Emergence of localized patterns in globally coupled networks of relaxation oscillators with heterogeneous connectivity

Relaxation oscillators may exhibit small amplitude oscillations (SAOs) in addition to the typical large amplitude oscillations (LAOs) as well as abrupt transitions between them (canard phenomenon). Localized cluster patterns in networks of relaxation oscillators consist of one cluster oscillating in the LAO regime or exhibiting mixed-mode oscillations (LAOs interspersed with SAOs), while the other oscillates in the SAO regime. We investigate the mechanisms underlying the generation of localized patterns in globally coupled networks of piecewise-linear (PWL) relaxation oscillators where global feedback acting on the rate of change of the activator (fast variable) involves the inhibitor (slow variable). We also investigate of these patterns are affected by the presence of a diffusive type of coupling whose synchronizing effects compete with the symmetry breaking global feedback effects.