  During morphogenesis, a featureless convex cerebellum develops folds. As it
does so, the cortex thickness is thinnest at the crest (gyri) and thickest at
the trough (sulci) of the folds. This observation cannot be simply explained by
elastic theories of buckling. A recent minimal model explained this phenomenon
by modeling the developing cortex as a growing fluid under the constraints of
radially spanning elastic fibers, a plia membrane and a nongrowing sub-cortex
(Engstrom, et. al., PRX 2019). In this minimal buckling without bending
morphogenesis (BWBM) model, the elastic fibers were assumed to act linearly
with strain. Here, we explore how nonlinear elasticity influences shape
development within BWBM. The nonlinear elasticity generates a quadratic
nonlinearity in the differential equation governing the system's shape and
leads to sharper troughs and wider crests, which is an identifying
characteristic of cerebellar folds at later stages in development. As
developing organs are typically not in isolation, we also explore the effects
of steric confinement, and observe flattening of the crests. Finally, as a
paradigmatic example, we propose a hierarchical version of BWBM from which a
novel mechanism of branching morphogenesis naturally emerges to qualitatively
predict later stages of the morphology of the developing cerebellum.
