A Mathematical Model of a Midbrain Dopamine Neuron Identifies Two Slow Variables Likely Responsible for Bursts Evoked by SK Channel

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. Abstract Midbrain dopamine neurons exhibit a novel type of bursting that we call “inverted square wave bursting ” when exposed to Ca2+-activated small conductance (SK) K+ channel blockers in vitro. This type of bursting has three phases: hyper-polarized silence, spiking, and depolarization block. We find that two slow variables are required for this type of bursting, and we show that the three-dimensional bi-furcation diagram for inverted square wave bursting is a folded surface with up-per (depolarized) and lower (hyperpolarized) branches. The activation of the L-type Ca2+ channel largely supports the separation between these branches. Spiking is initiated at a saddle node on an invariant circle bifurcation at the folded edge of the lower branch and the trajectory spirals around the unstable fixed points on the upper branch. Spiking is terminated at a supercritical Hopf bifurcation, but the trajectory remains on the upper branch until it hits a saddle node on the up-per folded edge and drops to the lower branch. The two slow variables contribut