The human and mammalian cerebrum scale by

computational power and information resistance

Marc H.E. de Lussanet

October 29, 2018

Abstract

The cerebrum of mammals spans a vast range of sizes and yet hasa very regular structure. The amount of folding of the cortical surfaceand the proportion of white matter gradually increase with size, but theunderlying mechanisms remain elusive. Here, two laws are derived tofully explain these cerebral scaling relations. The first law holds thatthe long-range information flow in the cerebrum is determined by thetotal cortical surface (i.e., the number of neurons) and the increasinginformation resistance of long-range connections. Despite having just onefree parameter, the first law fits the mammalian cerebrum better thanany existing function, both across species and within humans. Accordingto the second law, the white matter volume scales, with a few minorcorrections, to the cortical surface area. It follows from the first law thatlarge cerebrums have much local processing and little global informationflow. Moreover, paradoxically, a further increase in long-range connectionswould decrease the efficiency of information flow.

1 Introduction

The mammalian cerebrum is a highly regular structure, having a cortex of greymatter on its surface which is wrapped around a core of white matter. Thecortex is built up of regular layers, which have specific afferent or efferent long-distance connections [1]. The white matter contains long-range axonal connec-tions. The white color comes from myelinization.

Despite this regular structure, the mammalian cerebrums span a tremendoussize range from 11 mm3 (11 µl, pigmy shrew) to 2.5 · 106 mm3 (2.5 l, elephant)[2]. Given these extreme size differences, measures of the cerebrum are usuallyplotted on a double-logarithmic graph.

There exists a vast literature on interspecific scaling relationships of anythinkable measure of the vertebrate body. What becomes clear on a glance isthat such relationships usually show a large variance, whereas narrow distribu-tions are rare. The reason for this is obvious: most scaling relationships aregoverned by many factors and parameters and thus lead to a trend with much

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scatter (typically in the order of magnitudes). On the other hand, if one doesfind a narrow distribution, the chances are good that its singular cause is asimple mechanism.

Interestingly, a number of such narrow distributions with little variabilityaround the general trend are known for the mammalian cerebrum [3]. Theseare usually described by a power relation (i.e., a linear relation on a double-logarithmic scale). However, not every relation that is well described by alinear regression on a double logarithmic scale underlies a power-relation, aspointed out by Zhang and Sejnowski [2] (see also Fig. S4 in Supplement 2).Theseauthors investigated the power-relationships between the volume of the cortex,the white matter and the grey matter. Each of the three relationships betweenthese parameters followed a simple power over the entire size range and withremarkably little deviation. These volumetric measures, but also the outersurface and the total cortical surface (including the inward folded surfaces) showpower-relations to the brain volume, and the latter also to the cortical volume[4]. However, as the sum of two different powers (e.g., x1 + x1.5 or x + 1) isnever a power relation, at least some of the above relations do not truly reflect apower law [2]. Similar relations with little variability from the trend have beenreported for the number of neurons in the cerebrum, the cerebellum and therest of the brain [5–8].

Two most prominent neuro-anatomic properties of the mammalian cerebrumis the division in grey and white matter and the convoluted (folded) surface,which increases gradually with brain size. Whereas the smallest mammalianbrains possess a lissencephalic (smooth) cerebral surface, larger cerebrums haveridges (gyri) and folds (sulci). The number and depth of the sulci increasemonotonically with brain size. Thus, since the outer shape changes with size,the mammalian cerebrums do not scale isometrically. According to Hofman [4]the first systematic approach was made already by Baillarger [9], who did mea-surements of the cortical surface of a number of mammals. According to Gross[10, p. 90], already Franz Joseph Gall (1758-1828) asked why the cerebral cor-tex is convoluted and proposed that the folds conserve space. Many studies anda number of theories has been proposed since the early approaches [2, 11–15].Unfortunately these theories make no or extremely vague quantitative predic-tions, are circular, or contain serious errors so the problem remains unsolved(see Supplement 2).

The goal of the present work is to derive scaling laws for the cerebral surface-volume relation and for the white-grey matter volume relation. The laws arederived on the basis of a simple hypothesis, i.e., that the two central functionsof the cerebrum, processing and transmission of information, are the forces thatshape the cerebrum. In the following it is shown that the volume-outer surfaceof the cerebrum scales isometrically. Subsequently, theoretical relations for thecerebral cortical surface (including the typical convolutions in large brains) tovolume relation as well as for the volume of the white matter are derived.

2

2 Isometric scaling law

2.1 Isometric scaling of the cerebrum

To investigate whether the mammalian cerebrum scales isometrically on a globalscale, we regard the scaling relation between the cerebral volume Vc and theouter cerebral surface Sc. For this, let us start with a sphere. The relationbetween the surface area and the volume of a sphere can be calculated from thestandard formula:

Vsphere = 34πr

3

Ssphere = 4πr2 (1)

Vsphere = σA3/2sphere

where

σ =Vsphere

S3/2sphere

=43π

(4π)3/2

=1

6√π≈ 0.094 (2)

σ =Ssphere

V2/3sphere

=3√

36π ≈ 4.84.

