  Understanding the mechanics of brain embryogenesis can provide insights on
pathologies related to brain development, such as lissencephaly, a genetic
disease which cause a reduction of the number of cerebral sulci. Recent
experiments on brain organoids have confirmed that gyrification, i.e. the
formation of the folded structures of the brain, is triggered by the
inhomo-geneous growth of the peripheral region. However, the rheology of these
cellular aggregates and the mechanics of lissencephaly are still matter of
debate. In this work, we develop a mathematical model of brain organoids based
on the theory of morpho-elasticity. We describe them as non-linear elastic
bodies, composed of a disk surrounded by a growing layer called cortex. The
external boundary is subjected to a tissue surface tension due the
intercellular adhesion forces. We show that the resulting surface energy is
relevant at the small length scales of brain organoids and significantly
affects the mechanics of cellular aggregates. We perform a linear stability
analysis of the radially symmetric configuration and we study the post-buckling
behaviour through finite element simulations. We find that the process of
gyrification is triggered by the cortex growth and modulated by the competition
between two length scales: the radius of the organoid and the capillary length
due to surface tension. We show that a solid model can reproduce the results of
the in-vitro experiments. Furthermore, we prove that the lack of brain sulci in
lissencephaly is caused by a reduction of the cell stiffness: the softening of
the organoid strengthens the role of surface tension, delaying or even
inhibiting the onset of a mechanical instability at the free boundary.
