**Research**
__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 i*mplicit*
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.