Ising-like dynamics in large-scale functional brain networks

Brain “rest” is defined—more or less unsuccessfully—as the state in which there is no explicit brain input or output. This work focuses on the question of whether such state can be comparable to any known dynamical state. For that purpose, correlation networks from human brain functional magnetic resonance imaging are contrasted with correlation networks extracted from numerical simulations of the Ising model in two dimensions at different temperatures. For the critical temperature $T_c$, striking similarities appear in the most relevant statistical properties, making the two networks indistinguishable from each other. These results are interpreted here as lending support to the conjecture that the dynamics of the functioning brain is near a critical point.

Figure 1

(Color online) Panel (a): brain’s FMRI correlation density distribution for each of the five control subjects recorded under resting conditions. Panel (b): Ising model correlation density distribution for $T=2$ (green, dashed line), $T=2.3$ (red, continuous line), and $T=3$ (black, dot dashed line). Insets in both panels show the logarithmic-linear plot of the same density distributions.

Figure 2

(Color online) Average degree, $⟨k⟩$, as a function of threshold $\rho$ for (a) positive and (c) negative correlation networks. Variance of degree, ${\sigma }_k^2$, as a function of $⟨K⟩$ for (b) positively and (d) negatively correlated networks. The dashed black line [in panels (b) and (d)] corresponds to the expected behavior of a Poisson distribution. In all graphs the Ising model data at three temperatures, $T=2$, $T=2.3$, and $T=3$, are presented, as well as the data from the five subject’s FMRI brain networks.

Figure 3

(Color online) Normalized network clustering [panel (a)] and path length [panel (b)] as a function of average degree. Solid lines denote results for the Ising model at $T_c$ and dashed lines denote the results for the brain.

Figure 4

(Color) Degree distribution for positively correlated networks. Top three panels depict the degree distribution for the Ising networks at $T=2$, $T=2.3$, and $T=3$ for three representative values of $⟨k⟩\approx 26$, 127, and 713. Bottom panel: degree distribution for positively correlated brain network for the same three values of $⟨k⟩$.

Figure 5

(Color) Degree distribution for negatively correlated networks. Top three panels depict the degree distribution for the Ising networks at $T=2$, $T=2.3$, and $T=3$, respectively, for three representative values of $⟨k⟩\approx 49$, 127, and 277. Bottom panel: degree distribution for negatively correlated brain network for the same three values of $⟨k⟩$.

Figure 6

(Color online) Brain network’s average degree distribution computed from five volunteers for $⟨k⟩\approx 127$.

Figure 7

(Color online) Top four panels: neighbor degrees correlation. Plot of the nearest-neighbor degree, $⟨k_1⟩$, as a function of own degree $k$ for the two type of networks extracted from the brain (left) and from the Ising model at $T_c$ (right). Bottom panels: clustering, $C$, as a function of the degree $k$ for positively correlated networks extracted from the brain (left) and from the Ising model at $T_c$: in all plots dots represent individual nodes and empty circles joined by lines represent averages. Positively correlated networks correspond to $⟨k⟩\approx 26$ and negatively correlated networks correspond to $⟨k⟩\approx 49$.

Figure 8

(Color online) Typical brain (top) and Ising (bottom) correlation profiles. Correlations [i.e., Eq. (2)] are computed between the site with the largest degree and the rest of the time series and plotted at its respective normalized distance.