Multi-timescale analysis of midbrain dopamine neuronal firing activities.

Midbrain dopamine (DA) neurons exhibit spiking and bursting patterns under physiological conditions. Based on the data on electrophysiological recordings, Yu et al. developed a 13-dimensional mathematical model to capture the detailed characteristics of the DA neuronal firing activities. We use the fitting method to simplify the original model into a 4-dimensional model. Then, the spiking-to-bursting transition is detected from a simple and robust mathematical condition. Physiologically, this condition is a balance of the restorative and the regenerative ion channels at resting potential. Geometrically, this condition imposes a transcritical bifurcation. Moreover, we combine singularity theory and singular perturbation methods to capture the geometry of three-timescale firing attractors in a universal unfolding of a cusp singularity. In particular, the planar description of the corresponding firing patterns can generate the corresponding firing attractors. This analysis provides a new idea for understanding the firing activities of the DA neuron and the specific mechanisms for the switching and dynamic regulation among different patterns.