The role of thickness inhomogeneities in hierarchical cortical folding.

The mammalian brain cortex is highly folded, with several developmental disorders affecting folding. On the extremes, lissencephaly, a lack of folds in humans, and polymicrogyria, an overly folded brain, can lead to severe mental retardation, short life expectancy, epileptic seizures, and tetraplegia. Not only a specific degree of folding, but also stereotyped patterns, are required for normal brain function. A quantitative model on how and why these folds appear during the development of the brain is the first step in understanding the cause of these conditions. In recent years, there have been various attempts to understand and model the mechanisms of brain folding. Previous works have shown that mechanical instabilities play a crucial role in the formation of brain folds, and that the geometry of the fetal brain is one of the main factors in dictating its folding characteristics. However, modeling higher-order folding, one of the main characteristics of the highly gyrencephalic brain, has not been fully tackled. The simulations presented in this work are used to study the effects of thickness inhomogeneity in the gyrogenesis of the mammalian brain in silico. Finite-element simulations of rectangular slabs are performed. These slabs are divided into two distinct regions, where the outer region mimicks the gray matter, and the inner region the underlying white matter. Differential growth is introduced by growing the top region tangentially, while keeping the underlying region untouched. The brain tissue is modeled as a neo-Hookean hyperelastic material. Simulations are performed with both homogeneous and inhomogeneous cortical thicknesses. Our results show that the homogeneous cortex folds into a single wavelength, as is common for bilayered materials, while the inhomogeneous cortex folds into more complex conformations. In the early stages of development of the inhomogeneous cortex, structures reminiscent of the deep sulci in the brain are obtained. As the cortex continues to develop, secondary undulations, which are shallower and more variable than the structures obtained in earlier gyrification stage emerge, reproducing well-known characteristics of higher-order folding in the mammalian, and particularly the human, brain.