A universal scaling law between gray matter and
white matter of cerebral cortex
Kechen Zhang* and Terrence J. Sejnowski*†‡
*Howard Hughes Medical Institute, Computational Neurobiology Laboratory, The Salk Institute for Biological Studies, 10010 North Torrey Pines Road,
La Jolla, California 92037; and†Department of Biology, University of California at San Diego, 9500 Gilman Drive, La Jolla, California 92093
Edited by Vernon B. Mountcastle, The Johns Hopkins University, Baltimore, MD, and approved February 29, 2000 (received for review November 18, 1999)
has a similar layered architecture in species over a wide range of
brain sizes. Larger brains require longer fibers to communicate
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.
local uniformity of the cortex and the requirement for compact
arrangement of long axonal fibers. The predicted power law with
cortex accurately accounts for empirical data spanning several
orders of magnitude in brain sizes for various mammalian species,
including human and nonhuman primates.
for instance, the combined volume of neocortical gray matter
and the adjacent white matter occupies no more than 10% to
20% of the total brain volume, whereas in humans it reaches
As illustrated schematically in Fig. 1, as brain size increases,
the volume of the white matter beneath the cortex tends to
increase faster than the volume of the cortical gray matter itself.
Whereas the gray matter consists of neuronal cell bodies, their
dendrites, and local ramifications of axons, plus glial cells and
blood vessels, the white matter consists mostly of bundles of
axons running a long distance. It is estimated that the great
majority of white matter fibers connect different cortical regions
rather than connect the cortex and subcortical structures (p. 106
in ref. 4 and p. 35 in ref. 5), although quantitative and compar-
ative data are not available.
Allman (1) recently described an interesting power law rela-
tion between cortical gray matter volume and white matter
volume in primates, based on an analysis of the anatomical data
close to 4?3, suggesting that some simple geometric factors may
be responsible for this regularity (1). This observation initiated
Brain scaling issues were studied by Dubois and his contem-
poraries (6). Danilewsky first demonstrated that larger brains
contain relatively more white matter, and de Vries argued that
the white matter should increase as the cube, and the gray matter
as the square, which would imply a power law with the exponent
3?2 (7). Schlenska (8) presented quantitative data from different
species and demonstrated power law relation between the vol-
umes of gray and white matters with an exponent around 1.22
(inferred by using Eqs. 16 below). Frahm et al. (9) found similar
results in a new data set, with an inferred exponent around 1.24.
Hofman (10, 11) proposed different empirical scaling laws for
different brain sizes, distinguishing the smooth (lissencephalic)
brains from the convoluted (gyrencephalic) ones. He also in-
he neocortex has expanded greatly during evolution of
mammalian brains (1, 2). In insectivores, such as hedgehogs,
law. Prothero and Sundsten (12) proposed explicit cubic geo-
metric models for the cortical folding and generated various
scaling laws numerically.
The purpose of this paper is to present an analytical theory for
the gross organization of cortical gray matter and the associated
white matter. The predicted relationships among anatomical
variables, confirmed partially by the existing anatomical data,
can account for the empirical scaling law and may serve as the
basis for new avenues of investigation.
2. Empirical Scaling Law
law can adequately describe the relationship between the volume
G of neocortical gray matter and the volume W of the adjacent
white matter (including the corpus callosum but excluding the
internal capsule), even though the neocortex itself may scale
differently with the rest of brains for different species (9). Here
G and W span more than five and six orders of magnitude,
respectively, whereas their ratio W?G ranges over only one order
of magnitude from about 0.06–0.07 to about 0.7–0.8.
The slope of the best fit line in Fig. 2 is 1.23 ? 0.01 by three
different methods, including the standard least squares, least
This paper was submitted directly (Track II) to the PNAS office.
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The publication costs of this article were defrayed in part by page charge payment. This
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Article published online before print: Proc. Natl. Acad. Sci. USA, 10.1073?pnas.090504197.
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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. 7, reoriented and
rescaled approximately by using the cat and puma brains at the University of
Wisconsin website www.neurophys.wisc.edu?brain.
Schematic section diagrams showing that the cerebral cortex of a
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squares of perpendicular distance, and least absolute deviations,
the third of which is more robust against outliers (13). The
standard deviations for the slope and intercept were estimated
directly for the first method and by bootstrap for the last two
methods (14). Bootstrap may help detect outliers in the data
because, when they are left out from a same-size resample, the
correlation coefficient often increases, which could be exploited
to improve estimation. Systematic bias caused by outliers was not
detected in Fig. 2.
