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form that can be analyzed via a slight adaptation of the approach developed by M. Krupa, N. Popovic, and N. Kopell (unpublished).
Krupa
Martin
M
Department of Mathematical Sciences, New Mexico State University, P.O. Box 30001, Department 3MB, Las Cruces, New Mexico 88003, USA. mkrupa@nmsu.edu
Popović
Nikola
N
Kopell
Nancy
N
Rotstein
Horacio G
HG
D000200:Action Potentials / Q000502:physiology*; D000465:Algorithms; D001683:Biological Clocks / Q000502:physiology*; D003198:Computer Simulation; D004298:Dopamine / Q000378:metabolism*; D008959:Models, Neurological; D009474:Neurons / Q000502:physiology*; D017711:Nonlinear Dynamics
D004298:Dopamine