Locomotion: exploiting noise for state estimation.

Running, walking, flying and swimming are all processes in which animals produce propulsion by executing rhythmic motions of their bodies. Dynamical stability of the locomotion is hardly automatic: millions of older people are injured by falling each year. Stability frequently requires sensory feedback. We investigate how organisms obtain the information they use in maintaining their stability. Assessing stability of a periodic orbit of a dynamical system requires information about the dynamics of the system off the orbit. For locomotion driven by a periodic orbit, perturbations that "kick" the trajectory off the orbit must occur in order to observe convergence rates toward the orbit. We propose that organisms generate excitations in order to set the gains for stabilizing feedback. We hypothesize further that these excitations are stochastic but have heavy-tailed, non-Gaussian probability distributions. Compared to Gaussian distributions, we argue that these are more effective for estimating stability characteristics of the orbit. Finally, we propose experiments to test the efficacy of these ideas.

Invariant manifolds and global bifurcations.

Invariant manifolds are key objects in describing how trajectories partition the phase spaces of a dynamical system. Examples include stable, unstable, and center manifolds of equilibria and periodic orbits, quasiperiodic invariant tori, and slow manifolds of systems with multiple timescales. Changes in these objects and their intersections with variation of system parameters give rise to global bifurcations. Bifurcation manifolds in the parameter spaces of multi-parameter families of dynamical systems also play a prominent role in dynamical systems theory. Much progress has been made in developing theory and computational methods for invariant manifolds during the past 25 years. This article highlights some of these achievements and remaining open problems.

Discovering the flight autostabilizer of fruit flies by inducing aerial stumbles.

Just as the Wright brothers implemented controls to achieve stable airplane flight, flying insects have evolved behavioral strategies that ensure recovery from flight disturbances. Pioneering studies performed on tethered and dissected insects demonstrate that the sensory, neurological, and musculoskeletal systems play important roles in flight control. Such studies, however, cannot produce an integrative model of insect flight stability because they do not incorporate the interaction of these systems with free-flight aerodynamics. We directly investigate control and stability through the application of torque impulses to freely flying fruit flies (Drosophila melanogaster) and measurement of their behavioral response. High-speed video and a new motion tracking method capture the aerial "stumble," and we discover that flies respond to gentle disturbances by accurately returning to their original orientation. These insects take advantage of a stabilizing aerodynamic influence and active torque generation to recover their heading to within 2 degrees in < 60 ms. To explain this recovery behavior, we form a feedback control model that includes the fly's ability to sense body rotations, process this information, and actuate the wing motions that generate corrective aerodynamic torque. Thus, like early man-made aircraft and modern fighter jets, the fruit fly employs an automatic stabilization scheme that reacts to short time-scale disturbances.

Paddling mode of forward flight in insects.

By analyzing high-speed video of the fruit fly, we discover a swimminglike mode of forward flight characterized by paddling wing motions. We develop a new aerodynamic analysis procedure to show that these insects generate drag-based thrust by slicing their wings forward at low angle of attack and pushing backwards at a higher angle. Reduced-order models and simulations reveal that the law for flight speed is determined by these wing motions but is insensitive to material properties of the fluid. Thus, paddling is as effective in air as in water and represents a common strategy for propulsion through aquatic and aerial environments.

Synaptic patterning of left-right alternation in a computational model of the rodent hindlimb central pattern generator.

Establishing, maintaining, and modifying the phase relationships between extensor and flexor muscle groups is essential for central pattern generators in the spinal cord to coordinate the hindlimbs well enough to produce the basic walking rhythm. This paper investigates a simplified computational model for the spinal hindlimb central pattern generator (CPG) that is abstracted from experimental data from the rodent spinal cord. This model produces locomotor-like activity with appropriate phase relationships in which right and left muscle groups alternate while extensor and flexor muscle groups alternate. Convergence to this locomotor pattern is slow, however, and the range of parameter values for which the model produces appropriate output is relatively narrow. We examine these aspects of the model's coordination of left-right activity through investigation of successively more complicated subnetworks, focusing on the role of the synaptic architecture in shaping motoneuron phasing. We find unexpected sensitivity in the phase response properties of individual neurons in response to stimulation and a need for high levels of both inhibition and excitation to achieve the walking rhythm. In the absence of cross-cord excitation, equal levels of ipsilateral and contralateral inhibition result in a strong preference for hopping over walking. Inhibition alone can produce the walking rhythm, but contralateral inhibition must be much stronger than ipsilateral inhibition. Cross-cord excitatory connections significantly enhance convergence to the walking rhythm, which is achieved most rapidly with strong crossed excitation and greater contralateral than ipsilateral inhibition. We discuss the implications of these results for CPG architectures based on unit burst generators.

