Switching to criticality by synchronized input

The concept of self-organized criticality (SOC) describes a variety of phenomena ranging from plate tectonics, the dynamics of granular media and stick-slip motion to neural avalanches [1]. In all these cases the dynamics is marginally stable and event sizes obey a characteristic power-law distribution.

It was previously shown that an extended critical interval can be obtained in a neural network by incorporation of depressive synapses [2]. In the present study we scrutinize a more realistic dynamics for the synaptic interactions that can be considered as the state-of-the-art in computational modeling of synaptic interaction (Figure 1) [2]. Interestingly, the more complex model does not exclude an analytical treatment and it shows a type of stationary state consisting of self-organized critical phase and a subcritical phase that has not been previously described. The phases are connected by first- or second-order phase transitions in a cusp bifurcation. Switching between phases can be induced by synchronized activity or by activity deprivation (Figure 2). We present exact analytical results supported by extensive numerical simulations.

Figure 1

The distribution of avalanche sizes changes in dependence on the interaction parameter α from subcritical (circles, α = 0.52) via critical (stars, α = 0.56) to supercritical (triangles, α = 0.59). The inset shows the hysteresis of the distribution at an exemplary avalanche size (L = 40). The circles result at increasing α, stars at decreasing α.

Figure 2

Dynamic transition from a critical (red circles) to a subcritical state (green circles) by short-term activity deprivation. The two stars represent the immediate effect of the deprivation in a particular trial.

By elucidating the relation between the elementary synaptic processes and the network dynamics, our mean-field approach revealed a macroscopic bifurcation pattern, which can be verified experimentally through predicted hysteresis. Furthermore it may be able to explain observations of up and down states in the prefrontal cortex [4] as well as the discrete changes in synaptic potentiation and depression [5] as network effects.