Collective rhythm design in coupled mixed-feedback systems through dominance and bifurcations

The theory of mixed-feedback systems provides an effective framework for the design of robust and tunable oscillations in nonlinear systems characterized by interleaved fast positive and slow negative feedback loops. The goal of this paper is to extend the mixed-feedback oscillation design framework to networks. To this aim, we introduce a network model of coupled mixed-feedback systems, ask under which conditions it exhibits a collective oscillatory rhythm, and if, and how, this rhythm can be shaped by network design. In the proposed network model, node dynamics are nonlinear and defined by a tractable realization of the mixed-feedback structure. Coupling between nodes is also nonlinear and defined by a tractable abstraction of synaptic coupling between neurons. We derive constructive conditions under which the spectral properties of the network adjacency matrix fully and explicitly determine both the emergence of a stable network rhythm and its detailed rhythmic profile, i.e., the pattern of relative oscillation amplitudes and phase differences. Our theoretical developments are grounded on ideas from dominant systems and bifurcation theory. They provide a new framework for the analysis and design of nonlinear network rhythms.