The exterior of the mammalian brain—the cerebral cortex—has a conserved layered structure whose thickness varies little across species. However, selection pressures over evolutionary time scales have led to cortices that have a large surface area to volume ratio in some organisms, with the result that the brain is strongly convoluted into sulci and gyri. Here we show that the gyrification can arise as a nonlinear consequence of a simple mechanical instability driven by tangential expansion of the gray matter constrained by the white matter. A physical mimic of the process using a layered swelling gel captures the essence of the mechanism, and numerical simulations of the brain treated as a soft solid lead to the formation of cusped sulci and smooth gyri similar to those in the brain. The resulting gyrification patterns are a function of relative cortical expansion and relative thickness (compared with brain size), and are consistent with observations of a wide range of brains, ranging from smooth to highly convoluted. Furthermore, this dependence on two simple geometric parameters that characterize the brain also allows us to qualitatively explain how variations in these parameters lead to anatomical anomalies in such situations as polymicrogyria, pachygyria, and lissencephalia.The mammalian brain is functionally and anatomically complex. Over the years, accumulating evidence (
1,
2) shows that there are strong anatomical correlates of its information-processing ability; indeed the iconic convoluted shape of the human brain is itself used as a symbol of its functional complexity. This convoluted (gyrified) shape is associated with the rapid expansion of the cerebral cortex. Understanding the evolutionary and developmental origins of the cortical expansion (
1–
6) and their mechanistic role in gyrification is thus an important question that needs to be answered to decipher the functional complexity of the brain.Historically there have been three broad hypotheses about the origin of sulci and gyri. The first is that gyri rise above sulci by growing more (
7), requiring the pattern of sulci and gyri to be laid down before the cortex folds, presumably by a chemical morphogen. There is no evidence for this mechanism. The second hypothesis considers that the outer gray matter consists of neurons, and the inner white matter is largely long thin axons that connect the neurons to each other and to other parts of the nervous system and proposes that these axons pull mechanically, drawing together highly interconnected regions of gray matter to form gyri (
8–
10). However, recent experimental evidence (
11) shows that axonal tension when present is weak and arises deep in the white matter and is thus insufficient to explain the strongly deformed gyri and sulci. The third hypothesis is that the gray matter simply grows more than the white matter, an experimentally confirmed fact, leading to a mechanical buckling that shapes the cortex (
11–
14). Evidence for this hypothesis has recently been provided by observations of mechanical stresses in developing ferret brains (
11), which were found to be in patterns irreconcilable with the axonal tension hypothesis. In addition, experiments show that sulci and gyri can be induced in usually smooth-brained mice by genetic manipulations that promote cortical expansion (
15,
16), suggesting that gyrification results from an unregulated and unpatterned growth of the cortex relative to sublayers.Nevertheless, there is as yet no explicit biologically and physically plausible model that can convincingly reproduce individual sulci and gyri, let alone the complex patterns of sulci and gyri found in the brain. Early attempts to mechanically model brain folding (
13) were rooted in the physics of wrinkling and assumed a thin stiff layer of gray matter that grows relative to a thick soft substrate of white matter. This model falls short in two ways. First, the gray matter is neither thin nor stiff relative to the white matter (
17,
18). Second, this model predicts smooth sinusoidal wrinkling patterns, sketched in , whereas even lightly folded brains have smooth gyri but cusped sulci. More complicated mechanical models including, e.g., elasto-plasticity and stress-related growth (
14,
19,
20), lead to varying morphologies, but all produced simple smooth convolutions rather than cusped sulci.
Open in a separate windowWrinkling 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.A fundamentally different mechanical instability that occurs on the surface of a uniformly compressed soft solid (
21,
22) has recently been exposed and clarified, theoretically, computationally, and experimentally (
23–
26). This sulcification instability arises under sufficient compression leading to the folding of the soft surface to form cusped sulci via a strongly subcritical transition. In , we show a geometry dual to that associated with wrinkling: A soft layer of gray matter grows on a stiff white-matter substrate. Unlike wrinkling, this instability can produce the cusped centers of sulci, but the flat bottom of the gray matter is not seen in the brain. This is a consequence of the assumption that the gray matter is much softer than the white matter—in reality the two have very similar stiffnesses (
17,
18). We are thus led to the final simple alternate, sketched in , where the stiffnesses of the gray and white matter are assumed to be identical. Such a system is subject to a cusp-forming sulcification instability discussed earlier, and can lead to an emergent pattern very reminiscent of sulci and gyri in the brain.
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