A comparison of resonance tuning with positive versus negative sensory feedback

The effect of sensory feedback topology on the entrainment of a neural oscillator with a compliant foil for swimming systems

The sensorimotor system of fish endows them with remarkable swimming performance that is unmatched by current underwater robotic vehicles. To close the gap between the capabilities of fish and the capabilities of underwater vehicles engineers are investigating how fish swim. In particular, engineers are exploring the sensorimotor systems of fish that control the motion of fins. It is generally accepted that specialized neural circuits (known as central pattern generators) within the sensorimotor system produce the periodic drive signal that is used to control the motion of fins. An important aspect of these circuits is that their output signal can be modified by sensory feedback. Specifically, the way in which sensory feedback signals are applied to a CPG (i.e. the sensory feedback topology) affects the CPG's entrainment characteristics. This has been shown in simulation but has not been investigated in a robot interacting in the real-world. Furthermore, CPG-based control has only limitedly been applied to fish like robots and many questions remain as to how it should be applied to these types of systems. In this work we examine the effect of sensory feedback topology on the entrainment characteristics of a CPG-based neural oscillator driving three different foils swimming in flow. Additionally, we investigate how sensory feedback should be acquired from a foil and applied to a neural oscillator to promote beneficial swimming characteristics.

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.

When animals move, they receive sensory inputs, which in turn are used to modulate the movement. Relatively little is known about how these inputs affect performance during steady locomotion. Using a computational model of a swimming lamprey, we investigated two different types of feedback, both consistent with experimental data. Both have strong, but different, effects on swimming speed and energy consumption, suggesting that sensory feedback is crucial not just for responding to perturbations, but also for high performance steady locomotion.

The role of mechanical resonance in the neural control of swimming in fishes

The bodies of many fishes are flexible, elastic structures; if you bend them, they spring back. Therefore, they should have a resonant frequency: a bending frequency at which the output amplitude is maximized for a particular input. Previous groups have hypothesized that swimming at this resonant frequency could maximize efficiency, and that a neural circuit called the central pattern generator might be able to entrain to a mechanical resonance. However, fishes swim in water, which may potentially damp out many resonant effects. Additionally, their bodies are elongated, which means that bending can occur in complicated ways along the length of the body. We review previous studies of the mechanical properties of fish bodies, and then present new data that demonstrate complex bending properties of elongated fish bodies. Resonant peaks in amplitude exist, but there may be many of them depending on the body wavelength. Additionally, they may not correspond to the maximum swimming speed. Next, we describe experiments using a closed-loop preparation of the lamprey, in which a preparation of the spinal cord is linked to a real-time simulation of the muscle and body properties, allowing us to examine resonance entrainment as we vary the simulated resonant frequency. We find that resonance entrainment does occur, but is rare. Gain had a significant, though weak, effect, and a nonlinear muscle model produced resonance entrainment more often than a linear filter. We speculate that resonance may not be a critical effect for efficient swimming in elongate, anguilliform swimmers, though it may be more important for stiffer carangiform and thunniform fishes.

Using computational and mechanical models to study animal locomotion

Recent advances in computational methods have made realistic large-scale simulations of animal locomotion possible. This has resulted in numerous mathematical and computational studies of animal movement through fluids and over substrates with the purpose of better understanding organisms' performance and improving the design of vehicles moving through air and water and on land. This work has also motivated the development of improved numerical methods and modeling techniques for animal locomotion that is characterized by the interactions of fluids, substrates, and structures. Despite the large body of recent work in this area, the application of mathematical and numerical methods to improve our understanding of organisms in the context of their environment and physiology has remained relatively unexplored. Nature has evolved a wide variety of fascinating mechanisms of locomotion that exploit the properties of complex materials and fluids, but only recently are the mathematical, computational, and robotic tools available to rigorously compare the relative advantages and disadvantages of different methods of locomotion in variable environments. Similarly, advances in computational physiology have only recently allowed investigators to explore how changes at the molecular, cellular, and tissue levels might lead to changes in performance at the organismal level. In this article, we highlight recent examples of how computational, mathematical, and experimental tools can be combined to ultimately answer the questions posed in one of the grand challenges in organismal biology: "Integrating living and physical systems."

A dynamical systems analysis of afferent control in a neuromechanical model of locomotion: I. Rhythm generation

Locomotion in mammals is controlled by a spinal central pattern generator (CPG) coupled to a biomechanical limb system, with afferent feedback to the spinal circuits and CPG closing the control loop. We have considered a simplified model of this system, in which the CPG establishes a rhythm when a supra-spinal activating drive is present and afferent signals from a single-joint limb feed back to affect CPG operation. Using dynamical system methods, in a series of two papers we analyze the mechanisms by which this model produces oscillations, and the characteristics of these oscillations, in the closed- and open-loop regimes. In this first paper, we analyze the phase transition mechanisms operating within the CPG and use the results to explain how afferent feedback allows oscillations to occur at a wider range of drive values to the CPG than the range over which oscillations occur in the CPG without feedback, and then to comment on why stronger feedback leads to faster oscillations. Linking these transitions to structures in the phase plane associated with the limb segment clarifies how increased weights of afferent feedback to the CPG can restore locomotion after removal of supra-spinal drive to simulate spinal cord injury.

