This note focuses on the analysis of emergent behaviors in networks of dynamical systems, where their interactions are of activation-type locally and inhibitory-type in the long-range. Such interplay between activation and inhibition is reminiscent of the classical formula for oscillations in lumped systems, i.e., local positive feedback and long-range negative feedback. Dominance tools are used for the analysis of the resulting behavior, obtaining certificates in the form of linear matrix inequalities that allow to study the existence of emergent static patterns or oscillations in the network. A numerical example illustrates the proposed approach.