Dynamics and stability of legged locomotion in the horizontal plane: a test case using insects

 Motivated by experimental studies of insects, we propose a model for legged locomotion in the horizontal plane. A three-degree-of freedom, energetically conservative, rigid-body model with a pair of compliant virtual legs in intermittent contact with the ground allows us to study how dynamics depends on parameters such as mass, moment of inertia, leg stiffness, and length. We find periodic gaits, and show that mechanics alone can confer asymptotic stability of relative heading and body angular velocity. We discuss the relevance of our idealized models to experiments and simulations on insect running, showing that their gait and force characteristics match observations reasonably well. We perform parameter studies and suggest that our model is relevant to the understanding of locomotion dynamics across species. Received: 17 April 2001 / Accepted in revised form: 20 November 2001     

An elastic rod model for anguilliform swimming

We develop a model for anguilliform (eel-like) swimming as an elastic rod actuated via time-dependent intrinsic curvature and subject to hydrodynamic drag forces, the latter as proposed by Taylor (in Proc Roy Proc Lond A 214:158–183, 1952). We employ a eometrically exact theory and discretize the resulting nonlinear partial differential evolution both to perform numerical simulations, and to compare with previous models consisting of chains of rigid links or masses connected by springs, dampers, and prescribed force generators representing muscles. We show that muscle activations driven by motoneuronal spike trains via calcium dynamics produce intrinsic curvatures corresponding to near-sinusoidal body shapes in longitudinally-uniform rods, but that passive elasticity causes Taylor’s assumption of prescribed shape to fail, leading to time-periodic motions and lower speeds than those predicted Taylor (in Proc Roy Proc Lond A 214:158–183, 1952). We investigate the effects of bending stiffness, body geometry, and activation patterns on swimming speed, turning behavior, and acceleration to steady swimming. We show that laterally-uniform activation yields stable straight swimming and laterally differential activation levels lead to stable turns, and we argue that tapered bodies with reduced caudal (tail-end) activation (to produce uniform intrinsic curvature) swim faster than ones with uniform activation.     

A hexapedal jointed-leg model for insect locomotion in the horizontal plane

We develop a simple model for insect locomotion in the horizontal (ground) plane. As in earlier work by Seipel et al. (Biol Cybern 91(0):76–90, 2004) we employ six actuated legs that also contain passive springs, but the legs, with “hip” and ‘knee’ joints, better represent insect morphology. Actuation is provided via preferred angle inputs at each joint, corresponding to zero torques in the hip and knee springs. The inputs are determined from estimates of foot forces in the cockroach Blaberus discoidalis via an inverse problem. The head–thorax–body is modeled as a single rigid body, and leg masses, inertia and joint dissipation are ignored. The resulting three degree-of-freedom dynamical system, subject to feedforward joint inputs, exhibits stable periodic gaits that compare well with observations over the insect’s typical speed range. The model’s response to impulsive perturbations also matches that of freely-running cockroaches (Jindrich and Full, J Exp Biol 205:2803–2823, 2002), and stability is maintained in the face of random foot touchdowns representative of real insects. We believe that this model will allow incorporation of realistic muscle models driven by a central pattern generator in place of the joint actuators, and that it will ultimately permit the study of proprioceptive feedback pathways involving leg force and joint angle sensing.     

A contact model to simulate human–artifact interaction based on force optimization: implementation and application to the analysis of a training machine

Musculoskeletal multibody models are increasingly used to analyze and optimize physical interactions between humans and technical artifacts. Since interaction is conveyed by contact between the human body and the artifact, a computationally robust modeling approach for frictional contact forces is a crucial aspect. In this contribution, we propose a parametric contact model and formulate an associated force optimization problem to simultaneously estimate unknown muscle and contact forces in an inverse dynamic manner from a prescribed motion trajectory. Unlike existing work, we consider both the static and the kinetic regime of Coulomb’s friction law. The approach is applied to the analysis of a leg extension training machine with the objective to reduce the stress on the tibiofemoral joint. The uncertainty of the simulation results due to a tunable parameter of the contact model is of particular interest.     

