Brain folding may play an important role in facilitating and inducing a variety of anatomical and functional organisation patterns (Welker, 1990; Toro, 2012). Among the many folded structures of the mammalian brain, the cortices of the cerebrum and the cerebellum are the two largest ones. In humans, the cerebellum has ~80% of the surface area of the cerebral cortex (Sereno et al., 2020), and contains ~80% of all brain neurons, although it represents only ~10% of the brain mass (Azevedo et al., 2009). Both cortices are organised in layers, with neuronal cell bodies occupying a superficial position and principal neurons – pyramidal neurons in the cerebral cortex, Purkinje cells in the cerebellum – sending axons towards an internal white matter. Many aspects of the organisation, development, and evolution of the cerebral and cerebellar cortex are, however, very different. While the cerebral cortex is unique to mammals, the cerebellum is present in all vertebrates, with cerebellum-like structures which can be even identified in invertebrates such as octopuses (Nieuwenhuys et al., 1998; Shigeno et al., 2018). While the main target of pyramidal cells is the cerebral cortex itself (through profuse cortico-cortical connections), Purkinje cell afferents are mostly feedforward (Ramnani, 2006; Nieuwenhuys et al., 2008; Yamamoto et al., 2012). While dendritic trees of pyramidal neurons can be found in all cortical layers, those of Purkinje cells are restricted to the molecular layer – their nuclei forming a boundary between the molecular and granular layers. While the cortex is only folded in relatively large mammals, the cerebellum is folded in all of them, and is also folded in birds (Cunha et al., 2020; Cunha et al., 2021), some fish (Yopak et al., 2007) but not in reptiles (Larsell, 1926; Nieuwenhuys et al., 1998). It is interesting then to compare folding in both structures as a first step towards understanding how the mechanics of folding could influence the organisation in these two different types of tissue (Franze, 2013; Kroenke and Bayly, 2018; Foubet et al., 2019; Heuer and Toro, 2019).

Here, we aimed at characterising the folding of the cerebellar cortex using histological data from a sample of 56 mammalian species. The analysis of cerebellar folding is made difficult by the small size of the folia. Magnetic resonance imaging, the main tool for studying the folding of the cerebral cortex, does not provide the resolution required to distinguish and reconstruct cerebellar folding. Histological data, even at low scanning resolution, can provide sufficient information to distinguish folia as well as cerebellar cortical layers. However, data is only 2D, and sectioning often induces non-trivial deformations which make it challenging to create full 3D reconstructions (see, however, the work of Sereno et al., 2020, and Zheng et al., 2023). Using histological data, Ashwell, 2020, was able to estimate the surface area in 90 species of marsupials and 57 species of eutherian mammals, but did not study folia, of which the measurement remains challenging.

In our analyses we manually segmented the surface of the cerebellum and developed a method to automatically detect the boundary between the molecular and granular layers, allowing us to estimate the median thickness of the molecular layer. We also developed a method to detect individual folia, and to measure their median width and perimeter. Buckling models of brain folding suggest an inverse relationship between folding frequency and cortical thickness, with thicker cortices leading to lower frequency folding (Toro and Burnod, 2005). This has been verified in the mammalian cerebrum (Mota and Herculano-Houzel, 2015), and we aimed at studying it in the cerebellum. The analysis of phylogenetic data is challenging, because different species share a varying degree of genetic relationship. We used phylogenetic comparative methods to condition our analyses on the phylogenetic tree and tested different evolutionary scenarios, which allowed us to study the evolution of the cerebellum and its relationship with the cerebrum and with body size.

Our results showed two very different groups of phenotypes: those like brain size, that changed over several orders of magnitude, and those like folial width which were much more conserved. We confirm the strong allometry between the size of the cerebellum and the cerebrum across mammals (Barton and Harvey, 2000; Barton, 2002; Whiting and Barton, 2003; Herculano-Houzel, 2010; Barton, 2012; Smaers et al., 2018; Ashwell, 2020; Magielse et al., 2023) and show similarly strong relationships for the width of cerebellar folds and the thickness of the molecular layer, although with a much narrower range of variation. This deeply conserved pattern suggests the presence of a common mechanism underlying the development of the cerebellum and cerebellar folding across mammals. All our data and code are openly available online.

The presentation of our results is structured as follows: (1) we validate our measurements by comparing them with the literature, (2) we show the results of the fit of the three different evolutionary models to our data, and decide on the one which will be used in the following analyses to constrain the variance structure, (3) we describe the structure of correlations among our phenotypes and present their main multivariate patterns of variation, two different sets of phenotypes appeared, a first set showing large variation across several orders of magnitude, and a second set which is much more conserved in comparison. (4) We present bivariate allometry results to illustrate these relationships, (5) finally we estimate the ancestral states of the two main patterns of variation in the data, which provides us with an idea of their evolutionary history.

Validation of measurements

Cerebellar and cerebral section area

Our measurements were a good proxy for total volume and correlated well with those reported in the literature. We segmented the coronal mid-section of the cerebellum and the cerebrum. These section areas should scale as the 2/3 power of the volume. We confirmed that this was the case by comparing our measurements with the volume measurements reported by Smaers et al., 2018: the correlation captured ~96% of the variance (Figure 6a and b). Smaers et al., 2018, report separate values for the medial cerebellum (the vermal region) (based on measurements from MacLeod et al., 2003; Maseko et al., 2012; Smaers et al., 2011). Our cerebellum section area measurement also correlated strongly with their medial cerebellum volume measurement, a fact that could be eventually used in the future for imputing missing data, although we do not use medial cerebellar measurements in our analyses. See Figure 6—figure supplement 1 for correlation with measurements in Ashwell, 2020.

Figure 6 with 1 supplement see all

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Folding frequency

There are no comparative analyses of the frequency of cerebellar folding, to our knowledge. The closest is the measurements reported by Sultan and Braitenberg, 1993, measuring the length of structures which may include several individual folia. Our measurements of cerebellar folial width and folial perimeter, however, do not correlate significantly with Sultan and Braitenberg’s measurements (Figure 6c). Ashwell, 2020, studied cerebellar folding through a foliation index, defined as the ratio of the cerebellar pial surface over the external cerebellar surface. This method, however, does not provide information about folding frequency, and would be unable to distinguish a cerebellum with many shallow folds from one with a few deep ones, which could both produce a similar foliation index (e.g., Figure 3d).

Molecular layer thickness

Although several previous works report volume of the molecular layer, we were unable to find data on molecular layer thickness across species. Zheng et al., 2023, reported 0.32 mm (±0.08) for humans, and our estimation was of 0.29 mm, which is well within the confidence interval.

Correlation structure

Anatomical phenotypes segregate into ‘diverse’ and ‘stable’ groups

All measurements were strongly positively correlated (Figure 7a). To better understand the global structure of these correlations, we computed partial correlations, which are the correlations between pairs of variables after controlling for the effect of all other variables. Two groups of phenotypes appeared: First, a group of ‘diverse’ phenotypes that varied widely across species, including body weight, brain weight, cerebellar, and cerebral section area and length, characterised by strong positive partial correlations in the range from 0.3 to 0.7. Cerebellar section area, for example, varies over ~2.5 orders of magnitude, and body weight over 10.8 orders of magnitude. Second, a group of ‘stable’ phenotypes that showed much less variation in comparison, including folial width, folial period, and thickness of the molecular layer, with positive partial correlations in the range from 0.4 to 0.6. There was only a <4-fold variation in these phenotypes across species (~0.5 order of magnitude).

Figure 7

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Phenotypes within the diverse and stable groups showed significant positive within-group partial correlations. Within the diverse group, the most important positive partial correlation between cerebellum and cerebrum was between cerebral section area (a 2D section of cerebral volume) and cerebellar section area (a 2D section of cerebellar volume). The diverse and the stable groups were linked through significant negative correlations, which were the largest we observed. They linked (1) cerebellar section length and folial perimeter and (2) cerebellar section length and thickness of the molecular layer (p-values <10^–2), indicating that larger cerebella tend to have relatively smaller and thinner folia than what could be expected for their size.

Our simulations showed that, because of the strength of the allometric pattern in our data, we had excellent statistical power to detect large and medium partial correlations (90–100%) and good statistical power to detect small partial correlations (~78%, see Supplementary file 2 for details).

The strongest pattern of variation concerned phenotypes from the diverse group, followed by those in the stable group

We performed a phylogenetic PCA to explore further the main patterns of neuroanatomical variability.

The phylogenetic PCA of all cerebellar and cerebral measurements produced a PC1 which captured the largest part of the variance: 96.4% (Figure 8b). PC1 described a strong concerted change in body size and brain size, cerebellar and cerebral section area and length, and loaded weakly on folial width, folial perimeter, and molecular layer thickness. PC1 describes the most important pattern of coordinated allometric variation (Jolicoeur, 1963), which will be studied further in the next subsection.

Figure 8

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PC2 conveys the main way in which individuals deviate from the dominant allometric pattern. The proportion of variance captured by PC2 was 2.6%. Because of the prominent role of body weight in PC1, PC2 mostly allowed uncoupling body size from changes in brain anatomy (Figure 8b). Plotting all species in PC1/PC2 space (Figure 8a) showed that humans were here an important exception, with a small body size relative to brain size. To some extent, this was also the case for other primate species in our sample (chimpanzee, rhesus macaque, etc.).

To better understand the patterns of neuroanatomical variation, we performed an additional PCA excluding body and brain weight. This is because the large variation in body weight, in particular, obscured patterns of variation specific to the brain. This time, PC1 captured 92.1% of the variance while PC2 captured 3.0%. The allometric pattern described by PC1 was similar to the previous one, with strong concerted changes in cerebellar and cerebral section area and length, and less strong changes in folial width, folial perimeter, and molecular layer thickness. PC2 was different, and allowed to uncouple changes in folial width and folial perimeter from changes in molecular layer thickness. Here, humans appeared among the species having the highest frequency of cerebellar folia (smaller folial width and perimeter).

Allometry

Allometry is the study of the relationships between size and shape. As we mentioned earlier, when objects change only in size without changing shape, the length, surface area, and volume of their parts change in a characteristic way: length as the square root of surface area, surface area as the ⅔ power of the volume, etc. The multivariate allometry pattern of all our measurements, given by PC1, is shown in Table 3 (also shown in Figure 8b). Pairwise allometric scaling factors can be obtained from this table as the ratio between the loadings of the respective measurements.

Table 3

Body weight	Brain weight	Cerebellar section area	Cerebral section area	Cerebellar section length	Cerebral section length	Folial width	Folial period	Thickness
0.7923	0.5453	0.1519	0.1563	0.1138	0.1117	0.0237	0.0187	0.0336

The following analyses use bivariate allometry to focus into particular pairs of phenotypes. First, we focus on those belonging to the ‘diverse’ group, and then on those belonging to the ‘stable’ group.

Cerebellar folding increases with cerebellar size in a similar manner as cerebral folding increases with cerebral size

The bivariate analyses Figures 9 and 10 zoom into the allometry of cerebellar and cerebral folding. In both cases, our measurement of length – a section of the cerebellar and cerebral surface – correlated strongly with the corresponding measurement of section area – a section of cerebellar and cerebral volume. As we indicated previously, their partial correlations were also strong, suggesting a direct link between both phenotypes. The scaling slopes were clearly higher than 0.5 (the slope of isometric scaling of length versus area), indicating that cerebellar and cerebral cortices are increasingly folded when larger (hyper-allometric scaling). The increase of folding with size followed a similar law for both structures, with a scaling slope of 0.75–0.77 and overlapping 95% confidence intervals. The multivariate allometric slopes were similar to the bivariate slopes: 0.75 for the cerebellum and 0.72 for the cerebrum. In the case of the cerebrum, we confirmed that the Florida manatee was exceptionally unfolded: three times less cortical length than expected. For the cerebellum, we observed that humans had a particularly folded cerebellum: 1.5 times more cerebellar length than expected.

