We have developed several innovative designs for a new kind of robot that uses a continuous wave of peristalsis for locomotion, the same method that earthworms use, and report on the first completed prototypes. This form of locomotion is particularly effective in constrained spaces, and although the motion has been understood for some time, it has rarely been effectively or accurately implemented in a robotic platform. As an alternative to robots with long segments, we present a technique using a braided mesh exterior to produce smooth waves of motion along the body of a worm-like robot. We also present a new analytical model of this motion and compare predicted robot velocity to a 2D simulation and a working prototype. Because constant-velocity peristaltic waves form due to accelerating and decelerating segments, it has been often assumed that this motion requires strong anisotropic ground friction. However, our analysis shows that with smooth, constant velocity waves, the forces that cause accelerations within the body sum to zero. Instead, transition timing between aerial and ground phases plays a critical role in the amount of slippage, and the final robot speed. The concept is highly scalable, and we present methods of construction at two different scales.

Novel clinical treatments to target peripheral nerves are being developed which primarily use electrical current. Recently, infrared (IR) light was shown to inhibit peripheral nerves with high spatial and temporal specificity. Here, for the first time, we demonstrate that IR can selectively and reversibly inhibit small-diameter axons at lower radiant exposures than large-diameter axons. We provide a mathematical rationale, and then demonstrate it experimentally in individual axons of identified neurons in the marine mollusk Aplysia californica, and in axons within the vagus nerve of a mammal, the musk shrew Suncus murinus. The ability to selectively, rapidly, and reversibly control small-diameter sensory fibers may have many applications, both for the analysis of physiology, and for treating diseases of the peripheral nervous system, such as chronic nausea, vomiting, pain, and hypertension. Moreover, the mathematical analysis of how IR affects the nerve could apply to other techniques for controlling peripheral nerve signaling.

Motor systems must adapt to perturbations and changing conditions both within and outside the body. We refer to the ability of a system to maintain performance despite perturbations as “robustness,” and the ability of a system to deploy alternative strategies that improve fitness as “flexibility.” Different classes of pattern-generating circuits yield dynamics with differential sensitivities to perturbations and parameter variation. Depending on the task and the type of perturbation, high sensitivity can either facilitate or hinder robustness and flexibility. Here we explore the role of multiple coexisting oscillatory modes and sensory feedback in allowing multiphasic motor pattern generation to be both robust and flexible. As a concrete example, we focus on a nominal neuromechanical model of triphasic motor patterns in the feeding apparatus of the marine mollusk Aplysia californica. We find that the model can operate within two distinct oscillatory modes and that the system exhibits bistability between the two. In the “heteroclinic mode,” higher sensitivity makes the system more robust to changing mechanical loads, but less robust to internal parameter variations. In the “limit cycle mode,” lower sensitivity makes the system more robust to changes in internal parameter values, but less robust to changes in mechanical load. Finally, we show that overall performance on a variable feeding task is improved when the system can flexibly transition between oscillatory modes in response to the changing demands of the task. Thus, our results suggest that the interplay of sensory feedback and multiple oscillatory modes can allow motor systems to be both robust and flexible in a variable environment.

In this work, we present a dynamic simulation of an earthworm-like robot moving in a pipe with radially symmetric Coulomb friction contact. Under these conditions, peristaltic locomotion is efficient if slip is minimized. We characterize ways to reduce slip-related losses in a constant-radius pipe. Using these principles, we can design controllers that can navigate pipes even with a narrowing in radius. We propose a stable heteroclinic channel controller that takes advantage of contact force feedback on each segment. In an example narrowing pipe, this controller loses 40% less energy to slip compared to the best-fit sine wave controller. The peristaltic locomotion with feedback also has greater speed and more consistent forward progress

Selective control of individual neurons could clarify neural functions and aid disease treatments. To target specific neurons, it may be useful to focus on ganglionic neuron clusters, which are found in the peripheral nervous system in vertebrates. Because neuron cell bodies are found primarily near the surface of invertebrate ganglia, and often found near the surface of vertebrate ganglia, we developed a technique for controlling individual neurons extracellularly using the buccal ganglia of the marine mollusc Aplysia californica as a model system. We experimentally demonstrated that anodic currents can selectively activate an individual neuron and cathodic currents can selectively inhibit an individual neuron using this technique. To define spatial specificity, we studied the minimum currents required for stimulation, and to define temporal specificity, we controlled firing frequencies up to 45 Hz. To understand the mechanisms of spatial and temporal specificity, we created models using the NEURON software package. To broadly predict the spatial specificity of arbitrary neurons in any ganglion sharing similar geometry, we created a steady-state analytical model. A NEURON model based on cat spinal motorneurons showed responses to extracellular stimulation qualitatively similar to those of the Aplysia NEURON model, suggesting that this technique could be widely applicable to vertebrate and human peripheral ganglia having similar geometry.

