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, so simulation and analysis of these models is used 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.

We have also begun to work with Ole Kiehn of the Karolinska Institute in Stockholm, Sweden, on modeling neural networks in mouse spinal cord that control locomotion. Our focus is on the role of intercomissural neurons, neurons whose axons cross the spinal cord. These neurons play a fundamental role in coordinating motion of the  right and left limbs of the animal. Rhythmic behavior ("ficitive locomotion") is  elicited and recorded in a spinal cord preparation by bath application of neuromodulators. We use genetic techniques to identify neurons in the spinal network, investigate their electrophysiological properties and determine the effects of their modification on the network rhythms. Our experimental data provides the foundation for  network models that we use to gain insight into how the rhythmic activity of this network arises from the intrinsic properties of its individual neurons and their synaptic interactions with each other.

I am also particpating in an NSF sponsored FIBR project led by Robert Full to study animal locomotion. The other senior investigators are Philip Holmes, Dan Koditeschek and John Miller. Our goal is to discover  dynamical principles that are used in locomotion, using the cockroach as an experimental organism.We have formulated a set of hypotheses about locomotion that we are testing using animals, robots and computer models. Our starting point is the observation many animals have evolved behaviors that resemble a biped whose legs act like pogo sticks, or spring loaded inverted pendulua. Insects with six legs frequently run with a gait in which tripods of legs move together.  To the extent that these tripods are coordianted to act like inverted pendula, there is a dimension reduction that makes the system easier to control.  We hypothesize that neural control acts on this mechanical plant to modify its intrinsic behavior as contrasted with directing all aspects of the motion of legs as is done with many legged robots.

Periodic orbits are fundamental objects within dynamical systems. They represent cyclic processes in dynamical models of biological systems. Attracting periodic orbits can be found by simulation; i.e., solving initial value problems for long times. However, simulation alone is insufficient to obtain all of the information that we would like to have about periodic orbits. There are many circumstances in which it is desirable to have methods that compute periodic orbits directly. The algorithms used by such methods are examples of boundary value solvers. In addition to solving for individual periodic orbits, the methods enable one to determine local stability information, compute sensitivity information that describes how the orbit changes with  perturbations of parameters, identify parameter values at which bifurcations occur and track families of periodic orbits as parameters vary. Our contributions in this area have centered on the use of a technique called automatic differentiation in the solution of the boundary value problems. Automatic differentiation enables the computation of  derivatives of functions without the truncation errors that arise from the use of finite differences.  This facilitates the implementation of methods of very high order that are conceptually simple and utilize coarse meshes in discretizing the periodic orbits. Another aspect of our research on periodic orbits has been to generate rigorous proofs of the qualitative properties of numerically computed dynamical systems. 

Dynamical 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 implicitalgorithms 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.  Much of this work has been a collaborative effort with Kathleen Hoffman and Warren Weckesser. We have studied the classical forced van der Pol systems intensively. The pheomenon of chaos in dissipative synamical systems was first discovered and analyzed in this example by Cartwright and Littlewood during the 1940's and 1950's. Our work and that of my former student Radu Haiduc simplifies and extends the results of Cartwright and Littlewood. Haiduc's thesis establishes parameter values for which the forced van der Pol equation has a chaotic invariant set and is structurally stable. In joint workwith Martin Wechselberger and Lai-Sang Young, I have explored the occurrence of chaotic attractors in concrete vector fields. We develop a general theory relating "Henon-like" attractors of two dimensional diffeomorphisms to three dimensional flows with two slow and one fast variable. We illustrate  the theory with a modification of the forced van der Pol system. Models of neural systems frequently have multiple tiem scales; my research has examined several phenomena that have been discovered in this context.  

Invariant Manifolds

 Computation of invariant manifolds of dynamical systems with dimension larger than one has proved to be a challenging problem. There are two sources of the difficulty in computing these manifolds. First, they have an intricate geometric structure with folds and spiraling structures. Second there is "geometric stiffness" reflected in trajectories that evolve with very different speeds. Thus, attempts to compute the manifolds by solving initial value problems for a few trajectories tend to produce poor results because the trajectories don't spread uniformly, and where they do spread they diverge rapidly enough  to leave large gaps in the manifolds. Alex Vladimirsky and I have used "ordered upwinding" methods from the  numerical analysis of partial differential equations to develop and implement efficient methods for computing these manifolds. The methods as we have developed them are low order, but very fast.  Several research groups have introduced different approaches to teh computation of invariant two dimensional stable and unstable manifolds of equilibrium points of vector fields. Krauskopf et al. give a survey of these methods usingcomputation of the unstable manifold of the origin in the Lorenz systems as a benchmark problem.