Fig. 2.

Illustration of the diverse phases emerging in the model (case A). The baseline of synaptic resources, $\xi$, increases from top to bottom: $\xi =0.4$ (down state), $\xi =1.2$ (synchronous regime), $\xi =2.47$ (critical point for the considered size $N={128}^2$), $\xi =2.7$ (asynchronous phase), and $\xi =5$ (active phase). The first column shows snapshots of typical configurations; the color code represents the level of activity at each unit as shown in the scale. The network spiking or synchronous irregular phase is characterized by waves of activity growing and transiently invading the whole system before extinguishing the resources and coming to an end. However, in the nested oscillation or asynchronous irregular (AI) regime, multiple traveling waves coexist, interfering with each other. In the up state, waves are no longer observed, and a homogeneous state of self-sustained activity is observed (Movie S1). The second column shows the time series of the overall activity averaged over the whole network. In the down state, activity is almost vanishing. In the synchronous phase, macroscopic activity appears in the form of almost synchronous bursts interspersed by almost silent intervals. At the critical point, network spikes begin to superimpose, giving rise to complex oscillatory patterns (nested oscillations) and marginally self-sustained global activity all across the asynchronous regime; finally, in the up state, the global activity converges to steady state with small fluctuations. The third column shows steady-state probability distribution $P(\rho )$ for the global activity: in the down state and the network spiking regime, the distributions are shown in a double-logarithmic scale; observe the approximate power law for very small values of $\rho$ stemming from the presence of multiplicative noise (10). The fourth column shows an illustration of the different levels of synchronization across phases: a sample of $200$ randomly chosen units is mapped into oscillators using their analytic signal representation (Materials and Methods); the plot shows the time evolution of their corresponding phases ${\varphi }_k^{\mathcal{A}}$. Observe the almost periodic behavior in the synchronous phase, which starts blurring at the critical point and progressively vanishes as the control parameter is further increased. Parameter values: $a=1,b=1.5,{\tau }_R={10}^3,{\tau }_D={10}^2,h={10}^{-7}$.