Figure 9

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Figure 10

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The bivariate plot of cerebellar versus cerebral section area (Figure 9c) showed a strong correlation, however, their moderate partial correlation indicates that an important part of this correlation is mediated (including a significant negative partial correlation between cerebellar section area and cerebral section length, Figure 7c). The allometric scaling slope suggested that the cerebellum may be slightly proportionally smaller in species with large brains. In the case of isometry, the scaling slope should be 1. The observed scaling slope was slightly smaller than the isometric slope which was nevertheless at the margin of the 95% confidence interval (grey region in the figures), indicating a potential hypo-allometry of the cerebellum.

Folial width, folial perimeter, and the thickness of the molecular layer increase slightly with cerebellar size

Folial width, folial perimeter, and molecular layer thickness are all measurements of length which should also scale as the square root of cerebellar section area in case of isometry. We observed a much smaller scaling factor, indicating that when comparing small and large brains, those measurements increase substantially less than what could be expected from increases in cerebellar size (hypo-allometric scaling). The width and perimeter of cerebellar folia increased only slightly with cerebellar size (Figure 10). The scaling was markedly hypo-allometric: 0.10 for folial perimeter and 0.14 for folial width where isometric scaling should be 0.5. The scaling of the molecular layer’s thickness was also hypo-allometric, but with a slightly higher scaling slope of 0.2. Folial width and perimeter appeared to be more heterogeneous than molecular layer thickness: linear regression with cerebellar section area captured only 28% of folial width variance, and 14% of folial perimeter variance, while it captured 64% of molecular layer thickness variance (Figure 10c).

Ancestral phenotype estimation

Our ancestral phenotype estimations suggest evolutionary trajectories for the increases in brain size as well as for the decoupling between ‘diverse’ phenotypes (cerebellar and cerebral section area and length) and ‘stable’ phenotypes (folial width, folial perimeter, and thickness of the molecular layer). We made estimations of the ancestral states of our phenotypes based on the OU model fit. Our findings are presented for the evolution of PC1 and PC2, which provide a condensed representation of the concerted changes in the nine phenotypes studied. As indicated previously, the large variations in body weight make it the dominating feature in PC1. To zoom into the main patterns of neuroanatomical variation, we also show estimations of ancestral states excluding body and brain weight.

The increase in body size and brain size have decoupled in primates

When body and brain weight are included, PC1 (capturing 96.4% of the variance) showed the evolution of increasing body weight together with the brain phenotypes in the ‘diverse’ group (Figure 11a). These increases were estimated to appear first among the clade Scrotifera (harbour seal, zebu, pig, etc.) and later among primates. This set of phenotypes also decreased relative to the predicted common ancestor, for example, among Glires (rabbit, guinea pig, etc.) and Pilosa (eastern mole, common shrew, etc.). PC2 described the main deviations from this pattern, highlighting very particularly the group of primates, where brain sizes increased while body sizes stayed small. The common ancestor of mammals is predicted to have had a folded cerebrum with a correspondingly folded cerebellum, similar to that of a Rock hyrax.

Figure 11

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Diverse and stable phenotypes have decoupled in several branches of the phylogenetic tree

When excluding body and brain weight, PC1 loaded strongly on brain phenotypes from the ‘diverse’ group (Figure 11b). The ancestral estimation of PC1 (92.8% of the variance) appeared very similar to the previous one, highlighting the early concerted increase in cerebellar and cerebral size among Scrotifera, followed later by primates. The pattern of PC2, however, was now different from the previous PC2. It described an early decoupling of the group of ‘diverse’ and ‘stable’ brain phenotypes among primates, as well as a more recent decoupling among Scrotifera, especially Mustelida (harbour seal, ringtail cat, etc.). The cerebrum of the manatee is a well-known exception to the allometric relationship between brain size and folding: despite its large volume, it is almost completely lissencephalic. The evolutionary trajectory of the increase in manatee brain size (cerebrum and cerebellum) is clear in both PC1 ancestral estimations, however, the decoupling between diverse and stable phenotypes is not present.

‘Despite the impressive beauty of its wiring diagrams, the “neuronal machine” concept of the cerebellum remained vaguely defined as “a relatively simple machine devoted to some essential information processing.” I was frustrated enough at the Salishan meeting to ask what else experimentalists would need to uncover before we would be able to understand the meaning of these wiring diagrams. Someone equally frustrated replied that the available diagrams were too simple to construct even a primitive radio, so more information was urgently needed before any meaningful model could be conceived.’- M Ito, The Cerebellum (2011)

Whereas the circuits of the cerebral cortex are characterised by the presence of profuse re-entering loops (most connections of the cerebral cortex originate in other regions of the same cerebral cortex), the cerebellum exhibits an almost perfect feedforward structure, with an organisation whose regularity and simplicity has baffled generations of experimentalists and theoreticians. The impressive multi-scale regularity of cerebellar structure led to the idea of the ‘neuronal machine’ (Eccles et al., 1967). The quote from Yamamoto et al., 2012, illustrates the challenge of imagining how the obstinate repetition of a single interconnection pattern among a reduced number of components could lead to the complex cerebellar function. However, its impressive evolutionary conservation suggests that its function has to fulfil an important role, for motion and cognition (Whiting and Barton, 2003; Ramnani, 2006; Barton and Venditti, 2014; Magielse et al., 2022).

One possibility is that the complexity of cerebellar function should not be found at the level of the individual circuits – as would be the case for a radio – but that those circuits would provide the substrate over which complexity would emerge, as waves in the ocean or patterns in a vibrating Chladni plate. A deeper understanding of the mesoscopic and macroscopic scales of organisation would be probably more appropriate to understanding phenomena unique to this level of organisation, complementing the extensive exploration of microscopic circuits and molecular properties of cerebellar cells.

Our results provide a closer look into the nature of macroscopic cerebellar anatomy, its relationship with cerebral anatomy, its diversity across mammalian species, and its evolution. First, we showed that our computational neuroanatomy workflow, for which code and data have been made available open source, produces reliable results comparable with those of the literature. Second, we showed that the OU model, often used to model stabilising selection, is the one that fits phenotypic diversity the best. The OU model tended to dampen extreme variation, driving phenotypes towards the mean faster than the BM model would. Third, we showed that brain phenotypes segregated into a group of ‘diverse’ phenotypes, which increased over several orders of magnitude together with body size, and a second group of ‘stable’ phenotypes, which increased only slightly with brain size. Both groups were linked by significant negative partial correlations. The stable phenotypes were related to the local shape of cerebellar folding: folial width, folial perimeter, and thickness of the molecular layer. They changed over only ~0.5 order of magnitude between species with differences in brain size of ~2.5 orders of magnitude and difference in body size of ~11 orders of magnitude. Fourth, our allometric analyses confirmed the strong correlation between cerebellar and cerebral size (Barton and Harvey, 2000; Barton, 2002; Whiting and Barton, 2003; Herculano-Houzel, 2010; Barton, 2012; Smaers et al., 2018; Ashwell, 2020), and we extended these results to show that the same strong relationship holds for cerebellar folding: larger cerebella appear to be disproportionately more folded than smaller ones. Additionally, the allometric law for the increase of cerebellar folding was the same as for the increase of cerebral folding. Fifth, we estimated the evolutionary trajectories which may have led to the current phenotypic diversity. The main evolutionary trajectory showed the concerted increases and decreases in body size, brain size, cerebellar and cerebral size. Primates appeared as an outlier to this trend, with large brains relative to their body sizes. Focusing on brain phenotypes, our results showed a decoupling between diverse and stable phenotypes operating in several branches of the phylogenetic tree. This second pattern was, however, much more subtle than the global concerted increase in body and brain size.

Our results are influenced by a series of methodological choices. We will discuss the most important ones. First, we use a single cerebellar section and a single cerebral section for each species. More sections would increase the accuracy of our measurements, but would require a substantial manual segmentation effort. For our research question, having global measurements for many species was more relevant than having dense measurements for a few species. The previous work of Sultan and Braitenberg, 1993, is remarkable in this respect, providing beautiful representations of the unfolded cerebellar surface for 15 different species. However, their method produced smaller cerebellar surface areas than in the same species using a more precise method, suggesting that it underestimates the actual values (Sereno et al., 2020; Zheng et al., 2023), and hence the magnitude of cerebellar folding. Producing complete reconstructions of the cerebellar surface remains challenging, and as far as we know, only three cerebella have been precisely reconstructed: two from humans and one from a macaque (Sereno et al., 2020; Zheng et al., 2023). Second, we decided to use global measurements of brain anatomy. Although we computed folial width and folial perimeter for each folium (Figure 3a–c), and molecular layer thickness at numerous points for each brain section (Figure 4), we condensed all these local measurements into three median values. The cerebrum and the cerebellum show significant regional variability, but using local measurements would increase the number of phenotypes, largely exceeding the number of species. Fitting phylogenetic models with a significantly greater number of phenotypes than species is challenging (in particular, there is an ongoing debate regarding the use of OU models, see the work of Cooper et al., 2016; Grabowski et al., 2023). The use of penalisation techniques and constraints in the model’s structure holds great promise and will enable the exploration of new hypotheses regarding the evolution of complex phenotypes (Clavel et al., 2015; Clavel et al., 2019). Finally, we decided to measure the raw material without aiming at correcting for shrinkage. Our reasoning was that if the relationships were sufficiently robust, we should still be able to see them despite shrinkage. This appeared to be the case: accounting for shrinkage should enhance the accuracy of our measurements making our correlations even stronger and our estimations more precise, but should not change our conclusions. Shrinkage correction is often performed using correction factors obtained from the literature, which introduces a new source of methodological variability. A more precise account of shrinkage would require, for example, acquiring MRI data before and after fixation for each species (see Wehrl et al., 2015). Sections could be then registered to the undeformed MRI, allowing us to estimate deformation due to slicing and shrinkage.

All data analysed during this study is openly available at https://microdraw.pasteur.fr/project/brainmuseum-cb. Code for reproducing our analyses and our figures is openly available at https://github.com/neuroanatomy/comp-cb-folding (copy archived at Heuer et al., 2023b).

1. 1. Heuer K
   2. Traut N
   3. de Sousa AA
   4. Valk S
   5. Clavel J
   6. Toro R

   (2023) MicroDraw

   ID /project/brainmuseum-cb. Cerebellar and cerebral contours.

   https://microdraw.pasteur.fr/project/brainmuseum-cb

1. 1. Amunts K
   2. Lepage C
   3. Borgeat L
   4. Mohlberg H
   5. Dickscheid T
   6. Rousseau MÉ
   7. Bludau S
   8. Bazin PL
   9. Lewis LB
   10. Oros-Peusquens AM
   11. Shah NJ
   12. Lippert T
   13. Zilles K
   14. Evans AC

   (2013) BigBrain: an ultrahigh-resolution 3D human brain model

   Science 340:1472–1475.

   https://doi.org/10.1126/science.1235381

   • PubMed
   • Google Scholar
2. 1. Ashwell KWS

   (2020) Quantitative analysis of cerebellar morphology in monotreme, metatherian and eutherian mammals

   Zoology 139:125753.

   https://doi.org/10.1016/j.zool.2020.125753

   • PubMed
   • Google Scholar
3. 1. Azevedo FAC
   2. Carvalho LRB
   3. Grinberg LT
   4. Farfel JM
   5. Ferretti REL
   6. Leite REP
   7. Jacob Filho W
   8. Lent R
   9. Herculano-Houzel S

   (2009) Equal numbers of neuronal and nonneuronal cells make the human brain an isometrically scaled-up primate brain

   The Journal of Comparative Neurology 513:532–541.

   https://doi.org/10.1002/cne.21974

   • PubMed
   • Google Scholar
4. 1. Ballarin C
   2. Povinelli M
   3. Granato A
   4. Panin M
   5. Corain L
   6. Peruffo A
   7. Cozzi B

   (2016) The brain of the domestic bos taurus: Weight, encephalization and cerebellar quotients, and comparison with other domestic and wild cetartiodactyla

   PLOS ONE 11:e0154580.

   https://doi.org/10.1371/journal.pone.0154580

   • PubMed
   • Google Scholar
5. 1. Barton RA
   2. Harvey PH

   (2000) Mosaic evolution of brain structure in mammals

   Nature 405:1055–1058.

   https://doi.org/10.1038/35016580

   • PubMed
   • Google Scholar
6. 1. Barton RA

   (2002) How did brains evolve?

