Mammalian Brain Surface Scales One-to-One to Body Surface
Mark D. Reid
(Copyright May 02, 2002 by Mark D. Reid)
Department of Philosophy, University of Illinois Urbana-Champaign, 810 South Wright Street, Champaign Illinois USA
A deduction is produced from three allometric scaling laws, showing cortical surface area scales one-to-one with body surface area as a general but, ex hypothesi, highly consistent principle across Mammalia. Cortical surface area (CSA) scales to brain volume (BrainV) with a measured exponent of 8/9. BrainV scales to body volume (BodyV) with a measured exponent of 3/4. Body surface (BodyS) scales to BodyV at 2/3. A simple deduction:
(1) CSA/BrainV by BrainV/BodyV, (or 8/9 * 3/4) yields: CSA/BodyV = 2/3
(2) BodyS/BodyV inversion yields: BodyV/BodyS = 3/2
(3) CSA/BodyV by BodyV/BodyS (or 2/3 * 3/2) yields: CSA/BodyS = 1
The first says cortical surface area scales to body volume with exponent 2/3. The second says body volume scales to body surface with exponent 3/2. The third says cortical surface area scales to body surface area one-to-one.
Of the three scaling relations involved in this deduction, some are more complicated or controversial than others. Starting with the simplest, the 2/3 scaling exponent for body surface area to volume is a fact of geometry that applies universally to all things, without question, without exception, provided the shape remains isomorphic. The brain volume to body volume exponent of 3/4 has withstood the test of decades of empirical replication across millions of animals, thousands of species, and dozens of orders. There are differences between mammalian Orders, when measured separately. Many Orders scale to 3/4, as does Mammalia as a group. Concerning the 8/9 exponent, if the cortex were nothing but gyri, and if all gyri universally had absolute widths, the exponent would be 1. Conversely, if the outer surface of the brain were perfectly smooth, the exponent would be 2/3. The condition of lissencephaly-the absence of convolutions-is found in small brains presumably under the natural limits for gyri widths as total brain widths and in manatees. For the rest of Mammalia, the relation between cortical surface and brain volume is mathematically speaking twice as near to the first scenario than to the second, or 1/9 versus 2/9.
One advantage with this scaling exponent of 1 is that it is likely to have broad and consistent applicability for even some of the most "scaling exceptional" Orders. One example is Cetacea. In Cetacea, CSA/BrainV = 8/7, while BrainV/BodyV = 1/2 to 5/9 (see Notes  and  respectively). These measures yield a scaling exponent for cortical surface to body volume between 4/7 and 5/8. Assuming the (inverted) 3/2 volume to surface law, the exponent for cortical surface to body surface is predicted between 6/7 and 15/16. So, rather large differences in Cetacean scaling laws-up a 1/4 difference, in each of the two exponents, when compared to Mammalia-is absent from the scaling law presented here. The difference is 1/7 at most and 1/16 at least. Mathematical predictions are invariably accommodated with practical considerations, which here-seen in the downscaling of fins and flukes-could cover this differential.
Correspondence should be addressed to Mark D Reid (e-mail: firstname.lastname@example.org).
 Changizi, M. (2001). Principles underlying mammalian neocortical scaling. Biological Cybernetics, 84, 207-15.
 Martin, R. D. (1981). Relative brain size and basal metabolic rate in terrestrial vertebrates. Nature 293, 57-60.
 Ridgway, S. H. & Brownson, R. H. (1984). Relative brain size and cortical surface areas in odontocetes. Acta Zoolologica Fennica 172, 149-152. They provide only the nonlogarithmic equation CSA = 330.4 + 2.17*BrainW. When values respecting both this equation and their graph are plotted logarithmically, the exponent of CSA/BrW exponent is 1.14.
 Marino, L. A (1998). Comparison of Encephalization between Odontocete Cetaceans and Anthropoid Primates. Brain, Behavior, & Evolution, 51, 230–238.