Neuroscience and Locomotion

I studied 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 investigated 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 also worked on modeling neural networks in mouse spinal cord that control locomotion. Our focus was 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. The laboratory used genetic techniques to identify neurons in the spinal network, investigate their electrophysiological properties and determine the effects of their modification on the network rhythms. Experimental data provided the foundation for  network models that we used 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 participated in an NSF sponsored FIBR project led by Robert Full to study animal locomotion. The other senior investigators were Philip Holmes, Dan Koditeschek and John Miller. The project investigated dynamical principles used by cockroaches to control locomotion.  A set of hypotheses about locomotion were proposed that we studied using animals, robots and computer models. Our starting point was 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 coordinated to act like inverted pendula, there is a dimension reduction that makes the system easier to control.  We hypothesized 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. Shai Revzen, a former student in the Full lab, and I analyzed experimental data to assess whether cockroaches stabilize their movement around a periodic orbit as predicted by the theory.

Moritz Maus, Shai Revzen and I developed dynamical models that employed motion capture data for humans running on a treadmill. The data was reduced to body positions and velocities at discrete times of maximum height in flight and minimum height in stance. Our goal was to produce near optimal predictions of foot placement and leg stiffness as represented in spring loaded pendula models. Principal components analysis gave estimates for the minimal dimension of the phase space of models that produce near optimal predictions of foot placement, and indicated that incorporating positional information from prior steps did not improve the predictions. Fits of the data to a return map of the pendula models produced an unstable periodic orbit, suggesting that that "active" control of foot placement and leg stiffness using body position and velocity on the current step is essential. We studied several low dimesnional extensions of the pendula models that gave predictions of foot placement and return map eigenvalues almost as good as those which used all of the motion capture data. We hope that these results can be used to design effective controllers for legged robots.

The following mathematical/statistical issue arose in the work on running described above. The eigenvalues associated to the periodic orbit of "perfect running" were estimated from fluctuations of the observed trajectory away from the periodic orbit fit to the data. We modeled the observations by trajectories of a stochastic differential equation with a Brownian noise term. We noted that the variance of the eigenvalues estimated from trajectory segments of fixed length had a positive lower floor as the variance of the Brownian innovations tended to zero. This surprsing result can be understood as a consequence of the fact that the return map of a periodic orbit is a derivative, i.e., a ratio. When the variances in the numerator and denominator of the derivative are proportional, the variance in the ratio has a scale invariance. Aurya Javeed gave a rigorous demonstration that this phenomenon is an example of a statistical "information inequality." As is typically the case for Gaussian random variables, the variance of the return map and its eigenvalues decays as the length $T$ of the trajectory grows, but only slowly like 1/√ T  . This is a conundrum: if the parameters of a controller depend upon the return map, how long must the system be observed for the controller to stabilize the orbit. Runners are likely to fall if this time is longer than a characteristic time associated to unstable eigenvalues of the uncontrolled system. Numerical experiments indicate that "heavy tailed" noise distributions of a stochastic system with a periodic orbit can reduce the time needed to parameterize a stabilizing controller. Analyzing data to test this hypothesis is a topic of continuing interest.

I joined Jane Wang and Itai Cohen in investigations of insect flight, primarily with fruit flies of the genus Drosophila. HIgh spped films are made in Cohen's laboratory of freely flying insects. In some experiments, the insect is stimulated, for example by gluing a small iron thread to the animal and giving a brief magnetic pulse that exerts a torque via the iron thread. A goal of the research is to infer the mechanical forces that the insect uses in executing the maneuver that follows a stimulus. Typically, a fly reorients its direction to the original flight path within about one tenth of a second. Our analysis draws upon two advances that were developed by the group before I began working with them. First, Jane developed a quasisteady model for the forces acting on a plate moving through a fluid. These models approximate the fluid forces acting on the plate as a function of the veolcity and angular velocity of the plate. Second, automated tracking algorithms  were created to reconstruct the motion of a fly as a rigid body with two rigid wings attached to the body via universal joints. We model the fly as such a finite degree of freedom mechanical device. It executes a maneuver by exerting torques on its wings with its muscles. A key part of the analysis is to infer these torques by applying the quasisteady model to the measured flight paths of flies, using an inverse dynamics calculation to determine the combined fluid forces and muscle torques that produced the observed motion. We find that small asymmetries of the order of ten degrees in variables such as wing pitch or stroke amplitude are sufficient to produce the observed maneuvers.

Periodic Orbits and Their Bifurcations

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 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.  Much of this work began as a collaborative effort with Kathleen Hoffman and Warren Weckesser. We studied the classical forced van der Pol system intensively. The pheomenon of chaos in dissipative dynamical 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 explored the occurrence of chaotic attractors in concrete vector fields. We developed a general theory relating "Henon-like" attractors of two dimensional diffeomorphisms to three dimensional flows with two slow and one fast variable, and then illustrated  the theory with a modification of the forced van der Pol system.

Mathieu Desroches, Bernd Krauskopf, Christian Kuehn, Hinke Osinga, Martin Wechselberger and I have systematically brought multiple time scale methods to bear on understanding mixed mode oscillations. There is no precise definition of mixed mode oscillations, but they have been identified in systems of chemical reactors, neural oscillations and in other application domains. We give examples of slow-fast systems where the interactions of slow and fast time scales leads to small oscillations that are not apparent in the dynamics of either the slow or the fast time scales individually. This work analyzes several example more extensively than previous studies and includes a comprehensive list of  references  about mixed mode oscillations.

