Neuroscience

I study dynamical models of a small neural system, the stomatogastric ganglion of crustaceans, in collaboration with the laboratory of Ronald Harris-Warrick.  The STG is a central pattern generator, a group of neurons that control movement. Through this research, we hope to learn more about neuromodulation, the ways in which the rhythmic output of the STG is modified by chemical and electrical inputs. The models we investigate are systems of differential equations that describe the currents contributing to the membrane potential of each neuron in the network. The interaction of these currents is complex , making it difficult to predict what the effects of varying the conductances or other properties of the currents will be on the oscillations of an individual cell or the entire network.

Algorithms for Periodic Orbits

Periodic orbits are fundamental objects within dynamical systems. Attracting periodic orbits can be found by simulation; i.e., solving initial value problems for long times. There are many circumstances in which it is desirable to have methods that compute periodic orbits directly. I have been constructing a new family of algorithms that do this with very high accuracy.  The algorithms employ  automatic differentiation,, a technique for computing the derivatives of elementary functions that avoids truncation errors of finite differences.  This facilitates the implementation of methods of very high order that are conceptually simple and can utilize coarse meshes in discretizing the periodic orbits. One objective of this research is to automatically compute bifurcations of periodic orbits. A second objective is to generate rigorous computer proofs of the qualitative properties of numerically computed dynamical systems. 

Dynamics in systems with Multiple Time Scales

Multiple time scales present challenges in the simulation of dynamical systems. Resolving the dynamics of fast time scales while computing system behavior for long times on slower time scales requires special implicit algorithms that make assumptions about the character of the fast time dynamics. There are also qualitative features of the dynamics in systems with multiple time scales that do not appear in systems with single time scales. My research is directed at extending the qualitative theory of dynamical systems to apply to systems with multiple time scales.  The examples that I have studied  arise from models of neural systems and from "switching" controllers.