3. Theory of Scaling
small piece of cortex of unit area, regardless of its thickness and
the overall brain size, sends and receives about the same total
cross-sectional area of long-distance connection fibers to and
from other cortical regions. Second, we assume that the global
geometry of the cortex minimizes the average length of the
The second assumption follows from Ramon y Cajal’s prin-
ciple for conservation of space, conduction time, and cellular
materials (Chap. V in ref. 15). This principle has been explored
more recently as the principle of minimal axon length (16–18).
Consistent with previous observations on the basic uniformity of
the cortex (19–21), the first assumption is supported loosely by
the evidence that the total number of neurons beneath a unit
cortical surface area is about 105?mm2across different cortical
regions for several species, from mouse to human (22) (after
shrinkage correction). But there are exceptions, including the
higher density in striate cortex of primates (22, 23), the lower
density in dolphin cortex (24), and the variability observed in cat
cortex (25). The number of axons leaving or entering the
gray–white boundary per unit cortical area should be compara-
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. 3 were taken, including 23 primates, 2 tree shrews, and 13
insectivores. Another 11 species were taken from table 2 in ref. 8, including 3 primates, 2 carnivores, 4 ungulates, and 2 rodents. Five additional species came
from table 1 in ref. 11, including 1 elephant and 4 cetaceans. The data point for the mouse (G ? 112 mm3and W ? 13 mm3) was based on ref. 30, and that for
the rat (G ? 425 mm3and W ? 59 mm3) was measured from the serial sections in a stereotaxic atlas (42). The estimates for the fisherman bat (Noctilio leporinus,
G ? 329 mm3and W ? 43 mm3) and the flying fox (Pteropus lylei, G ? 2,083 mm3and W ? 341 mm3) were based on refs. 43 and 44, 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 mm3and W ? 56,100
mm3) were measured from the serial sections at the website given in the legend to Fig. 1, with shrinkage correction.
www.pnas.org Zhang and Sejnowski
ble to the number of cells, because the majority of cortical
neurons are pyramidal cells and because most of them send
axons into white matter, whereas most nonpyramidal cells do not
(26). The numbers of efferent and afferent corticocortical fibers
should roughly match up. Another piece of evidence consistent
callosum, the collection of fibers connecting the two hemi-
spheres, is proportional to the total cortical surface area for
different species (21, 27). We emphasize that the uniformity
assumption involves only the total cross-sectional area of white
matter fibers, without directly specifying other factors, such as
the total numbers of neurons and axons per unit area, the
percentage of neurons that project into white matter, the thick-
ness of axons, their branching, and the relative proportion of
efferent and afferent fibers.
Our goal is to show that these two assumptions and constraints
from geometry yield the empirical power law. The total volume
of gray matter (G) is given by
G ? ST,
where S is the total area of the cortical surface, which can be
convoluted, and T is the average thickness of the cortex. Because
the thickness of a local cortical region often varies along the
convolution (28), this relation is only a global average and is
sometimes used to define the average thickness.
The volume of the white matter (W) that connects different
cortical areas can be expressed as
where p is the fraction of the surface area occupied by the cross
section of the axonal fibers entering or leaving the white matter
and L is the average length of the white matter fibers. To derive
this expression, imagine that the white matter is divided con-
area a. Area a is fixed but can be arbitrarily chosen, as long as
the fibers within the same bundle are of the same length. Axons
are allowed to run in different directions within the same
imaginary bundle; occasional branching is allowed as long as the
cross section of the bundle is conserved. Suppose there are a
total of n such bundles and let Libe the length of the ith bundle,
then the total volume of the white matter should be the sum of
the volumes of all these bundles:
The second step is an identity, with 1?n ?i?1
length of the fibers in white matter, and na the total cross-
two distinct cortical areas, imagine that every bundle is painted
red at one end and green at the other end. Consider all of the
red ends only, which should contact half of the total cortical
surface area S?2. The total cross-sectional area at the red ends
is still na. This area, by assumption, is related to the contacted
half of cortex by the factor p, namely, na ? pS?2, which leads
to Eq. 2.
The uniformity assumption implies that the areal wiring
fraction p should be constant across species. In addition, p ? 1
is expected because axons leaving or entering the gray matter
often run at an angle other than perpendicular to the cortical
surface, so that their total cross-sectional area must be less than
the cortical surface area.
The key question is how the gray matter volume is related to
the white matter volume. Because they share the same surface
Li? L the average
area S, up to a constant fraction p, the real unknown quantity is
the average fiber length L in white matter, which should be
somehow related to the volume of the gray matter. We postulate
G ? qL3,
where q is a dimensionless constant, i.e., a pure number inde-
pendent of the unit of length. In Section 5, we justify this
postulate by a theoretical argument and support it with anatom-
The scaling law between gray and white matters now follows.
