Activity propagation with Hebbian learning

Biological and social systems, including infection spreading, inter-regional brain activity propagation, and population spreading, exhibit learning across a broad range of scales. These applications of the contact process therefore call for an extension that incorporates local learning rules. Here we introduce learning as a positive (Hebbian) or negative (anti-Hebbian) reinforcement of the activation rate between a pair of sites after each successful activation event. We show that Hebbian learning leads to a rich class of emergent behaviors, where local incentives can produce opposite global effects. In general, positive reinforcement causes the loss of the active phase, while negative reinforcement can turn the inactive phase into a globally active phase. Our analytical and numerical results demonstrate that, in two dimensions and above, the effect of negative reinforcement is twofold: it promotes the spreading of activity while simultaneously generating effectively immune regions, leading to the emergence of two distinct critical points. By contrast, positive reinforcement can give rise to Griffiths effects with non-universal power-law scaling, a manifestation of the ‘ant-mill’ phenomenon. Applications like infection spreading, brain activity propagation and population spreading are calling for activity spreading models that incorporate local learning. We find that Hebbian learning leads to a rich class of emergent phenomena, depending on the microscopic details of the learning mechanism, as well as on the initial conditions.