On the fractal distribution of brain synapses

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 3-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.