Author Response
The following is the authors’ response to the original reviews.
Public Reviews:
Reviewer #1 (Public Review):
This study examined a universal fractal primate brain shape. However, the paper does not seem well structured and is not well written. It is not clear what the purpose of the paper is. And there is a lack of explanation for why the proposed analysis is necessary. As a result, it is challenging to clearly understand what novelty in the paper is and what the main findings are.
We have now restructured the paper, including a summary of the main purpose and findings as follows:
“Compared to previous literature, we can summarise our main contribution and advance as follows:
(i) We are showing for the first time that representative primate species follow the exact same fractal scaling – as opposed to previous work showing that they have a similar fractal dimension [Hofman1985, Hofman1991], i.e. slope, but not necessarily the same offset, as previous methods had no consistent way of comparing offsets.
(ii) Previous work could also not show direct agreement in morphometrics between the coarse-grained brains of primate species and other non-primate mammalian species.
(iii) Demonstrating in proof-of-principle that multiscale morphometrics, in practice, can have much larger effect sizes for classification applications. This moves beyond our previous work where we only showed the scaling law across [Mota2015] and within species [Wang2016], but all on one (native) scale with comparable effect sizes for classification applications [Wang2021].
In simple terms: we know that objects can have the same fractal dimension, but differ greatly in a range of other shape properties. However, we demonstrate here, that representative primate brains and mammalian brain indeed share a range of other key shape properties, on top of agreeing in fractal dimension. This suggests a universal blueprint for mammalian brain shape and a common set of mechanisms governing cortical folding. As a practical additional outcome of our study, we could show that our novel method of deriving multiscale metrics of brain shape can differentiate subtle shape changes much better than the metrics we have been using so far at a single native scale.”
We plan to use the second paragraph as a plain-language summary of our work.
Additionally, several terms are introduced without adequate explanation and contextualization, further complicating comprehension.
We have now made sure that potential jargon is introduced with context and explanation. For example in Introduction: “This scaling law, relating powers of cortical thickness and surface area metrics, […]”
Does the second section, "2. Coarse-graining procedure", serve as an introduction or a method?
We have now renamed this section to “Coarse-graining Method” to indicate that this is a section about methods. However, to describe the methods adequately, we also expanded this section with introductory texts around the history and motivation of the method to provide context and explanations, as the reviewer rightly requested.
Moreover, the rationale behind the use of the coarse-graining procedure is not adequately elucidated. Overall, it is strongly recommended that the paper undergoes significant improvements in terms of its structure, explanatory depth, and overall clarity to enhance its comprehensibility.
To specifically explain the rationale behind the coarse-graining method, we added several clarifications, including the following paragraph:
“As a starting point for such a coarse-graining procedure, we suggest to turn to a well-established method that measures fractal dimension of objects: the so-called box-counting algorithm [Kochunov2007, Madan2019]. Briefly, this algorithm fills the object of interest (say the cortex in our case) with boxes, or voxels of increasingly larger sizes and counts the number of boxes in the object as a function of box size. As the box size increases, the number of boxes decreases; and in a log-log plot, the slope of this relationship indicates the fractal dimension of the object. In our case, this method would not only provide us with the fractal dimension of the cortex, but, with increasing box size, the filled cortex would also contain less and less detail of the folded shape of the cortex. Intuitively, with increasing box size, the smaller details, below the resolution of a single box, would disappear first, and increasingly larger details will follow -- precisely what we require from a coarse-graining method. We therefore propose to expand the traditional box-counting method beyond its use to measure fractal dimension, but to also analyse the reconstructed cortices as different realisations of the original cortex at the specified spatial scale.”