A dynamical systems analysis of afferent control in a neuromechanical model of locomotion: II. Phase asymmetry

In this paper we analyze a closed loop neuromechanical model of locomotor rhythm generation. The model is composed of a spinal central pattern generator (CPG) and a single-joint limb, with CPG outputs projecting via motoneurons to muscles that control the limb and afferent signals from the muscles feeding back to the CPG. In a preceding companion paper (Spardy et al 2011 J. Neural Eng. 8 065003), we analyzed how the model generates oscillations in the presence or absence of feedback, identified curves in a phase plane associated with the limb that signify where feedback levels induce phase transitions within the CPG, and explained how increasing feedback strength restores oscillations in a model representation of spinal cord injury; from these steps, we derived insights about features of locomotor rhythms in several scenarios and made predictions about rhythm responses to various perturbations. In this paper, we exploit our analytical observations to construct a reduced model that retains important characteristics from the original system. We prove the existence of an oscillatory solution to the reduced model using a novel version of a Melnikov function, adapted for discontinuous systems, and also comment on the uniqueness and stability of this solution. Our analysis yields a deeper understanding of how the model must be tuned to generate oscillations and how the details of the limb dynamics shape overall model behavior. In particular, we explain how, due to the feedback signals in the model, changes in the strength of a tonic supra-spinal drive to the CPG yield asymmetric alterations in the durations of different locomotor phases, despite symmetry within the CPG itself.

Spikes alone do not behavior make: why neuroscience needs biomechanics

Neural circuits do not function in isolation; they interact with the physical world, accepting sensory inputs and producing outputs via muscles. Since both these pathways are constrained by physics, the activity of neural circuits can only be understood by considering biomechanics of muscles, bodies, and the exterior world. We discuss how animal bodies have natural stable motions that require relatively little activation or control from the nervous system. The nervous system can substantially alter these motions, by subtly changing mechanical properties such as body or leg stiffness. Mechanics can also provide robustness to perturbations without sensory reflexes. By considering a complete neuromechanical system, neuroscientists and biomechanicians together can provide a more integrated view of neural circuitry and behavior.

Smooth changes in the EMG patterns during gait transitions under body weight unloading

During gradual speed changes, humans exhibit a sudden discontinuous switch from walking to running at a specific speed, and it has been suggested that different gaits may be associated with different functioning of neuronal networks. In this study we recorded the EMG activity of leg muscles at slow increments and decrements in treadmill belt speed and at different levels of body weight unloading. In contrast to normal walking at 1 g, at lower levels of simulated gravity (<0.4 g) the transition between walking and running was generally gradual, without systematic abrupt changes in either intensity or timing of EMG patterns. This phenomenon depended to a limited extent on the gravity simulation technique, although the exact level of the appearance of smooth transitions (0.4-0.6 g) tended to be lower for the vertical than for the tilted body weight support system. Furthermore, simulations performed with a half-center oscillator neuromechanical model showed that the abruptness of motor patterns at gait transitions at 1 g could be predicted from the distinct parameters anchored already in the normal range of walking and running speeds, whereas at low gravity levels the parameters of the model were similar for the two human gaits. A lack of discontinuous changes in the pattern of speed-dependent locomotor characteristics in a hypogravity environment is consistent with the idea of a continuous shift in the state of a given set of central pattern generators, rather than the activation of a separate set of central pattern generators for each distinct gait.

Computer simulations of neural mechanisms explaining upper and lower limb excitatory neural coupling

Background

When humans perform rhythmic upper and lower limb locomotor-like movements, there is an excitatory effect of upper limb exertion on lower limb muscle recruitment. To investigate potential neural mechanisms for this behavioral observation, we developed computer simulations modeling interlimb neural pathways among central pattern generators. We hypothesized that enhancement of muscle recruitment from interlimb spinal mechanisms was not sufficient to explain muscle enhancement levels observed in experimental data.

Methods

We used Matsuoka oscillators for the central pattern generators (CPG) and determined parameters that enhanced amplitudes of rhythmic steady state bursts. Potential mechanisms for output enhancement were excitatory and inhibitory sensory feedback gains, excitatory and inhibitory interlimb coupling gains, and coupling geometry. We first simulated the simplest case, a single CPG, and then expanded the model to have two CPGs and lastly four CPGs. In the two and four CPG models, the lower limb CPGs did not receive supraspinal input such that the only mechanisms available for enhancing output were interlimb coupling gains and sensory feedback gains.

Results

In a two-CPG model with inhibitory sensory feedback gains, only excitatory gains of ipsilateral flexor-extensor/extensor-flexor coupling produced reciprocal upper-lower limb bursts and enhanced output up to 26%. In a two-CPG model with excitatory sensory feedback gains, excitatory gains of contralateral flexor-flexor/extensor-extensor coupling produced reciprocal upper-lower limb bursts and enhanced output up to 100%. However, within a given excitatory sensory feedback gain, enhancement due to excitatory interlimb gains could only reach levels up to 20%. Interconnecting four CPGs to have ipsilateral flexor-extensor/extensor-flexor coupling, contralateral flexor-flexor/extensor-extensor coupling, and bilateral flexor-extensor/extensor-flexor coupling could enhance motor output up to 32%. Enhancement observed in experimental data exceeded 32%. Enhancement within this symmetrical four-CPG neural architecture was more sensitive to relatively small interlimb coupling gains. Excitatory sensory feedback gains could produce greater output amplitudes, but larger gains were required for entrainment compared to inhibitory sensory feedback gains.

