Abstract
Collective dynamics of networks of excitable neurons can be considered as the emergence of ordering from microscopic self-organization at the macroscopic scale. Sustained oscillation can emerge on networks of neurons even if a single neuron is dynamical excitable and non-oscillatory. Fundamental ingredients of networks such as loops, trees, and hubs, play distinct roles in supporting, propagating and impeding sustained oscillations. In this paper, we explore the mechanism of collective self-sustained oscillations on neuron networks by analyzing the functions of different topologies in shaping the oscillatory patterns on excitable neuron networks. The Winfree loops are revealed to be responsible for generating collective oscillations as the oscillation core, and other neurons act as the propagating paths. The existence of large numbers of loops in a network indicates potential competitions of the formation of collective oscillatory dynamics. The roles of loop-loop competition in homogeneous networks and loop-hub competition in heterogeneous networks are extensively discussed.