Reviewer #2 (Public Review):
In this manuscript, Wang and colleagues analyze the shapes of cerebral cortices from several primate species, including subgroups of young and old humans, to characterize commonalities in patterns of gyrification, cortical thickness, and cortical surface area. The work builds on the scaling law introduced previously by co-author Mota, and Herculano-Houzel. The authors state that the observed scaling law shares properties with fractals, where shape properties are similar across several spatial scales. One way the authors assess this is to perform a "cortical melting" operation that they have devised on surface models obtained from several primate species. The authors also explore differences in shape properties between the brains of young (~20 year old) and old (~80) humans. My main criticism of this manuscript is that the findings are presented in too abstract a manner for the scientific contribution to be recognized.
We recognise that our work is at the intersection of complex mathematical concepts and a perplexing biological phenomenon. Therefore, our paper has to strike a balance between scientifically accurate and succinct descriptions whilst giving sufficient space to provide context and explanations.
Throughout, we have now added text to provide more context, but also repeat key statements in plain-english terms.
For example, we added the following text to highlight our key contributions.
“In simple terms: we know that objects can have the same fractal dimension, but differ greatly in a range of other shape properties. However, we demonstrate here, that representative primate brains and mammalian brain indeed share a range of other key shape properties, on top of agreeing in fractal dimension. This suggests a universal blueprint for mammalian brain shape and a common set of mechanisms governing cortical folding. As a practical additional outcome of our study, we could show that our novel method of deriving multiscale metrics of brain shape can differentiate subtle shape changes much better than the metrics we have been using so far at a single native scale.”
(1) The series of operations to coarse-grain the cortex illustrated in Figure 1, constitute a novel procedure, but it is not strongly motivated, and it produces image segmentations that do not resemble real brains.
To specifically explain the rationale behind the coarse-graining method, we added several clarifications, including the following paragraph:
“As a starting point for such a coarse-graining procedure, we suggest to turn to a well-established method that measures fractal dimension of objects: the so-called box-counting algorithm [Kochunov2007, Madan2019]. Briefly, this algorithm fills the object of interest (say the cortex in our case) with boxes, or voxels of increasingly larger sizes and counts the number of boxes in the object as a function of box size. As the box size increases, the number of boxes decreases; and in a log-log plot, the slope of this relationship indicates the fractal dimension of the object. In our case, this method would not only provide us with the fractal dimension of the cortex, but, with increasing box size, the filled cortex would also contain less and less detail of the folded shape of the cortex. Intuitively, with increasing box size, the smaller details, below the resolution of a single box, would disappear first, and increasingly larger details will follow -- precisely what we require from a coarse-graining method. We therefore propose to expand the traditional box-counting method beyond its use to measure fractal dimension, but to also analyse the reconstructed cortices as different realisations of the original cortex at the specified spatial scale.”
We also note in several places in the text that the coarse-grained brains are not to be understood as exact reconstructions of actual brains, but serve the purpose of a model:
“[…] nor are the coarse-grained versions of human brains supposed to exactly resemble the location/pattern/features of gyri and sulci of other primates. The similarity we highlighted here are on the level of summary metrics, and our goal was to highlight the universality in such metrics to point towards highly conserved quantities and mechanisms.”
“Note, of course, that the coarse-grained brain surfaces are an output of our algorithm alone and not to be directly/naively likened to actual brain surfaces, e.g. in terms of the location or shape of the folds. Our comparisons here between coarse-grained brains and actual brains is purely on the level of morphometrics across the whole cortex.”
The process to assign voxels in downsampled images to cortex and white matter is biased towards the former, as only 4 corners of a given voxel are needed to intersect the original pial surface, but all 8 corners are needed to be assigned a white matter voxel (section S2). This causes the cortical segmentation, such as the bottom row of Figure 1B, to increase in thickness with successive melting steps, to unrealistic values. For the rightmost figure panel, the cortex consists of several 4.9-sided voxels and thus a >2 cm thick cortex. A structure with these morphological properties is not consistent with the anatomical organization of a typical mammalian neocortex.
