Dynamics and synchronization patterns in oscillator networks with heterogeneous inputs

In networks of nonlinear oscillators, complex dynamics emerge as a function of the interplay between network structure and distributions of external input. We study the dynamics and synchronization patterns that emerge in networks of coupled neuronal oscillators described by the FitzHugh-Nagumo (FN) model, a two-dimensional reduction of the higher-dimensional Hodgkin-Huxley model for neuronal membrane potential dynamics. In the uncoupled setting, the FN model exhibits three qualitatively distinct input-dependent regimes in its dynamical behavior: quiescence, firing, and saturation. When multiple FN oscillators are connected through diffusive coupling, the resulting dynamics exhibit complex behaviors, which include mixed-mode oscillations and asymptotically periodic dynamics. Using techniques from bifurcation theory and singular perturbation theory and leveraging multiple time scales in the dynamics, we identify the possible behavioral regimes of the oscillators in the network and characterize the ones where the system exhibits complex behavior. We further explore how transitions between regimes depend on heterogeneous external inputs, coupling strength, and time-scale separation parameters. Our work furthers the understanding of the dynamics of coupled oscillatory systems.