Eliminating S from Eqs. 1 and 2 gives W ?1
substituting L in terms of G by using Eq. 4 yields
2pGL?T, and then
scaling law predicts that the white matter volume increases
exactly as the 4?3 power of the gray matter volume. In reality, the
thickness increases slowly with the gray matter volume, so that
the effective exponent should be slightly smaller than 4?3.
4. Cortical Thickness
The slow thickening of cortex for larger brains can be fitted by
with previous results, direct power law regression of cortical
thickness against gray matter volume yielded T ? G0.10?0.02, with
a moderate correlation coefficient r ? 0.81 in log–log space,
based on 22 species from the data in table 1 in ref. 11. The five
smallest brains with the inferred cortical thickness all smaller
than 0.7 mm were left aside because they were too thin to fit
comfortably into the same power law. For comparison, the
average thickness of mouse cortex is 0.8–0.9 mm (30). Combin-
ing this result with Eq. 5, one expects that the white matter
volume should scale as
W ? G4?3?G0.10?0.02? G1.23?0.02,
which naturally accounts for the empirical power law with the
exponent 1.23 ? 0.01 in Fig. 2.
Power law scaling for cortical thickness may be justified
theoretically (19). Hofman (11) proposed that a logarithm
function would fit the thickness data for the smallest brains
better than a power law, but his analysis included inferred
thickness estimates, not all from direct measurement. For our
current purposes, the thickening of cortex with brain size is
treated simply as a small but noisy empirical correction to the
dominant 4?3 exponent. From the perspective of dimensional
consistency (31), neither a power law with a small exponent nor
variables from the same brain, although they are useful as
5. The Cubic Postulate: Theory and Evidence
In the theoretical treatment above, the cubic relation between
the gray matter volume G and the average length L of white
matter fibers was taken as a postulate. This possibility is dimen-
sionally consistent, but alternative relations also have consistent
units, for instance, G ? qL2T and G ? qL1/2T5/2, where T is the
cortical thickness. Here, we prove that the cubic relation allows
the shortest average fiber length among other possible relations
Zhang and SejnowskiPNAS ?
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of the general product form
G ? qL?l1
?2· · · ,
where q is a dimensionless constant and where l1, l2, . . . are
additional length variables that might describe, for example, the
thickness, the folding or gyri of the cortex, and so on, and are
assumed to be smaller than the average fiber length L, namely,
li? L; all of the exponents are assumed to be nonnegative: ? ?
0, ?i? 0, and should sum up to the correct dimension:
? ? ?1? ?2? · · · ? 3.
Under these assumptions, we have
G ? qL3?
· · ·
where the first equality follows directly from Eqs. 8 and 9 and
where the inequality holds because each ?i? 0 and li?L ? 1.
Thus, we always have G ? qL3, regardless of the exact values of
the auxiliary length variables and the choice of all of the
exponents. Given a fixed G, the minimum length of L is achieved
when the equality holds, which requires ? ? 3 and ?1? ?2? ? ? ?
? 0, so that all other variables disappear except for a cubic term
of L, which is the original postulate.
Therefore, when the average fiber length has the longest
length scale, the cubic relation should be optimal in reducing the
length of white matter fibers. Indeed, when the total exponent
of 3 is shared by shorter lengths instead of being occupied
exclusively by the average fiber length L, then L must be longer
to reach the same gray matter volume G. Although the constant
q was fixed when comparing different possible products, this
condition can be relaxed by allowing q to vary within fixed
bounds. An even more general formulation is as follows. The
most general relation between the volume G and the length
variables L, l1, l2, . . . can be written as
G ? L3Q?
L, · · ·?,
where Q is an arbitrary function whose value is independent of
the unit used for length measurement. This result may be
regarded as a consequence of Buckingham’s ?-theorem in
dimensional analysis (32).
A necessary and sufficient condition for the optimality of the
cubic relation is that the function Q is bounded, namely, Q ? q
for some fixed q, and the equality can be achieved within the
allowed ranges of variables and other hidden parameters (such
as the exponents ?iin the product case). Once we obtain G ?
L3Q ? qL3, the previous optimality argument following Eq. 11
applies without change.
