Folding of the cerebellar cortex is clade-specific in form, but universal in degree
Like the cerebral cortex, the surface of the cerebellum is repeatedly folded. Unlike the cerebral cortex, however, cerebellar folds in a given brain are much thinner and more numerous; repeat themselves largely along a single direction, forming long strips transverse to the mid-sagittal plane, like an accordion; and occur in the smallest of cerebella, including those of lissencephalic mammals and non-mammal vertebrates. We have shown previously that while the location of folds in mammalian cerebral cortex is clade-specific, the overall degree of folding strictly follows a universal power law relating cortical thickness, and the exposed and total surface areas. This law is derived from a statistical-physics model for gyrification that postulates that folding results from the interplay between axonal elongation dynamics and the self-avoiding nature of the expanding cortical surfaces. Since both aspects are present in the cerebellum, we hypothesize that a similar relation across species also exists therein. Furthermore, given the modular organization of cerebellar architecture and circuitry, as well as the transverse orientation of the folia, it is plausible that this relation is reflected in the degree of folding of the mid-sagittal section of the cerebellum, which greatly facilitates analysis. Here we show that a strict universal scaling law does apply to the folding of the mid-sagittal sections of the cerebellum of 53 species belonging to six mammalian clades, spanning a large range of sizes and degrees of gyrification. This folding is hierarchical and can be explicitly separated into branching orders, such that position of the 1st-order folds is largely stereotypical across all mammals examined. Subsequent nth-order folds become progressively less stereotypical, and folding within such cerebellar subsections scales with power laws whose exponents decrease monotonically with branching order, converging to the exponents predicted by a two-dimensional version of the same gyrification model that describes cortical folding. We propose that the changes in scaling exponent with branching order occurs as increasing amounts of white matter are included in the folding volume of the cerebellum, reflecting the difference between the outside-in development of the cerebellar cortex around a preexisting core of already connected white matter, compared to the inside-out development of the cerebral cortex with a white matter volume that develops as the cerebral cortex itself gains neurons. Our data strongly indicate that the mammalian cerebellum folds as a multi-fractal object, emerging from the interplay between clade-specificity and universality, and between phylogenetical contingency and the physics of self-organization. Thus, repeated folding, one of the most recognizable features of biology, can arise simply from the universal applicability of physical principles, without the need for invoking selective pressures in evolution; and diversity arises within the constraints imposed by physics.