Figures and data
Coarse-graining a cortex at different scales.
A: Example original pial surface from a healthy human viewed from varying angles. B: Same example cortical surface from A, coarse-grained to different spatial scales. Top row shows resulting pial surfaces (with a small amount of smoothing applied for visualisation purposes here). Bottom row shows the corresponding voxelisation at each scale through the slice indicated by blue plane in panel A.
Universal scaling law for 11 coarse-grained primate brains.
A: Coarse-grained primate brains are shown in terms of their relationship between 
Trajectories of coarse-grained primate cortices and other mammalian and human brains in K × S plane.
A: Straight trajectories indicate self-similarity (described by a scaling law). In particular, the black line here indicates objects with Ae = At for all scales, such as the box of finite thickness with a fractal dimension df = 1 (grey data points). This line is reproduced in all subpanels for reference. B: Morphological trajectories of multiple hypothetical fractal objects are shown which. A flat trajectory (constant K) correspond to df = 2.5 in this space. However, these objects are clearly different fractals with different values of K. C: Hypothetical objects with overlapping straight morphological trajectories indicate multiple realisations of the same fractal object. Flat trajectories (constant K) correspond to df = 2.5. The two hypothetical objects with a decreasing K(S) correspond to 2 < df < 2.5 D: Projecting our actual data into the normalised K × S plane showing the coarse-grained primate brains (same as in Fig. 2) as data point connected with solid lines (colour-code same as Fig. 2). Different mammalian brains are shown as grey scatter points, and adult human data points are blue.
Human ageing shows differential effects depending on spatial scale.
A: Top: At (λ) is shown for a group of 20-year-olds (red, n=27) and a group of 80-year-olds (blue, n=86). Mean and standard deviation are shown as the solid line and the shaded area respectively. Bottomml: Effect size (measured as ranksum z-values) between the older and younger groups at each scale. Positive effect indicates a larger value for the younger group. Blue arrows indicate the effect size at “native scale” i.e. using the original freesurfer grey and white matter meshes. B: Coronal slices of the pial surface of an example 20-year-old subject and an 80-year-old subject at different scales (columns).
Effect of using different original image resolution in five example HCP subjects.
In all plots, red lines indicate an original image resolution of 0.7mm isotropic, whereas blue line indicate an original image resolution of 1mm isotropic. Both hemispheres are shown for each subject.
Unscaled quantities cannot be used to verify scaling law.
A: The exposed surface area Ae barely changes across scales, and At only changes minimally. B: Plotted in the scaling law plane (as Fig. 2) the data points barely have any variance, and overlap each other substantially. Even after zooming in, the trace for the the coarse-graining procedure (solid line) is virtually a vertical line (with a small artifactual ‘tail’ coming from very large voxel sizes).
lr is a fixed factor for each cortical hemisphere, and does not change with λ. We use it to systematically shift all the data points within a range, such that the re-scaled quantities are not larger than those from the original cortical meshes. One can easily verify that lr will not change the slope or offset of any scaling law, but simply represents a constant shift to all data points. In our data, some of the re-scaled quantities would indeed be larger than those from the original cortical meshes, as the voxel size we choose is limited at the smaller end only by computational resources. In other words, we can use very small voxel sizes (relative to the mesh), which after re-sizing would yield very large values of Ar, Ar, and Tr. To avoid this, we chose lr simply as the ratio of the I (isometric term) of the mesh at the smallest scale we used relative to the original mesh, divided by the λ of the smallest scale:
Detailed scaling plots for each species.
Filled circle is the original mesh, empty circle are data points from the coarse-graining algorithm. Empty circles are connected by a line for visualisation. Two hemispheres are analysed separated for each species, although data points overlap substantially in plot. Slope estimates are given at top left corner in each plot.
Scaling behaviour of various shapes in K × S space.
Thick black line indicates 
Randomising grid for coarse-graining yields very consistent results within individuals in K × S space.
Thick black line indicates 
All morphological variables across scales in the 20 y.o. and 80 y.o. cohort.
Each panel shows a morphological metric, and data is shown for a group of 20-year-olds (red, n=27) and a group of 80-year-olds (blue, n=86). Mean and standard deviation are shown as the solid line and the shaded area respectively.
Effect size between the 20 y.o. and 80 y.o. cohort in all morphological variables.
Each panel shows a morphological metric, and data is shown for a group of 20-year-olds (red, n=27) and a group of 80-year-olds (blue, n=86). Effect size is measured as the ranksum z statistic between the two groups.
All morphological variables across scales in the 20 y.o. and 80 y.o. cohort in a separate (NKI) dataset.
Each panel shows a morphological metric, and data from an independent dataset (NKI) is shown for a group of 20-year-olds (red, n=10) and a group of 80-year-olds (blue, n=10).
Effect size between the 20 y.o. and 80 y.o. cohort in all morphological variables in a separate (NKI) dataset.
Effect size is measured as the ranksum z statistic between the two groups. Mean and standard deviation are shown as the solid line and the shaded area respectively.
