Evidence for Quasicritical Brain Dynamics

Much evidence seems to suggest the cortex operates near a critical point, yet a single set of exponents defining its universality class has not been found. In fact, when critical exponents are estimated from data, they widely differ across species, individuals of the same species, and even over time, or depending on stimulus. Interestingly, these exponents still approximately hold to a dynamical scaling relation. Here we show that the theory of quasicriticality, an organizing principle for brain dynamics, can account for this paradoxical situation. As external stimuli drive the cortex, quasicriticality predicts a departure from criticality along a Widom line with exponents that decrease in absolute value, while still holding approximately to a dynamical scaling relation. We use simulations and experimental data to confirm these predictions and describe new ones that could be tested soon.

Figure 1

Exponent relations and dynamical scaling line, of slope $\gamma$, for the various physical hypotheses (see text).

Figure 2

Logarithmically binned avalanche size, and duration, probability distributions [22] for (a) CBM simulation with $p_s={10}^{-3}$, (b) Mouse data 23, (c) Mouse data 18, (d) rat data. Red circles were selected for fitting.

Figure 3

Predictions of quasicriticality. (a) Simulations and data follow the scaling relation as predicted by quasicriticality. As $p_s$ is increased (blue arrow) exponents (${\tilde{\tau }}_T$,${\tilde{\tau }}_S$) decrease in the model (blue diamonds for $p_s={10}^{-5},{10}^{-4},{10}^{-3}$) while still approximately holding to the scaling relation (dashed line). Note that susceptibility curves A, B, and C in panel (c) correspond to these $p_s$ values. Increasing ${\tau }_r$ (orange arrow) increases the magnitude of the effective exponents (the orange squares correspond to ${\tau }_r=5$ for $p_s={10}^{-3}$ and ${10}^{-4}$, respectively). Actual datasets (red circles) also lie near the line, and have reduced values of susceptibility and branching ratio ${\kappa }_w$, in agreement with the simulations. Our scaling line has a slope of 1.45(4), in agreement with our data and differs from the slope in Ref. [11] shown in the inset. Note that model parameters can be easily changed to match any slope, but that the trends in susceptibility and ${\kappa }_w$ as $p_s$ is increased will not change and constitute the specific predictions of the theory tested here. (b) Phase diagram showing how increased external drive leads to the Widom line; the $y$ axis is the (time-averaged) density of active nodes (order parameter), the $x$ axis is the branching ratio, $\kappa$ (control parameter). With minimal external drive ($p_s={10}^{-5}$), the solid black line at A comes near the critical point. As the external drive is increased ($p_s={10}^{-4}$), the dashed line follows the solid black line, but rises above the critical point, with a gradual bend in the curve at B. Further increases in the external drive ($p_s={10}^{-3}$) cause the curve to rise more, bending near C. The thin gray line connecting A, B, C is the Widom line. Note that it tilts to the left, predicting that ${\kappa }_w$ should decrease as $p_s$ increases. (c) Susceptibility is maximal, but not divergent, along this line. Three plots of the susceptibility produced by simulations with different values of $p_s$. Note that increases in $p_s$ cause peak susceptibility to decline, leading to flattened curves.