Abstract. The spring-loaded inverted pendulum (SLIP), or monopedal hopper, is an archetypal model for running in numerous animal species. Although locomotion is generally considered a complex task requiring sophisticated control strategies to account for coordination and stability, we show that stable gaits can be found in the SLIP with both linear and "air" springs, controlled by a simple fixed-leg reset policy. We first derive touchdown-to-touchdown Poincaré maps under the common assumption of negligible gravitational effects during the stance phase. We subsequently include and assess these effects and briefly consider coupling to pitching motions. We investigate the domains of attraction of symmetric periodic gaits and bifurcations from the branches of stable gaits in terms of nondimensional parameters.Key words. legged locomotion, spring-loaded inverted pendulum, periodic gaits, bifurcation, stability AMS subject classifications. 34C23, 37J20, 37J25, 37J60, 70Hxx, 70K42, 70K50PII. S11111111024083111. Introduction. Locomotion, "moving the body's locus," is among the most fundamental of animal behaviors. A large motor science literature addresses gait pattern selection [1], energy expenditure [2], underlying neurophysiology [3], and coordination in animals and machines [4]. In this paper, we explore the stabilizing effect of a very simple control policy on a very simple running model.Legged locomotion is generally considered a complex task [5] involving the coordination of many limbs and redundant degrees of freedom [6]. In [7], Full and Koditschek note that "locomotion results from complex, high-dimensional, non-linear, dynamically coupled interactions between an organism and its environment." They distinguish locomotion models simplified for the purpose of task specification (templates) from more kinematically and dynamically accurate representations of the true body morphology (anchors). A template is a formal reductive model that (1) encodes parsimoniously the dynamics of the body and its payload transport capability, using the minimum number of variables and parameters, and (2) advances an intrinsic hypothesis concerning the control strategy underlying the achievement of this task. Anchors are not only more elaborate dynamical systems grounded in the morphology and physiology

SUMMARYTreatment of a pathogenic disease process is interpreted as the optimal control of a dynamic system. Evolution of the disease is characterized by a non-linear, fourth-order ordinary differential equation that describes concentrations of pathogens, plasma cells, and antibodies, as well as a numerical indication of patient health. Without control, the dynamic model evidences sub-clinical or clinical decay, chronic stabilization, or unrestrained lethal growth of the pathogen, depending on the initial conditions for the infection. The dynamic equations are controlled by therapeutic agents that affect the rate of change of system variables. Control histories that minimize a quadratic cost function are generated by numerical optimization over a fixed time interval, given otherwise lethal initial conditions. Tradeoffs between cost function weighting of pathogens, organ health, and use of therapeutics are evaluated. Optimal control solutions that defeat the pathogen and preserve organ health are demonstrated for four different approaches to therapy. It is shown that control theory can point the way toward new protocols for treatment and remediation of human diseases.

Abstract. We adapt the generic three-dimensional bursting neuron model derived in the companion paper [SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 636-670] to model central pattern generator interneurons and slow and fast motoneurons in insect locomotory systems. Focusing on cockroach data, we construct a coupled network that retains sufficient detail to allow investigation and prediction of biophysical parameter changes. We show that the model can encompass stepping frequency, duty cycle, and motoneuron output variations observed in cockroaches, and we reduce it to an analytically tractable symmetric network of coupled phase oscillators from which general principles can be extracted. The model's modular form allows dynamical analyses of individual components and the addition of other components, so we expect it to be more generally useful. 1. Introduction. Central pattern generators (CPGs) are networks of functionally distinguishable neurons, located in the vertebrate spinal cord or in invertebrate thoracic ganglia, capable of generating and regulating the spatio-temporal activity of motoneurons in the absence of sensory input (e.g., [2,3,4]). Over forty years of in vitro and in vivo studies of network architectures, intrinsic membrane properties, and neuromodulators (e.g., [5,6,7,4,8]) have firmly established their importance in motor behavior. CPG dynamics depends on intracellular, synaptic, and network level phenomena and can display remarkable richness and flexibility.In this paper, using the reduced bursting neuron ODEs derived and studied in the preceding paper [1], we develop a model of the CPG and associated bursting motoneurons for insect locomotion. We draw on data from the death's head and American cockroaches Blaberus discoidalis and Periplaneta americana and focus on rapid running, a regime in which preflexive feedforward control [9, 10] appears to dominate and reflexive feedback plays a less important role [11,12,13] than in, e.g., stick insects [14] that use more varied gaits and leg placement strategies. We include enough ionic current and conductance detail to reveal how modulation

Abstract. After reviewing the Hodgkin-Huxley ionic current formulation, we introduce a three-variable generic model of a single-compartment neuron comprising a two-dimensional fast subsystem and a very slow recovery variable. We study the effects of fast and slow currents on the existence and stability of equilibria and periodic orbits for the fast subsystem, presenting a classification of currents and developing graphical tools that aid in the analysis and construction of models with specified properties. We draw on these to propose a minimal model of a bursting neuron, identifying biophysical parameters that can shape and regulate key characteristics of the membrane voltage pattern: bursting frequency, duty cycle, spike rate, and the number of action potentials per burst. We present additional examples from the literature for comparison and illustration, and in a companion paper [SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 671-700], we construct a model of an insect central pattern generator using these methods.Key words. bursting neurons, motoneurons, fast-slow systems, bifurcation, stability AMS subject classifications. 34C15, 34C25, 34C29, 37Gxx, 92B05, 92B20, 92C20DOI. 10.1137/030602307 Introduction. In this and a companion paper [1]we develop and analyze a generic model of a bursting neuron and assemble a set of such models, suitably adapted to interneurons and motoneurons, to model a central pattern generator (CPG) for insect locomotion. We have two main goals: to integrate and extend a body of work, largely in theoretical and mathematical neuroscience, that enables (semi-) analytical studies of bursting neurons, while maintaining sufficient biophysical detail for comparisons with experimental data; and to use this to derive a model of a CPG that reveals how key locomotive properties may be determined by individual neurons and the network as a whole. In this first paper we show how complex models can be reduced and develop the analytical methods; in [1] we construct the CPG model.The first dynamical neural model based on biophysical data was due to Hodgkin and Huxley [2], and their description of the action potential (AP) and ionic currents in the giant axon of the squid has been vastly extended and generalized in the half century since. Detailed axonal and dendritic geometry can be included, for example, at the unicellular level. However,

stengel@princeton.edu; rghiglia@princeton.edu; nkulkarn@princeton.edu