Mechanical models for insect locomotion: active muscles and energy losses

 We extend the analysis of simple, energy-conserving models for the dynamics of insect locomotion in the horizontal plane developed in Schmitt and Holmes (2000a,b, 2001), where gaits characteristic of steady cockroach running and turning were evoked. In this paper, we include dissipation and energy inputs via active “muscles” in three forms: via prescribed torques at the “hip” pivot, via an active spring element of variable length, and via a pair of Hill-type muscle models representing an extensor/flexor system. Due to mechanical feedback of passive elastic forces, the stable gaits of the conservative models are preserved, and now energy input and absorption balances to additionally stabilize a preferred speed, with only modest neural sensing and feedback being required. However, these bipedal models still cannot simultaneously match observed moment-yaw magnitudes and fore-aft dynamics. Received: 17 September 2001 / Accepted: 20 February 2003 / Published online: 20 May 2003 Correspondence to: P. Holmes (e-mail: pholmes@math.Princeton.EDU) Acknowledgements. This work was supported by DARPA/ONR: N00014-98-1-0747 and DoE: DE-FG02-95ER25238. John Schmitt was partially supported by a DoD Graduate Fellowship, a Wu Fellowship of the School of Engineering and Applied Science, and a George Van Ness Lothrop Honorific Fellowship of the Graduate School at Princeton University. We thank Kenneth Meijer for allowing us to use his muscle model in Sect. 4 and Bob Full and Dan Koditschek for numerous helpful suggestions.     

Contact dynamics during keratocyte motility

BACKGROUND: Keratocytes are specialised, rapidly moving cells that generate substantial contractile force perpendicular to their direction of locomotion. Potential roles for contractile force in cell motility include cell-body transport, regulation of adhesion, and retraction of the cell's trailing edge. RESULTS: To investigate contact dynamics, we used simultaneous confocal fluorescence and interference reflection microscopy to image keratocytes injected with fluorescent vinculin. We found that contacts formed behind the leading edge and grew beneath both the lamellipodium and the cell body. Contacts in the middle of the cell remained stationary relative to the substrate and began to disassemble as the cell body passed over them. In contrast, contacts in the lobes of the cell grew continuously and more rapidly, incorporated more vinculin, and slid inwards towards the sides of the cell body. Contact sliding often led to merging of contacts before their removal from the substrate. CONCLUSIONS: We suggest a synthesis of two existing, apparently conflicting models for keratocyte motility, in which network contraction progressively reorients actin filaments using the contacts as pivots, forming bundles that then generate lateral tension by a sliding-filament mechanism. Contact dynamics vary between the middle of the cell and the lobes. We propose that laterally opposed contractile forces first enhance contact growth and stability, but escalating force eventually pulls contacts from the substrate at the back of the cell, without interfering with the cell's forward progress.     

Lyapunov functions for Lotka–Volterra predator–prey models with optimal foraging behavior

 The theory of optimal foraging predicts abrupt changes in consumer behavior which lead to discontinuities in the functional response. Therefore population dynamical models with optimal foraging behavior can be appropriately described by differential equations with discontinuous right-hand sides. In this paper we analyze the behavior of three different Lotka–Volterra predator–prey systems with optimal foraging behavior. We examine a predator–prey model with alternative food, a two-patch model with mobile predators and resident prey, and a two-patch model with both predators and prey mobile. We show that in the studied examples, optimal foraging behavior changes the neutral stability intrinsic to Lotka–Volterra systems to the existence of a bounded global attractor. The analysis is based on the construction and use of appropriate Lyapunov functions for models described by discontinuous differential equations. Received: 23 March 1999     

Body size and food web structure: testing the equiprobability assumption of the cascade model

