Universality in human cortical folding in health and diseaseeprint.ncl.ac.uk/file_store/production/228440/7C3B... · to the interspecies scaling law. To test if both the previously

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Universality in human cortical folding in healthand diseaseYujiang Wang ∗, Joe Necus †, Marcus Kaiser ∗ † , and Bruno Mota ‡

∗Interdisciplinary Computing and Complex BioSystems (ICOS), School of computing Science, Newcastle University, Newcastle upon Tyne NE1 7RU, UK,†Institute of Neuro-

science, Newcastle University, Newcastle upon Tyne NE2 4HH, UK, and ‡Instituto de Fisica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

Submitted to Proceedings of the National Academy of Sciences of the United States of America

The folding of the cortex in mammalian brains across species hasrecently been shown to follow a universal scaling law that can bederived from a simple physics model. However, it was yet to be de-termined whether this law also applies to the morphological diversityof different individuals in a single species, in particular with respectto factors such as age, gender and disease.To this end, we derived and investigated the cortical morphologyfrom magnetic resonance imaging (MRI) of over 1000 healthy hu-man subjects from three independent public databases.Our results show that all three MRI datasets follow the scalinglaw obtained from the comparative neuroanatomical data, whichstrengthens the case for the existence of a common mechanism forcortical folding. Additionally, for comparable age groups, both maleand female brains scale in exactly the same way, despite systematicdifferences in size and folding. Furthermore, age introduces a sys-tematic shift in the offset of the scaling law. In the model, thiscan be interpreted as changes in the mechanical forces acting onthe cortex. We also applied this analysis to a dataset derived fromcomparable cohorts of Alzheimer’s patients and healthy subjects ofsimilar age. We demonstrate a systematically lower offset, and apossible change in the exponent for Alzheimer’s subjects comparedto the control cohort.Finally, we discuss implications of the changes in offset and exponentin the data, and relate it to existing literature. We thus provide apossible mechanistic link between previously independent observa-tions.

brain morphogenesis, cortical gyrification, folding, aging, Alzheimer’s disease

Abbreviations: MRI, Magnetic resonance imaging; MRE, Magnetic Resonance Elas-

tography; CSF, cerebral spinal fluid;

IntroductionThe expansion of the cerebral cortex is the most obvious fea-ture of mammalian brain evolution, and is generally accom-panied by increasing degrees of folding of the cortical surface.The mechanisms that drive gyrification have been a matter ofintense research interest lately [1, 2, 3, 4], with a number ofproposals being put forward to explain it. Most such stud-ies have focused on human cortices, using detailed MRI datato postulate folding as driven by the (possibly differential [5]or multilayered [6]) expansion of the cortical surface. In con-trast, we have recently proposed a model [7] in which foldingis a consequence of the dynamics of surface expansion and self-avoidance coupled with a negative tension term. This modelwas partly inspired by Van Essens axonal tension hypothesis,[8] and the statistical physics of membranes [9]. By assum-ing that healthy adult mammalian cortices have a shape thatminimizes an effective free energy term that takes into accountthese effects, this model predicts a power-law relation betweencortical average thickness T , exposed area Ae and total areaAt, namely

AtT1/2 = kA5/4

e . [1]

The only free parameter is k, or offset, a dimensionlesscoefficient that is presumed to be related to both the ax-onal tension and the pressure of cerebral-spinal fluid (see also

suppl. text in [7]). In geometric terms, the variables T ,At and Ae associated to each cortex define a point in thelog T × logAt × logAe space, and eq. (1) constrains thesepoints to a plane: 1

2log T + logAt − 5

4logAt = log k In effect,

this eliminates the degree of freedom associated with the di-

rection ~k = { 12, 1, 5

4}. In other words, this approach essentially

provides a mapping to a new, more natural, set of variableswith which to describe cortical morphology.

