Physical mimic and numerical simulation of tangential cortical expansion.

a, Gyrification of the human brain during the latter half of gestation (photographs from ref. 1, adapted with permission from Elsevier). b, A 3D-printed model of the brain is produced from a 3D MRI image of a smooth fetal brain and then used to create a pair of negative silicone moulds for casting. To mimic the constrained growth of the cortex, a replicated gel-brain (white matter) is coated with a thin layer of gel (cortex) that swells by absorbing a solvent (hexanes) over time t (t1 ≍ 4 min, t2 ≍ 9 min, t3 ≍ 16 min). c, The layered gel progressively evolves into a complex pattern of sulci and gyri during the swelling process. d, A simulation starting from a smooth fetal brain shows gyrification as a result of uniform tangential expansion of the cortical layer. The brain is modelled as a soft elastic solid and a relative tangential expansion is imposed on the cortical layer as shown at left, and the system allowed to relax to its elastic equilibrium.

2 Gyrification from constrained cortical expansion

Wrinkling and sulcification in a layered material subject to differential growth. (A) If the growing gray matter is much stiffer than the white matter it will wrinkle in a smooth sinusoidal way. (B) If the gray matter is much softer than the white matter its surface will invaginate to form cusped folds. (C) If the two layers have similar moduli the gray matter will both wrinkle and cusp giving gyri and sulci. Physical realizations of A, B, and C, based on differential swelling of a bilayer gel (Materials and Methods), confirm this picture and are shown in D, E, and F, respectively.

Formation of a minimal sulcus. The 2D sulci with tangential expansion ratio of (A) g = 1.30 and (B) g = 2.25 of the gray matter (Eq. 2 and Fig. S1). Coloring shows radial and circumferential tensile stress in the left and right sulci, respectively. The stress is compressive in the noncolored areas. Grid lines correspond to every 20 rows or columns of the numerical discretization with nodes. The width W, depth D, and thickness of the gray matter in the sulcus (Ts) and gyrus (Tg) are indicated in B. For comparison with observations of brains, we also show sections of porcupine and cat brains, taken from (C) Scaled dimensions of the simulated sulcus (solid lines) as a function of g compared with those in porcupine (triangles), cat (dots), and human (squares) show that our model can capture the basic observed geometry. Width and depth are given relative to the undeformed thickness T of the gray matter (for details of the measurements and error bars

Known empirical scaling laws for gray-matter volume and thickness are mapped on a g2 vs. R/T diagram. Corresponding simulations for spherical brain configurations, with images shown at a few points, show that the surface remains smooth for the smallest brains, but becomes increasingly folded as the brain size increases. We also show patterns for ellipsoidal configurations (major axis = 1.5 × minor axes) that lead to anisotropic gyrification. Images of rat, lemur, wolf, and human brains illustrate the increasingly prominent folding with increasing size in real brains. Also shown are images of our physical mimic of the brain using a swelling bilayer gel of PDMS immersed in hexanes. The smooth initial state gives rise to gyrified states for different relative sizes of the brain R/T = 10, 15 (see also Fig. 5). All of the brain images are from

(A) Sections of a simulated brain (section planes indicated at right) are compared with coronal sections of a raccoon brain (from Cuts through the center of the brain (Upper) and the off the center (Lower) show that we can capture the hierarchical folds but emphasize how misleading sections can be in characterizing the sulcal architecture. (B) Confining our simulations with a uniform pressure of 0.7μ to mimic the meninges and skull leads to a familiar flattened sulcal morphology. (C) Changing the gray matter thickness in a small patch of the growing cortex leads to morphologies similar to polymicrogyria in our simulations. Here g2 = 5 and R/T = 20 except in the densely folded region where R/T = 40. (D) A simulated brain of same physical size as that in C but with a thickened cortex (R/T = 12) and reduced tangential expansion (g2 = 2) displays wide gyri and shallow sulci reminiscent of pachygyric brains.

A physical model of brainlike instability. To mimic the growth of the gray matter in the brain, a hemispherical elastomer (radius r, shear modulus μ0c) is coated with a top elastomer layer (thickness T0, shear modulus μ0t) that swells by absorbing solvent over time t. Representative images of a bilayer specimen in the initial (dried) state and swollen state (modulus ratio μt/μc ≈ 1) are shown at right

Cross-section views of 3D simulation geometries for small and large brains in their initial undeformed states. The gray-matter thickness T, brain radius R, and boundary conditions are indicated. A detailed image of the regular mesh structure of the large brain domain shows the reflection symmetry between every pair of elementary cubes that share a face.

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