Figure 2.
Measuring cortical complexity in three dimensions. Measuring cortical complexity in three dimensions avoids biases with GIs that depend on the orientation in which the brain is sliced. a, The idea behind gyrification indices, which measure cortical folding based on a series of MRI sections [adapted from Zilles et al. (1988)]. The GI compares the perimeter of the inner contour of the cortex, following sulcal crevices, with the perimeter of the cortical convex hull, which is the convex curve with smallest area that encloses the cortex. The ratio of these is computed and expressed as a weighted mean across slices. Instead, our approach computes complexity from a spherical surface mesh that is deformed onto the cortex (b). The cortex is then mathematically regridded at successively decreasing frequencies (c), such that smoother cortices have less surface area. By plotting the observed surface area versus the cutoff spatial frequency in the surface representation, on a log-log plot (d), more complex objects have greater gradients. This plot is called a multifractal plot: the x-axis represents the log of number of nodes in the surface grid (here denoted by ln N), and the y-axis measures the log of the surface area of the resulting mesh [here denoted by ln A(M(N)), where A is the area function and M(N) is the surface mesh with N nodes]. For nonflat surfaces, this plot has a positive slope, because the surface area increases as more nodes are included in the mesh. The slope of this plot is added to 2 to get the fractal dimension of the surface (Thompson et al., 1996) [b and c were adapted from Gu et al. (2003)]. Adding the gradient of the multifractal plot to 2 is a convention used when computing fractal dimensions for surfaces. It ensures that the computed fractal dimension of a flat 2D plane agrees with its Euclidean dimension, which is 2, because the surface is 2D (for details, see Materials and Methods).