Using data from different mammalian species, we had pre-viously verified that this relation is closely followed by gyrifiedand lissencephalic cortices. In adaptive terms, cortical foldingscales universally across clades and species, implying a singleconserved mechanism throughout evolution. Indeed, compar-ing AtT

1/2 with Ae for all 51 land mammals in our data showsan excellent fit for AtT

1/2 = kA1.305±0.007e . The empirical ex-

ponent for Ae, 1.305, is very close to but statistically distinctfrom the predicted 1.25. We previously discussed that thesource of this may have a number of reasons [7] (indeed weprovide a possible new explanation in our Discussion).

Regardless, the existence of such precise regularity as pre-dicted by model derived from simple assumptions is quite re-markable and revealing. This does not at all imply that otheromitted details are not important; but rather that they simplyare not the main drivers of the coarse-grained cortical mor-phology. At this whole-cortex level, gyrification appears to bedetermined by an extremely limited set of degrees of freedom,described by a simple scaling law. The exact explanation ofthis relation in terms of our model has also been questioned[10, 11] and defended [12]. However, even if one is not con-vinced about its proposed explanation, the fact remains that

Significance

Despite of the enormous diversity in size and function of themammalian cerebral cortex, it has been shown that the cortexof different species fold according to a simple universal law. Inthis study we investigate if this law also applies to variationwithin a single species, our own. Specifically, we examine howthe law is affected by gender, age or the presence of Alzheimer’sdisease. By investigating and quantifying what remains invari-ant and what changes in each case, we shed some light onthe underlying mechanisms through which the cortex changesin health and disease, and argue that morphological complexitycould emerge from a few simple rules.

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a strong and seemingly universal empirical relation exists be-tween the coarse-grained morphological features of cortices ofdifferent mammalian species. It is therefore worthwhile to in-vestigate whether the same relation holds for individuals of asingle species.

Previous studies of cortical folding in a single species mainlyfocused on MRI data in humans, and examined the changesin gyrification for example with age [13], gender [14] and indementia [15]. We now seek to test our model against suchhuman cortical surface reconstructions from MRI data, andexamine its applicability over variations across different hu-man subjects in gender, age and health or disease. This allowsus to delve deeper into the factors that neither a purely com-parative approach, with one or two specimens for each species,nor a descriptive detailed analysis of one or a few human cor-tices, can examine.

Specifically, we investigate how diversity in cortical mor-phology between species (as reported by comparative neu-roanatomy [7]) relates to diversity within a species (as reportedby human MRI). We examine how the factors that drive thelatter affect the model parameters, both those constrained (ex-ponents) and unconstrained (offset) by theory. For instance, itis known that the cortex generally shrinks and slightly changesin folding with age [16, 17, 13, 18], and that males have onaverage larger brain than females, with a slightly increaseddegree of folding [19, 17, 14]. However, it was unknown if andhow these changes affect the scaling law.

ResultsHuman brains fold like other mammalian cortices. Previously,Mota & Herculano-Houzel [7] demonstrated that althoughmammalian cortices vary greatly in size, thickness, and thedegree of folding, empirically they all closely follow a simpleuniversal scaling law. Here, we investigate if such an invariantrelation also exists within and across cohorts of human sub-jects. Not only is this testing the previously proposed theoryin an intraspecies study (in contrast to the previous inter-species study), but the range over which the measurementsof total cortical surface area, cortical thickness, and exposedsurface area vary is also dramatically reduced. Finally, themethod of data acquisition differs: here we use MRI-derivedmeasures of cortical surface area (At), cortical thickness (T ),and exposed surface area (Ae), rather than manual tracing ofscanned cortical slices. Given these major differences in dataacquisition, species, and range of measurements, it is not im-mediately obvious that the human cohort data will conformto the interspecies scaling law.

To test if both the previously published interspecies dataand our human MRI derived data are comparable, we showboth in Fig. 1 in the same plot. We used three independenthealthy datasets which were acquired on separate scanners,using different acquisition protocols. These datasets are HCP,OASIS and NKI (see Methods for more details). Indeed, theMRI derived datasets align well with the previously obtainedmammalian dataset, despite the fundamentally different meth-ods of obtaining them. In particular, the adult human dataset(HCP, see Fig. 1 left column) shows a very good agreementwith the regression based on the previously published mam-malian dataset, apart from a slight systematic offset.

