Neuro-evolutionary evidence for a universal fractal primate brain shape

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. Additionally, several terms are introduced without adequate explanation and contextualization, further complicating comprehension. Does the second section, "2. Coarse-graining procedure", serve as an introduction or a method? 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.

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.

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. 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.

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.

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. 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.

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. 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.

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?

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?

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).
- The rather small sample sizes, counterbalanced by the strength of the effect demonstrated.
- The use of either averaged or individual brains for the different sub-studies could be made clearer.
- 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).

Author Response:

We thank all reviewers for their comments and effort to improve our paper. We appreciate that the writing can be clarified overall, and some sections need more elaboration. We will provide these in the next revision within the coming months. Particularly, we will focus on some common themes identified by all reviewers:

1. We will clarify 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 analysis purely focuses on the likeliness in terms of whole-brain morphometrics between actual brains and coarse-grained brains. Specifically on the point of “thickening” of the brain: this is anatomically well-founded, as less folded brains have a “thicker” cortex than more folded brains, when they are all normalised to the same size. This is fundamentally why the universal scaling law also applies to these coarse-grained brains. We will provide more detail to highlight this.

2. We will clarify the motivation behind our coarse-graining procedure better: mathematically, this is directly inspired by box-counting algorithms in fractal geometry; but this algorithm also has elegant parallels with other algorithms which we will highlight.

3. The age effects are demonstrated here in a small sample as a proof-of-principle, but we will update our latest results using ~100 subjects from the CamCAN data demonstrating the same effect. We have additionally described and verified these age effects in more detail in a separate preprint (https://arxiv.org/abs/2311.13501) with ~1500 subjects, and additionally showed that scale-dependent metrics substantially improve understanding and applications such as brain age prediction.

4. We have independently also received the feedback that we need to clarify how our method interacts with different resolution of the original MRI. We will add this as a new set of results, demonstrating that the MRI acquisition resolution (within a reasonable range) has a very small effect, as our method takes the reconstructed surfaces as a starting point.

5. 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. We will clarify that 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 approx 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 are describing in the human ageing results. These changes are within the range of K values described in the across species results.

6. Finally, we will also make sure to summarise our specific contributions beyond existing work:

   (i) 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, 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 and within species, but all on one (native) scale with comparable effect sizes for classification applications.