Criticality in Spreading Processes without Timescale Separation and the Critical Brain Hypothesis

Abstract

Spreading processes on networks are ubiquitous in both human-made and natural systems. Understanding their behavior is of broad interest: from the control of epidemics to understanding brain dynamics. While in some cases there exists a clear separation of timescales between the propagation of a single spreading cascade and the initiation of the next - such that spreading can be modeled as directed percolation or a branching process - there are also processes for which this is not the case, such as spiking cascades in neural networks. For a large class of relevant network topologies, we show here that in such a scenario the nature of the overall spreading fundamentally changes. This change manifests itself in a transition between different universality classes of critical behavior, which determines the onset and the properties of neural activity turning epileptic, for example. We present analytical results in the mean-field limit, giving the critical line along which scale-free behavior can be observed. The two limits of this critical line correspond to the universality classes of directed and undirected percolation, respectively. Outside these two limits, this duality manifests itself in the appearance of critical exponents from the universality classes of both directed and undirected percolation. We find that the transition between these exponents is governed by a competition between merging and propagation of activity, and we identify an appropriate scaling relationship for the transition point. Finally, we show that commonly used measures, such as the branching ratio and dynamic susceptibility, fail to establish criticality in the absence of timescale separation, calling for a reanalysis of criticality in the brain.