The cerebral hemispheres deviate from a perfect sphere, so we can expressthe cerebral volume Vc as a function of the outer cerebral surface area, Sc:

Vc = σsS3/2c , (3)

where the sphericity factor s expresses how much the outer cerebral surface areadiffers from that of a sphere. Since the hemispheres are not exactly spherical,s < 1. It can be computed directly from the outer cerebral surface, Sc, whichis reported for part of the dataset of Hofman [4]:

s =Vc

σS3/2c

= 0.54± 0.09 (mean± standard deviation)

s =Sc

σV2/3c

= 1.54± 0.19 (mean± standard deviation)

with N = 32. However, the mean can only be used if Sc and Vc scale iso-metrically. To test this, a linear regression on the log-transformed data wasperformed, which gives Vc = −1.32S1.52

c (R2 = 0.998). Since the power is veryclose to the 3

2 power predicted for isometric scaling, and since the effect of thesmall deviation over the size range of ∼ 5 powers is smaller than the stan-dard deviation, we can reasonably assume that the sphericity factor s is indeedscale-invariant.

Thus, the cerebrum passes the first test for isometric scaling excellently, sincethe relation between the outer surface and the volume scales like the surface andvolume of a perfect sphere with a constant sphericity factor s.

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2.2 The concept of equivalent thickness

A central observation of the gross anatomy of the cerebrum is that the thicknessof the grey-matter cortex tends to saturate with cerebral size. This has typicallybeen quantified by simply dividing the grey matter volume by the cortical surfacearea [16, 17]. This measure is not the average thickness, because it neglects thatthe cerebrum is convex so that the surface area of the grey-white matter borderis smaller than the outer surface (i.e., the equivalent thickness is smaller thanthe geometric thickness).

The equivalent thickness is a measure of the dimensionality of the corticalsurface. For a sphere, the equivalent thickness is proportional to the radius.For a basket filled with laundry, the total surface of the laundry is proportionalto the volume of the basket, because the [equivalent] thickness of the laundryis independent of the volume of the basket. Thus, the observation that thethickness of the grey matter seems to saturate with brain size, is an indicationthat the cortical surface tends to scale proportionally to the cerebral volume forlarge cerebrums.

This can be expressed formally as:

Vc = TcAc. (4)

where Tc is the equivalent cerebral thickness (i.e., the sum of grey and whitematter), and Ac the total cortical surface area (i.e., including the cortical areathat is located inside the sulci). In this relation, Tc = Vc/Ac indeed saturates forlarge cerebrums (see Fig. S2 of Supplement 1).

Similarly, the equivalent thickness in terms of the outer surface Sc is:

Ts =VcSc

=Vc

σsV2/3c

=3√Vcσs

. (5)

2.3 Neurons in the cortex

The total cerebral surface is composed of the cortex of grey matter. The cortexis the region where the vast majority of cerebral neurons are located. Sincethe computational power of the cortex will depend largely on the number ofneurons, the distribution of neurons is highly relevant for scaling relations ofthe cerebrum.

The number of neurons per unit cerebral cortical surface is constant, 105

neurons/mm2. This was first estimated by Bok [18], and confirmed using cellcounts [19, 20]. In both latter studies, cell counts (neurons and glia cells) wereperformed for five regions on the cerebral cortex that differ strongly in thicknessand composition of the layers. Moreover, these samples were taken from fourmammals of very different brain size (mouse, rat, cat, monkey). Both studiesfound not only the same number of 105 neurons per mm2 for the differentsamples of each species, but also for each of the four species tested. The averagecortical surface per neuron is thus k = 10−5 mm2, irrespective of the cerebralsize and the location of the cortical surface.

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2.4 Information conduction and axonal diameter

The diameter of axons in the white matter increases with cerebral size [3]. It isknown that both, the average spike rate and the transmission velocity increaselinearly with axonal diameter [21, 22]. Based on theoretical considerations, anincrease of the signal band width will not increase the information rate in aproportional manner [23, 24]. Measurements on the optic tract have indicatedthat the higher spiking rate in wide axons also leads to a stronger correlationof subsequent spikes [25]. The latter study estimated that the information rateper axon is independent of its diameter. Thus, increasing the diameter leads toa quadratic increase of its cross-sectional area, but only a linear increase in itstransduction velocity.

Assuming that the axonal diameter is scaled such transmission time is ap-proximately size-independent, means that the information rate in a processingloop will decrease quadratically with the length of the connection.

2.5 Scaling law for the cortical surface

We can now derive the scaling law for the relation between the cortical surfaceand the cerebral volume. The two central functions of the cerebrum are compu-tational power and information transfer. Assuming that the number of neuronsin the cortex determines computational power, and given that the the numberof neurons per cortical surface area is constant, the scaling parameter for thecerebral volume Vc is cortical surface area, Ac.

There are two components to the information transfer in a neuron: long-range afferent axonal connections and efferent dendritic trees. The length ofaxonal connections in the cerebrum has a typical, gamma-like distribution, andincreases with the size of the cerebrum [27, 28]. Dendritic trees on the otherhand, have a finite size. That this is indeed so, is reflected by the fact that thenumber of neurons per cortical surface area is constant (cf. section 2.3), and theestablished concept of the cortical column [29, 30]. These two properties havebeen described as long-range scale-free connectedness [31, 32] and small-scale,small-world [33] properties [34]. In other words, the network of the cerebrum hasto different scales: the small scale is manifested on the level the cortical columnsand limited size of dendritic trees, whereas the large scale is manifested in thelong-range axonal connections.

