Adaptive reconfiguration of fractal small-world human brain functional networks

Abstract

Brain function depends on adaptive self-organization of large-scale neural assemblies, but little is known about quantitative network parameters governing these processes in humans. Here, we describe the topology and synchronizability of frequency-specific brain functional networks using wavelet decomposition of magnetoencephalographic time series, followed by construction and analysis of undirected graphs. Magnetoencephalographic data were acquired from 22 subjects, half of whom performed a finger-tapping task, whereas the other half were studied at rest. We found that brain functional networks were characterized by small-world properties at all six wavelet scales considered, corresponding approximately to classical delta (low and high), , alpha, beta, and gamma frequency bands. Global topological parameters (path length, clustering) were conserved across scales, most consistently in the frequency range 2-37 Hz, implying a scale-invariant or fractal small-world organization. Dynamical analysis showed that networks were located close to the threshold of order/disorder transition in all frequency bands. The highest-frequency gamma network had greater synchronizability, greater clustering of connections, and shorter path length than networks in the scaling regime of (lower) frequencies. Behavioral state did not strongly influence global topology or synchronizability; however, motor task performance was associated with emergence of long-range connections in both beta and gamma networks. Long-range connectivity, e.g., between frontal and parietal cortex, at high frequencies during a motor task may facilitate sensorimotor binding. Human brain functional networks demonstrate a fractal small-world architecture that supports critical dynamics and task-related spatial reconfiguration while preserving global topological parameters.

Fig. 1.

Scale-invariance of global topological and dynamical parameters of brain functional networks. Each image summarizes the group mean parameter values over all wavelet scales and in both resting (red) and motor (blue) states; error bars represent 95% confidence interval; blue and red bars below the x axis indicate the extent of the scaling regime for each parameter in both resting (red) and motor (blue) states. (A) Average path length, L. (B) Clustering, C. (C) Sigma, σ. (D) Synchronizability, S. (E) Characteristic length, ζ (mm). (F–H) Parameters of an exponentially truncated power law degree distribution of the form P(k) ≈ A kλ⁻¹e^k/kc. (F) Coefficient, A. (G) Power law exponent, λ. (H), Exponential cut-off degree, k_c.

Fig. 2.

Self-similarity of spatial distribution of highly connected network nodes or “hubs” in the frequency range 2–38 Hz (64). Each column shows the surface distribution of the degree of network nodes in frequency bands β to δ: red represents nodes with high degree. The last column shows the spatial distribution of degree averaged over these four frequency bands, which emphasizes the similarity of spatial configurations across scales. See SI Fig. 5 for the hub distributions in both states at all frequency bands.

Fig. 3.

State-related differences in spatial configuration of the highest frequency γ network. The top row shows the degree distribution and betweenness scores for the resting state γ network; the middle row shows the same maps for the motor γ network; the bottom row shows the between-state differences in degree and betweenness. It is clear that motor task performance is associated with emergence of greater connectivity in bilateral prefrontal and premotor nodes, and appearance of topologically pivotal nodes (with high betweenness scores) in medial premotor, right prefrontal, and parietal areas. See SI Fig. 7 for the betweenness distributions in both states at all frequency bands.

Fig. 4.

Change of provincial, connector, and kinless hubs with neighborhood size. As the radius r is increased, the proportion of provincial nodes is increased, and the proportions of kinless and connector hubs are decreased. The radius at which the proportion of connector and provincial hubs is equal, denoted ζ, is a measure of the characteristic length scale of connections between nodes in the network. (Left) Number of provincial, connector, and kinless hubs as functions of radius r in high-frequency γ networks acquired during motor task performance, ζ ≈ 225. (Right) Number of provincial, connector, and kinless hubs as functions of radius r in high-frequency γ networks acquired during the resting state, ζ ≈ 75. ζ is significantly smaller for the γ, β, and α bands in the resting state than in the tapping state, indicating that performance of the motor task is associated with emergence of longer range connectivity in the brain functional network. See SI Fig. 6 for provincial and connector hub distributions in both states at all frequency bands.