Bursting transition in a linear self-exciting point process

Self-exciting point processes describe the manner in which every event facilitates the occurrence of succeeding events. By increasing excitability, the event occurrences start to exhibit bursts even in the absence of external stimuli. We revealed that the transition is uniquely determined by the average number of events added by a single event, $1-1/\sqrt{2} \approx 0.2929$, independently of the temporal excitation profile. We further extended the theory to multi-dimensional processes, to be able to incite or inhibit bursting in networks of agents.