## Abstract

The cerebrum of large mammals is convoluted, whereas that of small mammals is smooth. Mota and Herculano-Houzel (Reports, 3 July 2015, p. 74) inspired a model on an old theory that proposed a fractal geometry. I show that their model reduces to the product of gray-matter proportion times the folding index. This proportional relation describes the available data even better than the fractal model.

A question that has inspired generations of neuroanatomists is why our cerebrum is walnut-shaped—and why that of small mammals is lissencephalic (smooth), whereas that of the largest ones is even more strongly convoluted than our own. Mota and Herculano-Houzel claim to have solved this mystery (*1*).

At least two earlier models exist that make quantitative predictions for the scaling of the mammalian cerebrum. Braitenberg (*2*) used a small data set and found that his model fits relatively small cerebra but not the human data. A larger data set (*3*) has confirmed that Braitenberg’s model fails dramatically for all large brains. Zhang and Sejnowski (*4*) did validate their model using a large data set, with very good results. Unfortunately, a closer look shows that they assumed that the cortical thickness scales with cerebral size by a power law, which is not the case (*5*). It can be shown that correcting for this is fatal for Zhang and Sejnowski’s model (*6*).

Thus, it is great news if a scaling relation is found that does have a theoretical and quantitative biological underpinning. Mota and Herculano-Houzel developed a 25-year-old hypothesis that the folding pattern of the mammalian cerebrum follows a fractal geometry and scaling (*7*). Fractal geometry is the self-similarity over a large range of scales. Classical examples from nature are coastal lines (*8*) and clouds (*9*). Mota and Herculano-Houzel derived a scaling relation using a model that was based on the tension hypothesis by D. C. Van Essen (*10*).

The authors did not acknowledge that the fractal hypothesis was first formulated by Hofman (*7*) [although they did use his data set (*11*)]; moreover, the tension model was only referred to in the supplement, and none of the existing quantitative models for cerebral scaling relations were cited.

The scaling relation derived by Mota and Herculano-Houzel is (1)with the exposed cortical surface area, the total cortical surface area, and cortical thickness *T*. Let us call the dimensionless constant *k* the “fractal constant.” Mota and Herculano-Houzel defined cortical thickness as “equivalent thickness,” which is the ratio of gray-matter volume to the cortical surface area . (*12*)

Given the huge size range of mammalian cerebra, relations are typically plotted on a double logarithmic scale, which has the disadvantage that deviations appear to be very small and provide values that are very close to 1. A standard method to display the goodness of fit is to plot the residuals (Fig. 1A).

Given that , the fractal model can be written as (2)Taking the square and solving to *k* gives (3)For better comparability, *k*^{2}, rather than *k*, will be used in the following. Obviously, the fractal constant is not nearly constant but increases with the exposed surface area (Fig. 1B), consistent with the trend in the fitting residual (Fig. 1A). In the original manuscript (*1*), this fact was hidden: Rather than showing the model fit, the authors chose to fit not only *k* but also the power of *A*_{E}, and concluded that the power (1.29) was “close to” 5/4. However, this seemingly small difference between fitted and predicted power is statistically highly significant: *t*(56) = 8.2, *P* < 0.0001.

An implicit assumption of Mota and Herculano-Houzel is that the overall shape of the cerebrum scales in an isometric manner (i.e., that the overall, “exposed” shape is independent of the size of the cerebrum). This assumption can be made explicit, as follows. For a perfect sphere, the relation between the exposed surface *A*_{sphere} and the total volume *V*_{sphere}, is , with . Because a cerebral hemisphere is not a perfect sphere, its volume is smaller than that of a perfect sphere that has the same surface area. We can therefore write for the relation between the exposed surface and the cerebral volume, , with *s *< 1 being a constant that expresses the sphericity of the cerebrum. Thus, substituting Eq. 3 with gives (4)with the “folding constant” kappa, a new dimensionless cerebral parameter (Fig. 1C). Thus, what looked like a fractal dimension of the outer surface in Eq. 1 is simply the product of the proportion of gray-matter volume *V*_{G}/*V*_{T} times the folding index *I* = *A*_{T}/*A*_{E}.