   Nature 415:134–135.

   https://doi.org/10.1038/415134a

   • PubMed
   • Google Scholar
7. 1. Barton RA

   (2012) Embodied cognitive evolution and the cerebellum

   Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences 367:2097–2107.

   https://doi.org/10.1098/rstb.2012.0112

   • PubMed
   • Google Scholar
8. 1. Barton RA
   2. Venditti C

   (2014) Rapid evolution of the cerebellum in humans and other great apes

   Current Biology 24:2440–2444.

   https://doi.org/10.1016/j.cub.2014.08.056

   • PubMed
   • Google Scholar
9. 1. Blomberg SP
   2. Garland T
   3. Ives AR

   (2003) Testing for phylogenetic signal in comparative data: behavioral traits are more labile

   Evolution; International Journal of Organic Evolution 57:717–745.

   https://doi.org/10.1111/j.0014-3820.2003.tb00285.x

   • PubMed
   • Google Scholar
10. 1. Borrell V

    (2018) How cells fold the cerebral cortex

    The Journal of Neuroscience 38:776–783.

    https://doi.org/10.1523/JNEUROSCI.1106-17.2017

    • Google Scholar
11. 1. Burger JR
    2. George MA
    3. Leadbetter C
    4. Shaikh F

    (2019) The allometry of brain size in mammals

    Journal of Mammalogy 100:276–283.

    https://doi.org/10.1093/jmammal/gyz043

    • Google Scholar
12. 1. Cavalli-Sforza LL
    2. Edwards AW

    (1967)

    Phylogenetic analysis. Models and estimation procedures

    American Journal of Human Genetics 19:233–257.

    • PubMed
    • Google Scholar
13. 1. Clavel J
    2. Escarguel G
    3. Merceron G

    (2015) mvMORPH: an R package for fitting multivariate evolutionary models to morphometric data

    Methods in Ecology and Evolution 6:1311–1319.

    https://doi.org/10.1111/2041-210X.12420

    • Google Scholar
14. 1. Clavel J
    2. Aristide L
    3. Morlon H

    (2019) A penalized likelihood framework for high-dimensional phylogenetic comparative methods and an application to new-world monkeys brain evolution

    Systematic Biology 68:93–116.

    https://doi.org/10.1093/sysbio/syy045

    • PubMed
    • Google Scholar
15. 1. Cooper N
    2. Thomas GH
    3. Venditti C
    4. Meade A
    5. Freckleton RP

    (2016) A cautionary note on the use of ornstein uhlenbeck models in macroevolutionary studies

    Biological Journal of the Linnean Society. Linnean Society of London 118:64–77.

    https://doi.org/10.1111/bij.12701

    • PubMed
    • Google Scholar
16. 1. Cunha F
    2. Racicot K
    3. Nahirney J
    4. Heuston C
    5. Wylie DR
    6. Iwaniuk AN

    (2020) Allometric scaling rules of the cerebellum in galliform birds

    Brain, Behavior and Evolution 95:78–92.

    https://doi.org/10.1159/000509069

    • PubMed
    • Google Scholar
17. 1. Cunha F
    2. Gutiérrez-Ibáñez C
    3. Racicot K
    4. Wylie DR
    5. Iwaniuk AN

    (2021) A quantitative analysis of cerebellar anatomy in birds

    Brain Structure & Function 226:2561–2583.

    https://doi.org/10.1007/s00429-021-02352-2

    • PubMed
    • Google Scholar
18. Book

    1. Eccles JC
    2. Ito M
    3. Szentágothai J

    (1967) The Cerebellum as a Neuronal Machine

    Springer-Verlag.

    https://doi.org/10.1007/978-3-662-13147-3

    • Google Scholar
19. 1. Felsenstein J

    (1985) Phylogenies and the comparative method

    The American Naturalist 125:1–15.

    https://doi.org/10.1086/284325

    • Google Scholar
20. 1. Felsenstein J

    (1988) Phylogenies and quantitative characters

    Annual Review of Ecology and Systematics 19:445–471.

    https://doi.org/10.1146/annurev.es.19.110188.002305

    • Google Scholar
21. 1. Fernández V
    2. Borrell V

    (2023) Developmental mechanisms of gyrification

    Current Opinion in Neurobiology 80:102711.

    https://doi.org/10.1016/j.conb.2023.102711

    • Google Scholar
22. 1. Fischl B
    2. Dale AM

    (2000) Measuring the thickness of the human cerebral cortex from magnetic resonance images

    PNAS 97:11050–11055.

    https://doi.org/10.1073/pnas.200033797

    • PubMed
    • Google Scholar
23. 1. Foubet O
    2. Trejo M
    3. Toro R

    (2019) Mechanical morphogenesis and the development of neocortical organisation

    Cortex; a Journal Devoted to the Study of the Nervous System and Behavior 118:315–326.

    https://doi.org/10.1016/j.cortex.2018.03.005

    • PubMed
    • Google Scholar
24. 1. Franze K

    (2013) The mechanical control of nervous system development

    Development 140:3069–3077.

    https://doi.org/10.1242/dev.079145

    • PubMed
    • Google Scholar
25. 1. Germanaud D
    2. Lefèvre J
    3. Toro R
    4. Fischer C
    5. Dubois J
    6. Hertz-Pannier L
    7. Mangin JF

    (2012) Larger is twistier: spectral analysis of gyrification (SPANGY) applied to adult brain size polymorphism

    NeuroImage 63:1257–1272.

    https://doi.org/10.1016/j.neuroimage.2012.07.053

    • PubMed
    • Google Scholar
26. 1. Germanaud D
    2. Lefèvre J
    3. Fischer C
    4. Bintner M
    5. Curie A
    6. des Portes V
    7. Eliez S
    8. Elmaleh-Bergès M
    9. Lamblin D
    10. Passemard S
    11. Operto G
    12. Schaer M
    13. Verloes A
    14. Toro R
    15. Mangin JF
    16. Hertz-Pannier L

    (2014) Simplified gyral pattern in severe developmental microcephalies? New insights from allometric modeling for spatial and spectral analysis of gyrification

    NeuroImage 102 Pt 2:317–331.

    https://doi.org/10.1016/j.neuroimage.2014.07.057

    • PubMed
    • Google Scholar
27. 1. Grabowski M
    2. Pienaar J
    3. Voje KL
    4. Andersson S
    5. Fuentes-González J
    6. Kopperud BT
    7. Moen DS
    8. Tsuboi M
    9. Uyeda J
    10. Hansen TF

    (2023) A Cautionary Note on “A Cautionary Note on the Use of Ornstein Uhlenbeck Models in Macroevolutionary Studies.”

    Systematic Biology 72:955–963.

    https://doi.org/10.1093/sysbio/syad012

    • PubMed
    • Google Scholar
28. 1. Hansen TF
    2. Martins EP

    (1996) Translating between microevolutionary process and macroevolutionary patterns: The correlation structure of interspecific data

    Evolution; International Journal of Organic Evolution 50:1404–1417.

    https://doi.org/10.1111/j.1558-5646.1996.tb03914.x

    • PubMed
    • Google Scholar
29. 1. Harmon LJ
    2. Losos JB
    3. Jonathan Davies T
    4. Gillespie RG
    5. Gittleman JL
    6. Bryan Jennings W
    7. Kozak KH
    8. McPeek MA
    9. Moreno-Roark F
    10. Near TJ
    11. Purvis A
    12. Ricklefs RE
    13. Schluter D
    14. Schulte Ii JA
    15. Seehausen O
    16. Sidlauskas BL
    17. Torres-Carvajal O
    18. Weir JT
    19. Mooers AØ

    (2010) Early bursts of body size and shape evolution are rare in comparative data

    Evolution; International Journal of Organic Evolution 64:2385–2396.

    https://doi.org/10.1111/j.1558-5646.2010.01025.x

    • PubMed
    • Google Scholar
30. 1. Herculano-Houzel S

    (2010) Coordinated scaling of cortical and cerebellar numbers of neurons

    Frontiers in Neuroanatomy 4:12.

    https://doi.org/10.3389/fnana.2010.00012

    • PubMed
    • Google Scholar
31. 1. Heuer K
    2. Gulban OF
    3. Bazin PL
    4. Osoianu A
    5. Valabregue R
    6. Santin M
    7. Herbin M
    8. Toro R

    (2019) Evolution of neocortical folding: A phylogenetic comparative analysis of MRI from 34 primate species

    Cortex; a Journal Devoted to the Study of the Nervous System and Behavior 118:275–291.

    https://doi.org/10.1016/j.cortex.2019.04.011

    • PubMed
    • Google Scholar
32. 1. Heuer K
    2. Toro R

    (2019) Role of mechanical morphogenesis in the development and evolution of the neocortex

    Physics of Life Reviews 31:233–239.

    https://doi.org/10.1016/j.plrev.2019.01.012

    • PubMed
    • Google Scholar
33. Software

    1. Heuer K
    2. Toro R
    3. Traut N

    (2023a) microdraw.py, version 887cb11

    GitHub.

    https://github.com/neuroanatomy/microdraw.py
34. Software

    1. Heuer T
    2. Traut N
    3. de Sousa AA
    4. Valk S
    5. Toro R

    (2023b) Comp-CB-folding, version swh:1:rev:8b603fb0bf711ece57ddb658d39d3fd5e01ff20e

    Software Heritage.

    https://archive.softwareheritage.org/swh:1:dir:bbc49898f53f9c083146be0cf689a86edf70b0e1;origin=https://github.com/neuroanatomy/comp-cb-folding;visit=swh:1:snp:09ee4d5660874454751a049493ef06dffe5e399d;anchor=swh:1:rev:8b603fb0bf711ece57ddb658d39d3fd5e01ff20e
35. 1. Iwaniuk AN
    2. Hurd PL
    3. Wylie DRW

    (2007) Comparative morphology of the avian cerebellum: II. Size of folia

    Brain, Behavior and Evolution 69:196–219.

    https://doi.org/10.1159/000096987

    • PubMed
    • Google Scholar
36. 1. Jolicoeur P

    (1963) 193. note: the multivariate generalization of the allometry equation

    Biometrics 19:497.

    https://doi.org/10.2307/2527939

    • Google Scholar
37. 1. Kaneko M
    2. Yamaguchi K
    3. Eiraku M
    4. Sato M
    5. Takata N
    6. Kiyohara Y
    7. Mishina M
    8. Hirase H
    9. Hashikawa T
    10. Kengaku M
    11. Chédotal A

    (2011) Remodeling of monoplanar purkinje cell dendrites during cerebellar circuit formation

    PLOS ONE 6:e20108.

    https://doi.org/10.1371/journal.pone.0020108

    • PubMed
    • Google Scholar
38. 1. Kroenke CD
    2. Bayly PV

    (2018) How forces fold the cerebral cortex

    The Journal of Neuroscience 38:767–775.

    https://doi.org/10.1523/JNEUROSCI.1105-17.2017

    • PubMed
    • Google Scholar
39. 1. Kumar S
    2. Stecher G
    3. Suleski M
    4. Hedges SB

    (2017) Timetree: a resource for timelines, timetrees, and divergence times

    Molecular Biology and Evolution 34:1812–1819.

    https://doi.org/10.1093/molbev/msx116

    • Google Scholar
40. 1. Larsell O

    (1926) The cerebellum of reptiles: Lizards and snake

    The Journal of Comparative Neurology 41:59–94.

    https://doi.org/10.1002/cne.900410103

    • Google Scholar
41. 1. Lawton AK
    2. Engstrom T
    3. Rohrbach D
    4. Omura M
    5. Turnbull DH
    6. Mamou J
    7. Zhang T
    8. Schwarz JM
    9. Joyner AL

