Figure 1.
Top: random networks with the same edge weight distribution as the brain. Middle: random networks with the same weight distribution as well as weighted node degree distribution as the brain. Bottom: random networks with the same weight distribution as well as topology as the brain. Wiring cost of the real brain network is shown by the red vertical bar for comparison.
Figure 2.
Left: Normalized wiring cost and path length of cheaper-than-brain networks, obtained from 10 runs of Algorithm A4. Cheaper wiring comes with reduced path length. The jump in path length is a result of the network splitting into two pieces. Right: Wiring cost, total inter-hemispheric connection weight and average path length of the brain and two contrived networks with cheaper wiring cost. Cheaper wiring is achieved by redirecting inter-hemispheric connections to sub-cortical ones, which results in higher path length. The wiring cost and connection weights are in units of millions.
Figure 3.
Convergence performance of wiring cost minimization algorithm.
Part (a) shows cost function of combined healthy network after random starting configurations, (b) shows cost over individual subjects' networks, after 10 random initializations, and (c) shows the mean (bold curve), upper and lower quartiles (dotted lines) of (b). Both the wiring cost (in blue, (2)) and a measure of similarity to the brain anatomic configuration (in red) are shown. The y-axis is in arbitrary units, after normalizing each quantity by the value at the initial random configuration. Note that although the algorithm only minimizes for wiring cost, the similarity measure is also getting optimized.
Figure 4.
Brain configurations using various algorithms.
Point cloud of the brain (left) and the wiring-optimally configuration (middle) are shown, color coded by lobe and sized according to node strength. For comparison the eigen-vector method [11] is also being shown (right).
Figure 5.
3D rendered unit sphere surface of the brain and the wiring-optimal configuration.
This is an approximately coronal view overlooking the parietal lobe, color coded by lobe.
Figure 6.
Sagittal view of previous figure.
This is an approximately sagittal view showing the intersection of the parietal, temporal and frontal lobes. The dataset being shown is the same as Figure 5.
Figure 7.
3D surface color coded by cortical region.
The coronal view and the dataset is identical to that of Figure 5.
Figure 8.
Effect of random perturbations.
(a) Minimization of the wiring cost and similarity measure for a randomly perturbed connectivity matrix - 5% rewiring. (b) Corresponding optimal surface map, color coded by cortical region. There is a noticeable mismatch to the brain sphere map.
Figure 9.
Wiring cost minimization for 1000 random networks.
Histogram of wiring cost of random networks at the brain's configuration (blue) and after placement optimization (red) for 1000 random networks, with the same edge weight distribution as the brain (top), and the same weighted degree distribution (bottom). Wiring cost of the real brain network is shown by the dotted vertical bar for comparison. Placement optimization greatly reduces wiring cost for any random network, sometimes giving cheaper-than-brain wiring cost (bottom).