The volume to a given cortical surface area is thus determined by the flowinformation processed by the neurons, and the transfer of this information on thelocal- and global-scale. This can be easiest be expressed in terms of informationresistance: how the flow of information [bit/s] divides among local and globalnetworks depends on how well each route transfers information. Assuming thatthe resistance to information depends linearly on the length of the connection(see Discussion), it becomes clear that the relative contribution of the small-scale and large-scale connections depends directly on the size of the cerebrum(Ts: eq. (5)). Thus, expressing information resistance in terms of equivalentconnection length, and subsequently substituting equation (5) we get:

5

(a)

3 3.22 2.2 2.41.8 2.6 2.8

0.9

1.0

0.8

0.7

1.4

1.1

1.2

1.3

model (eq. 7)1.6

Cortical surface Ac [105 mm2]

Cer

ebra

l vol

ume

V c [1

06 m

m3 ]

tlocalAc

(d)

model (eq. 7)tlocalAc

3/2σsAc

103 104 105 106102101100

103

104

105

102

101

106

107

Cortical surface Ac [mm2]

Cer

ebra

l vol

ume

V c [m

m3 ]

101

Surface Ac [mm2] Surface Ac [mm2]

model (eq. 7) regression

Erro

r

(b) (c)

104102 106103 105 101 104102 106103 105

Figure 1: (a) The model fit (eq. (7)) on the dataset of Hofman [4], N=37.Continuous line: the model fit; dashed lines: the asymptotic relations withpower slopes of 1 (cf. eq. (4)) and 1.5 (cf. eq. (3)). Square: human data point.Red cloud: the data of panel d (not included in the fit). (b) Error for the modelfit of panel a. (c) The error for a linear regression of the same, log-transformed,data. Note that the error of the regression (Panel c, two parameters) shows aninverted U-shaped trend but the model (Panel b, one parameter) does not. Thevertical axis depicts one order of magnitude in panels b and c. (d) The modelfit on the human data of Toro et al [26] on a linear scale. As a reference, thehuman data point of panel (a) is shown as an open square (not included in thefit).

6

1

RI,total=

1

RI,global+

1

RI,local

1

T 2c

=1

T 2s

+1

t2local

=

(σs3√Vc

)2

+1

t2local, (6)

where the constant tlocal expresses the scale-invariant information resistance ofthe local networks [m]. By substitution of equation (4) the volume to corticalsurface is obtained as:

(σsV

2/3c

)2

+

(1

tlocalVc

)2

= A2c . (7)

A remarkable property of this surface-volume relation is that it has just asingle free parameter: tlocal.

The fit of equation (7) on the log-transformed data [4] resulted in tlocal = 3.5mm (N = 37, R2 = 0.997). This result is presented in Figure 1a. Even thoughthe model has just a single free parameter, it describes the data much betterthan a conventional linear regression on the log-transformed data (which yieldslog(Vc) = 0.67 + 1.24 log(Ac) with an R2 = 0.98). This is illustrated in panelsb and c of Figure 1. If the model describes the data well, the error shouldbe homogeneous over the entire range of the data. Whereas the errors of themodel-fit are homogeneous (Fig. 1b), those of the regression show an invertedU-shaped trend (Fig. 1c). (See also Supplement 1 for a sensitivity analysis).

Since the wide availability of neuroimaging facilities, it has become possibleto measure the gross anatomy of large samples of cerebrums with standardizedand automized procedures. Such a data set of human subjects [35] was analyzedby Toro et al. [26]. These data (314 subjects, 164 females and 150 males of 12-20 years old) are presented in Figure 1d. The data were fitted using the samesphericity factor (s), because the human data point in the Hofman data hadexactly the same sphericity as the average of all mammals. The fit resulted intlocal = 4.8 mm (R2 = 0.86), which was exactly as good as the linear regressionon the log-transformed data (which resulted in a power slope of 1.0, intercept0.54 and R2 = 0.87).

Thus, the model not only describes the inter- but also the intraspecific vari-ation with high accuracy.

3 How much white matter does the cerebrumneed?

After the scaling relations of cerebral volume to outer surface and of cerebralvolume to cortical surface, the scaling of the cerebral white matter volume

7

Vw remains to be solved. The white matter volume scales in a highly regularmanner with the cerebral size as shown by [2], i.e., the proportion of whitematter increases monotonously with cerebral size.

We saw that the number of neurons is constant per unit cortical surface(section 2.3). The white matter consists mainly of the axons that either originatefrom or insert on these neurons. This leads to the prediction that the volumeof white matter may depend directly on the cortical surface. To confirm thatsuch a 3

2 power in the model is indeed reasonable, a linear regression on thelog-transformed data was performed, which gave a power of 1.51, R2 = 0.987.We thus have the following model for the white matter volume:

Vw = σsPwA3/2c , (8)

with σ and s as in equation (3). Fitting resulted in Pw = 5% (R2 = 0.987).Notice that this relation also includes the convoluted cerebrums. That meansthat, according to the model, the white matter volume is always 5% of thevolume that a lissencephalic cerebrum with the same cortical surface area wouldhave.

This simple, single parameter model does have a trend in the errors (Fig.2c), which indicates that it is not perfect. It is possible to derive an even moreaccurate model, when considering that the white matter reflects the myeliniza-tion level of long-range axonal connections, and therefore can be expected to bescale-dependent.

Since in very small brains the connections are short, it is to be expected thatthe white matter volume is disproportionally small in very small cerebrums. Ifwe assume that only axons of a length of at least Lax,min become myelinized,we get the following relation for lissencephalic brains:

Vw,liss = 43πr

2(r − Lax,min) (9)

= σsPw,lissA3/2c − 1

3Lax,minAc,

where σ and s are as above (Section 2.1); Lax,min = 0.04 mm; and Ac =Sc. Pw,liss = 6% expresses the upper limit percentage of (equivalent) cerebralvolume that is white matter (compare with the 5% of the regression fit above).This relation is shown by the left curve in Figure 2a.