Specifically on the point on increasing cortical thickness with increased level of coarse-graining, we have now added the following paragraph:
“The observation that with increasing voxel sizes, the coarse-grained cortices tend to be smoother and thicker is particularly interesting: the scaling law in Eq. 1 can be understood as thicker cortices (T) form larger folds (or are smoother i.e. less surface area At) when brain size is kept constant (Ae). This way of understanding has also been vividly illustrated by using the analogy of forming paper balls with papers of varying thickness in [Mota2015]: to achieve the same size of a paper ball (Ae), the one that uses thicker paper (T) will show larger folds (or is smoother i.e. less surface area At) than the one using thinner paper. The scaling law can therefore be understood as a physically and biologically plausible statement, and here, we are encouraged that our algorithm yields results in line with the scaling law.”
(2) For the comparison between 20-year-old and 80-year-old brains, a well-documented difference is that the older age group possesses more cerebral spinal fluid due to tissue atrophy, and the distances between the walls of gyri becomes greater. This difference is born out in the left column of Figure 4c. It seems this additional spacing between gyri in 80-year-olds requires more extensive down-sampling (larger scale values in Figure 4a) to achieve a similar shape parameter K as for the 20-year-olds. A case could be made that the familiar way of describing brain tissue - cortical volume, white matter volume, thickness, etc. - is a more direct and intuitive way to describe differences between young and old adult brains than the obscure shape metric described in this manuscript. At a minimum, a demonstration of an advantage of the Figure 4a and 4b analyses over current methods for interpreting age-related differences would be valuable.
We have demonstrated the utility of our new shape metrics in a separate paper [Wang2021]. However, we agree with the reviewer that, in this specific instance, it is much easier to understand the key message without considering the less traditional metrics. We have therefore completely revised this part of the Results section to highlight the advantage of multiscale morphometrics, and used the traditional metric of surface area to illustrate the point. The reasoning in surface area is much easier to follow, both visually and conceptually, exactly as the reviewer described.
(3) In Discussion lines 199-203, it is stated that self-similarity, operating on all length scales, should be used as a test for existing and future models of gyrification mechanisms. First, the authors do not show, (and it would be surprising if it were true) that self-similarity is observed for length scales smaller than the acquired MRI data for any of the datasets analyzed. The analysis is restricted to coarse (but not fine)-graining.
To clarify this point, we have added a supplementary section and the following sentence:
“Note this method has also no direct dependency on the original MR image resolution, as the inputs are smooth grey and white matter surface meshes reconstructed from the images using strong (bio-)physical assumptions and therefore containing more fine-grained spatial information than the raw images (also see Suppl. Text 3).”
We are indeed sampling at resolutions down to 0.2mm, which is below MR image resolution. The reviewer is, however, correct that we are only coarse-graining, not “fine-graining”. Coarse-graining, here, relates to more coarse than the smooth surface meshes though, not the MR image.
Therefore, self-similarity on all length scales would seem to be too strong a constraint. Second, it is hard to imagine how this test could be used in practice. Specific examples of how gyrification mechanisms support or fail to support the generation of self-similarity across any length scale, would strengthen the authors' argument.
We agree that spatial scales much below 0.2mm resolution may not be of interest, as these scales are only measuring the fractal properties, or “bumpiness”, of the surface meshes at the vertex level. We have therefore revised our statement in Discussion and clarified it with an example:
“Finally, this dual universality is also a more stringent test for existing and future models of cortical gyrification mechanisms at relevant scales, and one that moreover is applicable to individual cortices. For example, any models that explicitly simulate a cortical surface could be directly coarse-grained with our method and compared to actual human and primate data provided here.”
Some additional, specific comments are as follows:
(4) The definition of the term A_e as the "exposed surface" was difficult to follow at first. It might be helpful to state that this parameter is operationally defined as the convex hull surface area.
We agree and introduced this term now at first use: “The exposed surface area can be thought of as the surface area of a piece of cling film wrapped around the brain. Mathematically, for the remaining paper it is the convex hull of the brain surface.”
