Dynamical systems with multiple time scales are also called singularly perturbed systems.or slow-fast systems. The theory of systems with multiple time scales has been approached from at least three complementary viewpoints: nonstandard analysis, "classical" asymptotic methods, and geometric methods that have come to be called geometric singular perturbation theory.  The research has been shaped both by a few seminal examples and a desire to characterize typical or generic behavior in these systems. These efforts have been lmitied by difficulty in computing unstable trajectory segments, called canards, and by difficulty in comprehending the complex geometry of solutions. My research seeks to advance the frontiers in both these directions.

Working with Kathleen Hoffman and Warren Weckesser, I have investigated a slow-fast system with two slow and two fast variables that is a caricature of a pair of neurons coupled by reciprocal inhibition. We have used the computer program AUTO to investigate periodic solutions in this system. The results of this investigation are intriguing and perplexing. With one parameter allowed to vary, AUTO computes a continuous family of periodic orbits. This family encounters numerous canards and bifurcations. Thus far we have examined a few of the orbits in detail, characterizing the different types of canards they encounter. The image below shows a three dimensional projection of one of these trajectories onto a coordinate subspace with the two fast and one slow variable.

There are three canards in this trajectory, created by two different mechanisms. The manuscript Numerical Computation of Canards  contains a description of the model and technical details of this work.

I have also been interested in the role of bifurcations in shaping the firing properties of neurons that have complex rhythmic properties. Rinzel pioneered the interpretation of bursting oscillations of membrnae potential in terms of dynamical systems with multiple time scales. My work with Allan Willms, Ronald Harris-Warrick  and Jack Peck  (Bifurcation,bursting and spike frequency adaptation, J. Computational Neuroscience, 4, 257-277, 1997) seeks to quantify such analysis and apply it to the analysis of spike trains from a neuron in the stomatogastric ganglion. Spike frequency adaptation describes the phenomenon that occurs when the frequency of a neuron firing action potentials slows from its inception. This widely observed phenomenon can lead to the "death of periodicity" with the frequency slowing to the point at which the firing ceases. Viewing the oscillations as those of a dynamical system with a slowly varying parameter, we associate the death of periodicity with bifurcations that occur in the "frozen" system in which the evolution of the slow parameter is halted. Different types of bifurcations yield different asymptotic rates at which frequency of the oscillation tends to zero. We demonstrate that these predictions are valid in model systems and discuss a new type of bifurcation that leads to the death of periodicity. Finally the analysis is applied to spike train data from the LP neuron of the STG. The image below sketches four different types of bifurcations that lead to the death of periodicity. Families of periodic orbits to the left of each panel terminate in a bifurcation as a parameter varies along the horizontal axis.