For the white matter of convoluted cerebrums (in the dataset: Ac > 2000mm2), one can distinct between intra- and extra-gyral long-range connections.Since the number of gyri increases with the size of the cerebrum, the volumeof intra-gyral connections per unit cortical surface area can be expected to beindependent of cerebral size. The extra-gyral connections the other hand willdepend on the cerebral size. Thus, for convoluted cerebrums we expect thewhite matter volume to be the sum of two factors, one scaling with a power of1 and the other with a power of 3

2 of the cerebral surface area:

Vw,conv = Tw,maxAc + σsPw,convA3/2c , (10)

8

(a)

100

101

102

103

104

105

106

Cortical surface Ac [mm2]

Whi

te m

atte

r vol

ume

V w [m

m3 ]

101 102 103 104 105 106

volum

e of s

phere

volum

e of c

erebru

m

Vw,liss (eq. 7)

Vw,conv (eq. 8)

Cetacea

Homo

unlabeled

PrimataSoricomorpha

Erinaceomorpha

Surface Ac [mm2] Surface Ac [mm2]

eqn (7) & (8) eqn (6)

Erro

r

(b) (c)

101 102 103 104 105 106 101 102 103 104 105 106

Figure 2: (a) The relation between the cortical surface, Ac and white mattervolume, Vw. Dashed and dash-dotted curves: volume of the cerebrum and ofa sphere (cf. Fig. 1a). Note, that the white matter volume is for most of therange only a small fraction of the total volume. See text for an explanation ofthe fitted curves. (b) Errors for the model (eq. (9) and (10)). (c) Errors fora linear regression on the log-transformed data. Notice that error of the linearregression shows a clear trend. The vertical axis is one order of magnitude inpanels b and c. Data are taken from [4]

where the equivalent white matter thickness Tw,max = 0.4 mm and the minimalpercentage of white matter for convolute cerebrums Pw,conv = 3%. The equationfits the data well, as shown in Figure 2a,b. The values may seem small, butthat is because they are expressed to equivalent volume, i.e., the volume of alissencephalic cerebrum of the given cortical surface. The total R2 = 0.996,which is better than the simple, single-parameter model (eq. (8)), but whichis not too surprising, given that four parameters were fitted for the extendedmodel.

The discontinuity of equations (9) and (10) might seem to contradict the

9

main surface-volume relation (eq. (7)). However, the white matter representsonly a small fraction of the volume.

Notice that the single-parameter model (eq. (8)) is in fact a first-order ap-proximation of the complex model (eqs. (9) and (10)). The linear terms in theequations represent relatively minor corrections to the overall 3

2 power scalingwith the cortical surface.

4 Discussion

Compared to other scaling relations known for animals over a wide size range,the scaling relations of the mammalian brain are remarkably regular and re-markably linear on a double-logarithmic scale. This has been known for a longtime but the reasons have remained elusive, because the power laws that areobtained by such linear regressions on log-transformed data are incompatiblewith isometric scaling laws. Here, for the first time, a theory is developed thatfully explains these narrow scaling relations. Moreover it predicts these relationswith parameters that are physiologically meaningful, determined by the centralfunctions of the cerebrum –processing and transmitting information– and thatcan in part be derived from independent measurements.

These parameters are s, tlocal, and the quadratic relation between connectionlength and information resistance. The scale-invariant sphericity s is assumedto be constant, though, as indicated from the standard deviation, it shows con-siderable variation. The effect of such variations mainly affects the predictionsfor small cerebrums, as shown by the sensitivity analysis (Supplement 1: Fig.S3).For lissencephalic cerebrums, the deviations from the scaling relation arewell explained by the variations in the sphericity factor. It should be notedthat the scale invariance of s does not imply true isometric scaling, becauseeach value of the sphericity parameter (except the perfect sphere itself) can beachieved with a whole family of shapes. The empirical value s ≈ 1

2 thus onlytells that the cerebral volume is about half of the volume expected for a perfectsphere of the same outer surface.

The tlocal parameter expresses the information resistance on the local scale,but is also a length measure for the local networks. The measure expressesequivalent length (cf. section 2.2). That is, even if 3.5 mm may appear very little,it is equivalent to the size of the largest lissencephalic cerebrums. This is in theorder of the size of the gyral lobes of convoluted brains. The largest cerebrumsclosely approximate the limit relation tlocal. That means, such large cerebrumseffectively function more like a communicating cluster of local processing centersthan as a global processor. Following the sensitivity analysis, the confidencemargins of tlocal may be as much as ±1 mm. Further, the model is only a firstorder approximation in so far, that axonal lengths show a wide distributionwithin a cerebrum [27, 28]. Further the regional properties are likely to vary [1].Finally, there is evidence that the transition from lissencephalic to convolutedbrains differs between Glires and Primates [7]. This would indicate that thetlocal is smaller for Primates than for Glires, leading to a smoother surface in

10

the latter.Third, it was estimated that the information resistance increases quadrat-

ically with connective distance, but this may not be entirely accurate. As acomparison, the scaling law was also estimated with a linear scaling of informa-tion resistance with connective distance (Supplement 1: Fig. S3).The quality ofthe fit is remarkably insensitive to such a seemingly grave difference.

