Evaluation of the Phase-Dependent Rhythm Control of Human Walking Using Phase Response Curves

Bayesian estimation of trunk-leg coordination during walking using phase oscillator models

In human walking, the legs and other body parts coordinate to produce a rhythm with appropriate phase relationships. From the point of view for rehabilitating gait disorders, such as Parkinson Disorders, it is important to understand the control mechanism of the gait rhythm. A previous study showed that the antiphase relationship of the two legs during walking is not strictly controlled using the reduction of the motion of the legs during walking to coupled phase oscillators. However, the control mechanisms other than those of the legs remains unknown. In particular, the trunk moves in tandem with the legs and must play an important role in stabilizing walking because it is above the legs and accounts for more than half of the mass of the human body. This study aims to uncover the control mechanism of the leg-trunk coordination in the sagittal plane using the coupled phase oscillators model and Bayesian estimation. We demonstrate that the leg-trunk coordination is not strictly controlled, as well as the interleg coordination.

Interlimb coordination is not strictly controlled during walking

In human walking, the left and right legs move alternately, half a stride out of phase with each other. Although various parameters, such as stride frequency and length, vary with walking speed, the antiphase relationship remains unchanged. In contrast, during walking in left-right asymmetric situations, the relative phase shifts from the antiphase condition to compensate for the asymmetry. Interlimb coordination is important for adaptive walking and we expect that interlimb coordination is strictly controlled during walking. However, the control mechanism remains unclear. In the present study, we derived a quantity that models the control of interlimb coordination during walking using two coupled oscillators based on the phase reduction theory and Bayesian inference method. The results were not what we expected. Specifically, we found that the relative phase is not actively controlled until the deviation from the antiphase condition exceeds a certain threshold. In other words, the control of interlimb coordination has a dead zone like that in the case of the steering wheel of an automobile. It is conjectured that such forgoing of control enhances energy efficiency and maneuverability. Our discovery of the dead zone in the control of interlimb coordination provides useful insight for understanding gait control in humans.

Phase autoencoder for limit-cycle oscillators

We present a phase autoencoder that encodes the asymptotic phase of a limit-cycle oscillator, a fundamental quantity characterizing its synchronization dynamics. This autoencoder is trained in such a way that its latent variables directly represent the asymptotic phase of the oscillator. The trained autoencoder can perform two functions without relying on the mathematical model of the oscillator: first, it can evaluate the asymptotic phase and the phase sensitivity function of the oscillator; second, it can reconstruct the oscillator state on the limit cycle in the original space from the phase value as an input. Using several examples of limit-cycle oscillators, we demonstrate that the asymptotic phase and the phase sensitivity function can be estimated only from time-series data by the trained autoencoder. We also present a simple method for globally synchronizing two oscillators as an application of the trained autoencoder.

Excitatory and inhibitory interactions affect the balance of chorus activity and energy efficiency in the aggregations of male frogs: Numerical simulations using a hybrid dynamical model

We numerically study the role of excitatory and inhibitory interactions in the aggregations of male frogs. In most frogs, males produce sounds to attract conspecific females, which activates the calling behavior of other males and results in collective choruses. While the calling behavior is effective for mate attraction, it requires high energy consumption. In contrast, satellite behavior is an alternative mating strategy in which males deliberately stay silent in the vicinity of a calling male and attempt to intercept the female attracted to the caller, allowing the satellite males to reduce their energy consumption while having a chance of mating. Here we propose a hybrid dynamical model in which male frogs autonomously switch among three behavioral states (i.e., calling state, resting state, and satellite state) due to the excitatory and inhibitory interactions. Numerical simulations of the proposed model demonstrated that (1) both collective choruses and satellite behavior can be reproduced and (2) the satellite males can prolong the energy depletion time of the whole aggregation while they split the maximum chorus activity into two levels over the whole chorusing period. This study highlights the importance of the multiple behavioral types and their transitions for the performance of the whole aggregation.

