Optimization for bursting neural models

This thesis concerns parameter estimation for bursting neural models. Pa-rameter estimation for dierential equations is a dicult task due to complicated objective function landscapes and numerical challenges. These diculties are par-ticularly salient in bursting models and other multiple time scale systems. Here we make use of the geometry underlying bursting by introducing dening equations for burst initiation and termination. Fitting the timing of these burst events sim-pli es objective function landscapes considerably. We combine this with automatic dierentiation to accurately compute gradients for these burst events, and imple-ment these features using standard unconstrained optimization algorithms. We use trajectories from a minimal spiking model and the Hindmarsh-Rose equations as test problems, and bursting respiratory neurons in the preBotzinger complex as an application. These geometrical ideas and numerical improvements signicantly enhance algorithm performance. Excellent ts are obtained to the preBotzinger data both in control conditions and when the neuromodulator norepinephrine is added. The results suggest dierent possible neuromodulatory mechanisms, and help analyze the roles of dierent currents in shaping burst duration and period