The present approach to isometric scaling differs in two ways from earlierones. First, instead of allowing for scaling relations of any power (which isimplicit in the method of linear regression on the log-transformed data), onlynatural scaling powers were applied. The justification for this is simple: anydeviation from a natural power relation is by definition not isometric scaling,and therefore unlikely to produce a simple, narrow scaling relation. Second, ontheoretical considerations the volume is to be regarded to depend on corticalsurface. To the knowledge of the author, all approaches so far have sponta-neously assumed that the cortical surface is a function of the volume (i.e., is thedependent variable). Although this makes sense in many organs where surfaceis needed for efficient exchange, think of lungs and gills for oxygen and carbondioxide or body surface for heat exchange, this is not sensible for the corticalsurface.

Several studies have developed a theoretical approach to explain the empir-ical scaling relations [2, 12, 14, 15, 36]. Different to these studies, the presentapproach started from the notion that the computational power and informa-tion transfer are the most likely candidates as basic parameters in shaping thecortex. Since computational power depends directly on the number of neurons,which is again constant per cortical surface, it follows that cortical surface is tobe taken as the independent parameter.

Still, the relations are mainly based on relatively scarce data from heteroge-nous sources as collected by [4] and [2]. It would therefore be highly valuable toimprove the measurements, and with modern imaging techniques it is feasibleto measure much larger samples within species in a much more automated man-ner [26]. Such databases with larger samples should also enable to test morespecific patterns such as developmental and gender patterns [37] as well as localdifferences between regions of the cerebrum.

The strength of the new scaling relations is that they started from the sim-plest assumptions and the fewest number of parameters possible. Still, even ifthe presented scaling laws describe the scaling relations far better than earlierapproaches, this is not a proof that the laws are correct. There are differentfunctions with remarkably similar shapes that would fit the data equally well.Thus, it is to be preferred to derive the scaling relation on the basis of theoreticalconsiderations and then to test it to the available data.

The result suggests that the structure of our cerebrum has landed us in alocal optimum, because the wrinkled surface structure makes that many dis-tances are longer than necessary (think of the two flanking sides of a sulcus).In a nuclear structure such as the birds’ cerebrum this problem does probablynot exist [38].

In conclusion, the general basis for scaling of the cerebrum is the cortical

11

surface area, Ac. The equivalent volume is the volume the cerebrum would haveif its surface were lissencephalic (so for convoluted cerebrums the equivalentvolume is much larger than the actual volume).

Second, the amount of white matter in the cerebrum forms only a smallfraction in most brains, roughly 5% of the equivalent volume. The exact amountof white matter slightly deviates from this for brains of small size, and due tothe complex shape of large convoluted brains.

Third, the relation between cortical surface and cortical volume could befitted using just a single parameter, tlocal = 3.5 mm. The underlying theoryis based on efficient information processing and transmission. The tlocal of 3.5mm is consistent with an average gyral width of about 10 mm, and a maximallissencephalic radius of 10 mm (≈ 2 · 102 mm3 volume).

The theoretical foundation of the scaling laws suggests that the functions ofthe cerebrum, i.e., processing and transmission of information also are central indefining its macroscopic shape. This in turn suggests that the local structures inthe cerebrum are also strongly related to their function, which is consistent withthe fact that primary sulci tent to be located at the primary sensory and motorregions. Also the theory driven approach allows to make specific predictions forthe remaining interspecific variations. It seems likely that the same scaling lawshould also be applicable to the cerebellum, with the difference that the tlocalwill be smaller. Also, it seems likely that there are regional variations of tlocal,especially in large cerebrums [39].

Thus, the volume to surface relation does seem to be optimal. The opti-mization criterion is derived directly from the central function of the cerebrum:the processing and transfer of information. This result also has a direct and im-portant consequence for the way the brain works, because it means that smallcerebrums work fundamentally differently to large cerebrums. Whereas a in asmall cerebrum (say, a mouse’s) processes a unit, our cerebrum functions moreas a cluster of local processing units (the gyri) between which relatively littleinformation is exchanged. This is important, e.g., when comparing the behaviorof mice and man.

Acknowledgements

The author is supported by the German Federal Ministry of Education andResearch, BMBF [grant 01EC1003A]. I thank Heiko Wagner, Kim Bostrom,Markus Lappe, and Thomas Wulff for discussions and comments. I thank OnurGunturkun for his invitation to solve the problem.

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[15] Mota, B. & Herculano-Houzel, S. 2012 How the cortex gets its folds: Aninside-out, connectivity-driven model for the scaling of Mammalian corticalfolding. Front. Neuroanat., 6, 3. (doi:10.3389/fnana.2012.00003)

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[16] Hofman, M. A. 1988 Size and shape of the cerebral cortex in mammals. II.The cortical volume. Brain Behav. Evol., 32, 17–26.

