Abstract: 
Efforts to gain insight into the complex dynamics of the brain benefit from a detailed understanding of neurons and their interactive dynamics. Systems of two coupled model neurons exhibit rich dynamical patterns that can represent large networks comprised of two distinct clusters. In this talk we will focus on a system of two coupled neuronal oscillators, described by the FitzHugh-Nagumo (FN) model, where the first neuron receives a constant external input and the second neuron receives an input from the first neuron. 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. 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.