Estimating asymptotic phase and amplitude functions of limit-cycle oscillators from time series data

We propose a method for estimating the asymptotic phase and amplitude functions of limit-cycle oscillators using observed time series data without prior knowledge of their dynamical equations. The estimation is performed by polynomial regression and can be solved as a convex optimization problem. The validity of the proposed method is numerically illustrated by using two-dimensional limit-cycle oscillators as examples. As an application, we demonstrate data-driven fast entrainment with amplitude suppression using the optimal periodic input derived from the estimated phase and amplitude functions.

Contribution of Phase Resetting to Statistical Persistence in Stride Intervals: A Modeling Study

Stride intervals in human walking fluctuate from one stride to the next, exhibiting statistical persistence. This statistical property is changed by aging, neural disorders, and experimental interventions. It has been hypothesized that the central nervous system is responsible for the statistical persistence. Human walking is a complex phenomenon generated through the dynamic interactions between the central nervous system and the biomechanical system. It has also been hypothesized that the statistical persistence emerges through the dynamic interactions during walking. In particular, a previous study integrated a biomechanical model composed of seven rigid links with a central pattern generator (CPG) model, which incorporated a phase resetting mechanism as sensory feedback as well as feedforward, trajectory tracking, and intermittent feedback controllers, and suggested that phase resetting contributes to the statistical persistence in stride intervals. However, the essential mechanisms remain largely unclear due to the complexity of the neuromechanical model. In this study, we reproduced the statistical persistence in stride intervals using a simplified neuromechanical model composed of a simple compass-type biomechanical model and a simple CPG model that incorporates only phase resetting and a feedforward controller. A lack of phase resetting induced a loss of statistical persistence, as observed for aging, neural disorders, and experimental interventions. These mechanisms were clarified based on the phase response characteristics of our model. These findings provide useful insight into the mechanisms responsible for the statistical persistence of stride intervals in human walking.

Exercise Promotes Neurite Extensions from Grafted Dopaminergic Neurons in the Direction of the Dorsolateral Striatum in Parkinson's Disease Model Rats

Background:

Cell transplantation is expected to be a promising treatment for Parkinson’s disease (PD), in which re-innervation of the host striatum by grafted dopamine (DA) neurons is essential. In particular, the dorsolateral part of the striatum is important because it is the target of midbrain A9 DA neurons, which are degenerated in PD pathology. The effect of exercise on the survival and maturation of grafted neurons has been reported in several neurological disease models, but never in PD models.

Objective:

We investigated how exercise influences cell transplantation for PD, especially from the viewpoint of cell survival and neurite extensions.

Methods:

Ventral mesencephalic neurons from embryonic (E12.5) rats were transplanted into the striatum of adult 6-OHDA-lesioned rats. The host rats then underwent treadmill training as exercise after the transplantation. Six weeks after the transplantation, they were sacrificed, and the grafts in the striatum were analyzed.

Results:

The addition of exercise post-transplantation significantly increased the number of surviving DA neurons. Moreover, it promoted neurite extensions from the graft toward the dorsolateral part of the striatum.

Conclusions:

This study indicates a beneficial effect of exercise after cell transplantation in PD.

Interaction mechanisms quantified from dynamical features of frog choruses

We employ a mathematical model (a phase oscillator model) to describe the deterministic and stochastic features of frog choruses in which male frogs attempt to avoid call overlaps. The mathematical model with a general interaction term is identified using a Bayesian approach, and it qualitatively reproduces the stationary and dynamical features of the empirical data. In addition, we quantify the magnitude of attention paid among the male frogs from the identified model, and then analyse the relationship between attention and behavioural parameters using a statistical approach. Our analysis demonstrates a negative correlation between attention and inter-frog distance, and also suggests a behavioural strategy in which male frogs selectively attend to a less attractive male frog (i.e. a male producing calls at longer intervals) in order to more effectively advertise their superior relative attractiveness to females.