The cascade model successfuly predicts many patterns in reported food webs. A key assumption of this model is the existence of a predetermined trophic hierarchy; prey are always lower in the hierarchy than their predators. At least three studies have suggested that, in animal food webs, this hierarchy can be explained to a large extent by body size relationships. A second assumption of the standard cascade model is that trophic links not prohibited by the hierarchy occur with equal probability. Using nonparametric contingency table analyses, we tested this ”equiprobability hypothesis” in 16 published animal food webs for which the adult body masses of the species had been estimated. We found that when the hierarchy was based on body size, the equiprobability hypothesis was rejected in favor of an alternative, ”predator-dominance” hypothesis wherein the probability of a trophic link varies with the identity of the predator. Another alternative to equiprobabilty is that the probability of a trophic link depends upon the ratio of the body sizes of the two species. Using nonparametric regression and liklihood ratio tests, we show that a size-ratio based model represents a significant improvement over the cascade model. These results suggest that models with heterogeneous predation probabilities will fit food web data better than the homogeneous cascade model. They also suggest a new way to bridge the gap between static and dynamic food web models. Received: 3 February 1999 / Accepted: 26 October 1999     

Neural network firing-rate models on integral form

Firing-rate models describing neural-network activity can be formulated in terms of differential equations for the synaptic drive from neurons. Such models are typically derived from more general models based on Volterra integral equations assuming exponentially decaying temporal coupling kernels describing the coupling of pre- and postsynaptic activities. Here we study models with other choices of temporal coupling kernels. In particular, we investigate the stability properties of constant solutions of two-population Volterra models by studying the equilibrium solutions of the corresponding autonomous dynamical systems, derived using the linear chain trick, by means of the Routh–Hurwitz criterion. In the four investigated synaptic-drive models with identical equilibrium points we find that the choice of temporal coupling kernels significantly affects the equilibrium-point stability properties. A model with an α-function replacing the standard exponentially decaying function in the inhibitory coupling kernel is in most of our examples found to be most prone to instability, while the opposite situation with an α-function describing the excitatory kernel is found to be least prone to instability. The standard model with exponentially decaying coupling kernels is typically found to be an intermediate case. We further find that stability is promoted by increasing the weight of self-inhibition or shortening the time constant of the inhibition.     

Navigating Diagnoses: Understanding Mind–Body Relations, Mental Health, and Stigma in Nepal

Anthropologists and psychiatrists traditionally have used the salience of a mind–body dichotomy to distinguish Western from non-Western ethnopsychologies. However, despite claims of mind–body holism in non-Western cultures, mind–body divisions are prominent in non-Western groups. In this article, we discuss three issues: the ethnopsychology of mind–body dichotomies in Nepal, the relationship between mind–body dichotomies and the hierarchy of resort in a medical pluralistic context, and, finally, the role of mind–body dichotomies in public health interventions (biomedical and psychosocial) aimed toward decreasing the stigmatization of mental illness. We assert that, by understanding mind–body relations in non-Western settings, their implications, and ways in which to reconstitute these relations in a less stigmatizing manner, medical anthropologists and mental health workers can contribute to the reduction of stigma in global mental health care.

Predator-prey models with delay and prey harvesting

It is known that predator-prey systems with constant rate harvesting exhibit very rich dynamics. On the other hand, incorporating time delays into predator-prey models could induce instability and bifurcation. In this paper we are interested in studying the combined effects of the harvesting rate and the time delay on the dynamics of the generalized Gause-type predator-prey models and the Wangersky-Cunningham model. It is shown that in these models the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities, while the harvesting rate has a stabilizing effect on the equilibrium if it is under the critical harvesting level. In particular, one of these models loses stability when the delay varies and then regains its stability when the harvesting rate is increased. Computer simulations are carried to explain the mathematical conclusions. Received: 1 March 2000 / Revised version: 7 September 2000 /?Published online: 21 August 2001     

Stabilizing function of antagonistic neuromusculoskeletal systems: an analytical investigation

