**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.[1] BrainV scales to body volume (BodyV) with a
measured exponent of 3/4.[2] 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,[3] while BrainV/BodyV = 1/2 to 5/9 (see Notes [3]
and [4]
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*: markreid@uiuc.edu).

[1] Changizi, M.
(2001). Principles underlying mammalian neocortical scaling. *Biological
Cybernetics, 84, *207-15.

[2] Martin, R.
D. (1981). Relative brain size and basal metabolic rate in terrestrial
vertebrates. *Nature 293,* 57-60.

[3] 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.

[4] Marino, L. A
(1998). Comparison of Encephalization
between Odontocete Cetaceans and Anthropoid Primates. *Brain, Behavior, &
Evolution, 51, *230–238.