How can the cubic scaling be tested, given that the average
axonal length L in white matter is typically unknown? We may
infer its value indirectly from Eq. 2, namely,
L ? 2
This formula provides new information because the value of the
surface area S has not been explicitly used above (33, 34). The
value of the constant p for areal wiring fraction has no effect
on the exponent of the power law and will be estimated in
As shown in Fig. 3, the data are noisy but support the cubic
relation within the margin of error. Part of the noise may have
arisen because the volume and surface data were compiled from
different sources and different individual animals (11). Several
smaller brains were not used here because the inferred values of
L seemed too small to fit into the same power law. However, for
the mouse brain, which is also quite small, an estimate of fiber
length in white matter gives the average L ? 3 mm (figure 62 in
ref. 30). Together with G ? 112 mm3, this data point would be
close to the regression line in Fig. 3. Therefore, more experi-
mental data would be required to test whether the cubic relation
may fail for the smallest brains.
6. Arbitrary Dimensionless Product
be a dimensionless constant across species, independent of the
choice of length unit. An arbitrary product W?G?T?is dimen-
sionless whenever 3? ? 3? ? ? ? 0. Can such products that are
constant across species be discovered directly from the data?
function is also dimensionless, although larger powers tend to
amplify the noise in the data. To compare with the theoretical
formula, we set ? ? 1, leaving only one degree of freedom in
choosing a dimensionless product. So we need only consider
c??? ? WG??T3??3
with an arbitrary ?. Eq. 5 corresponds to the case ? ? 4?3.
To characterize the constancy of c(?) across species for each
given ?, we use the ratio of the standard deviation over the mean,
because both quantities may vary over several orders of magni-
tude as ? changes. In Fig. 4, the best constancy achieved for ? ?
1.36 ? 0.06 agrees with the theoretical value 4?3 ? 1.33.
Therefore, the dimensionless product predicted by the theory,
although still quite variable across species, is about the most
constant among arbitrary dimensionless products.
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. 11, where cortical surface area and thickness
are related rigorously by Eq. 1.
Testing the cubic postulate between the gray matter volume and the
www.pnas.orgZhang and Sejnowski
7. Limitation of Power Law Scaling
A problem arises when comparing Fig. 2 with Fig. 5 (8, 9), where
the total volume
V ? G ? W
is defined as the sum of gray and white matter volumes. Because
we already have the power law W ? G?with ? ? 1 (Fig. 2), it
is mathematically impossible to accommodate two additional
power laws G ? V?and W ? V?without violating Eq. 15. An
inescapable conclusion is that at least two of the power laws, and
possibly all three of them, must be approximate, even in theory.
Suppose one of them is genuine and known, then how can the
other two approximate power laws be justified? (We would
choose W ? G?as genuine, given our theory. Consistently, the
power law G ? V?in Fig. 5 is unlikely to be exact, despite its high
correlation coefficient, because of the small but systematic
deviations.) Given any one exponent, the other two exponents
can be determined as a linear approximation by solving the
?? ? ?,
?Gˆ? ?Wˆ? Gˆ? Wˆ,
where Gˆ? exp(?ln G?) and Wˆ? exp(?ln W?) are averages
performed in log space. For the data in Figs. 1 and 5, we have
Gˆ? 5200 mm3and Wˆ? 1248 mm3, and the three exponents
determined directly by least squares are ? ? 1.2284, ? ? 0.9549,
and ? ? 1.1739. Given ?, for instance, both ? and ? can be
predicted within 0.3% of error by Eqs. 16.
The effect of adding power laws has been studied in allometry
(35) and brain scaling (C. F. Stevens, personal communication).
Brain volume has many components, such as cell bodies, den-
drites, axons, blood vessels, and glia cells. Unless all components
scale with the same exponent, it is logically inconsistent to
assume rigorous power law relations among different compo-
nents and their sums. Although power laws are often excellent
approximations, a theory for power laws should be expected only
to reveal the leading contributing factors. Additional informa-
tion may be obtained by examining the relationships between the
residuals from the fits (36).
8. Determining the Constants
The theory presented in this paper contains a total of three free
parameters, namely, p, q, and c, all of which are dimensionless
constants. Their values are not constrained by the theory and
have to be estimated separately. Because they are related by Eq.
6, only two are independent.
The constant p is the fraction of the cortical surface area that
is occupied by the total cross section of the fibers in white matter
p ? ?a ? ,
wherea ? istheaveragecross-sectionalareaofasingleaxonalfiber
either entering or leaving the white matter, and ? is the total
number of these axons per a unit surface area of the cortex. The
diameters of axons, which may vary over an order of magnitude,
have an average value of roughly 1 ?m in corpus callosum (37,
38).Aroughestimatea ? ?0.8?m2fortheaveragecross-sectional
area of a single fiber, corresponding to a diameter about 1 ?m,
(26) estimated that the density of the axons crossing the gray–
white boundary in cat cortex is about ? ? 105?mm2, the same
order of magnitude as neuron density per unit area (22, 28, 30).