This alignment can be quantified by comparing the quan-tity k′ = AtT

1/2A−1.305e derived from the fit for the compar-

ative neuroanatomy dataset (grey line in Fig. 1). We obtaink′ComparativeNeuroanatomy = 0.1861±0.0184, k′HCP = 0.1822±0.0057, k′OASIS = 0.1669±0.0086 and k′NKI = 0.1786±0.0109.The variance in the comparative neuroanatomy dataset is

somewhat larger to that of the human MRI datasets, whilethe means in the latter all fall within one standard deviationof the former, indicating that the linear fit in the compara-tive neuroanatomy dataset is also a similarly good fit for thehuman MRI dataset, except for some change in the offset.Strikingly, the HCP dataset has a much smaller variance thanthe other human MRI datasets and we will see later that thisis due to the narrow age range in the HCP data.

The systematic differences in offset between the humanMRI datasets are probably attributable to different imag-ing protocols, scan resolution, processing pipelines, and fieldstrength, which are known to produce systematically differ-ent measured values, most notably for the cortical thickness[20, 21, 22, 23, 24]. Indeed, we show that cortical thickness dif-fers systematically between the datasets (SI Appendix, TextS1). The three different datasets also contain different ageranges and gender distributions. Thus, they are not aggre-gated in the following analyses, and we will investigate theeffect of gender and age on the parameters of the scaling lawseparately.

Male and female brains fold in the same way, despite dif-ferences in average sizes & folding index. In the same agegroup, on average females have smaller brains with a smallercerebrum volume and a smaller total cortical surface area[25, 19, 17]. We reproduced these findings in all our MRIderived datasets and also show that there are no gender dif-ferences in cortical thickness, but a slight difference in thegyrification index (At/Ae). Fig. 2 shows an example fromthe HCP dataset, in the age range of 22-25 years old. Wechose a very narrow age range in adulthood to eliminate theeffect of age on the results. The exposed and total surfaceareas are significantly smaller in females compared to males(p� 0.001), and the effect size is very large (Cohen’s D > 1.5)(see Fig. 2 (a,b)). However, there is no difference in averagecortical thickness between male and females (p = 0.59, Co-hen’s D = −0.09) (see Fig. 2 (c)). To test if both surfaceareas changed systematically we also tested for differences ingyrification index. We found it varies significantly betweenmales and females (p � 0.001), with a moderate effect size(Cohen’s D = 0.69) (Fig. 2 (d)). This essentially indicatesthat female brain are slightly less folded than male brains ofthe same age.

To test if a single invariant relationship exists for both malesand females despite the difference in the gyrification index, weperformed a multiple linear regression analysis with gender asa categorical variable on the same HCP dataset, in the agerange of 22-25. Indeed, the same scaling law is obeyed byboth males and females, with no significant differences in off-set or slope (Fig. 2 (e)). In fact the regression lines almostoverlap for the range of the measurement. The previously ob-served differences between males and females are still visible(females tend to be found near the bottom left, males near thetop right in Fig. 2 (e)), but in a way that is perfectly alignedalong the direction of the scaling law. We repeated this anal-ysis on all the other age groups and cohorts and come to thesame conclusion (SI Appendix, Text S2). In other words, de-spite differences between males and females in terms of totalgrey matter surface areas, exposed surface areas, and degreesof gyrification, the manner in which these quantities changeis perfectly constrained by the previously proposed invariantrelationship; both obey the same principles in how the corticalfolding occurs.

We conclude that there is a single scaling law relating corti-cal average thickness, total and exposed areas, for both maleand female subjects of the same age. Thus, we will not differ-

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entiate between genders in the following analysis of the effectof age.

Throughout the healthy lifespan, the scaling changes in offsetbut not in slope. We examined the effect of age on the scalinglaw, as it is also well documented that cortical thickness, totaland exposed grey matter surface area, and the gyrification in-dex change with age [16, 17, 13]. In terms of the scaling law wefound no change in the slope, but a systematic decrease in theoffset with increasing age (Fig. 3) in all datasets. With age,average values additionally shift systematically (illustrated bythe black line in Fig 3 (a) indicating the age group average,which moves in negative y direction, and in negative x direc-tion), as the brain shrinks with increasing age.