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 the computation of invariant two dimensional stable and unstable manifolds of equilibrium points of vector fields. Krauskopf et al. give a survey of these methods using computation of the unstable manifold of the origin in the Lorenz systems as a benchmark problem.

Invariant manifolds also play a key role in systems with multiple time scales. Christian Kuehn and I developed an algorithm for computing slow manifolds of saddle-type. These manifolds are the subject of the Exchange Lemma for slow-fast systems, and they are an important component of classes of homoclinic orbits of systems with multiple time scales. Homoclinic orbits arise as traveling wave profiles of partial differential equations. Traveling waves of the FitzHugh-Nagumo model for action potentials have been investigated intensively. Our algorithm yields improved calculations of these periodic orbits. This work also highlights how tangential intersections of invariant manifolds underlie bifurcations that create both homoclinic orbits and mixed mode oscillations.

El Niño and the Southern Oscillation

El Niño is a phenomenon in the equatorial Pacific ocean. In "normal" conditions, there is a gradient of sea surface temperature with much warmer waters in the western Pacific and colder surface water off the coast of South America. Surface winds in these equatorial latitudes blow from east to west in contrast to prevailing westerlies in mid latitudes that are driven by the coriolos force accompanying the earth's rotation. Every few years, this pattern of winds and waves changes: the winds diminish and surface waters of the eastern Pacific warm. This is El Niño. Its strength and timing is highly irregular, with exceptionally strong events ocurring on a decadal time scale. The strong event affect weather around the globe, including in the US. Atmospheric scientists have invested increasing effort in monitoring and forecasting this Southern Oscillation. Models that have been developed ranges from detailed coupled ocean-atmosphere climate models, to small systems of ordinary or delayed differential equation models that attempt to capture the large scale phenomena underlying El Niño.

Andrew Roberts and I, together with Henk Djikstra, Axel Timmermann, Esther Widiasih and Chris Jones, invesstigated a three dimensional vector field formulated by Fei Fei Jin in the 1990's as one of the smallest models of the Southern Oscillation. Its phase space variables represent the sea surface temperatures in the eastern and western equatorial Pacific and the depth of the thermocline separating cold, deep water form warmer surface waters. I joined Roberts, Jones and Widiasih in applying multiple time scale methods to analyze the dynamics of the Jin model. Though the model was not formulated as a slow-fast sytem with multiple time scales, we observed that it displays dynamics as if it were. We found oscillations within the model that resemble strong El Niño events. Furthermore, there are parameter regimes in which the timing of these events is highly unpredictable. This suggests that the irregularity of the strong El Niño events may be an intrinsic feature of the large scale dynamics of the Pacific ocean.

Lines of Curvature on Embedded Surfaces

The differential geometry of smooth surfaces embedded in three space is a classical mathematical subject. At each point of a surface, there are orthogonal directions along which normal slices of the surface have the largest and smallest curvature. These principal directions form fields on the surface that are similar to vector fields, but they are not orientable. Lines of curves are characterized by having tangents in the principal directions. They are analogous to the trajectories of a vector field and together comprise the principal foliations of the surface, analogous to the phase portrait of a vector field. Incontrast to vector fields, there are few examples of surfaces whose principal foliations have been determined. The example displayed in most differential geometry texts are the principal foliations of a triaxial ellipsoid which were found by Monge already in the eighteenth century.

One of the highlights of dynamical systems theory is the characterization of structurally stable vector fields on compact surfaces. The geometric perspective of the theory was initiated by Poincaré and further developed by Andronov and Pontryagin in the 1930s. The Poincaré-Bendixson Theorem immplies that the limit set of all trajectories of a vector field on the two sphere is either a periodic orbit or contains an equilibrium point. Structurally stable vector fields on the two spheres are characterized byhaving hyperbolic equilibrium points and periodic orbits and no trajectories that connect saddle points. Sotomayor and Gutierrez developed an analogous theory for principal foliations. The singular points are umbiics where the curvatures of all normal slices are the same. Generic umbiilic points were classified by Darboux over a century ago: the three types are now called stars, monstars and lemons.

Software for visualizing the phase portraits of two dimensional vector fields is plentiful. Software for visualizing principal foliations is not. I have begun to address this gap, and the results are surprising. Sotomayor and Gutierrez realized that the Poincaré-Bendixson Theorem fails for compact convex surfaces and gave examples of surfaces with dense lines of curvature. They posed the problem of determining the principal foliations of perturbations of the triaxial ellipsoid in their lecture notes. I have solved this exercise, demonstrating that these principal foliations can be "lifted" to vector fields on the two torus which have no equilibrium points. Return maps of these flows are diffeomorphisms of the circle, a subject that has been studied extensively -- again beginning with Poincaré. He defined a rotation number for such diffeomorphisms, and Denjoy proved that smooth diffeomorphisms with irrational rotation number have dense trajectories. Seminal work by Herman and Yoccoz established that irrational rotation numbers occur for positive measure sets of parameters in one parameter families of circle diffeomorphisms. Applying this theory to principal foliations, I have characterized the principal foliations for perturbations of triaxial ellipsoids. Many have dense lines of curvature.

Last Update: February 10, 2020