[17] Harman, P. J. 1947 On the significance of fissuration of the isocortex. J.Comp. Neurol., 87(2), 161–168. (doi:10.1002/cne.900870206)

[18] Bok, S. T. 1929 Der Einfluß der in den Furchen und Windungenauftretenden Krummungen der Großhirnrinde auf die Rindenarchitek-tur. Zeitschr. gesamte Neur. Psych., 121(1), 682–750. (doi:10.1007/BF02864437)

[19] Rockel, A. J., Hiorns, R. W. & Powell, T. P. S. 1980 The basic uniformityin structure of the neocortex. Brain, 103(2), 221–244. (doi:10.1093/brain/103.2.221)

[20] Carlo, C. N. & Stevens, C. F. 2013 Structural uniformity of neocortex,revisited. Proc. Natl. Acad. Sci. USA, 110(4), 1488–1493. (doi:10.1073/pnas.1221398110)

[21] Perge, J. A., Koch, K., Miller, R., Sterling, P. & Balasubramanian, V. 2009How the optic nerve allocates space, energy capacity, and information. J.Neurosci., 29(24), 7917–7928. (doi:10.1523/JNEUROSCI.5200-08.2009)

[22] Perge, J. A., Niven, J. E., Mugnaini, E., Balasubramanian, V. & Sterling,P. 2012 Why do axons differ in caliber? J. Neurosci., 32(2), 626–638.(doi:10.1523/jneurosci.4254-11.2012)

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[26] Toro, R., Perron, M., Pike, B., Richer, L., Veillette, S., Pausova, Z. & Paus,T. 2008 Brain size and folding of the human cerebral cortex. Cereb. Cortex,18(10), 2352–2357. (doi:10.1093/cercor/bhm261)

[27] Jones, E. G. & Wise, S. P. 1977 Size, laminar and columnar distributionof efferent cells in the sensory-motor cortex of monkeys. J. Comp. Neurol.,175(2), 391–438. (doi:10.1002/cne.901750403)

[28] Kaiser, M., Hilgetag, C. C. & van Ooyen, A. 2009 A simple rule foraxon outgrowth and synaptic competition generates realistic connectionlengths and filling fractions. Cereb. Cortex, 19(12), 3001–3010. (doi:10.1093/cercor/bhp071)

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[29] Mountcastle, V. B. 1997 The columnar organization of the neocortex.Brain, 120(4), 701–722. (doi:10.1093/brain/120.4.701)

[30] Roerig, B. & Chen, B. 2002 Relationships of local inhibitory and excitatorycircuits to orientation preference maps in ferret visual cortex. CerebralCortex, 12(2), 187–198. (doi:10.1093/cercor/12.2.187)

[31] Eguıluz, V. M., Chialvo, D. R., Cecchi, G. A., Baliki, M. & Apkarian, A. V.2005 Scale-free brain functional networks. Phys. Rev. Lett., 94, 018 102.(doi:10.1103/PhysRevLett.94.018102)

[32] Markov, N. T., Ercsey-Ravasz, M., Lamy, C., Ribeiro Gomes, A. R., Ma-grou, L., Misery, P., Giroud, P., Barone, P., Dehay, C. et al. 2013 Therole of long-range connections on the specificity of the macaque interarealcortical network. Proc. Natl. Acad. Sci. USA, 110(13), 5187–5192. (doi:10.1073/pnas.1218972110)

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[35] Pausova, Z., Paus, T., Abrahamowicz, M., Almerigi, J., Arbour, N.,Bernard, M., Gaudet, D., Hanzalek, P., Hamet, P. et al. 2007 Genes, mater-nal smoking, and the offspring brain and body during adolescence: Designof the Saguenay Youth Study. Hum. Brain Mapp., 28(6), 502–518. (doi:10.1002/hbm.20402)

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Supplement 1: Properties and sensitivity of thescaling law

Properties of the scaling law

The limit behavior of equation (7) is visualized in Figure 1 on a double-log-arithmic scale. For clarity, the data are plotted on a linear scale in Figure S1.The linear presentation of the data stresses that vast range of cerebral sizes,considering that the bottom panels of the figure present a range (indicated bysmall squares) that is hardly visible on the top panels. The figure also gives abetter impression of the non-explained variance, and shows why it is necessaryto estimate the quality of the model on a logarithmic scale.

The model also predicts the equivalent cerebral thickness Tc, in equation (6).This relation is presented in Figure S2. It shows again the very good fit of themodel to the data.

Sensitivity of the scaling law

The model was based on a quadratic relation between brain size and informationresistance for long-range axonal connections (see section 2.4), but there are noreliable data to tell whether this really is the correct relation. A linear relationwould result in the following model:

σsV2/3c +

1

tlocalVc = Ac. (s1)

This equation would make the model prediction slightly less convex (see Fig.S3), but the prediction almost as good as the original model (Tmax = 4.9 mm;R = 0.996).

As shown in Figure 1b, the fit of the scaling law (eq. (7)) does not makesystematic errors. Provided that the model is correct, the errors that do remaincan be caused by two main types of causes: measurement errors and true varia-tions of the model parameters. The measurement errors are difficult to estimatebecause Hofman’s dataset is based on a large number of studies from differ-ent authors. Moreover, each data point represents just a single sample from aspecies, so intraspecific variability is not known. Instead we can estimate theinfluence of variability of the model parameters.

The sphericity parameter, s, estimated from 23 measurements of the datasetof Hofman [1] had a standard deviation (SD) of 0.09. This variability may againpartially reflect measurement errors. The range of the SD is indicated in FigureS3. This shows that variations in the sphericity parameter mainly affect themodel predictions for the small range of the cerebrums. In this range, most ofthe variability of the data is within the model range of s± 1 SD.

To estimate the sensitivity of the model to variations in the only fittedparameter, the maximal equivalent thickness tlocal. Changes in tlocal affect themodel in the range of large cerebrums. To illustrate this, the total range of

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s ± 1 SD and tlocal ± 1 mm is underlaid in red in Figure S3. Including thesevariations, 26 of 37 (70%) of the data fall within the modeled range.

