Universal and nonuniversal neural dynamics on small world connectomes: A finite-size scaling analysis

Evidence of critical dynamics has been found recently in both experiments and models of large-scale brain dynamics. The understanding of the nature and features of such a critical regime is hampered by the relatively small size of the available connectome, which prevents, among other things, the determination of its associated universality class. To circumvent that, here we study a neural model defined on a class of small-world networks that share some topological features with the human connectome. We find that varying the topological parameters can give rise to a scale-invariant behavior either belonging to the mean-field percolation universality class or having nonuniversal critical exponents. In addition, we find certain regions of the topological parameter space where the system presents a discontinuous, i.e., noncritical, dynamical phase transition into a percolated state. Overall, these results shed light on the interplay of dynamical and topological roots of the complex brain dynamics.

Figure 1

Finite-size behavior for $\pi =0.6$ and $\langle k\rangle =5$ corresponding to the segregated regime of the model where phase transition is absent for any $T$. (a) Order parameter ${\overline{S}}_1/N$. (b) Second largest cluster relative sizes. Notice that they are of the same order of magnitude as the largest clusters shown in (a) and also that ${\overline{S}}_2/N\to 0\forall T$ when $N\to \infty$. (c) Average clusters size $\langle s\rangle$ as a function of the threshold $T$ for different sizes $N$.

Figure 2

Finite-size behavior for $\pi =0.6$ and $\langle k\rangle =10$, corresponding to the critical regime (MFC). (a) Order parameter. (b) Second largest cluster size ${\overline{S}}_2$. The inset shows the maximum of ${\overline{S}}_2$ as a function of $N$, where the solid line is a power-law fitting giving a critical exponent $d_f/d=0.67\pm 0.02$. (c) Average clusters size $\langle s\rangle$. The inset shows the maximum of $\langle s\rangle$ as a function of $N$, where the solid line is a power-law fitting giving a critical exponent $\gamma /\nu d=0.35\pm 0.02$.

Figure 3

Finite-size behavior for $\pi =0.6$ and $\langle k\rangle =30$, corresponding to the discontinuous regime. (a) Order parameter. The inset shows the autocorrelation function of the average total activity $AC$ close to the critical value $T_c$. (b) Second largest cluster size ${\overline{S}}_2$. The inset shows the maximum of ${\overline{S}}_2$ as a function of $N$, where the solid line is a power-law fitting giving a critical exponent $d_f/d=0.29\pm 0.02$. (c) Average clusters size $\langle s\rangle$. The inset shows the maximum of $\langle s\rangle$ as a function of $N$, where the solid line is a power-law fitting giving a critical exponent $\gamma /\nu d=0.05\pm 0.02$. The inset shows the autocorrelation function of the average activity for three values of $T$ very close to $T_c$.

Figure 4

Parameter space for the network topology values $(\pi ,\langle k\rangle )$ in the large-$N$ limit. As a function of the node threshold $T$, the system can exhibit a percolationlike dynamical phase transition for parameter values above the black line. Below the black line there is no dynamical phase transition (NT). Between the black line and the red dashed line (MFC) the behavior becomes scale invariant, at a certain critical threshold $T$, with exponents consistent with the mean-field percolation universality class. Farther above, in the region denoted by NUC, the behavior can be still scale invariant and critical, i.e., the transition is still second order, but without universal exponents. Finally, above the green dash-dotted line, the transition becomes discontinuous (Disc.) and the dynamic is short-range correlated and oscillatory. The triangle, circle, and star at $\pi =0.6$ correspond to the parameter values used for the statistics presented in Figs. 5, 5, and 5, respectively. In the inset the same data are plotted in on a double logarithmic axis to best denote the relative sizes of each dynamical regime.

Figure 5

(a) The CCDF of the cluster size distribution for $\langle k\rangle =10, \pi =0.6$, and different system sizes. The dashed line corresponds to a linear fitting of the central part of the distribution with a power law $\sim s^{-1.58}$. The inset shows the cutoff $S^{*}$ as a function of $N$; the solid line is a power-law fitting giving an exponent $0.62\pm 0.05$. (b) The CCDF of the cluster size distribution for $\langle k\rangle =15, \pi =0.6$, and different system sizes. The dashed line corresponds to a linear fitting of the central part of the distribution with a power law $\sim s^{-1.54}$. The inset shows the cutoff $S^{*}$ as function of $N$; the solid line is a power-law fitting giving an exponent $0.60\pm 0.03$. (c) The CCDF of the cluster size distribution for $\langle k\rangle =30, \pi =0.6$, and different system sizes.

Figure 6

Critical exponents obtained by finite-size scaling analysis as a function of $\langle k\rangle$ for the range of the rewiring probability $\pi$ values given in the legend. (a) Critical exponent of the maximum of ${\overline{S}}_2$. The dashed line corresponds to the mean-field value $d_f/d=2/3$. (b) Critical exponent of the maximum of $\langle s\rangle$. The dashed line corresponds to the mean-field $\mathrm{value}\gamma /\nu d\approx 1/3$.