Fruit flies modulate passive wing pitching to generate in-flight turns.

Flying insects execute aerial maneuvers through subtle manipulations of their wing motions. Here, we measure the free-flight kinematics of fruit flies and determine how they modulate their wing pitching to induce sharp turns. By analyzing the torques these insects exert to pitch their wings, we infer that the wing hinge acts as a torsional spring that passively resists the wing's tendency to flip in response to aerodynamic and inertial forces. To turn, the insects asymmetrically change the spring rest angles to generate asymmetric rowing motions of their wings. Thus, insects can generate these maneuvers using only a slight active actuation that biases their wing motion.

Lateral stability of the spring-mass hopper suggests a two-step control strategy for running.

This paper investigates the control of running gaits in the context of a spring loaded inverted pendulum model in three dimensions. Specifically, it determines the minimal number of steps required for an animal to recover from a perturbation to a specified gait. The model has four control inputs per step: two touchdown angles (azimuth and elevation) and two spring constants (compression and decompression). By representing the locomotor movement as a discrete-time return map and using the implicit function theorem we show that the number of recovery steps needed following a perturbation depends upon the goals of the control mechanism. When the goal is to follow a straight line, two steps are necessary and sufficient for small lateral perturbations. Multistep control laws have a larger number of control inputs than outputs, so solutions of the control problem are not unique. Additional constraints, referred to here as synergies, are imposed to determine unique control inputs for perturbations. For some choices of synergies, two-step control can be expressed as two iterations of its first step policy and designed so that recovery occurs in just one step for all perturbations for which one-step recovery is possible.

Parameter estimation for bursting neural models.

This paper presents work on parameter estimation methods for bursting neural models. In our approach we use both geometrical features specific to bursting, as well as general features such as periodic orbits and their bifurcations. We use the geometry underlying bursting to introduce defining equations for burst initiation and termination, and restrict the estimation algorithms to the space of bursting periodic orbits when trying to fit periodic burst data. These geometrical ideas are combined with automatic differentiation to accurately compute parameter sensitivities for the burst timing and period. In addition to being of inherent interest, these sensitivities are used in standard gradient-based optimization algorithms to fit model burst duration and period to data. As an application, we fit Butera et al.'s (Journal of Neurophysiology 81, 382-397, 1999) model of preBötzinger complex neurons to empirical data both in control conditions and when the neuromodulator norepinephrine is added (Viemari and Ramirez, Journal of Neurophysiology 95, 2070-2082, 2006). The results suggest possible modulatory mechanisms in the preBötzinger complex, including modulation of the persistent sodium current.

Estimating the phase of synchronized oscillators.

The state of a collection of phase-locked oscillators is determined by a single phase variable or cyclic coordinate. This paper presents a computational method, Phaser, for estimating the phase of phase-locked oscillators from limited amounts of multivariate data in the presence of noise and measurement errors. Measurements are assumed to be a collection of multidimensional time series. Each series consists of several cycles of the same or similar systems. The oscillators within each system are not assumed to be identical. Using measurements of the noise covariance for the multivariate input, data from the individual oscillators in the system are combined to reduce the variance of phase estimates for the whole system. The efficacy of the algorithm is demonstrated on experimental and model data from biomechanics of cockroach running and on simulated oscillators with varying levels of noise.

Manipulating the edge of instability.

