Mixed-mode oscillations in a three time scale system of ODEs motivated by a neuronal model

International audienc

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

Item does not contain fulltextMixed-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form that can be analyzed via a slight adaptation of the approach developed by M. Krupa, N. Popovic, and N. Kopell (unpublished)

On the Role of Theory and Modeling in Neuroscience

In recent years, the field of neuroscience has gone through rapid experimental advances and a significant increase in the use of quantitative and computational methods. This growth has created a need for clearer analyses of the theory and modeling approaches used in the field. This issue is particularly complex in neuroscience because the field studies phenomena that cross a wide range of scales and often require consideration at varying degrees of abstraction, from precise biophysical interactions to the computations they implement. We argue that a pragmatic perspective of science, in which descriptive, mechanistic, and normative models and theories each playa distinct role in defining and bridging levels of abstraction, will facilitate neuroscientific practice. This analysis leads to methodological suggestions, including selecting a level of abstraction that is appropriate for a given problem, identifying transfer func-tions to connect models and data, and the use of models themselves as a form of experiment