Brain structure reflects the influence of evolutionary processes that pit the costs of its anatomical wiring against the computational advantages conferred by its complexity. We show that cost-neutral ‘mutations’ of the human connectome almost inevitably degrade its complexity and disconnect high-strength connections to prefrontal network hubs. Conversely, restoring the peripheral location and strong connectivity of empirically observed hubs confers a wiring cost that the brain appears to minimize. Progressive cost-neutral randomization yields daughter networks that differ substantially from one another and results in a topologically unstable phenomenon consistent with a phase transition in complex systems. The fragility of hubs to disconnection shows a significant association with the acceleration of gray matter loss in schizophrenia. Together with effects on wiring cost, we suggest that fragile prefrontal hub connections and topological volatility act as evolutionary influences on brain networks whose optimal set point may be perturbed in neuropsychiatric disorders.
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The authors thank the chief investigators and manager of the ASRB: V. Carr, U. Schall, R. Scott, A. Jablensky, B. Mowry, P. Michie, S. Catts, F. Henskens, and C. Loughland. The authors acknowledge the support of the National Health and Medical Research Council of Australia (APP1110975 to L.L.G.; APP1145168 and APP1144936 to J.R.; APP1037196, APP1118153, and APP1095227 to M.B.; APP1047648 to A.Z.; and ID1105825 to C.P.) and the Australian Research Council (CE140100007). The Australian Schizophrenia Research Bank (ASRB) is supported by the NHMRC (enabling grant 386500), the Pratt Foundation, Ramsay Health Care, the Viertel Charitable Foundation, and the Schizophrenia Research Institute.
The authors declare no competing interests.
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Integrated supplementary information
Supplementary Figure 1 Schematic representation of the connectivity matrices for the different surrogate networks.
Matrices show the logarithm of the edge weights for these fully connected (unthresholded) networks.
Supplementary Figure 2 Wiring length as a function of the fraction of randomized edges for the different algorithms.
Same as Figure 2 but for fiber length instead of wiring cost. Mean fiber length as a function of the fraction of randomized edges. A: Weight-preserving geometric surrogate Gw (blue). B: Strength-preserving geometric surrogate Gs (black). C: Strength-sequence-preserving geometric surrogate Gss (red). D: Geometry-ignoring random surrogates Rw (blue), Rs (black), and Rss (red). Connectomes at the top of each panel illustrate one case in which all edges have been randomized. Shaded areas indicate the standard deviation over n=100 trials.
Supplementary Figure 3 Probabilities of obtaining connectomes with reduced wiring costs and fiber lengths.
A, Probability of obtaining surrogate networks with reduced wiring cost (Pwc) as a function of the fraction of randomized edges (frac). B, Same as (A) but for a zoom for small values of frac. C, Probability of obtaining surrogate networks with reduced fiber length (Pfl) as a function of frac. D, Same as (C) but for zoom for small values of frac. Blue curves correspond to weight-preserving geometric surrogate Gw, black curves correspond to strength preserving geometric surrogate Gs, and red curves correspond to strength-sequence preserving geometric surrogate Gss. For all panels probabilities are estimated from n=10,000 independent trials.
A, Connectome representation of Gss as a function of the density of connections. B, Connectivity matrices lighter colors represent stronger connections. C, Increase in fiber length for the Gss connectome for different densities. D, Increase in wiring cost for the Gss connectome for different densities. Shaded regions correspond to the standard deviation. E, Maximum increase in fiber length for Gss surrogates as a function of connectome density. F, Maximum increase in wiring cost for Gss surrogates as a function of connectome density. Results of are an average of n=100 independent trials.
A, Proportion of inter-hemispheric connections as a function of connectome density for Gs (black) and Gss (red). B, Proportion of homologous connections relative to the proportion of homologous connections in a geometry-ignoring random network (nHrand). C, Same as Fig. 3D but for clustering instead of small-world index. D, Same as Fig. 3D but for path length. Figure generated with n=100 independent trials. Shaded regions correspond to the standard deviation.
A, Normalized network susceptibility after randomizing the empirical brain using geometry-ignoring randomization Rw. B: Normalized network susceptibility after randomizing the fully randomized Gs connectome using geometry-ignoring randomization Rw. C, Normalized network susceptibility after randomizing the empirical brain using strength-preserving geometry-ignoring randomization Rs. D: Normalized network susceptibility after randomizing the fully randomized Gs connectome using strength-preserving geometry-ignoring randomization Rs. Figure generated with n=100 trials.
Supplementary Figure 7 Relationship between fragility and the sum of the first two principal gradients shown in Fig. 5g.
Scatter plot of fragility and the sum of the first two gradients shown in Fig. 5G. Black line shows the linear regression between fragility and the sum of these gradients (r=0.43, p=0.00012, n=75 hubs).
Supplementary Figure 8 Hub strength is not a reliable predictor for gray matter deterioration in schizophrenia.
Same as Fig. 6 but for hub strength instead of fragility. Hub strength does not reach FDR-corrected significance (p<0.05, Pearson correlation, n=75 hubs).
Supplementary Figures 1–8, Supplementary Tables 1 and 2
- One illustrative randomization trial using the geometry-preserving algorithm Gs. Connectomes as a function of the fraction of randomized edges (frac). Hubs are represented as large dots, inter-hub connections are shown in green, and other connection to non-hub regions are in gray. The empirical connectome (starting point) has hubs distant from each other. Hubs move towards the center of the brain as the fraction of randomized edges increases.
- An average over 100 independent randomization trials using the geometry-preserving algorithm Gs. The empirical connectome (starting point) has hubs distant from each other, which move towards the center of the brain as the fraction of randomized edges increases. Frontal and posterior hubs are disconnected earlier in the process.
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Gollo, L.L., Roberts, J.A., Cropley, V.L. et al. Fragility and volatility of structural hubs in the human connectome. Nat Neurosci 21, 1107–1116 (2018). https://doi.org/10.1038/s41593-018-0188-z
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