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

Midbrain dopamine neurons exhibit a novel type of bursting that we call "inverted square wave bursting" when exposed to Ca(2+)-activated small conductance (SK) K(+) channel blockers in vitro. This type of bursting has three phases: hyperpolarized 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 bifurcation diagram for inverted square wave bursting is a folded surface with upper (depolarized) and lower (hyperpolarized) branches. The activation of the L-type Ca(2+) 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 upper folded edge and drops to the lower branch. The two slow variables contribute as follows. A second, slow component of sodium channel inactivation is largely responsible for the initiation and termination of spiking. The slow activation of the ether-a-go-go-related (ERG) K(+) current is largely responsible for termination of the depolarized plateau. The mechanisms and slow processes identified herein may contribute to bursting as well as entry into and recovery from the depolarization block to different degrees in different subpopulations of dopamine neurons in vivo.