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. 2000 May 9;97(10):5621-6.
doi: 10.1073/pnas.090504197.

A universal scaling law between gray matter and white matter of cerebral cortex

K Zhang  1 , T J Sejnowski
Affiliations

A universal scaling law between gray matter and white matter of cerebral cortex

K Zhang et al. Proc Natl Acad Sci U S A. .

Abstract

Neocortex, a new and rapidly evolving brain structure in mammals, has a similar layered architecture in species over a wide range of brain sizes. Larger brains require longer fibers to communicate between distant cortical areas; the volume of the white matter that contains long axons increases disproportionally faster than the volume of the gray matter that contains cell bodies, dendrites, and axons for local information processing, according to a power law. The theoretical analysis presented here shows how this remarkable anatomical regularity might arise naturally as a consequence of the local uniformity of the cortex and the requirement for compact arrangement of long axonal fibers. The predicted power law with an exponent of 4/3 minus a small correction for the thickness of the cortex accurately accounts for empirical data spanning several orders of magnitude in brain sizes for various mammalian species, including human and nonhuman primates.

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Figures

Figure 1
Figure 1
Schematic section diagrams showing that the cerebral cortex of a larger brain tends to have disproportionally more long-distance connection fibers or white matter (dark regions) than the gray matter (folded outer surface). Adapted from a drawing by E. de Vries in ref. , reoriented and rescaled approximately by using the cat and puma brains at the University of Wisconsin website www.neurophys.wisc.edu/brain.
Figure 2
Figure 2
Cortical white and gray matter volumes of various species (n = 59) are related by a power law that spans five to six orders of magnitude. Most data points are based on measurement of a single adult animal. The line is the least squares fit, with a slope around 1.23 ± 0.01 (mean ± SD). The average and median deviations of the white matter volumes from the regression line are, respectively, 18% and 13% on a linear scale. Sources of data: If the same species appeared in more than one source below, the one mentioned earlier was used. All 38 species in table 2 in ref. were taken, including 23 primates, 2 tree shrews, and 13 insectivores. Another 11 species were taken from table 2 in ref. , including 3 primates, 2 carnivores, 4 ungulates, and 2 rodents. Five additional species came from table 1 in ref. , including 1 elephant and 4 cetaceans. The data point for the mouse (G = 112 mm3 and W = 13 mm3) was based on ref. , and that for the rat (G = 425 mm3 and W = 59 mm3) was measured from the serial sections in a stereotaxic atlas (42). The estimates for the fisherman bat (Noctilio leporinus, G = 329 mm3 and W = 43 mm3) and the flying fox (Pteropus lylei, G = 2,083 mm3 and W = 341 mm3) were based on refs. and , with the ratios of white and gray matters estimated roughly from the section photographs in the papers. The sea lion data (Zalophus californianus, G = 113,200 mm3 and W = 56,100 mm3) were measured from the serial sections at the website given in the legend to Fig. 1, with shrinkage correction.
Figure 3
Figure 3
Testing the cubic postulate between the gray matter volume and the average length of fibers in white matter, which was inferred from Eq. 13 assuming p = 0.08. The theoretical slope of 3 (dashed line) is reasonably close to the slope of 3.3 estimated by least squares as shown, or the slope of 3.2 estimated by robust estimation with absolute deviations (not shown). Data were taken from table 1 in ref. , where cortical surface area and thickness are related rigorously by Eq. 1.
Figure 4
Figure 4
The coefficient of variation (standard deviation/mean) of the dimensionless number c(δ) plotted as a function of the test exponent δ in Eq. 14. The theoretical optimal value 4/3, as indicated by the dashed line, is close to the empirical minimum (•) reached at δ = 1.36 (±0.06, SD by bootstrap). Data were from table 1 in ref. (27 species).
Figure 5
Figure 5
The same data as in Fig. 1 plotted against the total volume of gray matter and white matter, showing the approximate nature of power laws for additive components. Of the three power laws shown here and in Fig. 1, at most one can be assumed to be rigorously true in theory without logical inconsistency.
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