    (2019) Cerebellar folding is initiated by mechanical constraints on a fluid-like layer without a cellular pre-pattern

    eLife 8:e45019.

    https://doi.org/10.7554/eLife.45019

    • PubMed
    • Google Scholar
42. 1. Leto K
    2. Arancillo M
    3. Becker EBE
    4. Buffo A
    5. Chiang C
    6. Ding B
    7. Dobyns WB
    8. Dusart I
    9. Haldipur P
    10. Hatten ME
    11. Hoshino M
    12. Joyner AL
    13. Kano M
    14. Kilpatrick DL
    15. Koibuchi N
    16. Marino S
    17. Martinez S
    18. Millen KJ
    19. Millner TO
    20. Miyata T
    21. Parmigiani E
    22. Schilling K
    23. Sekerková G
    24. Sillitoe RV
    25. Sotelo C
    26. Uesaka N
    27. Wefers A
    28. Wingate RJT
    29. Hawkes R

    (2016) Consensus paper: Cerebellar development

    Cerebellum 15:789–828.

    https://doi.org/10.1007/s12311-015-0724-2

    • PubMed
    • Google Scholar
43. 1. Lisney TJ
    2. Yopak KE
    3. Montgomery JC
    4. Collin SP

    (2008) Variation in brain organization and cerebellar foliation in chondrichthyans: batoids

    Brain, Behavior and Evolution 72:262–282.

    https://doi.org/10.1159/000171489

    • Google Scholar
44. 1. Llinares-Benadero C
    2. Borrell V

    (2019) Deconstructing cortical folding: genetic, cellular and mechanical determinants

    Nature Reviews. Neuroscience 20:161–176.

    https://doi.org/10.1038/s41583-018-0112-2

    • PubMed
    • Google Scholar
45. 1. MacLeod CE
    2. Zilles K
    3. Schleicher A
    4. Rilling JK
    5. Gibson KR

    (2003) Expansion of the neocerebellum in Hominoidea

    Journal of Human Evolution 44:401–429.

    https://doi.org/10.1016/s0047-2484(03)00028-9

    • PubMed
    • Google Scholar
46. 1. Magielse N
    2. Heuer K
    3. Toro R
    4. Schutter D
    5. Valk SL

    (2022) A comparative perspective on the cerebello-cerebral system and its link to cognition

    Cerebellum 1495.

    https://doi.org/10.1007/s12311-022-01495-0

    • PubMed
    • Google Scholar
47. Preprint

    1. Magielse N
    2. Toro R
    3. Steigauf V
    4. Abbaspour M
    5. Eickhoff SB
    6. Heuer K
    7. Valk SL

    (2023) Primate cerebellar scaling in connection to the cerebrum: a 34-species phylogenetic comparative analysis

    bioRxiv.

    https://doi.org/10.1101/2023.03.15.532597

    • Google Scholar
48. 1. Maseko BC
    2. Spocter MA
    3. Haagensen M
    4. Manger PR

    (2012) Elephants have relatively the largest cerebellum size of mammals

    The Anatomical Record 295:661–672.

    https://doi.org/10.1002/ar.22425

    • PubMed
    • Google Scholar
49. 1. Mora T
    2. Boudaoud A

    (2006) Buckling of swelling gels

    The European Physical Journal. E, Soft Matter 20:119–124.

    https://doi.org/10.1140/epje/i2005-10124-5

    • PubMed
    • Google Scholar
50. 1. Mota B
    2. Herculano-Houzel S

    (2015) Cortical folding scales universally with surface area and thickness, not number of neurons

    Science 349:74–77.

    https://doi.org/10.1126/science.aaa9101

    • Google Scholar
51. Book

    1. Nieuwenhuys R
    2. ten Donkelaar HJ
    3. Nicholson C

    (1998) The Central Nervous System of Vertebrates

    Springer Berlin, Heidelberg.

    https://doi.org/10.1007/978-3-642-18262-4

    • Google Scholar
52. Book

    1. Nieuwenhuys R
    2. Voogd J
    3. van Huijzen C

    (2008) The Human Central Nervous System

    Steinkopff Heidelberg.

    https://doi.org/10.1007/978-3-540-34686-9

    • Google Scholar
53. 1. Ramnani N

    (2006) The primate cortico-cerebellar system: anatomy and function

    Nature Reviews. Neuroscience 7:511–522.

    https://doi.org/10.1038/nrn1953

    • PubMed
    • Google Scholar
54. 1. Rash BG
    2. Arellano JI
    3. Duque A
    4. Rakic P

    (2023) Role of intracortical neuropil growth in the gyrification of the primate cerebral cortex

    PNAS 120:e2210967120.

    https://doi.org/10.1073/pnas.2210967120

    • PubMed
    • Google Scholar
55. 1. Revell LJ

    (2012) phytools: an R package for phylogenetic comparative biology (and other things)

    Methods in Ecology and Evolution 3:217–223.

    https://doi.org/10.1111/j.2041-210X.2011.00169.x

    • Google Scholar
56. 1. Schaer M
    2. Cuadra MB
    3. Tamarit L
    4. Lazeyras F
    5. Eliez S
    6. Thiran JP

    (2008) A surface-based approach to quantify local cortical gyrification

    IEEE Transactions on Medical Imaging 27:161–170.

    https://doi.org/10.1109/TMI.2007.903576

    • PubMed
    • Google Scholar
57. 1. Sereno MI
    2. Diedrichsen J
    3. Tachrount M
    4. Testa-Silva G
    5. d’Arceuil H
    6. De Zeeuw C

    (2020) The human cerebellum has almost 80% of the surface area of the neocortex

    PNAS 117:19538–19543.

    https://doi.org/10.1073/pnas.2002896117

    • PubMed
    • Google Scholar
58. 1. Shigeno S
    2. Andrews PLR
    3. Ponte G
    4. Fiorito G

    (2018) Cephalopod brains: an overview of current knowledge to facilitate comparison with vertebrates

    Frontiers in Physiology 9:952.

    https://doi.org/10.3389/fphys.2018.00952

    • PubMed
    • Google Scholar
59. 1. Smaers JB
    2. Steele J
    3. Zilles K

    (2011) Modeling the evolution of cortico-cerebellar systems in primates

    Annals of the New York Academy of Sciences 1225:176–190.

    https://doi.org/10.1111/j.1749-6632.2011.06003.x

    • PubMed
    • Google Scholar
60. 1. Smaers JB
    2. Turner AH
    3. Gómez-Robles A
    4. Sherwood CC

    (2018) A cerebellar substrate for cognition evolved multiple times independently in mammals

    eLife 7:e35696.

    https://doi.org/10.7554/eLife.35696

    • PubMed
    • Google Scholar
61. 1. Smaers JB
    2. Rothman RS
    3. Hudson DR
    4. Balanoff AM
    5. Beatty B
    6. Dechmann DKN
    7. de Vries D
    8. Dunn JC
    9. Fleagle JG
    10. Gilbert CC
    11. Goswami A
    12. Iwaniuk AN
    13. Jungers WL
    14. Kerney M
    15. Ksepka DT
    16. Manger PR
    17. Mongle CS
    18. Rohlf FJ
    19. Smith NA
    20. Soligo C
    21. Weisbecker V
    22. Safi K

    (2021) The evolution of mammalian brain size

    Science Advances 7:eabe2101.

    https://doi.org/10.1126/sciadv.abe2101

    • PubMed
    • Google Scholar
62. Book

    1. Stanković L
    2. Sejdić E

    (2019) Vertex-Frequency Analysis of Graph Signals

    Springer Cham.

    https://doi.org/10.1007/978-3-030-03574-7

    • Google Scholar
63. 1. Sultan F
    2. Braitenberg V

    (1993)

    Shapes and sizes of different mammalian cerebella: a study in quantitative comparative neuroanatomy

    Journal Fur Hirnforschung 34:79–92.

    • PubMed
    • Google Scholar
64. 1. Tallinen T
    2. Chung JY
    3. Biggins JS
    4. Mahadevan L

    (2014) Gyrification from constrained cortical expansion

    PNAS 111:12667–12672.

    https://doi.org/10.1073/pnas.1406015111

    • PubMed
    • Google Scholar
65. 1. Toro R
    2. Burnod Y

    (2005) A morphogenetic model for the development of cortical convolutions

    Cerebral Cortex 15:1900–1913.

    https://doi.org/10.1093/cercor/bhi068

    • PubMed
    • Google Scholar
66. 1. Toro R
    2. Perron M
    3. Pike B
    4. Richer L
    5. Veillette S
    6. Pausova Z
    7. Paus T

    (2008) Brain size and folding of the human cerebral cortex

    Cerebral Cortex 18:2352–2357.

    https://doi.org/10.1093/cercor/bhm261

    • PubMed
    • Google Scholar
67. 1. Toro R
    2. Chupin M
    3. Garnero L
    4. Leonard G
    5. Perron M
    6. Pike B
    7. Pitiot A
    8. Richer L
    9. Veillette S
    10. Pausova Z
    11. Paus T

    (2009) Brain volumes and Val66Met polymorphism of the BDNF gene: local or global effects?

    Brain Structure & Function 213:501–509.

    https://doi.org/10.1007/s00429-009-0203-y

    • PubMed
    • Google Scholar
68. 1. Toro R

    (2012) On the possible shapes of the brain

    Evolutionary Biology 39:600–612.

    https://doi.org/10.1007/s11692-012-9201-8

    • Google Scholar
69. 1. van der Walt S
    2. Schönberger JL
    3. Nunez-Iglesias J
    4. Boulogne F
    5. Warner JD
    6. Yager N
    7. Gouillart E
    8. Yu T
    9. scikit-image contributors

    (2014) scikit-image: image processing in Python

    PeerJ 2:e453.

    https://doi.org/10.7717/peerj.453

    • PubMed
    • Google Scholar
70. 1. Van Essen DC
    2. Donahue CJ
    3. Glasser MF

    (2018) Development and Evolution of Cerebral and Cerebellar Cortex

    Brain, Behavior and Evolution 91:158–169.

    https://doi.org/10.1159/000489943

    • PubMed
    • Google Scholar
71. 1. Van Essen DC

    (2020) A 2020 view of tension-based cortical morphogenesis

    PNAS 117:32868–32879.

    https://doi.org/10.1073/pnas.2016830117

    • PubMed
    • Google Scholar
72. 1. Wehrl HF
    2. Bezrukov I
    3. Wiehr S
    4. Lehnhoff M
    5. Fuchs K
    6. Mannheim JG
    7. Quintanilla-Martinez L
    8. Kohlhofer U
    9. Kneilling M
    10. Pichler BJ
    11. Sauter AW

    (2015) Assessment of murine brain tissue shrinkage caused by different histological fixatives using magnetic resonance and computed tomography imaging

    Histology and Histopathology 30:601–613.

    https://doi.org/10.14670/HH-30.601

    • PubMed
    • Google Scholar
73. Book

    1. Welker W

    (1990) Cerebral cortex

    In: Jones EG, Peters A, editors. Why Does Cerebral Cortex Fissure and Fold? Cerebral Cortex. Springer US. pp. 3–136.

    https://doi.org/10.1007/978-1-4615-3824-0

    • Google Scholar
74. 1. Whiting BA
    2. Barton RA

    (2003) The evolution of the cortico-cerebellar complex in primates: anatomical connections predict patterns of correlated evolution

    Journal of Human Evolution 44:3–10.

    https://doi.org/10.1016/s0047-2484(02)00162-8

    • PubMed
    • Google Scholar
75. Book

    1. Whittaker J

    (1990)

    Graphical Models in Applied Multivariate Statistics

    John Wiley and Sons.

