A computational study of the mechanisms of growth-driven folding patterns on shells, with application to the developing brain

We consider the mechanisms by which folds, or sulci (troughs) and gyri (crests), develop in the brain. This feature, common to many gyrencephalic species including humans, has attracted recent attention from soft matter physicists. It occurs due to inhomogeneous, and predominantly tangential, growth of the cortex, which causes circumferential compression, leading to a bifurcation of the solution path to a folded configuration. The problem can be framed as one of buckling in the regime of linearized elasticity. However, the brain is a very soft solid, which is subject to large strains due to inhomogeneous growth. As a consequence, the morphomechanics of the developing brain demonstrates an extensive post-bifurcation regime. Nonlinear elasticity studies of growth-driven brain folding have established the conditions necessary for the onset of folding, and for its progression to configurations broadly resembling gyrencephalic brains. The reference, unfolded, configurations in these treatments have a high degree of symmetry--typically, ellipsoidal. Depending on the boundary conditions, the folded configurations have symmetric or anti-symmetric patterns. However, these configurations do not approximate the actual morphology of, e.g., human brains, which display unsymmetric folding. More importantly, from a neurodevelopmental standpoint, many of the unsymmetric sulci and gyri are notably robust in their locations. Here, we initiate studies on the physical mechanisms and geometry that control the development of primary sulci and gyri. In this preliminary communication we carry out computations with idealized geometries, boundary conditions and parameters, seeking a pattern resembling one of the first folds to form: the Central Sulcus.