How Mathematics Can Reveal Brain Structure and Diagnose Brain Health

Recent research indicates that a healthy mouse brain has its synapses (certain points of neural connection) distributed according to a dimension that is not three, but rather more like 2.8. A random cloud of points inside a skull would measure out at dimension three (that is, the random points are uniformly filling a 3-dimensional volume), but in the actual mouse-brain data analyzed, one finds that because of delicate structure, the dimension is consistently less than three.

In fact, this "2.8" fractal dimension oscillates meaningfully as one goes deeper down into the brain tissue.  In this way, detailed fractal measurements reveal a dimension that fluctuates around 2.8 according to what are called "neural layers."  Both mouse brains and human brains exhibit neural layers.


array tomography brain
Actual array-tomography data from mouse brain.  The synapses analyzed in the
new research paper are red dots. The "gumstick" at upper left represents
the full brain section that was assessed; the small rectangle in the gumstick is the
section currently being blown up and visualized.
Image Credit: Morgan Grace, PSI Press.

So now there is a way to detect the healthy  occurrences---and even the positions of---neural layers via these fractal techniques.

The actual brain data, on which fractal analysis was performed, was from the so-called "whisker sector" of mouse brain.  This is the brain region where sensations via the mouse's whiskers are processed.  (Mice do not see well; indeed they use whiskers to navigate, such as in dark spaces…thus the whisker sector is highly developed.)

"It is remarkable, that 3-dimensional brain data can actually be measured in the laboratory.  When colleagues at the Stanford Medical School were able to send my Oregon lab the precise locations of more than 1 million brain synapses, I knew that something wonderful just had to come of it." says the author, Prof. Richard Crandall, Center for Advanced Computation, Reed College.

fractal analysis of brain
Crandall's mathematical techniques allow measurement of the fractal dimension of a point-cloud.  If the right-hand cloud here is a set of brain synapses from the laboratory, the so-called "space-filling curve" at the left is made to wind through the cloud.  Then the distribution of intercepts on the left-hand space-filling curve can be used to get a mathematical value for fractal dimension. Image Credit: Morgan Grace, PSI Press.

The brain data was obtained via "array tomography," whereby ultrathin slices of brain matter are scanned, with structural bodies given pixel locations on each layer, all of this eventually assembled into a 3-dimensional cloud.  For neurobiologists, the next step is to understand structure.  For a researcher like Crandall---who is not a biologist by any means---the next steps became mathematical.  A subfield of pure mathematics called "box integration"---which field has blossomed over the last decade---turned out to be an ideal setting for this brain analysis.

In the words of Stephen J Smith, PhD, Professor of Molecular and Cellular Physiology, Stanford University School of Medicine: "We [at Stanford Medical School] are very excited that Crandall has come up with this incisive new metric of the brain's synaptic architecture.  In combination with array tomography, it is likely to be of value to both basic and applied neuroscience."

So, what good are these studies, that portray brain matter as having fractal structure, with dimension less than three?  The findings are likely to be important in diagnostics.  According to Smith, "This mathematical approach is especially relevant to research on mental and neurological disorders, including Alzheimer's and other neurodegenerative diseases."

Science news reference: 

Richard Crandall. On the fractal distribution of brain synapses. Computational and Analytical Mathematics, Proceedings, in Springer Series in Mathematics and Statistics, 2012. Free article link

Additional references: Smithlab brain video on YouTube

Full HD quality quicktime movie on the Smithlab website

Science news source: 

PSI Press