Biased excitable network model for non-periodic phenomena in recurrent dynamics

Non-periodic phenomena are common in a wide range of real-world recurrent dynamics, such as the occasional pandemic of seasonal influenza and the abrupt collapse of stock markets. In this paper we propose a biased excitable network model and illustrate the non-periodic phenomena as the collective response of a large amount of excitable individuals. In contrast with classic excitable networks, we introduce the bias of external stimuli that affects the exact behavior of each individual rather than its own inherent property. Based on the locally tree-like topology, we make a second order approximation on the network activity with diminishing stimuli. Result shows that the self-sustainment of network dynamics is determined by the largest eigenvalue λ of the weighted adjacent matrix. For λ>1 the network is self-sustained even if the stimuli intensity approaches zero. For λ<1, the system tends to end up in quiescent state. At critical condition λ=1, the dynamic range of the system, which is the range of stimuli intensity that is distinguishable according to network activity, reaches maximum value. We also find that the dynamic range can be further enhanced when nodes are more inclined to inhibitory state as a result of smaller stimuli bias. These results are well supported by numerical simulations on both synthetic and real-world networks. Based on the proposed model, we manage to reproduce similar non-periodic phenomena to those in real-world recurrent dynamics, even when the intensity of external stimuli remains constant. Our research shed light on the mechanism of non-periodic phenomena in recurrent dynamics, which can be applied to the prevention of epidemic outbreaks as well as financial crisis.