We investigate the integration of visual and tactile sensory input for dynamic manipulation. Our experimental data and computational modeling reveal that time-delays are as critical to task-optimal multisensory integration as sensorimotor noise. Our focus is a dynamic manipulation task "at the edge of instability." Mathematical bifurcation theory predicts that this system will exhibit well-classified low-dimensional dynamics in this regime. The task was using the thumbpad to compress a slender spring prone to buckling as far as possible, just shy of slipping. As expected from bifurcation theory, principal components analysis gives a projection of the data onto a low dimensional subspace that captures 91-97% of its variance. In this subspace, we formulate a low-order model for the brain+hand+spring dynamics based on known mechanical and neurophysiological properties of the system. By systematically occluding vision and anesthetically blocking thumbpad sensation in 12 consenting subjects, we found that vision contributed to dynamic manipulation only when thumbpad sensation was absent. The reduced ability of the model system to compress the spring with absent sensory channels closely resembled the experimental results. Moreover, we found that the model reproduced the contextual usefulness of vision only if we took account of time-delays. Our results shed light on critical features of dynamic manipulation distinct from those of static pinch, as well as the mechanism likely responsible for loss of manual dexterity and increased reliance on vision when age or neuromuscular disease increase noisiness and/or time-delays during sensorimotor integration.

Overexpression of a hyperpolarization-activated cation current (Ih) channel gene modifies the firing activity of identified motor neurons in a small neural network.

The hyperpolarization-activated cation current (Ih) is widely distributed in excitable cells. Ih plays important roles in regulation of cellular excitability, rhythmic activity, and synaptic function. We previously showed that, in pyloric dilator (PD) neurons of the stomatogastric ganglion (STG) of spiny lobsters, Ih can be endogenously upregulated to compensate for artificial overexpression of the Shal transient potassium channel; this maintains normal firing properties of the neuron despite large increases in potassium current. To further explore the function of Ih in the pyloric network, we injected cRNA of PAIH, a lobster gene that encodes Ih, into rhythmically active PD neurons. Overexpression of PAIH produced a fourfold increase in Ih, although with somewhat different biophysical properties than the endogenous current. Compared with the endogenous Ih, the voltage for half-maximal activation of the PAIH-evoked current was depolarized by 10 mV, and its activation kinetics were significantly faster. This increase in Ih did not affect the expression of IA or other outward currents. Instead, it significantly altered the firing properties of the PD neurons. Increased Ih depolarized the minimum membrane potential of the cell, reduced the oscillation amplitude, decreased the time to the first spike, and increased the duty cycle and number of action potentials per burst. We used both dynamic-clamp experiments, injecting the modeled PAIH currents into PD cells in a functioning STG, and a theoretical model of a two-cell network to demonstrate that the increased Ih was sufficient to cause the observed changes in the PD activity.

Controlling roll perturbations in fruit flies.

Owing to aerodynamic instabilities, stable flapping flight requires ever-present fast corrective actions. Here, we investigate how flies control perturbations along their body roll angle, which is unstable and their most sensitive degree of freedom. We glue a magnet to each fly and apply a short magnetic pulse that rolls it in mid-air. Fast video shows flies correct perturbations up to 100° within 30 ± 7 ms by applying a stroke-amplitude asymmetry that is well described by a linear proportional-integral controller. For more aggressive perturbations, we show evidence for nonlinear and hierarchical control mechanisms. Flies respond to roll perturbations within 5 ms, making this correction reflex one of the fastest in the animal kingdom.

Constructing predictive models of human running.

Running is an essential mode of human locomotion, during which ballistic aerial phases alternate with phases when a single foot contacts the ground. The spring-loaded inverted pendulum (SLIP) provides a starting point for modelling running, and generates ground reaction forces that resemble those of the centre of mass (CoM) of a human runner. Here, we show that while SLIP reproduces within-step kinematics of the CoM in three dimensions, it fails to reproduce stability and predict future motions. We construct SLIP control models using data-driven Floquet analysis, and show how these models may be used to obtain predictive models of human running with six additional states comprising the position and velocity of the swing-leg ankle. Our methods are general, and may be applied to any rhythmic physical system. We provide an approach for identifying an event-driven linear controller that approximates an observed stabilization strategy, and for producing a reduced-state model which closely recovers the observed dynamics.

Active and passive stabilization of body pitch in insect flight.

Flying insects have evolved sophisticated sensory-motor systems, and here we argue that such systems are used to keep upright against intrinsic flight instabilities. We describe a theory that predicts the instability growth rate in body pitch from flapping-wing aerodynamics and reveals two ways of achieving balanced flight: active control with sufficiently rapid reactions and passive stabilization with high body drag. By glueing magnets to fruit flies and perturbing their flight using magnetic impulses, we show that these insects employ active control that is indeed fast relative to the instability. Moreover, we find that fruit flies with their control sensors disabled can keep upright if high-drag fibres are also attached to their bodies, an observation consistent with our prediction for the passive stability condition. Finally, we extend this framework to unify the control strategies used by hovering animals and also furnish criteria for achieving pitch stability in flapping-wing robots.