    • Google Scholar
76. 1. Yamamoto R
    2. Takasuga S
    3. Yoshida Y
    4. Mafune S
    5. Kominami K
    6. Sutoh C
    7. Kato Y
    8. Yamauchi M
    9. Ito M
    10. Kanamura K
    11. Kinoshita M

    (2012) In vitro and in vivo transdermal iontophoretic delivery of naloxone, an opioid antagonist

    International Journal of Pharmaceutics 422:132–138.

    https://doi.org/10.1016/j.ijpharm.2011.10.042

    • PubMed
    • Google Scholar
77. 1. Yopak KE
    2. Lisney TJ
    3. Collin SP
    4. Montgomery JC

    (2007) Variation in brain organization and cerebellar foliation in chondrichthyans: sharks and holocephalans

    Brain, Behavior and Evolution 69:280–300.

    https://doi.org/10.1159/000100037

    • PubMed
    • Google Scholar
78. 1. Zheng J
    2. Yang Q
    3. Makris N
    4. Huang K
    5. Liang J
    6. Ye C
    7. Yu X
    8. Tian M
    9. Ma T
    10. Mou T
    11. Guo W
    12. Kikinis R
    13. Gao Y

    (2023) Three-dimensional digital reconstruction of the cerebellar cortex: lobule thickness, surface area measurements, and layer architecture

    Cerebellum 22:249–260.

    https://doi.org/10.1007/s12311-022-01390-8

    • PubMed
    • Google Scholar
79. 1. Zilles K
    2. Armstrong E
    3. Schleicher A
    4. Kretschmann HJ

    (1988) The human pattern of gyrification in the cerebral cortex

    Anatomy and Embryology 179:173–179.

    https://doi.org/10.1007/BF00304699

    • PubMed
    • Google Scholar
80. 1. Zilles K
    2. Armstrong E
    3. Moser KH
    4. Schleicher A
    5. Stephan H

    (1989) Gyrification in the cerebral cortex of primates

    Brain, Behavior and Evolution 34:143–150.

    https://doi.org/10.1159/000116500

    • PubMed
    • Google Scholar

Decision letter after peer review:

Thank you for submitting your article "Diversity and evolution of cerebellar folding in mammals" for consideration by eLife. Your article has been reviewed by 3 peer reviewers, including Jessica Dubois as the Reviewing Editor, and the evaluation has been overseen by Floris de Lange as the Senior Editor. The following individuals involved in the review of your submission have agreed to reveal their identity: Felipe Cunha (Reviewer # 1); Héloïse de Vareilles (Reviewer #2); Robert A Barton (Reviewer #3).

The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.

Essential revisions:

All reviewers acknowledged the importance and originality of your study. But they also raised a number of important concerns to consider.

The most critical one concerns the statistical analyses performed (see Reviewer #3's comments). Further evidence should be provided regarding the superiority of the Ornstein-Uhlenbeck model. Besides, the objectives and relevance of the partial correlations and principal component analyses to all included variables should also be better argued given the possible over-fitting and the underlying theoretical hypotheses which are not always clear.

The reviewers were also concerned about the global presentation of the manuscript. Additional description of some methods is required. The different results should be better linked, and potentially reordered. The discussion should also be reviewed in depth. Better highlighting the methodological novelties should also be considered.

We hope that you will be able to take all these general comments into account, as well as the more detailed comments of the reviewers (see below).

Reviewer #2 (Recommendations for the authors):

This study is both rich and generous, and I thank the authors for the beautiful work presented and shared with the community. The investigation of the cerebellum is both essential to our general understanding of the brain and under-explored, making the efforts developed in this article essential.

I find the analyses well-led. Nevertheless, I think the reach of the current work would gain from a better emphasis on why the analyses have been conducted and what they imply. There are also aspects of the structure of the article which I find unconventional and potentially confusing for the reader.

I will address my general recommendations following the current structure of the article.

I will not address more specific concerns (specific wording, punctuation marks…) that I have in this step of the review for two reasons:

1) I believe the more general recommendations may lead to a restructuring of the article which would make some of the specific concerns irrelevant.

2) The current draft does not include line numbers, making it challenging to address line-specific recommendations (please add line numbering to the next version if possible).

Introduction:

– I have identified that the last three paragraphs match an Aims (bottom of p.2) – Methods (first paragraph p.3) – Results (second paragraph p.3). Then, the buckling models consideration of the Methods paragraph is rather an aim than a method and should be addressed in the previous paragraph.

– In my opinion, the results should not be addressed in the introduction, as there is yet no information to support them. They should be summarized at the start of the discussion instead.

Methods:

– Histological data: the microdraw.pasteur.fr/project/brainmuseum-cb webpage was not publicly available at the time of writing of this review (Not authorized to view project – 14.03.23). Please ensure that you make it available.

– Histological data: what are the anatomical criteria to identify the mid-cerebellar slice? Do these criteria ensure that you are looking at comparable structures between species? (this can either be addressed in methods or in discussion).

– Histological data: exclusion threshold – It would be nice if you included it in supp. Mat. An example of images that are just under and just over this threshold for the reader to get a grasp of the image quality requirement of the study.

– Measurement of section area and length of the cerebral and cerebellar cortex: The implications of measurements not being modified to account for shrinkage should be briefly addressed in the discussion.

– Figure 2/Table 1: the cerebellar section area/length would gain from its boundary being more explicitly defined in Figure 2's legend: is it the black contour in both cases? Is there a difference between "cerebellar mid-slice section" and "cerebellar pial surface of mid-slice section" as a boundary for those two items?

– Measurement of molecular layer thickness: From "We applied a Laplacian smoothing […]" to "from the outside of the image." Is it the same concept with two different wordings? Maybe blend the two sentences together or delete one as it may appear that two different gradients are computed.

– Figure 4: I find image 4c confusing: is it related to images 4a and 4b? If so, why are the red dots creating such an angular curve when the black region's boundary appears very smooth on 4a and 4b? Please add more details to the legend.

– Please address all the numerical analyses led in the Results section within the methods section.

1) It would be helpful if some of the more complex types of results were introduced in the methods section, such as partial correlations conditional to all remaining measurements, or multivariate allometric scaling slopes.

2) It would be helpful if the impact of genetic model choice on the other analyses was clarified (as in, which subsequent analyses rely on this model selection).

Results:

I could not find the common thread linking all the results together. Can you either make the logical structure of the results explicit or change it so that it follows a logical order? As an example, it could go from simpler to more complex analyses.

In addition, I would move some results to the discussion. This could lead to the following structure: – statement of numerical results (including section areas if relevant and mean thickness of the molecular layer) – correlation structure – allometries – genetic models assessment – phylogenetic PCA – Ancestor character state estimation.

I suggest moving comparisons with anterior studies in the discussion, both for the very first and very last section of the results.

– Correlation structure: p.9 – please detail partial correlations and their meaning in methods.

– Correlation structure: p.9 – please detail "phylogenetic" PCA in methods.

– Correlation structure: p.10 – The PCA analyses should have their own subsection.

– Correlation structure: p.11 – please detail allometric slopes and multivariate allometric slopes in methods (are they used anywhere?).

– Figure 8: the method for creating coloured clusters for each group should be briefly explained, maybe in the methods section?

– Allometry: p.14 – you could re-quote the grey area in figure 10 when mentioning significance since it is not very visible and it may not be obvious to the reader that it is present in the figure.

Discussion:

In my opinion, the discussion needs to be thoroughly reworked. The current content of the discussion is very general and seems to address questions related to the work proposed rather than the work in itself. Discussing related topics is interesting and relevant, but should only take a subsection of the discussion.

Here is a list of subsections that I would expect in a discussion (the order can be changed):

– summary of the results demonstrated.

– considerations about the work in a link to the state of the art (maybe 1st section of the results, considerations about coherence with the Tallinen model, and considerations about limits of the Sultan and Braitenberg pioneer study).

– discussion of methodological decisions and potential limitations (not taking into account shrinkage, comparability of mid-cerebellar slice between species, the necessity of a sophisticated method for measurement of molecular layer thickness, …).

– discussion about the interpretation of the results, ideally creating an overall link about the different results and what they tell. This is essential to convey the overall meaning of the results reported.

– relation of the current report to broader questions and considerations (has to come after the interpretation of results, or it can be confusing for the reader).

Reviewer #3 (Recommendations for the authors):

Results/stats

– Please either remove the OU models or at least discuss these issues and justify conclusions about model fit with more caution.

– Please provide more justification for variable inclusion in the partial correlation and PC analyses, clarify the theoretical motivation for and inferences from the analyses, and discuss statistical issues of statistical power and collinearity.

– Please consider testing the hypothesis of cortico-cerebellar correlated evolution by running models something like [cerebellar measure ~ cortical measure + rest of the brain measure]; and/or if this is not possible, clarify the claims you are making about correlated evolution of cortex and cerebellum (why the patterns indicate something more specific than overall size).

– Figure 6 a: please at least present the CIs on the coefficients to show that the observed values include the predicted one.

Referencing/discussion/attribution of previous findings:

– Please revise/discuss/add citations as appropriate, based on my comments and references provided in the public review

[Editors’ note: further revisions were suggested prior to acceptance, as described below.]

Thank you for resubmitting your work entitled "Diversity and evolution of cerebellar folding in mammals" for further consideration by eLife. Your revised article has been evaluated by Floris de Lange (Senior Editor) and a Reviewing Editor.

The manuscript has been improved but there are some remaining issues that need to be addressed, as outlined below.

Among the reviewers' recommendations (detailed below), the main criticisms are as follows:

– Following the second comment from the third reviewer (see below), all reviewers and the editor agree that the partial correlation analyses may be problematic in view of the sample size, and that they need to be better justified and interpreted to guarantee the robustness of the results.

– The justification of the phylogenetic model still raises questions.

– The section about the measurement of molecular layer thickness could be further clarified.

Reviewer #2 (Recommendations for the authors):

I thank the authors for addressing most of the concerns issued thoroughly. Overall, the aims, methods and meanings of the results are clearer and my understanding was hence improved. In my opinion, only two points remain unresolved.

Unresolved points:

I. Accessibility of the Brain Museum Cerebellum project:

When trying to access it, I still get an error: "Not authorized to view project".

II. I still don't understand the section about the measurement of molecular layer thickness. (p.6)

This section has not been modified since the previous version of the article, and I find it challenging because it presents a rather complex methodology with not very clear context. In particular, the sentence "In mature cerebella, the molecular layer appears as a light band followed by the granular layer which appears as a dark band" with no additional comment is confusing in my opinion: are you dealing with only mature cerebella which makes the boundary between molecular and granular layer easy to detect or is it this not-so-obvious boundary that makes this complex method necessary? what does the boundary detection in Figure 4.c represent regarding Figure 4.b? I guess the molecular layer thickness results from averaging the length of the profile lines from figure 4.b?

To sum up, I have difficulties understanding what the problem is and hence what the method solves. I think this paragraph would gain from being more explicit in the reasons for both the whole and subparts of the process.

Apart from that, I found the current version much clearer than the previous one.

Reviewer #3 (Recommendations for the authors):

In general, I am happy with the responses of the authors and the changes they have made. There are still two statistical issues requiring a response:

1. The justification for the OU model. Grabowski et al. do not actually demonstrate that the critique by Cooper et al. is misleading – mainly they just show that Cooper et al. were correct! That is not to say that the OU model can never be justified (although some query its basis in biological reality), but that instances of its superiority over BM need to be treated very cautiously. I don't think their simulation solves the problem – the reason the OU is biased is because it does not conform to the data distributions assumptions of AIC (which has strong assumptions about normality or AICc) – this would be the same for other tree transforms like δ (Pagel). This results in the OU being statistically favoured when the 'pull' parameter is so weak it is essentially the same as the Brownian model.

