiler, S. Smith, and colleagues of Smithlab at Stanford Medical School, for their conceptual and algorithmic contributions to this project. T. Mehoke aided this research via statistical algorithms and preprocessing of synapse files. Mathematical colleague T. Wieting supported this research by being a selfless, invaluable resource for the more abstract fractal concepts. D. Bailey, J, Borwein, and M. Rose aided the author in regard to experimental mathematics on fractal sets. This author benefitted from productive discussions with the Advanced Computation Group at Apple, Inc.; in particular, D. Mitchell provided clutch statistical tools for various of the moment analyses herein.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Richard Crandall
    • 1
  1. 1.KelownaCanada

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