Also, for the pial surface, A_t, there are several who advocate instead for the analysis of a cortical mid-thickness surface area, as the pial surface area is subject to bias depending on the gyrification index and the shape of the gyri. It would be helpful to understand if the same results are obtained from mid-thickness surfaces.
This point is indeed being investigated independently of this study. Our provisional understanding is that in healthy human brains, at native scale, using the mid (or the white matter) surface introduced a systematic offset shift in the scaling law, but does not affect the scaling slope of 1.25. However, this requires a more in-depth investigation in a range of other conditions, and in the context of the coarse-grained shapes, which is on-going. Nevertheless, the scaling law, at first introduction already, has been using the pial surface area [Mota2015] and all subsequent follow-up studies followed this convention. To make our paper here accessible and directly comparable, we therefore used the same metric. Future work will investigate the utility of other metrics.
(5) In Figure 2c, the surfaces get smaller as the coarse-graining increases, making it impossible to visually assess the effects of coarse-graining on the shapes. Why aren't all cortical models shown at the same scale?
The purpose of rescaling the surfaces is to match the scaling plot (Fig 2A) directly, which are showing shrinking surface areas Ae and At with increasing coarse-graining. Here, we are effectively keeping the size of the box constant and resizing the cortical surface instead, which is mathematically equivalent to changing the box size and keeping the cortical surface constant.
An alternative interpretation of the “shrinking” is, therefore, that with increasingly smaller cortical surfaces, the folding details disappear, as we require from our coarse-graining method. This is also visually apparent, as the reviewer points out. We have added this to the explanation in the text.
If we, however, changed the box size instead, the scaling law plot would be meaningless: for example, Ae would barely change with coarse-graining. We would therefore have needed to introduce more complexity in our analysis in terms of how we can measure the scaling law. Thus, we opted to present the simpler method and interpretation here.
Nevertheless, we agree that a direct comparison would be beneficial and have thus added the videos for each species in supplementary under this link: https://bit.ly/3CDoqZQ
Upon completed peer-review, we hope to integrate these directly into eLife’s interactive displays for this figure.
(6) Text in Section 3.2 emphasizes that K is invariant with scale (horizontal lines in Figure 3), and asserts this is important for the formation of all cortices. However, I might be mistaken, but it appears that K varies with scale in Figure 4a, and the text indicates that differences in the S dependence are of importance for distinguishing young vs. old brains. Is this an inconsistency?
We agree that it may be confusing to emphasise a “constant K” in the first set of results across species, and then later highlight a changing K in the human ageing results. To clarify, in the first set of results, we find a constant K relative to a changing S: the range in K across melted primate brains is less than 0.1, whereas in S it is over 1.2. In other words, S changes are an order of magnitude higher than K changes. Hence, we described K as “constant” relative to S.
Nevertheless, K shows subtle changes within individuals, which is what we were describing in the human ageing results. These changes are within the range of K values described in the across species results.
However, in the interest of clarity, we followed the reviewer’s suggestion of simplifying the last set of results on human ageing and therefore the variable K in human ageing now only appears in Supplementary. We have now added clarifications to the supplementary on this point.
Reviewer #3 (Public Review):
Summary:
Through a detailed methodology, the authors demonstrated that within 11 different primates, the shape of the brain matched a fractal of dimension 2.5. They enhanced the universality of this result by showing the concordance of their results with a previous study investigating 70 mammalian brains, and the discordance of their results with other folded objects that are not brains. They incidentally illustrated potential applications of this fractal property of the brain by observing a scale-dependent effect of aging on the human brain.
Strengths:
- New hierarchical way of expressing cortical shapes at different scales derived from the previous report through the implementation of a coarse-graining procedure.
Positioning of results in comparison to previous works reinforcing the validity of the observation.
- Illustration of scale-dependence of effects of brain aging in the human.
