Effect of diversity distribution symmetry on global oscillations of networks of excitable units

We investigate the role of the degree of symmetry of the diversity distribution in shaping the collective dynamics of networks of coupled excitable units modeled by FitzHugh-Nagumo equations. While previous studies have focused primarily on the ratio between the numbers of individually oscillatory and excitable units, we show that the symmetry of the diversity distribution plays a fundamental role in the emergence of global network oscillations. By exploring various symmetric and asymmetric distributions and simulating network dynamics across various topologies, we demonstrate that symmetric distributions promote resonant collective oscillations even in the absence of oscillatory units. We propose two quantitative metrics, the normalized center of mass and the symmetry balance score, to assess the degree of symmetry and predict the presence or absence of global oscillations. By studying a minimal two-unit system and its effective pseudo-potential, we show that symmetry enables the formation of a landscape characterized by a cyclic valley supporting limit cycles, whereas asymmetry collapses the system into a single non-oscillatory equilibrium. These results provide a general mechanism by which network symmetry drives emergent synchronization in heterogeneous excitable systems.