Hysteresis and multistability in network spreading with neuronal activity feedback

Abstract

Spreading processes on networks often interact with other dynamics on the same nodes. Neurodegenerative disease provides one example: pathological proteins spread through anatomical connections, while neuronal activity influences and is altered by their spread, forming a spreading-activity feedback loop. However, models coupling pathological protein spreading and neuronal activity have largely focused on linear feedback between the two processes. Here we show that nonlinear feedback can fundamentally change the invasion dynamics in a susceptible--infected--susceptible spreading process coupled to a co-evolving activity process. On general weighted networks, the strength and shape of the feedback may generate finite-amplitude invasion thresholds, hysteresis, and endemic multistability. On regular graphs with homogeneous dynamics, we rule out periodic solutions and show that the degrees of polynomial coupling functions bound the number of endemic states, while monotone couplings require reinforcing feedback for multistability. We test these predictions in simulations of a stochastic spiking neuronal network described by quadratic integrate-and-fire dynamics, where we recover both finite-amplitude invasion thresholds and endemic bistability. These results show that feedback from activity processes can create hysteresis and multistability in network spreading dynamics. In neuroscientific applications, our work suggests that neuronal dynamics may act as a control point in neurodegenerative disease, with even transient changes in activity capable of tipping the brain between health and disease.