Self-organized critical branching in systems that violate conservation laws

Abstract

A non-conservative critical branching model is proposed to demonstrate that self-organized criticality (SOC) can occur in mean-field sandpiles that violate a conservation law. The critical state is characterized by avalanche sizes and lifetimes that obey an inverse power-law distribution with exponents τS = 3/2 and τT = 2, respectively. Criticality is achieved when the branching process is coupled to a background activity characterized by the spontaneous switching between refractoriness and quiescence among system components. The stationary state of the system is analysed mathematically and numerically, and is shown to exhibit a transition from a subcritical phase to a critical phase. SOC in sandpile models has been widely believed to occur only when grains are conserved during avalanches. However, such a conservation law is likely to be violated by open, non-equilibrium systems such as biological networks and socially interacting systems like animal groups. The model explores the role of dynamic synapses and synaptic plasticity in maintaining criticality of cortical networks. These brain networks have been found to display neuronal avalanches that obey a power-law distribution. The non-conservative model also emulates the main features of the size distributions of free-swimming tuna schools and red deer herds. Demonstrating criticality in self-organizing systems that violate conservation laws enhances the predictive ability of the theory of SOC in the arena of biocomplexity.