Finding the dimension of slow dynamics in a rhythmic system.

Dynamical systems with asymptotically stable periodic orbits are generic models for rhythmic processes in dissipative physical systems. This paper presents a method for reconstructing the dynamics near a periodic orbit from multivariate time-series data. It is used to test theories about the control of legged locomotion, a context in which time series are short when compared with previous work in nonlinear time-series analysis. The method presented here identifies appropriate dimensions of reduced order models for the deterministic portion of the dynamics. The paper also addresses challenges inherent in identifying dynamical models with data from different individuals.

Return maps of folded nodes and folded saddle-nodes.

Folded nodes occur in generic slow-fast dynamical systems with two slow variables. Open regions of initial conditions flow into a folded node in an open set of such systems, so folded nodes are an important feature of generic slow-fast systems. Twisting and linking of trajectories in the vicinity of a folded node have been studied previously, but their consequences for global dynamical behavior have hardly been investigated. One manifestation of the twisting is as "mixed mode oscillations" observed in chemical and neural systems. This paper presents the first systematic numerical study of return maps for trajectories that flow through a region with a folded node. These return maps are approximated by rank-1 maps, and the local twisting of trajectories near a folded node gives rise to multiple turning points in the approximating one dimensional maps. A variant of the forced van der Pol system is used here to illustrate that folded nodes can be a "chaos-generating" mechanism. Folded saddle-nodes occur in generic one-parameter families of slow-fast dynamical systems with two slow variables. These bifurcations give birth to folded nodes. Numerical simulations demonstrate that return maps of systems that are close to a folded saddle-node can be even more complex than those of folded nodes that are far from folded saddles.

Estimating effective degrees of freedom in motor systems.

Studies of the degrees of freedom and "synergies" in musculoskeletal systems rely critically on algorithms to estimate the "dimension" of kinematic or neural data. Linear algorithms such as principal component analysis (PCA) are the most popular. However, many biological data (or realistic experimental data) may be better represented by nonlinear sets than linear subspaces. We evaluate the performance of PCA and compare it to two nonlinear algorithms [Isomap and our novel pointwise dimension estimation (PD-E)] using synthetic and motion capture data from a robotic arm with known kinematic dimensions, as well as motion capture data from human hands. We find that PCA can lead to more accurate dimension estimates when considering additional properties of the PCA residuals, instead of the dominant method of using a threshold of variance captured. In contrast to the single integer dimension estimates of PCA and Isomap, PD-E provides a distribution and range of estimates of fractal dimension that identify the heterogeneous geometric structure in the experimental data. A strength of the PD-E method is that it associates a distribution of dimensions to the data. Since there is no a priori reason to assume that the sets of interest have a single dimension, these distributions incorporate more information than a single summary statistic. Our preliminary findings suggest that fewer than ten DOFs are involved in some hand motion tasks. Contrary to common opinion regarding fractal dimension methods, PD-E yielded reasonable results with reasonable amounts of data. Given the complex nature of experimental and biological data, we conclude that it is necessary and feasible to complement PCA with methods that take into consideration the nonlinear properties of biological systems for a more robust estimation of their DOFs.

The invariant constrained equilibrium edge preimage curve method for the dimension reduction of chemical kinetics.