Many behaviors require reliably generating sequences of motor activity while adapting the activity to incoming sensory information. This process has often been conceptually explained as either fully dependent on sensory input (a chain reflex) or fully independent of sensory input (an idealized central pattern generator, or CPG), although the consensus of the field is that most neural pattern generators lie somewhere between these two extremes. Many mathematical models of neural pattern generators use limit cycles to generate the sequence of behaviors, but other models, such as a heteroclinic channel (an attracting chain of saddle points), have been suggested. To explore the range of intermediate behaviors between CPGs and chain reflexes, in this paper we describe a nominal model of swallowing in Aplysia californica. Depending upon the value of a single parameter, the model can transition from a generic limit cycle regime to a heteroclinic regime (where the trajectory slows as it passes near saddle points). We then study the behavior of the system in these two regimes and compare the behavior of the models with behavior recorded in the animal in vivo and in vitro. We show that while both pattern generators can generate similar behavior, the stable heteroclinic channel can better respond to changes in sensory input induced by load, and that the response matches the changes seen when a load is added in vivo. We then show that the underlying stable heteroclinic channel architecture exhibits dramatic slowing of activity when sensory and endogenous input is reduced, and show that similar slowing with removal of proprioception is seen in vitro. Finally, we show that the distributions of burst lengths seen in vivo are better matched by the distribution expected from a system operating in the heteroclinic regime than that expected from a generic limit cycle. These observations suggest that generic limit cycle models may fail to capture key aspects of Aplysia feeding behavior, and that alternative architectures such as heteroclinic channels may provide better descriptions.

We describe the development of a course to teach modeling and mathematical analysis skills to students of biology and to teach biology to students with strong backgrounds in mathematics, physics, or engineering. The two groups of students have different ways of learning material and often have strong negative feelings toward the area of knowledge that they find difficult. To give students a sense of mastery in each area, several complementary approaches are used in the course: 1) a “live” textbook that allows students to explore models and mathematical processes interactively; 2) benchmark problems providing key skills on which students make continuous progress; 3) assignment of students to teams of two throughout the semester; 4) regular one-on-one interactions with instructors throughout the semester; and 5) a term project in which students reconstruct, analyze, extend, and then write in detail about a recently published biological model. Based on student evaluations and comments, an attitude survey, and the quality of the students' term papers, the course has significantly increased the ability and willingness of biology students to use mathematical concepts and modeling tools to understand biological systems, and it has significantly enhanced engineering students' appreciation of biology.

The asymptotic phase θ of an initial point x in the stable manifold of a limit cycle (LC) identifies the phase of the point on the LC to which the flow φt(x) converges as t → ∞. The infinitesimal phase response curve (iPRC) quantifies the change in timing due to a small perturbation of a LC trajectory. For a stable LC in a smooth dynamical system, the iPRC is the gradient ∇x(θ) of the phase function, which can be obtained via the adjoint of the variational equation. For systems with discontinuous dynamics, the standard approach to obtaining the iPRC fails. We derive a formula for the iPRCs of LCs occurring in piecewise smooth (Filippov) dynamical systems of arbitrary dimension, subject to a transverse flow condition. Discontinuous jumps in the iPRC can occur at the boundaries separating subdomains, and are captured by a linear matching condition. The matching matrix, M, can be derived from the saltation matrix arising in the associated variational problem. For the special case of linear dynamics away from switching boundaries, we obtain an explicit expression for the iPRC. We present examples from cell biology (Glass networks) and neuroscience (central pattern generator models). We apply the iPRCs obtained to study synchronization and phase-locking in piecewise smooth LC systems in which synchronization arises solely due to the crossing of switching manifolds.