2. The authors reject the criticism of overfitting by stating that their analyses do not involve fitting models. This is technically correct in strict terms but misleading. Exactly the same issues around small sample sizes per parameter (or variable) arise in partial correlation networks as in fitting a multiple regression model. Partial regression coefficients and partial correlation coefficients are susceptible to the same problem- neither can be estimated accurately without sufficient statistical power. Say you have 20 variables and 20 data points – no one would (hopefully) run a partial correlation on the 20 variables, or even 10 variables, or 5 – maybe just 2 could be justified. Here, the authors have 9 variables and 56 data points – with only 6 data points per variable, they don't have the statistical power to tease apart the relationships as they imply. Added to this, even with large samples, throwing in a bunch of variables without some explicit model that you are testing does indeed result in a correlational salad. Typically, such a partial correlation network would be liable to instability, and the more variables you add the harder it is to make sense of the individual relationships between them. Their explanation of how a negative correlation can arise between variables after controlling for their relationship to other variables is not necessary – this is obvious. What is not so obvious is why in principle there should be a negative (partial) correlation within their network – what is its interpretation? If they started with a model that predicted this or had some compelling explanation, I would be more impressed, but as it is I suspect it is an artefact for the reasons stated above.

Essential revisions:

All reviewers acknowledged the importance and originality of your study. But they also raised a number of important concerns to consider.

The most critical one concerns the statistical analyses performed (see Reviewer #3's comments). Further evidence should be provided regarding the superiority of the Ornstein-Uhlenbeck model.

As suggested by Reviewer #3, we evaluated our ability to decide correctly for the Brownian Motion model over the Ornstein-Uhlenbeck model by simulating multivariate correlated phenotypes (9 variables) for 56 species using our phylogenetic tree. Over 1000 simulations, the Brownian Motion model was correctly selected 99.3% of the times (see the answer to Reviewer #3 for more details). This additional analysis is added to the revised version of the manuscript.

Besides, the objectives and relevance of the partial correlations and principal component analyses to all included variables should also be better argued given the possible over-fitting

We have now added a more detailed description of the objectives of our analysis of partial correlations, principal component analysis, and allometry. We have aimed at providing an intuitive explanation. Neither partial correlations nor principal components rely on the fitting of a model, so there is no possible over-fitting. We hear, however, Reviewer #3’s concern about the number of phenotypes per species becoming too large for the complexity of the models, and the temptation to run arbitrary comparisons. We share that concern, and the primary objective of these two methods is to avoid the “correlation salad” evoked by the Reviewer by looking at global multivariate patterns of covariation. Our simulations also provide evidence that the complexity of our model remains adequate.

and the underlying theoretical hypotheses which are not always clear.

We have completely re-structured our manuscript to make our theoretical standpoint more clear, providing a lengthier discussion of our methodological approach, and a more clear structure of our results and discussion, as suggested by the Reviewers. Our approach is not to test particular hypotheses, but to provide a characterisation of the phenomenon of cerebellar brain folding and its relationship with the folding of the cerebrum, as we indicate in the introduction.

From this characterisation we were able to advance hypotheses (presented in the discussion) which could be tested in the future.

The reviewers were also concerned about the global presentation of the manuscript. Additional description of some methods is required.

The revised manuscript has an expanded Methods section including introductions to our methods as well as illustrations aiming at highlighting their meaning and relevance. We introduce, as suggested by the Reviewers, an introduction to the measurement of folding and the advantages of our method (illustrated in Figure 3d), an introduction to the phylogenetic comparative models that we used, including an illustration showing their importance for the study of comparative data (Figure 5b), an introduction to the analysis of partial correlations with an example that aims at making their meaning more intuitive, especially concerning negative partial correlations, an introduction to principal components analysis for phylogenetic data, and an introduction to the study of allometry for bivariate and multivariate data, with examples from the literature. We have aimed at finding a good compromise between making a clear exposition, while keeping the size of the methods section reasonable. The examples that we have introduced can be reproduced and explored further in Jupyter notebooks which are included in the accompanying Github repository.

The different results should be better linked, and potentially reordered.

We have re-structured the Results section and provided a more clear description of their links, following the advice of Reviewer #2. Our Results section starts now with a guiding paragraph, which explains the links between the results in their logical sequence.

The discussion should also be reviewed in depth.

This is now also the case. We have integrated the advice of all Reviewers, adapting our structure to the one they proposed, in particular, including a more detailed description of the implications and limitations of our methodological decisions, and a summary discussion of each of our results.

Better highlighting the methodological novelties should also be considered.

The methods section has been expanded to include more detail on our methodology, which serves to highlight the novelty and relevance of our approach.

We hope that you will be able to take all these general comments into account, as well as the more detailed comments of the reviewers (see below).

Reviewer #2 (Recommendations for the authors):

This study is both rich and generous, and I thank the authors for the beautiful work presented and shared with the community. The investigation of the cerebellum is both essential to our general understanding of the brain and under-explored, making the efforts developed in this article essential.

I find the analyses well-led. Nevertheless, I think the reach of the current work would gain from a better emphasis on why the analyses have been conducted and what they imply. There are also aspects of the structure of the article which I find unconventional and potentially confusing for the reader.

I will address my general recommendations following the current structure of the article.

I will not address more specific concerns (specific wording, punctuation marks…) that I have in this step of the review for two reasons:

1) I believe the more general recommendations may lead to a restructuring of the article which would make some of the specific concerns irrelevant.

2) The current draft does not include line numbers, making it challenging to address line-specific recommendations (please add line numbering to the next version if possible).

Unfortunately, there doesn’t seem to be a way of adding line numbers in Google docs, and moving to LibreOffice completely broke the layout… However, the information the Reviewer provided was very clear and we did not have difficulties finding the sections in the paper that were being discussed.

Introduction:

– I have identified that the last three paragraphs match an Aims (bottom of p.2) – Methods (first paragraph p.3) – Results (second paragraph p.3). Then, the buckling models consideration of the Methods paragraph is rather an aim than a method and should be addressed in the previous paragraph.

The revised version of the manuscript provides additional context which highlights the novelty of our approach, in particular concerning the measurement of folding and the use of phylogenetic comparative models. The limitations of the previous approaches are stated more clearly, and illustrated in Figures 3 and 5.

Following the Reviewer’s advice, we have thoroughly restructured our results and Discussion section.

– In my opinion, the results should not be addressed in the introduction, as there is yet no information to support them. They should be summarized at the start of the discussion instead.

The brief mention of buckling models at the end of the introduction provides our justification for our measurement of folial width and molecular layer thickness. The idea of that short summary of our results is to allow readers to decide if they want to keep reading further. They are not described in any detail, just listed. A more detailed summary is provided in the discussion, as suggested by the Reviewer.

Methods:

– Histological data: the microdraw.pasteur.fr/project/brainmuseum-cb webpage was not publicly available at the time of writing of this review (Not authorized to view project – 14.03.23). Please ensure that you make it available.

The MicroDraw project is now public.

– Histological data: what are the anatomical criteria to identify the mid-cerebellar slice? Do these criteria ensure that you are looking at comparable structures between species? (this can either be addressed in methods or in discussion).

We realised that the term mid-slice may have been confusing, and we now call it mid-section throughout the manuscript. It is simply the section in the middle of the cerebellum: there is one section that is the first one to contain the cerebellum, another one that is the last one to contain the cerebellum, this is the one that is in the middle (or the nicest of the 2 middle ones if there’s an even number of slices). The exact index of the sections is included in the repository with our data and code.

– Histological data: exclusion threshold – It would be nice if you included it in supp. Mat. An example of images that are just under and just over this threshold for the reader to get a grasp of the image quality requirement of the study.

Our whole project is now public, giving access to the raw images, and allowing for easy visual inspection of their differences.

– Measurement of section area and length of the cerebral and cerebellar cortex: The implications of measurements not being modified to account for shrinkage should be briefly addressed in the discussion.

As suggested by the Reviewer, the revised version of our manuscript includes a more detailed description of our methodological decisions and their limitations. We discuss in particular the issue of shrinkage:

“Finally, we decided to measure the raw material without aiming at correcting for shrinkage. Our reasoning was that if the relationships were sufficiently robust, we should still be able to see them despite shrinkage. This appeared to be the case: accounting for shrinkage should enhance the accuracy of our measurements making our correlations even stronger and our estimations more precise, but should not change our conclusions. Shrinkage correction is often performed using correction factors obtained from the literature, which introduces a new source of methodological variability. A more precise account of shrinkage would require, for example, acquiring MRI data before and after fixation for each species (see Wehrl et al. 2015). Sections could be then registered to the undeformed MRI, allowing us to estimate deformation due to slicing and shrinkage.”

– Figure 2/Table 1: the cerebellar section area/length would gain from its boundary being more explicitly defined in Figure 2's legend: is it the black contour in both cases? Is there a difference between "cerebellar mid-slice section" and "cerebellar pial surface of mid-slice section" as a boundary for those two items?

To avoid terminological confusion we now use mid-section throughout the manuscript. The contour we drew is the one that appears in black (we now indicate this in Figure 2’s legend), with the inside of the contour being filled in translucent red.

– Measurement of molecular layer thickness: From "We applied a Laplacian smoothing […]" to "from the outside of the image." Is it the same concept with two different wordings? Maybe blend the two sentences together or delete one as it may appear that two different gradients are computed.

We refer to 2 different things. Laplacian smoothing produces a potential field which has a gradual change. The gradient (the mathematical operation) gives the direction of maximum change along this potential field. We changed the formulation from “We applied a Laplacian smoothing to the masked image, which created a gradient going from (…)” to “We applied a Laplacian smoothing to the masked image, which created a gradual change in grey level going from…”

– Figure 4: I find image 4c confusing: is it related to images 4a and 4b? If so, why are the red dots creating such an angular curve when the black region's boundary appears very smooth on 4a and 4b? Please add more details to the legend.

The Figure 4a-b show a small excerpt of the whole contour, which may give the impression of regularity. Figure 4c, shows all the profiles of the section. The revised version of the manuscript includes a more detailed legend.

– Please address all the numerical analyses led in the Results section within the methods section.

1) It would be helpful if some of the more complex types of results were introduced in the methods section, such as partial correlations conditional to all remaining measurements, or multivariate allometric scaling slopes.

2) It would be helpful if the impact of genetic model choice on the other analyses was clarified (as in, which subsequent analyses rely on this model selection).

We now provide an introduction to partial correlations, with an example illustrating their meaning, especially for the negative ones. We also provide an introduction to allometry, phylogenetic models, and phylogenetic PCA. All this serves to clarify the impact and the necessity of phylogenetic models.

Results:

I could not find the common thread linking all the results together. Can you either make the logical structure of the results explicit or change it so that it follows a logical order? As an example, it could go from simpler to more complex analyses.

The Results section in the revised manuscript includes a paragraph providing a rationale for the presentation of the results, as suggested by the Reviewer.

In addition, I would move some results to the discussion. This could lead to the following structure: – statement of numerical results (including section areas if relevant and mean thickness of the molecular layer) – correlation structure – allometries – genetic models assessment – phylogenetic PCA – Ancestor character state estimation.

As suggested by the Reviewer, we have displaced the allometry subsection after the subsection on correlation structure. The new structure is as follows: (1) validation of measurements, (2) evolutionary model selection, (3) correlation structure, (4) allometry, (5) ancestral phenotype estimation. The rationale for this ordering is provided in the first paragraph of the Results section.

I suggest moving comparisons with anterior studies in the discussion, both for the very first and very last section of the results.

We followed the advice of the Reviewer and moved the consideration on mechanical models to the discussion. However, we believe that the “validation” section has to come at the beginning, because it aims at showing that our methodology of measurement is appropriate.

– Correlation structure: p.9 – please detail partial correlations and their meaning in methods.

An introduction to partial correlations is now included in the methods.

– Correlation structure: p.9 – please detail "phylogenetic" PCA in methods.

An introduction to phylogenetic models and in particular their use in phylogenetic principal component analysis is now included in the methods.

– Correlation structure: p.10 – The PCA analyses should have their own subsection.

PCA is a method for the study of correlation structure, and as such the correlation structure section is its natural place. We have aimed at making the link between PCA, partial correlations and correlations more clear in the revised text.

– Correlation structure: p.11 – please detail allometric slopes and multivariate allometric slopes in methods (are they used anywhere?).

We now include a more detailed introduction of allometry, bivariate and multivariate. We indicate that multivariate slopes are used for all our allometric analyses, and we included a new table, Table 3, with the loadings of PC1 which describe the multivariate allometric relationships.

