Effects of parcellation and threshold on brainconnectivity measures

Abstract

It is shown that the statistical properties of connections between regions of the brain and their dependence on coarse-graining and thresholding in published data can be reproduced by a simple distance-based physical connectivity model. This allows studies with differing parcellation and thresholding to be interrelated objectively, and for the results of future studies on more finely grained or differently thresholded networks to be predicted. As examples of the implications, it is shown that the dependences of network measures on thresholding and parcellation imply that chosen brain regions can appear to form a small world network, even though the network at finer scales, or ultimately of individual neurons, may not be small world networks themselves.

Fig 1. Simulated neural network.

Locations of the neurons (black crosses) in the simulated network on a square cortex, in 36 regions of 5 neurons each (red lines show the borders of each region, axes show spatial coordinates on the square cortex with side length normalized to 1).

Fig 2. Simulated neural connections.

Connections (blue lines, darker for one particular neuron) from neurons in one region to ones in the rest of the simulated network, with the same neurons and regions as Fig 1. Axes show spatial coordinates, as in Fig 1.

Fig 3. Thresholded neural connections.

Connections remaining after thresholding between the 36 regions in Fig 1 for three different thresholds. (a) Threshold of 1 connection. (b) 2 connections. (c) 3 connections. Axes show spatial coordinates, as in Fig 1.

Fig 4. Thresholded inter-regional connections.

Connections between regions for 36 regions of 100 neurons, for three thresholds. (a) Threshold of 200 connections. (b) 600 connections. (c) 1000 connections. Axes show spatial coordinates, as in Fig 1.

Fig 5. Clustering coefficient vs. θ.

C vs. θ for different values of R, where R = 84 is red, 91 is green, 230 is blue, 438 is orange, 890 is cyan, 1314 is magenta. (a) Results from the simulated networks. (b) Results from the observed networks, adapted from [8].

Fig 6. Clustering coefficient vs. θ.

Same as Fig 5 for the full range of thresholds, where lines correspond to the number of regions, R = 84, 91, 230, 438, 890, and 1314 from bottom to top. (a) Regions in a sphere. (b) Regions in a disk. (c) Regions on a spherical shell.

Fig 7. Simulated clustering coefficient vs. θ.

The median (black), one standard deviation from median (blue) and two standard deviations from median (red) clustering coefficients vs. threshold from 400 simulated regional networks, with R = 84.

Fig 8. Clustering coefficient vs. R and θ.

C vs. R and θ, with log scales on both axes and the color bar indicates the value of C. The white dots show the R and θ values for three selected studies. In order of decreasing R (top to bottom), they are from [3], [1], and [2].

Fig 9. Clustering coefficient vs. R and Rθ.

C vs. R and Rθ, with log scales on both axes and the color bar indicates the value of C. The white dots show the R and Rθ values for three selected studies, and the vertical red line is Rθ = 5. In order of decreasing R (top to bottom), they are from [3], [1], and [2].

Fig 10. Largest connected component.

Size of the largest connected component of the network as a percentage of the entire network vs. θ for the same networks as in Fig 5(a). The bottom-most curve is for R = 84, the middle is for R = 91, and the upper combines all the cases from with R ≥ 230, which are indistinguishable on this scale.

Fig 11. Path length vs. threshold.

L vs. θ for different values of R (R = 84 is red, 91 is green, 230 is blue, 438 is orange, 890 is cyan, 1314 is magenta). (a) Results from the simulated networks. (b) Results from the networks adapted from [8].

Fig 12. Path length vs. threshold.

L vs. θ as in Fig 11a, but extended to cover all θ. The inset shows a closeup for low θ and L.

Fig 13. Simulated path length scaling.

L⁻³ in the simulated regional network (averaged over 100 networks for each curve) vs. θ for changing R where R = 84 is red, 91 is green, 230 is blue, 438 is orange, 890 is cyan, 1314 is magenta. (a) Regions in a sphere. (b) Regions on a disk. (c) Regions on a spherical shell.

Fig 14. Experimental path length scaling.

L⁻³ vs. θ for same data as in Fig 11 for different values of R, R = 84 is red, 91 is green, 230 is blue, 438 is orange, 890 is cyan, 1314 is magenta.

Fig 15. Small-worldness vs. threshold.

σ vs. θ for the same data as Fig 6, where R = 84 is red, 91 is green, 230 is blue, 438 is orange, 890 is cyan, 1314 is magenta. Inset is zoomed to show low θ.

Fig 16. Small-worldness vs. threshold.

ω vs. θ for the same data as Fig 6, where R = 84 is red, 91 is green, 230 is blue, 438 is orange, 890 is cyan, 1314 is magenta. Inset is zoomed to show low |ω|).