Conclusions

Based on these simulations, symmetrical interlimb coupling can account for much, but not all of the excitatory neural coupling between upper and lower limbs during rhythmic locomotor-like movements.

A computational model for rhythmic and discrete movements in uni- and bimanual coordination

Current research on discrete and rhythmic movements differs in both experimental procedures and theory, despite the ubiquitous overlap between discrete and rhythmic components in everyday behaviors. Models of rhythmic movements usually use oscillatory systems mimicking central pattern generators (CPGs). In contrast, models of discrete movements often employ optimization principles, thereby reflecting the higher-level cortical resources involved in the generation of such movements. This letter proposes a unified model for the generation of both rhythmic and discrete movements. We show that a physiologically motivated model of a CPG can not only generate simple rhythmic movements with only a small set of parameters, but can also produce discrete movements if the CPG is fed with an exponentially decaying phasic input. We further show that a particular coupling between two of these units can reproduce main findings on in-phase and antiphase stability. Finally, we propose an integrated model of combined rhythmic and discrete movements for the two hands. These movement classes are sequentially addressed in this letter with increasing model complexity. The model variations are discussed in relation to the degree of recruitment of the higher-level cortical resources, necessary for such movements.

Altered Gravity Highlights Central Pattern Generator Mechanisms

In many nonprimate species, rhythmic patterns of activity such as locomotion or respiration are generated by neural networks at the spinal level. These neural networks are called central pattern generators (CPGs). Under normal gravitational conditions, the energy efficiency and the robustness of human rhythmic movements are due to the ability of CPGs to drive the system at a pace close to its resonant frequency. This property can be compared with oscillators running at resonant frequency, for which the energy is optimally exchanged with the environment. However, the ability of the CPG to adapt the frequency of rhythmic movements to new gravitational conditions has never been studied. We show here that the frequency of a rhythmic movement of the upper limb is systematically influenced by the different gravitational conditions created in parabolic flight. The period of the arm movement is shortened with increasing gravity levels. In weightlessness, however, the period is more dependent on instructions given to the participants, suggesting a decreased influence of resonant frequency. Our results are in agreement with a computational model of a CPG coupled to a simple pendulum under the control of gravity. We demonstrate that the innate modulation of rhythmic movements by CPGs is highly flexible across gravitational contexts. This further supports the involvement of CPG mechanisms in the achievement of efficient rhythmic arm movements. Our contribution is of major interest for the study of human rhythmic activities, both in a normal Earth environment and during microgravity conditions in space.

Rostral Versus Caudal Differences in Mechanical Entrainment of the Lamprey Central Pattern Generator for Locomotion

In fishes, undulatory swimming is produced by sets of spinal interneurons constituting a central pattern generator (CPG). The CPG generates waves of muscle activity that travel from head to tail, which then bend the body into wave shapes that also travel from head to tail. In many fishes, the wavelengths of the neural and mechanical waves are different, resulting in a rostral-to-caudal gradient in phase lag between muscle activity and bending. The neural basis of this phase gradient was investigated in the lamprey spinal cord using an isolated in vitro preparation. Fictive swimming was induced using d-glutamate and the output of the CPG was measured using suction electrodes placed on the ventral roots. The spinal cord was bent sinusoidally at various points along its length. First, the ranges of entrainment were estimated. Middle segments were able to entrain to frequencies approximately twice as high as those at end segments. Next, phase lags between centers of ventral root bursts and the stimulus were determined. Two halves of the cycle were identified: stretching and shortening of the edge of spinal cord on the same side as the electrode. Stimuli at rostral segments tended to entrain segmental bursting at the beginning of the stretch phase, almost 50% out of phase with previously measured in vivo electromyography data. Stimuli at caudal segments, in contrast, entrained segments at the end of stretch and the beginning of shortening, approximately the same phase as in vivo data.

Formal analysis of resonance entrainment by central pattern generator

The neuronal circuit controlling the rhythmic movements in animal locomotion is called the central pattern generator (CPG). The biological control mechanism appears to exploit mechanical resonance to achieve efficient locomotion. The objective of this paper is to reveal the fundamental mechanism underlying entrainment of CPGs to resonance through sensory feedback. To uncover the essential principle, we consider the simplest setting where a pendulum is driven by the reciprocal inhibition oscillator. Existence and properties of stable oscillations are examined by the harmonic balance method, which enables approximate but insightful analysis. In particular, analytical conditions are obtained under which harmonic balance predicts existence of an oscillation at a frequency near the resonance frequency. Our result reveals that the resonance entrainment can be maintained robustly against parameter perturbations through two distinct mechanisms: negative integral feedback and positive rate feedback.