We compared the slope of all age groups in all cohortsagainst each other and against the theoretically predictedvalue of 1.25 (Fig 3 (b)). Essentially there is no differencein the slope estimation, as all 95% confidence intervals of theslope overlap with each other, and with the predicted value of1.25. One exception is the NKI age group 35-45 years, whichfalls just short of 1.25. We show that this is likely due to an un-even distribution of age in the group (SI Appendix, Text S4).We also repeated the multiple linear regression analysis withage as a continuous regressor, and found in all datasets a sig-nificant age main effect (in all cohorts p� 0.001), but no ageinteraction effect on the slope (in all cohorts p > 0.01, also seeSI Appendix, Text S2); and the slope is not different to 1.25within a 95% confidence interval.

Finally, we estimated the rate of decrease in offset with age(Fig 3 (c)) in the different cohorts. The estimate was obtainedby assuming a slope of 1.25 (see Methods). Despite some sys-tematic differences between the cohorts, the evolution of theoffset with age appear very consistent across cohorts. Theoffset appears to decrease rapidly between the age of 4 and30 years (especially as seen from the NKI cohort) and moregradually afterwards.

In Alzheimer’s disease, the scaling behaviour is significantlyaltered. Finally, we investigated if the scaling behaviour shownin the previous sections is altered in brain diseases. Specifi-cally, we focused on Alzheimer’s disease, as there are knownchanges in cortical thickness, total brain volume, and gyrifi-cation [26, 27, 15]. To that end we use the publicly availableADNI database, which contains a big cohort of patients andcontrols.

As a first step, we repeated our previous analysis of gen-der differences in different age groups on the ADNI data.In the control group (CTRL), as expected, we detected nodifference in scaling between males and females, but in theAlzheimer’s patient group (AD) we detected significant differ-ences in slope and/or offset in two age groups (SI Appendix,Text S5). Hence, we performed our subsequent analysis andcomparison of patients vs. controls for different genders sepa-rately.

When we examined the slope of the regression, we foundno significant difference to the theoretically predicted slope of1.25 in the control male or female group (with the exceptionof 55-64 y.o. male CTRL, probably caused by the low samplesize of 4 subjects). In Alzheimer’s patients, we found a sig-nificantly different slope to 1.25 in both male and females inthe age group of 65-74 years old (see Fig. 4 (a)). In females,the slope is significantly higher than 1.25, and different to thecontrol group. In males, the slope is significantly lower than1.25, but not different to the control group. To test if thisslope change is genuine, we further restricted the age rangeto 70-74 years old, and detected essentially the same effect in

both males and females. We also performed a multiple linearregression, with age as a continuous regressor, and with genderand disease as categorical regressors. We found a significantinteraction effect of gender, disease, and gender:disease on theslope (p = 0.002, 0.005 and 0.01, respectively), meaning thatdepending on gender and disease the slope of regression maychange significantly. Finally, we repeated the entire analysiswith a subset of the data, where the patients and controlswere age and gender matched. The main findings highlightedhere essentially stay the same. Together, these results hint ata possible genuine slope difference between genders, patientsand controls.

Next, we turned our attention to the offset of the scalinglaw, which we showed in healthy controls to be a monoton-ically decreasing function of age. We repeated the analysisas before, by estimating the offset using the theoretically pre-dicted slope of 1.25. In the control group, we found that theestimated offset decreases with age for both males and females,as expected (Fig. 4 (b)). However, we did not find the sametrend in the Alzheimer’s patients. The offset in the AD groupstarts off lower than the CTRL group, but essentially stays atthis level throughout age. These observations were confirmedby performing a multiple linear regression analysis, with ageas a continuous regressor, and with gender and disease as cat-egorical regressors (significant main effects in age, disease, andage:disease with p� 0.001, and a main effect in gender:diseasewith p = 0.031).