 Under normal conditions human walking or running consists of stable cyclic movements. Minor perturbances such as a stone or a pothole do not disrupt the cycle, and the system returns to its prescribed trajectory. We investigated whether a pair of antagonistic muscles is able to stabilize the movement without neuronal feedback. The human is represented by a model consisting of a massless two-segment linkage system (leg) topped by a point mass. Both the extensor and flexor muscles are described by a Hill-type muscle model. Conditions for stability are calculated analytically based on the Ljapunov Theory and the results are illustrated by numerical examples. The activation functions of both the extensor and flexor muscles can be calculated for a prescribed trajectory to maintain the self-stabilizing ability of such a system. Experimental evidence supports the prediction. Our investigation shows that a moving center of rotation of the kneejoint, a biarticular flexor muscle group, the force-velocity relation, and the ascending limb of the force-length relation improves the self-stabilizing ability of human movement. Received: 24 March 1999 / Accepted: 20 February 2003 / Published online: 22 May 2003 Correspondence to: H. Wagner (e-mail: heiko.wagner@uni-jena.de, Tel.:+49-3641-945706) Acknowledgements. The authors wish to express their gratitude to Arnd Friedrichs and Lars Wendrock for their contribution to data acquisition and analysis. Veit Wank is acknowledged for making available for our use the x-ray photos of several knee joints. We would like to thank Anna N. Ahn for her very detailed and helpful comments. With support from DFG (Innovationskolleg “Motion Systems”).     

Numerical Equilibrium Analysis for Structured Consumer Resource Models

In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the (two-parameter) plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability (and their derivatives) as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for “Daphnia consuming algae” models in C-code. The results obtained by way of this implementation are shown in the form of graphs.     

Stability of discrete one-dimensional population models

We give conditions for local and global stability of discrete one-dimensional population models. We give a new test for local stability when the derivative is −1. We give several sufficient conditions for global stability. We use these conditions to show that local and global stability coincide for the usual models from the literature and even for slightly more complicated models. We give population models, which are in some sense the simplest models, for which local and global stability do not coincide.     

A self-consistent cell flux expression for simultaneous chemotaxis and contact guidance in tissues

During wound healing, both chemotaxis and contact guidance can contribute to the migration of blood and tissue cells to the wound. In order to understand the wound healing process, we must thus understand how cells respond to both these simultaneous directional cues, which are not necessarily coaligned. Although chemotaxis and contact guidance have been studied individually, the interaction between them has not been addressed. We extend a stochastic cell movement model, developed by Dickinson and Tranquillo (1995) [6] for individual cues, for simultaneous chemotaxis and contact guidance by a two-parameter perturbation analysis in terms of the two associated cues, a chemotactic factor gradient and aligned tissue fibers. We present results from analysis of the first-order perturbation, which includes the cell flux expression heuristically proposed by others, but reveals paradoxical results for other indices of cell movement, such as the mean-squared displacement. We then present second-order perturbation results that resolve these paradoxical results. Finally, we relate these results to a continuum mechanical model developed by Barocas and Tranquillo (1997) [3] that predicts fiber alignment due to cell traction induced tissue contraction. Received: 30 April 1999 / Revised version: 30 October 1999 / Published online: 14 September 2000     

Allee threshold and extinction threshold for spatially explicit metapopulation dynamics with Allee effects

In this paper, we investigate a spatially explicit metapopulation model with Allee effects. We refer to the patch occupancy model introduced by Levins (Bull Entomol Soc Am 15:237–240, 1969) as a spatially implicit metapopulation model, i.e., each local patch is either occupied or vacant and a vacant patch can be recolonized by a randomly chosen occupied patch from anywhere in the metapopulation. When we transform the model into a spatially explicit one by using a lattice model, the obtained model becomes theoretically equivalent to a “lattice logistic model” or a “basic contact process”. One of the most popular or standard metapopulation models with Allee effects, developed by Amarasekare (Am Nat 152:298–302, 1998), supposes that those effects are introduced formally by means of a logistic equation. However, it is easier to understand the ecological meaning of associating Allee effects with this model if we suppose that only the logistic colonization term directly suffers from Allee effects. The resulting model is also well defined, and therefore we can naturally examine it by Monte Carlo simulation and by doublet and triplet decoupling approximation. We then obtain the following specific features of one-dimensional lattice space: (1) the metapopulation as a whole does not have an Allee threshold for initial population size even when each local population follows the Allee effects; and (2) a metapopulation goes extinct when the extinction rate of a local population is lower than that in the spatially implicit model. The real ecological metapopulation lies between two extremes: completely mixing interactions between patches on the one hand and, on the other, nearest neighboring interactions with only two nearest neighbors. Thus, it is important to identify the metapopulation structure when we consider the problems of invasion species such as establishment or the speed of expansion.     