Taken together, we obtain a crude estimate p ? ?a ? ? 0.08.
Another independent estimate of p is possible by using
Greilich’s estimate of the distance of corticocortical connections
in mouse (30). From the histogram in figure 62 of ref. 30, we
estimate that in mouse the average axon length in white matter
is around L ? 3 mm. From Eqs. 1 and 2:
p ? 2WT
where D ? W?S ? WT?G can be interpreted as the average
thickness of the layer of white matter, with S being the cortical
surface area. Inserting the estimate of L and other known
quantities (gray matter volume G ? 112 mm3, white matter
sionless number c(?) plotted as a function of the test exponent ? in Eq. 14. The
theoretical optimal value 4?3, as indicated by the dashed line, is close to the
from table 1 in ref. 11 (27 species).
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
The same data as in Fig. 1 plotted against the total volume of gray
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volume W ? 13 mm3, cortical thickness T ? 0.85 mm) into Eq. Download full-text
18 yields p ? 0.07, close to the first estimate.
The constant q is the coefficient for the cubic relation between
gray matter volume and average axonal length. Assuming p ?
0.08 in Eq. 13 and by using Eq. 4, q can be estimated by averaging
G?L3over the same data as in Fig. 3. The result is q ? 10 ? 5.
The constant c is the only free parameter in the 4?3 scaling law
in Eq. 5. Averaging the product c ? WG?4/3T across the same
data in Fig. 4 yields c ? 0.02 ? 0.01. Alternatively, Eq. 6 with the
existing estimates p ? 0.08 and q ? 10 gives a consistent value
consistently with moderate accuracy from the existing data.
2pq?1/3? 0.02. Thus, all three constants can be estimated
The volumes of cortical gray matter and the adjacent white
matter are closely related by a power law across nearly the full
range of mammalian brain sizes (Fig. 2). The theoretical argu-
ments in this paper show quantitatively how this relationship
might follow as a necessary consequence of the basic uniformity
of the neocortex and the compact wiring of its interconnection.
The theory can account for the exponent of the empirical scaling
law (Fig. 2 and Eq. 7) and allows all free parameters to be
determined from the existing data (Section 8).
same total cross-sectional area of long-distance fibers regardless of
brain size, then these fibers would have to go longer distances to
connect different parts of a larger brain, leading to a dispropor-
tionate expansion of the white matter. Although the exact relation-
manner of cortical folding, we have avoided explicitly modeling
gray matter volume should be proportional to the cube of the
average fiber length in the white matter. The cubic relation is
theoretically optimal in some general situations and appears to be
valid empirically, at least for larger brains (Fig. 3). Taken together,
as shown here the amount of white matter needed to interconnect
the whole neocortex must follow a power law if cortical uniformity
and compact wiring are assumed.
The theory here does not presume the uniformity of the cortex
beyond the assumption that the total cross section of the axons
passing the gray–white matter boundary should occupy a con-
stant fraction of the cortical surface area across species. Al-
though we have reviewed evidence that makes the assumption
look plausible (Section 3), it remains to be determined to what
extent the areal wiring fraction is actually constant, and how it
compares with the neuron and axon densities per unit area, for
different cortical regions of the same animal and across different
Additional anatomical measurements are needed. For exam-
ple, it is uncertain exactly how many fibers in white matter
interconnect the cortex and how many connect the cortex with
various subcortical structures. We assumed here that the former
is much more numerous. One crucial variable used in this paper
is the average length of corticocortical fibers in the white matter,
which has been inferred theoretically (Fig. 3). Measuring the
fiber length in a wide range of species would provide valuable
information, as for mouse cortex (30). Such information would
be a key link between a global principle of minimal wiring, such
as the cubic postulate above, and the geometry of the brain that
actually embeds the hierarchical organization of the multiple
cortical areas, as illustrated by the primate visual systems
(39–41). The minimal wiring principle itself also needs to be
assessed quantitatively for its range of validity. Noninvasive
measurement with magnetic resonance imaging, by avoiding the
shrinkage and distortion caused by fixation, staining, and sec-
tioning, may yield useful geometric information about the cortex
and its associated fibers, sampling across more living animals,
and tracking the same individuals throughout development.
These measurements combined with theoretical analysis should
provide more accurate information on the basic constraints
underlying brain development and evolution.
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