The most striking difference between the AD and CTRLgroup is their difference in offset regardless of age and gender,most likely due the significantly lower cortical thickness in theAD group (also see SI Appendix, Text S6). Furthermore, thedecrease in offset with age, seen in other healthy cohorts, isalso found in the CTRL, but not AD group. Finally, thereare indications that males and females differ in the AD grouprelative to each other, as well as relative to the control groupin terms of offset and slope.

DiscussionIn conclusion, adult healthy human cortices conform to thesame theoretically predicted relation that also seems to applyto the full range of mammalian species. Analysed separately,healthy human adult MRI data from a variety of sources alsofit the predicted power law well, although different data setshave small systematic differences in the offset parameter. Eachdataset can be further segmented by age and gender. We showthat the gyrification of male and female cortices of similarages follow the same scaling rule, even though the latter areon average slightly smaller and less gyrified than the former.Over different age groups, however, we show that while theexponents do not vary significantly, the offset term decreasessmoothly. This suggests that the universal mechanisms thatshape the mammalian cortex continue to operate over the de-velopment of healthy adults, and that possibly all variationacross age in terms of these coarse-grained variables may beattributed to a monotonic change in the mechanical propertiesof the brain.

It is important to note that this relation between the threecoarse-grained variables, Ae, At and T , does not fully specifythe shape of the cortex, but rather constrains it by eliminat-ing one degree of freedom. Thus, these values can vary sig-nificantly between healthy individuals, and may change sig-nificantly with age. However, they are constrained in their

changes in such a way that the combination AtT1/2A

−5/4e =

k(age) remains almost constant for each age. This is the mainimplication of Fig. 3.

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Taken separately and disaggregated by age, the slopes of thescaling laws in all the MRI datasets agree well with the theo-retically predicted slope of 1.25. This is significant in anotherway, because for this value, and for this value only, one canscale all areas as multiples of the square of the average corticalthickness, and obtain a power law relation between an intrin-sic (At) and an extrinsic (Ae) measure of the cortical surface.The exponent of the latter is then simply half the fractal di-

mension of the surface: From eq. (1), AtT2 = k

(AeT2

)df/2, and

df = 5/2. Thus, empirically, the human cortical surface seemsto scale like a fractal of dimension 2.5 for typical lengths largerthan k−1/2T [28, 29, 27].

With age, the value of k decreases monotonically, and it ap-pears that the rate of decrease is initially (up to the early twen-ties) faster. There are a range of factors that could accountfor this observation. As the theory postulates, k is related tothe negative tension term of the cortical surface. Changes in kcould for example correlate with changes in the elasto-plasticproperties of underlying white matter, or it could correspondto changes in cerebral spinal fluid (CSF) pressure. In the liter-ature, there is evidence of slight changes of CSF pressure withage [30], but systematic studies are lacking, making conclusiveanswers impossible at this stage. In terms of the elasto-plasticproperties of the white matter, one study systematically mea-sured the stiffness of the white matter using Magnetic Res-onance Elastography (MRE) and found no changes with age[31]. However, another study, also using MRE, reported asystematic decrease in viscoelastic modulus with age [32]. In-terestingly, a significantly reduced stiffness was recorded inpatients with Alzheimer’s disease [33], where we also find adecreased k value. Altogether, both theory and observationseem to suggest that the value of k is a function of the inter-nal tensions and external pressure applied to the grey matter.This is one of the key contributions one can derive from thescaling law. It suggests, for the first time, the existence of alink between mechanical properties of the brain with measuresof brain morphology. In other words, we can hope to mecha-nistically relate MRI derived measures of the brain with MREor other mechanical measurements of the brain via the scalinglaw. To fully accomplish this goal, however, will probably takea more fine-grained iteration of our model of gyrification.

