An important implication of this model is that gyrencephaly is a generalized mechanical product of differential tangential surface expansion and is not reducible to a single evolutionary adaptation.
Here, we investigate an alternative hypothesis, namely, whether the differential tangential expansion of the cortex alone can account for the degree and pattern-specificity of gyrification.
In order to investigate this hypothesis, it is necessary to quantify both the degree of differential tangential expansion, as well as its pattern-specificity across the cortex, and compare these to the corresponding regional degree and pattern of gyrification.
In summary, regional variations in expansion due to development, cytoarchitecture or both, may give rise to a characteristic differential tangential expansion of the cortex, which in turn may induce pattern-specific gyrification.
Thus, the more mathematically fundamental nature of intrinsic curvature supports the hypothesis that cortical gyrification arises from differential tangential expansion of the cortex rather than intrinsic curvature arising as a consequence of extrinsic folding.
In order to relate this to the degree of differential tangential expansion of the cortex, we used intrinsic curvature as a marker of the latter (Todd 1985; Ronan et al. 2011).
Discrete approximations of several first- and second-order geometric differential operators, such as the tangential gradient operator, the second tangential operator, the Laplace Beltrami operator and the Giaquinta Hildebrandt operator, are utilized in the numerical integrations.
Besides, we denote the differential length in the tangential direction at any point on M as ∂l = aβ∂φ.
Using the Berreman 4 × 4 matrix method [19], the Maxwell equations are reduced to four ordinary differential equations in terms of tangential electric and magnetic field components.
The derived equations are fully coupled partial differential equations through the transverse, radial and tangential displacements.