Weaknesses:
- The impact of the contribution should be clarified compared to previous studies (implementation of new coarse graining procedure, dimensionality of primate brain vs previous studies, and brain aging observations).
We have now made these changes, particularly by adding two paragraphs to the start of Discussion. One summarising the main contributions above previous work, and one paraphrasing the former in plain English for accessibility.
- The rather small sample sizes, counterbalanced by the strength of the effect demonstrated.
We have now increased the sample size of the human ageing analysis substantially to over 100 subjects and observe the same trends, but with an even stronger effect. We therefore believe that this revision serves as an additional internal validation of our data and methods.
- The use of either averaged or individual brains for the different sub-studies could be made clearer.
We have now added this to our Suppl methods: with the exception of the Marmoset, all brain surface data were derived from healthy individual brains.
- The model discussed hypothetically in the discussion is not very clear, and may not be state-of-the-art (axonal tension driving cortical folding? cf. https://doi.org/10.1115/1.4001683).
We have now added this citation to our Discussion and given it context:
“Indeed, our previously proposed model [Mota2015] for cortical gyrification is very simple, assuming only a self-avoiding cortex of finite thickness experiencing pressures (e.g. exerted by white matter pulling, or by CSF pressure). The offset K, or 'tension term', precisely relates to these pressures, leading us to speculate that subtle changes in K correlate with changes in white matter property [Wang2016, Wang2021]. In the same vein of speculation, the scale-dependence of K shown in this work might therefore be related to different types of white matter that span different length scales, such as superficial vs. deep white matter, or U-fibres vs. major tracts. However, there are also challenges to the axonal tension hypothesis [Xu2010]. Indeed, white matter tension differentials in the developed brain may not explain location of folds, but instead white matter tension may contribute to a whole-brain scale 'pressure' during development that drives the folding process overall.”
Reviewer #3 (Recommendations For The Authors):
Many thanks to the authors for this elegant article. I will only report here on the cosmetics of the article.
We thank the reviewer for their kind words and attention to detail and have made all the suggested changes and revised the paper generally for readability, grammar and spelling.
p2: last line of abstract: 'for a range of conditions in the future'.
p3 l.37: I would not self-describe this method as elegant as this is a subjective property .
p3 l.38: 'that will render' -> I wouldn't use the future here.
p.4 l.59: double spacing before ref [9]?
p.6 l.99: 'approximate a fractal' -> why is 'a' italicized?
p.7 fig.2: I would expect the colours to be detailed in the legend. Are there two data points per species because both hemispheres are treated separately?
p.9 l.134-135: 'similar to and in terms of the universal law 'as valid as' -> please add commas for reading comfort: 'similar to, and, in terms of the universal law, 'as valid as'.
p.9 l. 141: For all the cortices we analysed.
p.9 Fig 3: I find the colours a bit confusing in Figs B and C. I find Fig C a bit confusing: what are all the lines representative of, and more specifically, the two lower lines with a different trajectory?
p.10 l.155: '1̃500' -> '~1500'.
p.13 l. 209: either 'speculate that' of 'wonder if'.
p.14 l.232: 'neuron numbers' -> 'number of neurons'.
p.26 S2 second paragraph: 'gryi' -> 'gyri'.
p.30 l.3: please refrain from starting a sentence with I.e..
p.30 last line before S3.2: 'The algorithmic implementation in MATLAB can be found on Zenodo: TBA' - I guess this is linked to you disclosing the code upon acceptance, but please complete before final submission.
p.34 middle/bottom of page: 'The scheme described in Sec. S3.1' -> double spacing before S3.1?
p.35 l.1: 'We simply replace' -> 'we simply replace' (no capital).
p.36 Fig S5.1: explicit the same colouring of the points and boxes in legend
p.38 Fig. S6.1: briefly describe the use of colours in the legend.
p.39 Fig. S7.1: detail colours in the legend.
p.41 Fig. S7.3: detail colours in the legend.