It is also worth noting that the change in k with age inhealthy subjects is not simply an effect of decreasing corticalthickness (see SI Appendix, Text S6), but rather an emergenteffect from correlated changes in all three primary variablesof cortical thickness, total surface area, and exposed surfacearea. Hence, k can be understood as a new, more naturalvariable, which on its own might correlate with biologicallyinterpretable properties of the brain. Indeed, we also showthat k might not be the only biologically relevant new variable(SI Appendix, Text S6). This reinterpreting of the previouslyproposed scaling law as a constraint is another contributionfrom this work. Even if one is not convinced about the sim-plifications and assumptions that have been made to deducethe proposed scaling law, it becomes clear that it captures aprincipal direction in the data, which other measures, such asthe gyrification index, do not.

In Fig. 3 we demonstrated that age-related changes intro-duced further variance in the data, which when not accountedfor, may skew the data fit. Indeed the direction of the agecomponent of the variance is towards negative x = log(Ae)

and negative y = log(√TAt). This might explain the pre-

viously found statistically significant difference between thetheoretical exponent (1.25) and empirical (1.305) exponent inthe comparative neuroanatomy data [7]. As age was not ac-

counted for in each species, the variance along the direction ofthe age-related change might incorrectly lead to a regressionthat gives a higher slope than expected. Similarly, it would beinteresting to investigate the interspecies variation for other,non-human species, and to check how much of the variabilityis due variations that are intraspecific, as opposed to inter-specific. If humans are typical, we may very well verify thatthere is more of the former than the latter, which would betruly remarkable. Similar to analogous studies on the sourceof variance in cortical area [34, 35], such a comparison mightalso hint at the evolutionary source of the variability.

In Alzheimer’s disease, we show a slope significantly differ-ent than 1.25 in the data for a subset of subjects. There areseveral potential reasons for this. It could be a genuine effectin that the slope is altered compared to the healthy brains,in which case, the proposed theory breaks down and cannotprovide any more insight. A more fine-grained analysis mightbe needed in this case to determine what the alterations are inAlzheimer’s disease. However, we might also be lumping a het-erogeneous cohort of subjects (e.g. different in disease severity,or duration, or time from onset) together in one group. Inde-pendent of this, the systematically low values of k, comparableto those of very old healthy cortices, tentatively suggests thatthe main morphological effect of Alzheimer’s disease is akin toa premature aging of the cortex.

More generally, in this and previous works we used coarse-grained measures for an entire hemisphere. Thus strong re-gional variations in e.g. cortical thickness may skew the dataaway from theoretical predictions. Future work could extendthe model to incorporate more fine-grained variables, such asfor example the systematic differences in cortical thickness be-tween gyri and sulci. Similarly, the segmentation of a hemi-sphere into e.g. lobes might yield additional regional insight.This approach may be particularly useful in the study of dis-eases that affect different cortical regions differently. Indeed,this is going to be the subject of a forthcoming paper.

We would also like to note that the term ‘universality’ refersspecifically to the universality of the scaling exponent. A keycontribution of this study is that we have shown that the offsetin the scaling law is not universal, and that it indeed changeswith age. One question that arises is if other measures couldalso show universal properties. For example, the gyrificationindex ( At

Ae) has been shown to be a useful measure in detect-

ing subtle changes in cortical folding (e.g. [36]). However, weshow in our SI Appendix, Text S6 that the gyrification indexdoes not appear to capture the trend in the data in a univer-sal manner. Further studies are needed to show if any otherbiologically interpretable measures also show universality.

There are also important caveats to the claim of universal-ity in the scaling law. Even for similar age groups, there is ofcourse still some residual variation for the value of the offsetaround the predicted ‘universal’ value. This indicates theremay be other neglected processes influencing the cortical ge-ometry. The theoretical model we use to predict the slopeis a coarse-grained approximation that relies on a number ofsimplifying assumptions. That it manages to capture so muchof its morphological outcome in such a simple framework isremarkable, but it cannot be regarded as a full or final de-scription.

