Ensembles of excitable two-state units with delayed feedback

Abstract

A two-state unit is considered as an abstract modification for an excitable system. Each state is characterized by a different waiting time distribution. This non-Markovian approach allows for a renewal process description of the system dynamics. Exact formulas for the interspike interval distribution and power spectral density are found. In the limit of an infinity ensemble of globally coupled units the mean-field equations for the populations of both states are derived. Depending on the coupling strength and on the noise intensity the ensemble undergoes saddle-node bifurcations and demonstrates bistability, while a pitchfork bifurcation emerges on a cusp point. The ensemble undergoes Hopf bifurcations and bulk oscillations emerge, in the onset of coherent activation events, only when the feedback affects individual units with a certain time delay.