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Supplement 2: Earlier theories

The theory of brain scaling has a long history. It is outside the scope to givea complete listing and I am not even sure to know all theories. A few areworth treating in in more detail though because they are cited and mentionedfrequently.

Surface-volume relations

The theories for the convoluted surface area of the cerebrum seem to have tolongest history Baillarger [2], Gross [10]. For a review, see [1]. For a long timeit has been thought that the folding is a result of a large cortical area beingfitted inside a comparably small space inside the skull. This model apparentlydates back to Le Gros Clark [4], and has been modeled numerically in a slightlymodified form [5, 17]. However, for this model to work, one has to assumethat the grey matter is incompressible, rubber-like in the dimensions parallelto the surface, and that the white matter is viscous. These are highly unlikelyproperties making the model as a whole questionable. Moreover, the model doesnot make any predictions as to the scaling relations of the cortical surface area.

The theory favored by Hofman, and that still seems to be popular was de-veloped by Prothero & Sundsten [6] in a number of works. According to thismodel, the gyral width and height are governed by the white matter of the long-range axonal connections. The strongest prediction of the model is that thereexists an upper limit to the size of the cerebral cortex. The strongest weaknessof the model is that it approximates the cerebrum with a cubical core of whitematter which has a surface of equally-shaped gyral ridges, separated by sulciof equal depth. In such a design, the gyri of a large cerebrum cannot containany white matter. However, in a true cerebrum, the depth of neighboring sulcishows strong differences, so in practice this is unlikely to present a real designproblem to the cerebrum.

According to one intuition the space required by the site matter is a shapingfactor for the gyri [6]. According to a second notion, the anisotropic materialproperties of the grey and white matter are essential shaping factors. For exam-ple, according to an early proposal, the cortex behaves like a thick, lubricatedrubber sheet that is packed inside a skull that is too small for its surface [4].It will fold (as Le Gros Clark demonstrated experimentally) because the sheetis hardy compressible in the tangential directions. This proposal has also beenmodeled numerically in a slightly modified form [5]. In this model, the corticalsheet did not grow against a skull, but against a compliant centripetal force.

A very interesting mechanism for the development of folding has been madeby Van Essen [7]. According to Van Essen, the mechanical stiffness of axonsmake that the cerebral white matter is a highly anisotropic material [8]. Itseems plausible that the mechanism of Van Essen accounts for the developmentof the relationship between cerebral volume and cortical surface.

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Grey-White matter volume

Zhang & Sejnowski [9] claimed that the relation between grey and white mattervolume in the cerebral cortex describes a monotonic power relation. However,in the same study, the authors acknowledged that the relations of the grey andwhite matter volumes (Vg, Vw) to the total volume (Vc = Vg+Vw) appear just asimpressively linear on a double logarithmic scale and that this is mathematicallyinconsistent: if Vw = kV 1.23

g then Vc = Vw + (kVw)1/1.23 and Vc = Vg + kV 1.23g ,

which are obviously not linear on a double logarithmic scale (Fig. S4). So whatis the problem here?

Part of the problem is, that in all but the largest brains the white mattervolume is almost negligible with respect to the grey matter volume. Thus,the grey to total matter volume has a slope of almost exactly 1 on the doublelogarithmic scale for most of the range and flattens off a little at the end of therange. The relations of Vw to Vg and Vc have slightly more variability, and soit is more difficult to decide how linear they really are. Thus, empirical doublelogarithmic plots are very fine, but the conclusion that they follow a simplepower relation on the basis that a relation is straight on the double-logarithmicscale, even over a wide range, must be taken with care.

A “general law”?

Zhang & Sejnowski [9] also derived what they called “a general law” to predictthe relation between the grey and white matter volume. Unfortunately, therewas a circularity in their equations as I demonstrate here. To avoid confusion,I will use symbols that are consistent with the present work.

Zhang and Sejnowski started with two assumptions. (1) That there is a directlinear relation between the cortical surface area (Ac) and the physiological cross-sectional area (PCSA) of the axons. (2) That the global geometry minimizes theaverage length of the axonal fibers (Lax). This latter assumption is somehowincomplete, because the shortest average length would obviously be zero.

The grey matter volume Vg is simply the surface Ac times the equivalentthickness Tg (cf. eq. (4)):

Vg = TgAc. (s2)

Given assumption no. 1, the white matter volume Vw is:

Vw = c1AcLax (s3)

Zhang and Sejnowski divided by 2 to acknowledge that axons have a start andan end, but a proportion of axons connects to basal ganglia and thalamus, so Isimply put in the constant c1 here and assume that it is scale-invariant.

The crucial step now is to obtain the relation between G and W . Zhangand Sejnowski simply postulated that the total grey matter volume Vg dependson the average axon length Lax. To match the dimensions they assumed thatVg = c2L

3. They tested this postulate against a family of alternative postulatesin which L3 is replaced by L3−nXn with n < 2 and X another length measure

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such as the cortical thickness T or average gyrus length, etc. They then showedthat Lax is smallest for the original postulate, assuming that all alternativeparameters X < L. Combining this postulate with equations (s2) and (s3)yields the “universal scaling law”:

Vw =c3TgG4/3. (s4)

This formulation is misleading because according to equation (s2) Vg equalsTgAc so “the law” really states:

Vw = c3T1/3g A

4/3c (s5)

so, in fact, the equation does not directly link W and G at all. If we replace Vwagain with equation (s3) we obtain again the postulation, so we have not learnedanything. The circularity in the derivation of their law occurred in their equation[13], where they substituted the axonal length from their equation [2] (eq. (s3)),which already been substituted in their equation [5] (i.e., the “universal scalinglaw”, eq. (s4)).