Finally, we would like to highlight that our proposed scalinglaw relationship, although simple in nature, is highly meaning-ful and not a mere linear regression analysis. As other stud-ies, our model also directly addresses the underlying mecha-nisms through its assumptions. Where other studies chooseto test assumptions through simulations to recreate a cortexthat is then compared to the shape of the brain (bottom-

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up approach), we test our assumptions through derivationsof properties that should be fulfilled given the assumptions(top-down approach). In our case, such a property is a relationbetween three (hitherto independent) morphological variables.The relation is derived from a theoretical model with a sin-gle free parameter (k), and was then verified empirically, withexponents that were very close (comparative neuroanatomy)or statistically the same (human MRI) as the expected val-ues. To our knowledge no other proposed mechanism has ei-ther predicted this regularity, or generated it numerically insimulations. Nevertheless, the fact remains that the field issplit between competing gyrification models and approaches[1], each of which may explain some but not all features ofcortical morphology. We believe that the bottom-up and thetop-down approaches are in fact complementary, and that onlywhen they are successfully combined, a more complete and de-tailed model will emerge.

Materials and Methods

Data sources & processing.

We used four publicly available datasets in this study. The HCP dataset has been

obtained from 500 subjects release from the Human Connectome Project (also see

[37]). The OASIS data has been obtained from the Open Access Series of Imaging

Studies project. We used the cross-sectional dataset [38], and only included the subset

of healthy subjects. The NKI data [39] has been obtained from Nathan Kline Institute

(NKI) Rockland Sample. The ADNI data has been obtained from the Alzheimer’s

Disease Neuroimaging Initiative. We included subjects from ADNI1, ADNI GO, and

ADNI2 [40], and selected all 3T sessions. More details for all the datasets, includ-

ing where our derived data is available for download, can be found in SI Appendix,

Text S7.

As we only used publicly available datasets, no informed consent procedure was

required. We confirm that we complied with all the data usage policies of each of the

datasets we used.

For all datasets we used Freesurfer for processing. We ran the Freesurfer recon-all

pipeline on all subjects in the NKI and ADNI datasets. For the HCP dataset, we used

the preprocessed package provided by the human connectome project, which already

includes the Freesurfer subjects. Details regarding the preprocessing can be found in

[37]. For the OASIS dataset we used the Freesurfer subjects provided.

To find the exposed surface, we ran the local gyrification pipeline in Freesurfer with

the standard settings, and use the surfaces *h.pial-outer-smoothed produced by that

pipeline.

We then calculated the cortical thickness, total, and exposed surface area from the

Freesurfer outputs, details can be found in SI Appendix, Text S7.

Statistical analysis.

To test for differences in surface area, cortical thickness, and gyrification index, we

used t-tests to determine the p-value, To determine effect size, Cohen’s D was used:

D = m2−m1s

, m1,m2 being the means of the two groups to be compared.

s =

√(n1−2)∗s21+(n2−1)∗s22)

n1+n2−2, with n1, n2 as the sample size for each group,

and s1, s2 is the standard deviation for each group.

To determine differences in the scaling law between gender or age groups we used

multiple linear regression with gender or age as a categorical variable. We report

p-value for the main and interaction effects.

Offset estimation.

To estimate the offset in the scaling law, we decided not to use a regression-based

approach, as the estimates for offsets can vary widely with small changes in the slope.

Instead, we chose an alternative approach:

Assuming a linear relationship with the slope of 1.25 as predicted by the theory,

we can rewrite the relationship y = 1.25x + c as c = y − 1.25x. Here,

x = log10(Ae), y = log10(At

√T ); and c = log(k) is the offset for each

pair of values of x and y.

ACKNOWLEDGMENTS. We thank Simon Keller, Peter Neal Taylor, RobertForsyth, Gavin Clowry, Cheol Han, Sol Lim, and Suzana Herculano-Houzel for fruitfuldiscussions.

YW is supported by the Newcastle University Santander Collaboration Fund, andthe Newton Mobility Award. JN is supported by the Reece Foundation. MK issupported by the Engineering and Physical Sciences Research Council of the UnitedKingdom (EP/K026992/1).

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Comp. neuroanat.