– Figure 8: the method for creating coloured clusters for each group should be briefly explained, maybe in the methods section?

The coloured clusters are only used to facilitate the visualisation of the groups. They are computed by the ggscatter package in R by fitting a normal distribution to the data points, and selecting a threshold. The colour scheme is indicated in each figure, and corresponds to the one used in Figure 5 and Figure 11.

– Allometry: p.14 – you could re-quote the grey area in figure 10 when mentioning significance since it is not very visible and it may not be obvious to the reader that it is present in the figure.

We have done as suggested by the Reviewer. We have also explicitly added the numeric CI ranges to all figures, in addition to their graphical representation.

Discussion:

In my opinion, the discussion needs to be thoroughly reworked. The current content of the discussion is very general and seems to address questions related to the work proposed rather than the work in itself. Discussing related topics is interesting and relevant, but should only take a subsection of the discussion.

The discussion has been completely revised and reorganised, addressing our questions, results and hypotheses more precisely.

Here is a list of subsections that I would expect in a discussion (the order can be changed):

– summary of the results demonstrated.

– considerations about the work in a link to the state of the art (maybe 1st section of the results, considerations about coherence with the Tallinen model, and considerations about limits of the Sultan and Braitenberg pioneer study).

– discussion of methodological decisions and potential limitations (not taking into account shrinkage, comparability of mid-cerebellar slice between species, the necessity of a sophisticated method for measurement of molecular layer thickness, …).

– discussion about the interpretation of the results, ideally creating an overall link about the different results and what they tell. This is essential to convey the overall meaning of the results reported.

– relation of the current report to broader questions and considerations (has to come after the interpretation of results, or it can be confusing for the reader).

Following the advice of the Reviewer, the discussion is now organised as follows: (1) general context for our question, (2) detailed summary of our results, (3) discussion on methodological decisions and limitations, (4) discussion on the measurement of folding, highlighting the relevance of our new approach, (5) discussion on models of cortical folding, stating our hypotheses and supporting them through our results, (6) a consideration of our results to the broader question of the potential role of cerebellar folding.

Reviewer #3 (Recommendations for the authors):

Results/stats

– Please either remove the OU models or at least discuss these issues and justify conclusions about model fit with more caution.

We have supported our result on the selection of the OU model with additional simulations.

– Please provide more justification for variable inclusion in the partial correlation and PC analyses, clarify the theoretical motivation for and inferences from the analyses, and discuss statistical issues of statistical power and collinearity.

We now provide a more detailed justification for our analyses including and excluding body weight and brain weight. Our introduction to partial correlations and PCA in the methods section should help clarify their utility for addressing the problem of collinearity.

– Please consider testing the hypothesis of cortico-cerebellar correlated evolution by running models something like [cerebellar measure ~ cortical measure + rest of the brain measure]; and/or if this is not possible, clarify the claims you are making about correlated evolution of cortex and cerebellum (why the patterns indicate something more specific than overall size).

Our introduction to partial correlation analysis should now clarify the similarity with the suggestion of the Reviewer (i.e., considering a bivariate relationship while covarying the rest).

– Figure 6 a: please at least present the CIs on the coefficients to show that the observed values include the predicted one.

95% confidence intervals have now been added for the slopes in the linear regressions and orthogonal regressions. We have also added p-values for all plots, as well as for the partial correlations in the diagram in Figure 7c.

Referencing/discussion/attribution of previous findings:

– Please revise/discuss/add citations as appropriate, based on my comments and references provided in the public review

We have added the references suggested by the Reviewer.

[Editors’ note: what follows is the authors’ response to the second round of review.]

The manuscript has been improved but there are some remaining issues that need to be addressed, as outlined below.

Among the reviewers' recommendations (detailed below), the main criticisms are as follows:

– Following the second comment from the third reviewer (see below), all reviewers and the editor agree that the partial correlation analyses may be problematic in view of the sample size, and that they need to be better justified and interpreted to guarantee the robustness of the results.

We have conducted additional analyses which demonstrate that our results using partial correlation analyses are robust. Our simulations show that we have excellent statistical power to detect even medium size partial correlations, due to the strength of the allometric pattern that we describe. This is referred to in the manuscript and its implementation is detailed in the Supplementary Methods.

– The justification of the phylogenetic model still raises questions.

We provide additional arguments to support our approach, and references to relevant publications. Despite the OU model being widely used and implemented by all major packages for phylogenetic comparative analyses, we recognise that it is important to expose the criticism that this approach has received, in particular in the literature mentioned by Reviewer #3. This is especially important given that many potential readers of our work may not be familiar with phylogenetic comparative methods. We have highlighted this in our revised manuscript.

– The section about the measurement of molecular layer thickness could be further clarified.

We have integrated the suggestions of Reviewer #2, and clarified the outstanding points.

Reviewer #2 (Recommendations for the authors):

I thank the authors for addressing most of the concerns issued thoroughly. Overall, the aims, methods and meanings of the results are clearer and my understanding was hence improved. In my opinion, only two points remain unresolved.

Unresolved points:

I. Accessibility of the Brain Museum Cerebellum project:

When trying to access it, I still get an error: "Not authorized to view project".

The project is publicly available now.

II. I still don't understand the section about the measurement of molecular layer thickness. (p.6)

This section has not been modified since the previous version of the article, and I find it challenging because it presents a rather complex methodology with not very clear context. In particular, the sentence "In mature cerebella, the molecular layer appears as a light band followed by the granular layer which appears as a dark band" with no additional comment is confusing in my opinion: are you dealing with only mature cerebella which makes the boundary between molecular and granular layer easy to detect or is it this not-so-obvious boundary that makes this complex method necessary?

We may have been providing irrelevant information, which may be the cause of the confusion. “Mature cerebella” makes reference to cerebellar development – which is not the focus of our manuscript. During cerebellar development, there is a transient layer of cell nuclei close to the surface, composed of migrating granular cells. Hence, during cerebellar development, there is not only one but two layers of neuronal nuclei (dark), one external and one internal. Migrating granule cells move from the external to the internal layer, which is why in mature cerebellar cortex only one layer of neurones is present (i.e., the internal dark band observed in histological data). The aim of our algorithm is then to detect the border of this dark layer. We have removed the mention to “mature cerebella” to prevent confusion.

The molecular layer of the cerebellum appears as a light band followed by the granular layer which appears as a dark band.

what does the boundary detection in Figure 4.c represent regarding Figure 4.b? I guess the molecular layer thickness results from averaging the length of the profile lines from figure 4.b?

To sum up, I have difficulties understanding what the problem is and hence what the method solves. I think this paragraph would gain from being more explicit in the reasons for both the whole and subparts of the process.

Apart from that, I found the current version much clearer than the previous one.

Figure 4a shows our manually drawn surface. Figure 4b shows profile lines going from the manually drawn surface in Figure 4a to the granular layer. Each of the columns in Figure 4c corresponds to one of the profile lines in Figure 4b. Figure 4b shows just an excerpt of the whole cerebellar slice, and then just a subset of all profile lines. Figure 4c, on the contrary, shows all the profile lines for that cerebellar slice. The vertical coordinate of each profile line in Figure 4c represents the distance from the cerebellar surface; the horizontal coordinate represents the gradient (in mathematical terms) of the profile grey level. When a value changes little, the gradient is small, but when there is a large change, the gradient is large. We use this to detect the border between the molecular layer (light) and the granular layer (dark), which produces a large change in gradient. In Figure 4c these points are indicated with a red spot. We have added additional precision to our explanations.

“We estimated the thickness of the molecular layer automatically (Figure 4). We processed the histological images to convert them to grey levels, denoise them and equalise their histograms, using functions from the Scikit-image package (van der Walt et al. 2014). Starting from the manual segmentation (blue line in Figure 4a), we created a binary mask and masked out the non-cerebellar regions (for example, the neighbouring cerebrum). The molecular layer of the cerebellum appears as a light band followed by the granular layer which appears as a dark band. We applied a Laplacian smoothing to the masked image, which created a gradual change in grey level going from white close to the surface of the cerebellum to black toward the granular layer. We computed the gradient of the image, which produced a vector field from the outside to the inside. We integrated this vector field to produce a series of lines, each for every vertex in the manual cerebellar segmentation. The vector field was integrated only if the grey values decreased, and if a minimum was reached, the vector field was continued linearly. At the end of this procedure, all lines had the same length, computed so as to cover the whole range from the surface of the cerebellum to the granular layer. The grey levels in the original image were sampled along each scan line, producing a grey level profile. The grey level profiles were derived, and a peak detection function was used to determine the point of maximum white/grey contrast (Figure 4c, where the red dot shows the detected boundary), indicating the boundary between the molecular and granular layers. Figure 4b shows the part of the profile lines starting at the cerebellar surface and ending at the detected boundary. For each scan line the corresponding thickness of the molecular layer was defined as the length from the manual cerebellar segmentation until the maximum contrast point (red point). Finally, for each species, a single thickness value was computed as the median of the thickness measured for each profile line, to make our estimation robust to outliers. See the accompanying source code for further implementation details.”

Reviewer #3 (Recommendations for the authors):

In general, I am happy with the responses of the authors and the changes they have made. There are still two statistical issues requiring a response:

1. The justification for the OU model. Grabowski et al. do not actually demonstrate that the critique by Cooper et al. is misleading – mainly they just show that Cooper et al. were correct! That is not to say that the OU model can never be justified (although some query its basis in biological reality), but that instances of its superiority over BM need to be treated very cautiously. I don't think their simulation solves the problem – the reason the OU is biased is because it does not conform to the data distributions assumptions of AIC (which has strong assumptions about normality or AICc) – this would be the same for other tree transforms like δ (Pagel). This results in the OU being statistically favoured when the 'pull' parameter is so weak it is essentially the same as the Brownian model.

The reviewer’s comment include 2 sub-points:

1) The meaning of Grabowski et al.’s work. The reviewer indicates that Grabowsky et al.’s work mainly shows that Cooper et al. were correct. It is not immediately evident that this is what Grabowsky et al. intended to convey. Referring to Cooper et al.’s work, they write: “we argue that these results have little relevance to the estimation of adaptation with Ornstein–Uhlenbeck models for three reasons.” which they detail. Further in the introduction they write “We disagree, however, with Cooper et al.’s (2016) claim to have detected specific problems with the Ornstein–Uhlenbeck method and we find their recommendations against the application of the method to be unfounded”. The reviewer’s comment points, however, to an important issue, which goes beyond the specific details of Cooper et al.’s and Grabowsky et al.’s papers: that OU models (as well as any other models) need to be appraised critically, and their results not taken at face value. This is especially relevant in the case of our work, which aims at being read by an audience that in many cases will not be familiar with phylogenetic comparative methods. In the revised version of our manuscript we have aimed at making this point more clear, pointing the readers to the relevant literature. We modified our Discussion section as follows:

“Fitting phylogenetic models with a significantly greater number of phenotypes than species is challenging (in particular, there is an ongoing debate regarding the use of OU models, see the work of Cooper et al. 2016, Clavel et al. 2019, and Grabowsky et al. 2023). The use of penalisation techniques and constraints in the model’s structure holds great promise and will enable the exploration of new hypotheses regarding the evolution of complex phenotypes (Clavel et al. 2015, 2019).”