We conclude, first, that what looked like a complex fractal relation between exposed surface, total cortical surface, and cortical thickness is really a simple proportional relation of gray-matter volume and the folding index. Consequently, there is a proportionate relation , provided that the exposed shape of the cerebrum is approximately invariant. Here lies the critical difference between Eqs. 1 and 4: Whereas Eq. 1 involves , Eq. 4 uses the product of *A*_{E}/*V*_{T}.

Second, *k*^{2} and κ are not simply equivalent because they differ by a factor σ*s* where the sphericity *s* of the cerebral hemispheres differs between species (*3*). Whereas *k* is a just a fitting constant without biological meaning (Eq. 1), κ has direct biological relevance, being the product of the folding index and the proportion of gray matter. Here, the data clearly speak in favor of the proportional relation (Eq. 4), because κ has a much smaller coefficient of variance (COV) than *k*^{2}. This is even clearer when comparing cerebra of similar exposed surface area (see Fig. 1, B and C). The difference between the variances is statistically reliable [F(41,41) = 0.4, *P* = 0.009, after removal of a linear trend].

This brings us to a third conclusion. Because κ shows extraordinary little variance for cerebra of similar size, the figure reveals that κ increases with cerebral size in a complex manner. Since κ ≈ 1 for *A*_{E} < 10^{3} mm^{2} but gradually increases for larger cerebra, the relation between κ and cerebral size cannot be explained by a simple power relation (*13*).

Mota and Herculano-Houzel motivated their supposed fractal relation with a developmental growth model that was inspired by the tension model (*10*). The present analysis suggests that their model fails to explain a trend in the data that cannot be mended in a simple manner, such as slightly changing the power relation as Mota and Herculano-Houzel did. The data also strongly suggest that the model addresses the wrong parameters ( instead of *A*_{E}*V*_{T}). Nevertheless, we cannot conclude that the cerebral surface does not have a fractal shape, because there is no evidence against that. However, at present there is no evidence, empirical nor theoretical, in favor of a clear fractal relation. A more promising strategy would probably be to return to the definition of fractal shapes and analyze the self-similarity of the cortical surface of a few, particularly strongly convoluted, cerebra across a range of scales, as well as to search for a folding rule, rather than plotting the whole variety of mammalian species.

Small mammalian cerebra consist almost entirely of gray matter (*4*) and are lissencephalic so that κ ≈ 1. Larger, convoluted, cerebra have an increasingly large proportion of white matter so that the ratio is less than one. Because, for these cerebra, the folding index increases disproportionately to the reduction in the fraction of gray-matter volume, κ is larger than one. Also, even though the cetaceans seem to be less extreme in terms of κ than in terms of *k*^{2}, they might still be systematically off the general trend for mammalian cerebra. Explanations range from the exclusively aquatic life mode, which seems to increase cortical thickness (i.e., *V*_{G}) as a result of lack of REM sleep (*14*), to increased glia for heating the brain (*15*).

However, the fractal model does not describe the data satisfactorily. The κ model describes the relation to the exposed surface area better, but, paradoxically, the systematic increase of κ with cerebral size shows even more convincingly that both models fail to explain the scaling relation of the cerebrum. Obvious “repairs” by fitting the power as in Mota and Herculano-Houzel (*1*), or as given in note (*13*), lack theoretical biological underpinning. Thus, the remarkably regular scaling relations of the mammalian cerebrum remain a highly intriguing but yet unsolved mystery.

## References and Notes

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- ↵Using the logarithmic relation of Hofman (
*5*) or the fit of the cortical thickness by de Lussanet [figure S1 in (*3*)] leads to large errors, especially for small cerebra. - ↵
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- ↵The equivalent thickness does not truly measure the cortical thickness because the cortical surface is not flat (hence, “equivalent” thickness).
- ↵One could get rid of the trend in κ if one would allow to scale the folding index (Eq. 4) by a power of 2/5 (i.e., =
*I*^{2/5}*V*_{G}/*V*_{T}). The resulting relation has a COV of just 0.08 and no systematic dependency of the exposed surface area. - ↵
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**Acknowledgments:**I thank K. Boström and H. Wagner for their inspiring discussions.