HCPOASISNKI

x=log10(Ae)

y=log 10(A

t T1/2 )

2 3 4 5

6

5

4

3

2

1

4.5 4.74.6

5.4

5.3

5.2

5.1

5.04.5 4.74.6

5.4

5.3

5.2

5.1

5.0

Fig. 1. Comparing MRI derived human datasets to the estab-lished neuroanatomy dataset for mammalian brains. The scaling law

for the comparative neuroanatomy data across different species is shown in grey. The grey

regression line is also obtained for this data, with a slope of 1.307. Overlaid are the human

MRI-derived datasets: green for HCP, blue for OASIS and magenta for NKI. The insets (red

frame) show the human MRI derived data in more detail relative to the regression line of the

inter-species data. Note the two grey dots in the zoom-ins are from the previous comparative

neuroanatomy dataset for humans .

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12

10

8

6

4

2

00 0.6 0.8 1 1.2 1.4x105[mm2]

Total grey matter surface area

d=1.6172p=1.1911e-18

Count

Count

10

8

6

4

2

0 3.2 3.6 4 4.4 4.8x104[mm2]

Exposed grey matter surface area

d=1.7368p=1.2062e-20

141210864220 2 2.5 3 3.5

Average cortical thickness

d=-0.0866p=0.5903

[mm]

Count

1412108642202 2.5 3 3.5

Gyrification index

d=0.6943p=2.7296e-5

Count

F: slope 1.1898M: slope 1.2033

5.35

5,3

5.25

5.2

5.15

5.14.5 4.52 4.54 4.56 4.58 4.6 4.62 4.64 4.66 4.68

x=log10(Ae)

y=log 10(A

t T1/2 )

(a) (b)

(c)

(e)

(d)

Fig. 2. Gender differences in exposed and total surface area,cortical thickness, gyrification index, and scaling behaviour. The

subjects are selected from a narrow adult age range (22-25 years old in the HCP dataset) to

exclude interaction effects with age. p-values are from a two tailed t-test, d represents effect

size measured by Cohens D (see Methods). SI Appendix, Text S2 also repeats this analysis for

other age groups and datasets.

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OASIS5.3

5.25

5.2

5.15

5.1

5.054.5 4.52 4.54 4.56 4.58 4.6 4.62 4.64 4.66 4.68

x=log10(Ae)

y=log 10(A

t T1/2 )

15-2425-3435-4445-5455-6465-7475-8475-84

HCPOASISNKI

1.6

1.4

1.2

1

0.8

0.6 15 25 35 45 55 65 75 85 95Age [years]

Slope

HCPOASISNKI

Offset log(k)

15 25 35 45 55 65 75 85 95Age [years]

-0.45

-0.5

-0.55

-0.6

(a)

(b)

(c)

Fig. 3. Change in the scaling behaviour with age. (a) Scaling law

for different age groups in the OASIS dataset using both males and females. Regression lines

are show in the corresponding colour as the age group. The age group average is shown as a

black solid line. (b) The scaling law slope estimate is shown with 95% confidence intervals

for each age group for all three datasets. The solid grey line indicates the 1.25 slope predicted

by the theory. (c) Using the predicted slope of 1.25, the average estimated offset is shown

for all three datasets over age (see Methods for details). The error bars indicate the standard

deviation. The original data for the estimated offset is shown in SI Appendix, Fig. S3

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ADCTRL

2

1.5

1

0.5

Slope

55 65 75 85 95Age [years]

ADCTRL

2

1.5

1

0.5

Slope

55 65 75 85 95Age [years]

-0.52

-0.6

-0.6655 65 75 85 95

Age [years]

Offset log(k)

-0.52

-0.6

-0.6655 65 75 85 95

Age [years]

Offset log(k)

ADCTRL

ADCTRL

(a)

(b)

Female

Female

Male

Male

Fig. 4. Scaling behaviour in Alzheimer’s disease. (a) Slope of scaling

law for females and males for different age groups of Alzheimer’s patients (AD) and healthy

controls (CTRL). The vertical dashed grey lines mark the boundaries used for the age groups.

The horizontal solid grey line is the theoretically predicted slope value of 1.25. (b) Mean and

standard deviations of the estimated offset are also shown. The offset was estimated assuming

the theoretically predicted slope value of 1.25.

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