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.
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Notes
- 1.
From Smithlab, of Stanford Medical School [15].
- 2.
Indeed, one motivation for high-level brain science in neurobiology laboratories is the understanding of such conditions as Alzheimer’s syndrome. One should not rule out the possibility of “statistical” detection of some brain states and conditions—at least, that is our primary motive for bringing mathematics into play.
- 3.
A cuboid being a parallelepiped with all faces rectangular—essentially a “right parallelepiped.”
- 4.
The present author devised this method in 1997, in an attempt to create “1∕f” noise by digital means, which attempt begat the realization that fractal dimension could be measured with a Hilbert space-fill.
- 5.
As in “sonogram”—which these days can be a medical ultrasound image, but originally was a moving spectrum, like a fingerprint of sound that would fill an entire sheet of strip-chart.
- 6.
Mathematically, the available fractal dimensions for the generalized Cantor fractals are dense in said interval.
- 7.
Synapses live on dendrites, exterior to actual neurons.
- 8.
Of course, the situation is different if hole existence is connected with microscopic synapse distribution, e.g., if synapses were to concentrate near surfaces of large bodies.
- 9.
Again, we are not constructing here a neurophysiological model; rather, a phenomenological model whose statistical measures have qualitative commonality with the given synapse data.
- 10.
The heuristic form of dimension δ here may not be met if there are not enough total points. This is because the fractal-slope paradigm has low-resolution box counts that depend also on parameters N 0,r.
References
Adesnik, H., Scanziani, M.: Lateral competition for cortical space by layer-specific horizontal circuits. Nature 464, 1155–1160 (2010)
Bailey, D., Borwein, J., Crandall, R.: Box integrals. J. Comput. Appl. Math. 206, 196–208 (2007)
Bailey, D., Borwein, J., Crandall, R.: Advances in the theory of box integrals. Math. Comp. 79, 1839–1866 (2010)
Bailey, D., Borwein, J., Crandall, R., Rose, M.: Expectations on fractal sets. Appl. Math. Comput. (2013)
Borwein, J., Chan, O-Y, Crandall, R.: Higher-dimensional box integrals. Exp. Math. 19(3), 431–445 (2010)
Clarkson, K.J.: Nearest-neighbor searching and metric space dimensions. In: Shakhnarovich, G., et al. (eds.) Nearest-Neighbor Methods in Learning and Vision: Theory and Practice. MIT Press, Cambridge (2005)
Fischler, M.: Distribution of minimum distance among N random points in d dimensions. Technical Report FERMILAB-TM-2170 (2002)
Georgiev, S., et al.: EEG fractal dimension measurement before and after human auditory stimulation. Bioautomation 12, 70–81 (2009)
Grski, A., Skrzat, J.: Error estimation of the fractal dimension measurements of cranial sutures. J. Anat. 208(3), 353–359 (2006)
Lynnerup, N., Jacobsen, J.C.: Brief communication: age and fractal dimensions of human sagittal and coronal sutures. Am. J. Phys. Anthropol. 121(4), 332–6 (2003)
Micheva, K.D., Smith, S.J.: Array tomography: A new tool for imaging the molecular architecture and ultrastructure of neural circuits. Neuron 55, 25–36 (2007)
Micheva, K.D., Busse, B., Weiler, N.C., O’Rourke, N., Smith, S.J.: Single-synapse analysis of a diverse synapse population: proteomic imaging methods and markers. Neuron 68(4), 639–53 (2010)
Micheva, K.D., O’Rourke, N., Busse, B., Smith, S.J.: Array tomography: high-resolution three-dimensional immunofluorescence. In: Imaging: A Laboratory Manual, Ch. 45, pp. 697–719, 3rd edn. Cold Spring Harbor Press, New York (2010)
Philip, J.: The probability distribution of the distance between two random points in a box. TRITA MAT 07 MA 10 (2007)
Smithlab: Array tomography. http://smithlab.stanford.edu/Smithlab/Array_Tomography.html
Spruston, N.: Pyramidal neurons: dendritic structure and synaptic integration. Nat. Rev. Neurosci. 9, 206–221 (2008)
Teich, M., et al.: Fractal character of the neural spike train in the visual system of the cat. J. Opt. Soc. Am. A 14(3), 529–546 (1997)
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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Communicated By Heinz H. Bauschke.
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Crandall, R. (2013). On the Fractal Distribution of Brain Synapses. In: Bailey, D., et al. Computational and Analytical Mathematics. Springer Proceedings in Mathematics & Statistics, vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7621-4_14
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DOI: https://doi.org/10.1007/978-1-4614-7621-4_14
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