A unified computational model of cortical folding for the cerebrum and the cerebellum
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Cortical folding is a hallmark of the mammalian brain, with the cerebrum displaying profuse gyri and sulci, and the cerebellum showing densely packed, thin folia. We hypothesise, in line with previous phylogenetic studies [1,2], that a common mechanism underlies folding in both structures: a buckling instability caused by tangential growth of an elastic outer layer on a viscoelastic inner layer. In the cerebrum, tangential expansion arises from dendritic growth of pyramidal neurons in the neocortex over white matter [3]. In the cerebellum, we conjecture an analogous role for dendritic proliferation of Purkinje cells in the molecular layer over the granular layer and white matter [2]. Building on this hypothesis, our computational model of cortical folding implements two stacked layers of isotropic hyper-viscoelastic material with distinct mechanical properties. Volumetric growth is described by a multiplicative decomposition of deformation into elastic and growth components. Elastic deformation in both layers generates stress via a fixed corotational strain-energy function [5]. Tangential, isotropic and anisotropic growth is included only in the outer layer. However, unlike previous approaches, we also include stress-induced growth through plastic relaxation in both layers, with a higher plasticity in the inner one. We employ the material point method to robustly capture self-collisions and large deformations [6]. Our simulations show several levels of nested folding, where early folds buckle again once a critical threshold is reached. They also recover a direct relationship between cortical thickness and folding wavelength: thicker outer layers yield broader, sparser convolutions similar to cerebral gyri and sulci, whereas thinner layers form finer, densely packed cerebellum-like folia. Furthermore, our results suggest that curvature and symmetry in the initial geometry provide a mechanical type of positional information that guides the development of the folding pattern.