2) Justification of the OU model. We have followed the reviewer’s advice and inspected our selection of the OU model over the BM model at greater detail. We did this using simulations, as suggested in the reviewer’s previous review. Our simulations aimed at replicating the procedure adopted by Cooper et al. – i.e., simulating BM data and evaluating the number of times the OU model is incorrectly selected – while using AICc for model selection which takes model complexity into account (unlike the comparison of log-likelihoods used by Cooper et al). The reviewer comment does not make it clear why our simulations would be incorrect. AIC and AICc criteria are grounded on information theory (they evaluate the amount of information lost by modelling) and do not make distribution assumptions. AIC and AICc require the models to be estimated by maximum likelihood because one can show that asymptotically the deviance plus twice the number of degrees of freedom is proportional to the Kullback-Leibler divergence between models and the observed data, for any probability distribution (Akaike 1974). Moreover, the OU process is a Gaussian diffusion process, and by definition it has a normal distribution. So, if normality were required (which is not the case) that would not violate it. If the “pull” parameter (α in our equations) were close to 0 the OU model would reduce to the BM model and should provide the same fit, however, it would also have more parameters which will result in a higher AIC (AIC = 2k – 2 ln (likelihood)) leading to the selection of the simpler model (BM over OU). Our simulations, which use the same number of variables and species as in our data, show this to be indeed the case. In addition to our simulations, the work of Clavel et al. (2019) provides a thorough exploration of the ability of the penalised likelihood framework implemented by mvMORPH to select the correct model in different scenarios. Finally, as Clavel et al. (2019) show, the instances where one would select OU rather than BM come from upward bias in estimating α, so that we would tend to detect it to be stronger than it is rather than having it shrinking to zero as the reviewer suggests. Again, our aim is not to dismiss the reviewer’s concern, but given the evidence at hand and the results of our analyses, we have no justification for selecting a BM model by principle when both our model selection methodology and our power analysis indicate that we should select the OU model. All our data and code are openly available for criticism, and we make clearer in the revised version of the manuscript that model selection is not without controversy.

“Fitting phylogenetic models with a significantly greater number of phenotypes than species is challenging (in particular, there is an ongoing debate regarding the use of OU models, see the work of Cooper et al. 2016, Clavel et al. 2019, and Grabowsky et al. 2023). The use of penalisation techniques and constraints in the model’s structure holds great promise and will enable the exploration of new hypotheses regarding the evolution of complex phenotypes (Clavel et al. 2015, 2019).”

2. The authors reject the criticism of overfitting by stating that their analyses do not involve fitting models. This is technically correct in strict terms but misleading. Exactly the same issues around small sample sizes per parameter (or variable) arise in partial correlation networks as in fitting a multiple regression model. Partial regression coefficients and partial correlation coefficients are susceptible to the same problem- neither can be estimated accurately without sufficient statistical power. Say you have 20 variables and 20 data points – no one would (hopefully) run a partial correlation on the 20 variables, or even 10 variables, or 5 – maybe just 2 could be justified. Here, the authors have 9 variables and 56 data points – with only 6 data points per variable, they don't have the statistical power to tease apart the relationships as they imply.

We apologise for misinterpreting the reviewer’s previous comment. We have addressed the reviewer’s criticism and performed simulations to estimate our statistical power to detect large, medium and small partial correlations. In general, the assessment of statistical power includes 4 main components: (1) sample size, (2) α value (significance threshold), (3) effect size, and (4) power level. The sample size on its own is not sufficient to estimate statistical power. Neither a sample size of N=20 nor one of N=1,000 are intrinsically small or large: that will depend on the effect size and the α value. N=1,000 may be a large sample size for estimating differences in height between males and females, but would be completely insufficient for a GWAS where effect sizes are infinitesimal. In our case, our effect sizes (the correlations we observe) are extremely large, and our sample size largely sufficient (to clarify, we have a sample of 56 species, with 9 – strongly correlated – variables measured for each species, resulting in 504 data points). We computed our statistical power to detect large (“3 asterisks”, approx. |r|>0.5), medium (“2 asterisks”, approx. |r|>0.34) and small (“1 asterisk”, approx. |r|>0.25) partial correlations. We had over 95% power to detect large partial correlations (α=0.05), over 90% power for medium partial correlations, and over 75% for small partial correlations (see Author response image 1a). For this, we generated 1000 simulations of multivariate data following our observed variance-covariance matrix (capturing the correlation structure across our 9 variables) and computed the number of true positives, false positives, true negatives and false negatives. Sensitivity (power) was computed as TP/(TP+FN). We repeated the same procedure for sample sizes going from 20 to 100 species. The reason for this is that the allometric pattern in our data is so strong (PC1 captures >90% of the variance) that large and medium partial correlations could be detected with 80% power even with less than 40 species. Indeed, modifying our observed variance-covariance matrix to weaken the strength of the allometric pattern (PC1) resulted, as expected, in a decreased statistical power level (Author response image 1b). The criticism of the reviewer would apply in the limit where all 9 variables would be independent.

Author response image 1

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We added the statistical power code to our GitHub repository, and the following to the Methods section:

“The statistical significance of each partial correlation being different from 0 was estimated using the edge exclusion test introduced by Whittaker (1990). We evaluated our statistical power to detect partial correlations of different strengths using simulations. Code is included in the accompanying source code along with a Jupyter notebook providing an executable version of our partial correlations example together with further details on our power analysis. A non-executable version of this notebook is provided as Supplementary file 2.”

And to the Results section:

“Our simulations showed that, because of the strength of the allometric pattern in our data, we had excellent statistical power to detect large and medium partial correlations (90-100%) and good statistical power to detect small partial correlations (~78%, see Supplemental Methods for details).”

Added to this, even with large samples, throwing in a bunch of variables without some explicit model that you are testing does indeed result in a correlational salad.

We understand that the reviewer favours a model-based approach to our exploratory approach. Both approaches are, however, equally valid and used, yet with different aims in mind. Here we interpret it as “an unprincipled selection of variables and their bivariate correlations”. First, our selection of variables is justified, and their more thorough presentation was already addressed following the previous comments from reviewers #1 and #2. Briefly, they cover different but complementary aspects of anatomical diversity (body size, brain size, section area and length) and stability (folial width, folial perimeter and thickness of the molecular layer). The first group of variables is widely used in the literature, and their inclusion facilitates comparison (some of these comparisons are the subject of our subsection on “validation”). The second group of variables is more infrequent, however, they are fully justified in theoretical terms, in particular given biomechanical models of folding (as described at length in our manuscript). Second, our methodology for the exploration of the correlation structure of the data is based on well established multivariate methods (principal component analysis, partial correlations analysis) whose aim is exactly to avoid an unprincipled choice of arbitrary bivariate correlations. Our analyses also take phylogenetic structure into account – in a multivariate manner. However, if “correlational salad” refers only to the presence of many correlations, then yes, our multivariate analysis of correlation structure could be seen as many bivariate correlations (although they all result from a single matrix decomposition operation in the case of PCA, and a single matrix inversion in the case of partial correlations), but this is a valid approach for data exploration that is able to provide helpful insight.

Typically, such a partial correlation network would be liable to instability, and the more variables you add the harder it is to make sense of the individual relationships between them.

Our simulations help address this concern, as they show that we have excellent statistical power to detect the partial correlations conveyed by the variance-covariance matrix of our variables. While we acknowledge that in many instances partial correlations may offer unreliable information, in the current scenario this is not applicable due to the exceptionally strong allometric pattern present in our data (strong effect size).

Their explanation of how a negative correlation can arise between variables after controlling for their relationship to other variables is not necessary – this is obvious. What is not so obvious is why in principle there should be a negative (partial) correlation within their network – what is its interpretation? If they started with a model that predicted this or had some compelling explanation, I would be more impressed, but as it is I suspect it is an artefact for the reasons stated above.

We apologise for having misunderstood the previous comment of the reviewer concerning the impossibility to interpret negative partial correlations. Our aim was to provide a simple example showing how they arise and which type of information they may convey. Partial correlation analysis is one of the basic tools for exploratory data analysis, and we believe their general usefulness is beyond doubt.

In the precise case of our results, partial correlation provides interesting perspectives on several aspects. For example, they suggest that the strong correlation between cerebellar and cerebral section area and length is not completely the reflection of a common correlation with body size or brain weight: cerebellar section area would be significantly correlated (Figure 7c, r=0.328) with cerebral section area even if the influence of all other variables were removed. Additionally, the two significant negative correlations (r<-0.34, r<-0.36, p<0.01) are also interesting, as they suggest a decoupling between diverse and stable phenotypes. This information is particularly relevant for the discussion concerning cerebral and cerebellar folding, as they point to a similar folding mechanism for both structures. We do not pretend to provide a definitive answer on the issue, but our results provide a compelling direction for future investigation. The statistical significance of our findings, reported in the previous revision, provided evidence on their robustness, which is now reinforced by our power analysis.

We thank the reviewer for his comments, which have motivated us in particular to perform additional analyses which demonstrate more clearly the validity of our results, while also stating more clearly their potential limitations. The reviewer had concerns about sample sizes, and the pertinence of OU models and partial correlation analyses which we have tried to operationalise and approach through simulations and analyses of statistical power. We hope the exposition of our results is now more clear, as well as the evidence of their robustness.

References

Akaike, H. (1974). A new look at the statistical model identification. In IEEE Transactions on Automatic Control (Vol. 19, Issue 6, pp. 716–723). Institute of Electrical and Electronics Engineers (IEEE). https://doi.org/10.1109/tac.1974.1100705

Clavel, J., Aristide, L., and Morlon, H. (2018). A Penalized Likelihood Framework for High-Dimensional Phylogenetic Comparative Methods and an Application to New-World Monkeys Brain Evolution. In L. Harmon (Ed.), Systematic Biology (Vol. 68, Issue 1, pp. 93–116).

Oxford University Press (OUP). https://doi.org/10.1093/sysbio/syy045

Cooper, N., Thomas, G. H., Venditti, C., Meade, A., and Freckleton, R. P. (2015). A cautionary note on the use of Ornstein Uhlenbeck models in macroevolutionary studies. In Biological Journal of the Linnean Society (Vol. 118, Issue 1, pp. 64–77). Oxford University Press (OUP). https://doi.org/10.1111/bij.12701

Grabowski, M., Pienaar, J., Voje, K. L., Andersson, S., Fuentes-González, J., Kopperud, B. T., Moen, D. S., Tsuboi, M., Uyeda, J., and Hansen, T. F. (2023). A Cautionary Note on “A Cautionary Note on the Use of Ornstein Uhlenbeck Models in Macroevolutionary Studies.” In Systematic Biology. Oxford University Press (OUP). https://doi.org/10.1093/sysbio/syad012

Author details

1. Katja Heuer

   Institut Pasteur, Université Paris Cité, Unité de Neuroanatomie Appliquée et Théorique, Paris, France

   Contribution

   Conceptualization, Resources, Data curation, Software, Formal analysis, Supervision, Funding acquisition, Validation, Investigation, Visualization, Methodology, Writing – original draft, Project administration, Writing – review and editing

   For correspondence

   katjaqheuer@gmail.com

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-7237-0196

2. Nicolas Traut

   Institut Pasteur, Université Paris Cité, Unité de Neuroanatomie Appliquée et Théorique, Paris, France

   Contribution

   Software, Methodology

   Competing interests

   No competing interests declared

3. Alexandra Allison de Sousa

   Centre for Health and Cognition, Bath Spa University, Bath, United Kingdom

   Contribution

   Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-2379-3894

4. Sofie Louise Valk

   1. Otto Hahn Group Cognitive Neurogenetics, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany
   2. Institute of Neuroscience and Medicine, Brain & Behaviour (INM-7), Research Centre Jülich, FZ Jülich, Jülich, Germany
   3. Institute of Systems Neuroscience, Medical Faculty, Heinrich Heine University Düsseldorf, Düsseldorf, Germany

   Contribution

   Data curation, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-2998-6849

5. Julien Clavel

   Université Lyon, Université Claude Bernard Lyon 1, CNRS, ENTPE, UMR 5023 LEHNA, Villeurbanne, France

   Contribution

   Software, Visualization, Methodology, Writing – review and editing

   Competing interests

   No competing interests declared

6. Roberto Toro

   Institut Pasteur, Université Paris Cité, Unité de Neuroanatomie Appliquée et Théorique, Paris, France

   Contribution

   Conceptualization, Resources, Data curation, Software, Formal analysis, Supervision, Funding acquisition, Validation, Investigation, Visualization, Methodology, Writing – original draft, Project administration, Writing – review and editing

   For correspondence

   rto@pasteur.fr

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-6671-858X

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