This work addresses the construction and use of low-dimensional invariant manifolds to simplify complex chemical kinetics. Typically, chemical kinetic systems have a wide range of time scales. As a consequence, reaction trajectories rapidly approach a hierarchy of attracting manifolds of decreasing dimension in the full composition space. In previous research, several different methods have been proposed to identify these low-dimensional attracting manifolds. Here we propose a new method based on an invariant constrained equilibrium edge (ICE) manifold. This manifold (of dimension nr) is generated by the reaction trajectories emanating from its (nr-1)-dimensional edge, on which the composition is in a constrained equilibrium state. A reasonable choice of the nr represented variables (e.g., nr "major" species) ensures that there exists a unique point on the ICE manifold corresponding to each realizable value of the represented variables. The process of identifying this point is referred to as species reconstruction. A second contribution of this work is a local method of species reconstruction, called ICE-PIC, which is based on the ICE manifold and uses preimage curves (PICs). The ICE-PIC method is local in the sense that species reconstruction can be performed without generating the whole of the manifold (or a significant portion thereof). The ICE-PIC method is the first approach that locally determines points on a low-dimensional invariant manifold, and its application to high-dimensional chemical systems is straightforward. The "inputs" to the method are the detailed kinetic mechanism and the chosen reduced representation (e.g., some major species). The ICE-PIC method is illustrated and demonstrated using an idealized H2O system with six chemical species. It is then tested and compared to three other dimension-reduction methods for the test case of a one-dimensional premixed laminar flame of stoichiometric hydrogen/air, which is described by a detailed mechanism containing nine species and 21 reactions. It is shown that the error incurred by the ICE-PIC method with four represented species is small across the whole flame, even in the low temperature region.

Dopamine modulation of two delayed rectifier potassium currents in a small neural network.

Delayed rectifier potassium currents [I(K(V))] generate sustained, noninactivating outward currents with characteristic fast rates of activation and deactivation and play important roles in shaping spike frequency. The pyloric motor network in the stomatogastric ganglion of the spiny lobster, Panulirus interruptus, is made up of one interneuron and 13 motor neurons of five different classes. Dopamine (DA) increases the firing frequencies of the anterior burster (AB), pyloric (PY), lateral pyloric (LP), and inferior cardiac (IC) neurons and decreases the firing frequencies of the pyloric dilator (PD) and ventricular dilator (VD) neurons. In all six types of pyloric neurons, I(K(V)) is small with respect to other K(+) currents. It is made up of at least two TEA-sensitive components that show differential sensitivity to 4-aminopyridine and quinidine, and have differing thresholds of activation. One saturable component is activated at potentials above -25 mV, whereas the second component appears at more depolarized voltages and does not saturate at voltage steps up to +45 mV. The magnitude of the components varies among cell types but also shows considerable variation within a single type. A subset of PY neurons shows a marked enhancement in spike frequency with DA; DA evokes a pronounced reversible increase in I(K(V)) conductance of < or = 30% in the PY neurons studied, and on average significantly increases both components of I(K(V)). The AB neuron also shows a reversible 20% increase in the steady state I(K(V)). DA had no effect on I(K(V)) in PD, LP, VD, and IC neurons. The physiological roles of these currents and their modulation by DA are discussed.

Activity-independent coregulation of IA and Ih in rhythmically active neurons.

The fast transient potassium or A current (IA) plays an important role in determining the activity of central pattern generator neurons. We have previously shown that the shal K+ channel gene encodes IA in neurons of the pyloric network in the spiny lobster. To further study how IA shapes pyloric neuron and network activity, we microinjected RNA for a shal-GFP fusion protein into four identified pyloric neuron types. Neurons expressing shal-GFP had a constant increase in IA amplitude, regardless of cell type. This increase in IA was paralleled by a concomitant increase in the hyperpolarization-activated cation current Ih in all pyloric neurons. Despite significant increases in these currents, only modest changes in cell firing properties were observed. We used models to test two hypotheses to explain this failure to change firing properties. First, this may reflect the mislocalization of the expressed shal protein solely to the somata and initial neurites of injected neurons, rendering it electrically remote from the integrating region in the neuropil. To test this hypothesis, we generated a multicompartment model where increases in IA could be localized to the soma, initial neurite, or neuropil/axon compartments. Although spike activity was somewhat more sensitive to increases in neuropil/axon versus somatic/primary neurite IA, increases in IA limited to the soma and primary neurite still evoked much more dramatic changes than were seen in the shal-GFP-injected neurons. Second, the effect of the increased IA could be compensated by the endogenous increase in Ih. To test this, we modeled the compensatory increases of IA and Ih with a cycling two-cell model. We found that the increase in Ih was sufficient to compensate the effects of increased IA, provided that they increase in a constant ratio, as we observed experimentally in both shal-injected and noninjected neurons. Thus an activity-independent homeostatic mechanism maintains constant neuronal activity in the face of dramatic increases in IA.