Subtle ‘boom and bust’ response of <Emphasis Type="Italic">Macquaria ambigua</Emphasis> to flooding in an Australian dryland river

The ecology of dryland rivers is driven by their highly variable hydrology, particularly flooding regimes, whereby intermittent floods typically generate ‘booms’ of primary and secondary productivity, including massive fish production. We tested these concepts in the Moonie River, Australia, using the percichthyid, Macquaria ambigua, a dryland river species known to display pronounced ‘boom and bust’ abundance patterns in response to floodplain inundation followed by extended periods of low to no channel flow. We expected that body condition (as measured by whole body lipid content) and biomass of M. ambigua would be related to prey biomass, and that these factors would all ‘spike’ following widespread flooding. Instead we found more subtle responses. There were ‘booms’ in biomass of Macrobrachium and zooplankton, two important food items, whereas M. ambigua maintained relatively low but sustained lipid and biomass levels following flooding. It appears that instead of a ‘boom’ in fish biomass, abundant invertebrate food resources and sustained lipid levels contributed to high survivorship of this species during the ‘bust’ period over cool dry months.     

Trade-off between current reproductive effort and delay to next reproduction in the leatherback sea turtle

The trade-off between current and future reproduction plays an important role in demographic analyses. This can be revealed by the relationship between the number of years without reproduction and reproductive investment within a reproductive year. However, estimating both the duration between two successive breeding season and reproductive effort is often limited by variable recapture or resighting effort. Moreover, a supplementary difficulty is raised when nonbreeder individuals are not present sampling breeding grounds, and are therefore unobservable. We used capture–recapture (CR) models to investigate intermittent breeding and reproductive effort to test a putative physiological trade-off in a long-lived species with intermittent breeding, the leatherback sea turtle. We used CR data collected on breeding females on Awa:la-Ya:lima:po beach (French Guiana, South America) from 1995 to 2002. By adding specific constraints in multistate (MS) CR models incorporating several nonobservable states, we modelled the breeding cycle in leatherbacks and then estimated the reproductive effort according to the number of years elapsed since the last nesting season. Using this MS CR framework, the mean survival rate was estimated to 0.91 and the average resighting probability to 0.58 (ranged from 0.30 to 0.99). The breeding cycle was found to be limited to 3 years. These results therefore suggested that animals whose observed breeding intervals are greater than 3 years were most likely animals that escaped detection during their previous nesting season(s). CR data collected in 2001 and 2002 allowed us to compare the individual reproductive effort between females that skipped one breeding season and females that skipped two breeding seasons. These inferences led us to conclude that a trade-off between current and future reproduction exists in leatherbacks nesting in French Guiana, likely linked to the resource provisioning required to invest in reproduction.     

Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example

We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration). As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023–1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman–Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254–274, 1984; de Roos et al. in J Math Biol 28:609–643, 1990) and a model introduced by Gurney–Nisbet (Theor Popul Biol 28:150–180, 1985) and Jones et al. (J Math Anal Appl 135:354–368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases.     

Competitive exclusion in a vector-host model for the dengue fever

 We study a system of differential equations that models the population dynamics of an SIR vector transmitted disease with two pathogen strains. This model arose from our study of the population dynamics of dengue fever. The dengue virus presents four serotypes each induces host immunity but only certain degree of cross-immunity to heterologous serotypes. Our model has been constructed to study both the epidemiological trends of the disease and conditions that permit coexistence in competing strains. Dengue is in the Americas an epidemic disease and our model reproduces this kind of dynamics. We consider two viral strains and temporary cross-immunity. Our analysis shows the existence of an unstable endemic state (‘saddle’ point) that produces a long transient behavior where both dengue serotypes cocirculate. Conditions for asymptotic stability of equilibria are discussed supported by numerical simulations. We argue that the existence of competitive exclusion in this system is product of the interplay between the host superinfection process and frequency-dependent (vector to host) contact rates. Received 4 December 1995; received in revised form 5 March 1996