The role of curvature feedback in the energetics and dynamics of lamprey swimming: A closed-loop model.

Like other animals, lampreys have a central pattern generator (CPG) circuit that activates muscles for locomotion and also adjusts the activity to respond to sensory inputs from the environment. Such a feedback system is crucial for responding appropriately to unexpected perturbations, but it is also active during normal unperturbed steady swimming and influences the baseline swimming pattern. In this study, we investigate different functional forms of body curvature-based sensory feedback and evaluate their effects on steady swimming energetics and kinematics, since little is known experimentally about the functional form of curvature feedback. The distributed CPG is modeled as chains of coupled oscillators. Pairs of phase oscillators represent the left and right sides of segments along the lamprey body. These activate muscles that flex the body and move the lamprey through a fluid environment, which is simulated using a full Navier-Stokes model. The emergent curvature of the body then serves as an input to the CPG oscillators, closing the loop. We consider two forms of feedback, each consistent with experimental results on lamprey proprioceptive sensory receptors. The first, referred to as directional feedback, excites or inhibits the oscillators on the same side, depending on the sign of a chosen gain parameter, and has the opposite effect on oscillators on the opposite side. We find that directional feedback does not affect beat frequency, but does change the duration of muscle activity. The second feedback model, referred to as magnitude feedback, provides a symmetric excitatory or inhibitory effect to oscillators on both sides. This model tends to increase beat frequency and reduces the energetic cost to the lamprey when the gain is high and positive. With both types of feedback, the body curvature has a similar magnitude. Thus, these results indicate that the same magnitude of curvature-based feedback on the CPG with different functional forms can cause distinct differences in swimming performance.

Characterization of the encoding properties of intraspinal mechanosensory neurons in the lamprey.

Proprioceptive sensory inputs are an integral part of the closed-loop system of locomotion. In the lamprey, a model organism for vertebrate locomotion, such sensory inputs come from intraspinal mechanosensory cells called "edge cells". These edge cells synapse directly onto interneurons in the spinal central pattern generator (CPG) circuit and allow the CPG to adjust the motor output according to how the body is bending. However, the encoding properties of the edge cells have never been fully characterized. To identify these properties and better understand edge cells' role in locomotion, we isolated spinal cords of silver lampreys (Ichthyomyzon unicuspis) and recorded extracellularly from the lateral tracts where edge cell axons are located. We identified cells that responded to mechanical stimuli and used standard spike sorting algorithms to identify separate units, then examined how the cells respond to bending rate and bending angle. Although some cells respond to the bending angle, as was previously known, the strongest and most common responses were to bending velocity. These encoding properties will help us better understand how lampreys and other basal vertebrates adapt their locomotor rhythms to different water flow patterns, perturbations, or other unexpected changes in their environments.

Entrainment ranges of forced phase oscillators.

In the vertebrate spinal cord, a neural circuit called the central pattern generator produces the basic locomotory rhythm. Short and long distance intersegmental connections serve to maintain coordination along the length of the body. As a way of examining the influence of such connections, we consider a model of a chain of coupled phase oscillators in which one oscillator receives a periodic forcing stimulus. For a certain range of forcing frequencies, the chain will match the stimulus frequency, a phenomenon called entrainment. Motivated by recent experiments in lampreys, we derive analytical expressions for the range of forcing frequencies that entrain the chain, and how that range depends on the forcing location. For short intersegmental connections, in which an oscillator is connected only to its nearest neighbors, we describe two ways in which entrainment is lost: internally, in which oscillators within the chain no longer oscillate at the same frequency; and externally, in which the the chain no longer has the same frequency as the forcing. By analyzing chains in which every oscillator is connected to every other oscillator (i.e., all-to-all connections), we show that the presence of connections with lengths greater than one do not necessarily change the entrainment ranges based on the nearest-neighbor model. We derive a criterion for the ratio of connection strengths under which the connections of length greater than one do not change the entrainment ranges produced in the nearest-neighbor model, provided entrainment is lost externally. However, when this criterion holds, the range of entrained frequencies is a monotonic function of forcing location, unlike experimental results, in which entrainment ranges are larger near the middle of the chain than at the ends. Numerically, we show that similar non-monotonic entrainment ranges are possible if the ratio criterion does not hold, suggesting that in the lamprey central pattern generator, intersegmental connection strengths are not a simple function of the connection length.

Analysis of SI models with multiple interacting populations using subpopulations.

Computing endemic equilibria and basic reproductive numbers for systems of differential equations describing epidemiological systems with multiple connections between subpopulations is often algebraically intractable. We present an alternative method which deconstructs the larger system into smaller subsystems and captures the interactions between the smaller systems as external forces using an approximate model. We bound the basic reproductive numbers of the full system in terms of the basic reproductive numbers of the smaller systems and use the alternate model to provide approximations for the endemic equilibrium. In addition to creating algebraically tractable reproductive numbers and endemic equilibria, we can demonstrate the influence of the interactions between subpopulations on the basic reproductive number of the full system. The focus of this paper is to provide analytical tools to help guide public health decisions with limited intervention resources.

Methods for determining stability in continuum elastic-rod models of DNA.

Elastic-rod models of DNA have offered an alternative method for studying the macroscopic properties of the molecule. An essential component of the modelling effort is to identify the biologically accessible, or stable, solutions. The underlying variational structure of the elastic-rod model can be exploited to derive methods that identify stable equilibrium configurations. We present two methods for determining the stability of the equilibria of elastic-rod models: the conjugate-point method and the distinguished-diagram method. Additionally, we apply these methods to two intrinsically curved DNA molecules: a DNA filament with an A-tract bend and a DNA minicircle with a catabolite gene activator protein binding site. The stable solutions of these models provide visual insight into the three-dimensional structure of the DNA molecules.

Link, twist, energy, and the stability of DNA minicircles.

We describe how the stability properties of DNA minicircles can be directly read from plots of various biologically intuitive quantities along families of equilibrium configurations. Our conclusions follow from extensions of the mathematical theory of distinguished bifurcation diagrams that are applied within the specific context of an elastic rod model of minicircles. Families of equilibria arise as a twisting angle alpha is varied. This angle is intimately related to the continuously varying linking number Lk for nicked DNA configurations that is defined as the sum of Twist and Writhe. We present several examples of such distinguished bifurcation diagrams involving plots of the energy E, linking number Lk, and a twist moment m3, along families of cyclized equilibria of both intrinsically straight and intrinsically curved DNA fragments.