# paper a day : power laws for white/gray matter ratios

I swear I’ll post more often starting soon — I just need to get back into the swing of things. In the meantime, here’s a short but fun paper.

A universal scaling law between gray matter and white matter of cerebral cortex
K. Zhang and T. Sejnowski
PNAS v.97 no. 10 (May 9, 2000)

This paper looks at the brain structure of mammals, and in particular the volumes of gray matter (cell bodies, dendrites, local connections) and white mattern (longer-range inter-area fibers). A plot of white matter vs. gray matter volumes showing different mammals, from a pygmy shrew to an elephant, show a really close linear fit on a log-log scale, with the best line having a slope of log(W)/log(G) = 1.23. This paper suggests that the exponent can be explained mathematically using two axioms. The first is that a piece of cortical area sends and receives ths same cross-sectional area of long-range fibers. The second more important axiom is that the geometry of the cortex is designed to minimize the average length of the long-distance fibers.

By using these heuristics, they argue that an exponent of 4/3 is “optimal” with respect to the second criterion. The difference of 0.10 can be explained by the fact that cortical thickness increases with the size of the animal, so they regressed cortical thickness vs. log(G) to get a thickness scaling of 0.10. It’s a pretty cute analysis, I thought, although it can’t really claim that minimum wiring is a principle in the brain so much as the way brains are is consistent with minimal wiring. Of course, I don’t even know how you would go about trying to prove the former statement — maybe this is why I feel more at home in mathematical engineering than I do in science…