Symmetry Breaking in Wrinkling Patterns: Gyri Are Universally Thicker than Sulci

Abstract

Wrinkling instabilities appear in soft materials when a flat elastic layer on an elastic substrate is sufficiently stressed that it buckles with a wavy pattern to minimize the energy of the system. This instability is known to play an important role in engineering, but it also appears in many biological systems. In these systems, the stresses responsible for the wrinkling instability are often created through differential growth of the two layers. Beyond the instability, the upper and lower sides of the elastic layer are subject to different forces. This difference in forces leads to an interesting symmetry breaking whereby the thickness becomes larger at ridges than at valleys. Here we carry out an extensive analysis of this phenomenon by combining analytical, computational, and simple polymer experiments to show that symmetry breaking is a generic property of such systems. We apply our idea to the cortical folding of the brain for which it has been known for over a century that there is a thickness difference between gyri and sulci. An extensive analysis of hundreds of human brains reveals a systematic region-dependent thickness variation. Our results suggest that the evolving thickness patterns during brain development, similar to our polymer experiments, follow simple physics-based laws: Gyri are universally thicker than sulci.

The role of thickness inhomogeneities in hierarchical cortical folding

“…The gyral crowns are not affected as strongly due to their relatively small curvature (Bok, 1929). Our simulations are consistent with this principle, showing that an initially homogeneous cortex can develop cortical inhomogeneities through buckling, coherent with prior observations on homogeneous systems (Holland et al, 2018;Riccobelli and Bevilacqua, 2020;Welker, 1990).…”

“…Soft layered systems can be realized experimentally by gel slabs coated with gels with different properties (Auguste et al, 2014;. These systems have been used as simulacra for brain folding (Holland et al, 2018;Tallinen et al, 2016), where they were able to mimic the folding a 3D-printed human brain. Thus, the production of samples with sinusoidal variation of the top-layer thickness would allow for the experimental testing of our predictions.…”

Mathematical Modeling of Cortical Neurogenesis Reveals that the Founder Population does not Necessarily Scale with Neurogenic Output

“…Specifically, it was shown that cortical folding approximately scales with surface area and thickness, and not with neuronal numbers ( Mota and Herculano-Houzel 2015 ). Additionally, through integration of imaging and mathematical modeling it was demonstrated that variations in cortical thickness within the cortex are caused by the physical forces emerging during the folding process ( Holland et al 2018 ). However, the problem of neocortex folding is a separate avenue of research, decoupled from the mechanisms that we want to investigate here.…”

Tug-of-war between stretching and bending in living cell sheets

“…The equilibrium value of ξ eq can be easily translated into a measured deflection through Eq. (11), which, at zeroth order in e/L, reads…”

“…Such activity is responsible for the appearance of a cleavage cytokinetic furrow driving the division of a single cell [2,3], or multicellular topological transitions during development such as mesoderm invagination in Drosophila [4,5] and inversion of the Volvox embryo [6], both involving hundreds of cells. The detailed patterns resulting from these mechanical interactions often involve instabilities where elastic stretching and bending deformations play an important role, such as in the rupture and subsequent curling of single red blood cells [7,8] or the formation of villi in the gut [9] and gyri and sulci in the brain [10][11][12] which shape entire organs.…”