This paper presents work on parameter estimation methods for bursting neural models. In our approach we use both geometrical features specific to bursting, as well as general features such as periodic orbits and their bifurcations. We use the geometry underlying bursting to introduce defining equations for burst initiation and termination, and restrict the estimation algorithms to the space of bursting periodic orbits when trying to fit periodic burst data. These geometrical ideas are combined with automatic differentiation to accurately compute parameter sensitivities for the burst timing and period. In addition to being of inherent interest, these sensitivities are used in standard gradient-based optimization algorithms to fit model burst duration and period to data. As an application, we fit Butera et al.'s (Journal of Neurophysiology 81, 382–397, 1999) model of preBötzinger complex neurons to empirical data both in control conditions and when the neuromodulator norepinephrine is added (Viemari and Ramirez, Journal of Neurophysiology 95, 2070–2082, 2006). The results suggest possible modulatory mechanisms in the preBötzinger complex, including modulation of the persistent sodium current.