Extended Data Fig. 6: Morphological simulation study of fractal dimensionality (FD) and surface-to-volume voxel ratios (SVR).

a, Illustration of a simulation run. The simulation starts from a binary matrix of 0s (100x100x100 voxels) filled with a plane of 1s (100x100x1) as the initial object (left panel). In every iteration, the FD and SVR of the simulated object are estimated as in the empirical data (see Methods). Subsequently, a surface voxel of the object is chosen at random and defined as the center of a 5x5x5 cube which is set to 1. Thus, the simulated object increasingly fills more of the embedding space, as it transforms from a Euclidean plane (theoretical FD = 2) into a fully filled cube (theoretical FD = 3, right panel). The simulation ran 100 times with 30,000 iterations each, and all runs arrived at the cube. b, Results of the simulation study. The upper row shows the FD of the simulated objects as they transition from a ‘plane-like’ to a ‘cube-like’ geometry, where the gray tile marks the numerical range observed empirically in neonatal brains. The inset on the right zooms in on this range. The lines mark the empirical group averages (dHCP data) for cortical gray matter and global white matter at 35 and 40 weeks, respectively. The simulation thus suggests that white matter (WM) starts out as a more ‘cube-like’ geometry in younger infants and develops into a more ‘plane-like’ geometry towards term maturity, while the opposite is true for cortical gray matter (GM). Furthermore, the simulation suggests that the geometric properties of GM and WM converge towards term maturity, reflected in a numerical convergence of GM- and WM-FD. The lower row shows the corresponding SVR of the simulated objects (left). The simulation suggests a strong inverse correlation between SVR and FD (right; two-tailed product-moment correlation test, P ≈ 0 within machine precision). These theoretical results from the simulation are tested empirically in Extended Data Figs. 7 and 8.

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