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 vs. log10(Ae). Each solid line indicates a cortical hemisphere from a primate species. Thin grey lines indicate a slope of 1 for reference. Filled circles mark the data points of the original cortical surfaces. Grey data points are plotted for reference and show the comparative neuroanatomy dataset across a range of mammalian brains [5]. B: Slopes (a) of the regression of data points in A for each species. C: Rescaled brain surfaces visualised for five example species at different levels of coarse-graining.

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: Multiple hypothetical flat trajectories are shown which correspond to df = 2.5 in this space. C: Hypothetical overlapping straight trajectories indicate multiple realisations of the same fractal object. 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. 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: K(λ) is shown for a group of 20-year-olds (red) and a group of 80-year-olds (blue). Individual data points show individual subjects. Mean and standard deviation are shown as the solid line and the shaded area respectively. Bottom: 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 grey and white matter meshes. B: Same as A, but for log10 At (without rescaling). C: Coronal slices of the pial surface of a 20-year-old and an 80-year-old at different scales (columns).

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).

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 and a gyrification index . Points above and below the line have, respectively, g > 1 and g < 1. Colour-coded boxes show the shape of the objects we analysed. For simplicity we show the outer “pial” surface of each object. In the case of the box, we indicated both outer and inner surfaces.

Randomising grid for coarse-graining yields very consistent results within individuals in K × S space.

Thick black line indicates and a gyrification index . Each colour in the plot indicates a human individual (HCP 103414,135225,138534,144832,148840 respectively). 30 jittered grid outputs are shown for each individual, lines connect datapoints of the same jittered version. Zoomed in panels show that the jittered outputs have far less variation within than between individuals.

All morphological variables across scales in the 20 y.o. and 80 y.o. cohort.

Effect size between the 20 y.o. and 80 y.o. cohort in all morphological variables.

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.

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.