Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron

Mixed- mode dynamics is a complex type of dynamical behavior that has been observed both numerically and experimentally in numerous prototypical systems in the natural sciences. The compartmental Wilson- Callaway model for the dopaminergic neuron is an example of a system that exhibits a wide variety of mixed- mode patterns upon variation of a control parameter. One characteristic feature of this system is the presence of multiple time scales. In this article, we study the Wilson- Callaway model from a geometric point of view. We show that the observed mixed- mode dynamics is caused by a slowly varying canard structure. By appropriately transforming the model equations, we reduce them to an underlying three- dimensional canonical form that can be analyzed via a slight adaptation of the approach developed by M. Krupa, N. Popovic, and N. Kopell ( unpublished ). (C) 2008 American Institute of Physics.