Other relations

The theoretical approach by Changizi [10] is interesting for its attempt to explaina whole set of scaling relations of the mammalian cerebrum at once, and thusaiming at a general theory, instead of focussing on just one relation at thetime. A weakness of his approach is that it is based on the power fitted fromdouble-logarithmic relations of a long, heterogenous list of studies, without anyreference to the quality of these fits. For example, he lists a power of 0.08-0.197for the relation between cortical thickness and grey matter volume althoughthis relation deviates strongly from a power relation (cf. Fig. S2). His modelconsists of two parts. The first part predicts that the number of synapsesper volume of dendritic tree is a scale-invariant constant. However he fails tomention the implicit assumption that the branching rate must be independentof dendritic length. The second part builds upon this, and a number of far-reaching assumptions. For example, it is assumed that each region of the brainconnects to a limited, fixed fraction of other regions, independent of the numberof regions in the cerebrum. Not only is the fraction of connected regions assumedconstant, also the fraction of neurons connected to in each connected area isassumed constant.

A recent modeling approach has been presented by Mota & Herculano-Houzel [11], to develop a computational framework for the data on cell countsin the cerebrum and other brain regions in a large range of mammals by SusanaHerculano-Houzel et al. [12–16]. This model treats the cerebral volume as thesum of grey and white matter. It has neurons, axons and glia as major param-eters, and is explorative in nature, meaning that it is an attempt to list andidentify probable parameters that underly the empirical scaling laws. The mainresult is that it is a complex matter.

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Supplement 3: Validation data

The data used for validating the scaling laws were taken from Toro [17], Zhang& Sejnowski [9], and Hofman [1].

References

[1] Hofman, M. A. 1989 On the evolution and geometry of the brain inmammals. Prog. Neurobiol., 32(2), 137–158. (doi:10.1016/0301-0082(89)90013-0)

[2] Baillarger, J. 1845 De l’etendue de la surface du cerveau. Gazette desHopiteaux, 18, 179.

[3] Gross, C. G. 1999 Brain, vision, memory: Tales in the history of neuro-science. Bradford books.

[4] Le Gros Clark, W. 1945 Deformation patterns on the cerebral cortex. InEssays on growth and form, pp. 1–22. Oxford: Oxford University Press.

[5] Toro, R. & Burnod, Y. 2005 A morphogenetic model for the developmentof cortical convolutions. Cereb. Cortex, 15(12), 1900–1913. (doi:10.1093/cercor/bhi068)

[6] Prothero, J. W. & Sundsten, J. W. 1984 Folding of the cerebral cortex inmammals: a scaling model. Brain Behav. Evol., 24(2-3), 152–167.

[7] Van Essen, D. C. 1997 A tension-based theory of morphogenesis and com-pact wiring in the central nervous system. Nature, 385(6614), 313–318.(doi:10.1038/385313a0)

[8] Peter, S. J. & Mofrad, M. R. K. 2012 Computational modeling of axonalmicrotubule bundles under tension. Biophysical J., 102(4), 749–757. (doi:10.1016/j.bpj.2011.11.4024)

[9] Zhang, K. & Sejnowski, T. J. 2000 A universal scaling law between graymatter and white matter of cerebral cortex. Proc. Natl. Acad. Sci. USA,97(10), 5621–5626. (doi:10.1073/pnas.090504197)

[10] Changizi, M. A. 2001 Principles underlying mammalian neocortical scaling.Biol. Cybern., 84(3), 207–215. (doi:10.1007/s004220000205)

[11] Mota, B. & Herculano-Houzel, S. 2012 How the cortex gets its folds: Aninside-out, connectivity-driven model for the scaling of Mammalian corticalfolding. Front. Neuroanat., 6, 3. (doi:10.3389/fnana.2012.00003)

[12] Gabi, M., Collins, C. E., Wong, P., Torres, L. B., Kaas, J. H. & Herculano-Houzel, S. 2010 Cellular scaling rules for the brains of an extended numberof primate species. Brain Behav. Evol., 76, 32–44. (doi:10.1159/000319872)

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[13] Herculano-Houzel, S. 2009 The human brain in numbers: A linearly scaled-up primate brain. Front. Hum. Neurosci., 3, 31. (doi:10.3389/neuro.09.031.2009)

[14] Herculano-Houzel, S., Mota, B., Wong, P. & Kaas, J. H. 2010 Connectivity-driven white matter scaling and folding in primate cerebral cortex.Proc. Natl. Acad. Sci. USA, 107(44), 19 008–19 013. (doi:10.1073/pnas.1012590107)

[15] Herculano-Houzel, S. 2011 Not all brains are made the same: New viewson brain scaling in evolution. Brain Behav. Evol., 78, 22–36. (doi:10.1159/000327318)

[16] Herculano-Houzel, S. 2012 The remarkable, yet not extraordinary, humanbrain as a scaled-up primate brain and its associated cost. Proc. Natl. Acad.Sci. USA, 109(S1), 10 661–10 668. (doi:10.1073/pnas.1201895109)

[17] Toro, R. 2012 On the possible shapes of the brain. Evol. Biol., 39(4),600–612. (doi:10.1007/s11692-012-9201-8)

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