Contribution of Phase Resetting to Adaptive Rhythm Control in Human Walking Based on the Phase Response Curves of a Neuromusculoskeletal Model

Humans walk adaptively in varying environments by manipulating their complicated and redundant musculoskeletal system. Although the central pattern generators in the spinal cord are largely responsible for adaptive walking through sensory-motor coordination, it remains unclear what neural mechanisms determine walking adaptability. It has been reported that locomotor rhythm and phase are regulated by the production of phase shift and rhythm resetting (phase resetting) for periodic motor commands in response to sensory feedback and perturbation. While the phase resetting has been suggested to make a large contribution to adaptive walking, it has only been investigated based on fictive locomotion in decerebrate cats, and thus it remains unclear if human motor control has such a rhythm regulation mechanism during walking. In our previous work, we incorporated a phase resetting mechanism into a motor control model and demonstrated that it improves the stability and robustness of walking through forward dynamic simulations of a human musculoskeletal model. However, this did not necessarily verify that phase resetting plays a role in human motor control. In our other previous work, we used kinematic measurements of human walking to identify the phase response curve (PRC), which explains phase-dependent responses of a limit cycle oscillator to a perturbation. This revealed how human walking rhythm is regulated by perturbations. In this study, we integrated these two approaches using a physical model and identification of the PRC to examine the hypothesis that phase resetting plays a role in the control of walking rhythm in humans. More specifically, we calculated the PRC using our neuromusculoskeletal model in the same way as our previous human experiment. In particular, we compared the PRCs calculated from two different models with and without phase resetting while referring to the PRC for humans. As a result, although the PRC for the model without phase resetting did not show any characteristic shape, the PRC for the model with phase resetting showed a characteristic phase-dependent shape with trends similar to those of the PRC for humans. These results support our hypothesis and will improve our understanding of adaptive rhythm control in human walking.

Phase resetting and intermittent control at the edge of stability in a simple biped model generates 1/f-like gait cycle variability

The 1/f-like gait cycle variability, characterized by temporal changes in stride-time intervals during steady-state human walking, is a well-documented gait characteristic. Such gait fractality is apparent in healthy young adults, but tends to disappear in the elderly and patients with neurological diseases. However, mechanisms that give rise to gait fractality have yet to be fully clarified. We aimed to provide novel insights into neuro-mechanical mechanisms of gait fractality, based on a numerical simulation model of biped walking. A previously developed heel-toe footed, seven-rigid-link biped model with human-like body parameters in the sagittal plane was implemented and expanded. It has been shown that the gait model, stabilized rigidly by means of impedance control with large values of proportional (P) and derivative (D) gains for a linear feedback controller, is destabilized only in a low-dimensional eigenspace, as P and D decrease below and even far below critical values. Such low-dimensional linear instability can be compensated by impulsive, phase-dependent actions of nonlinear controllers (phase resetting and intermittent controllers), leading to the flexible walking with joint impedance in the model being as small as that in humans. Here, we added white noise to the model to examine P-value-dependent stochastic dynamics of the model for small D-values. The simulation results demonstrated that introduction of the nonlinear controllers in the model determined the fractal features of gait for a wide range of the P-values, provided that the model operates near the edge of stability. In other words, neither the model stabilized only by pure impedance control even at the edge of linear stability, nor the model stabilized by specific nonlinear controllers, but with P-values far inside the stability region, could induce gait fractality. Although only limited types of controllers were examined, we suggest that the impulsive nonlinear controllers and criticality could be major mechanisms for the genesis of gait fractality.

Data-driven spectral analysis for coordinative structures in periodic human locomotion

Living organisms dynamically and flexibly operate a great number of components. As one of such redundant control mechanisms, low-dimensional coordinative structures among multiple components have been investigated. However, structures extracted from the conventional statistical dimensionality reduction methods do not reflect dynamical properties in principle. Here we regard coordinative structures in biological periodic systems with unknown and redundant dynamics as a nonlinear limit-cycle oscillation, and apply a data-driven operator-theoretic spectral analysis, which obtains dynamical properties of coordinative structures such as frequency and phase from the estimated eigenvalues and eigenfunctions of a composition operator. Using segmental angle series during human walking as an example, we first extracted the coordinative structures based on dynamics; e.g. the speed-independent coordinative structures in the harmonics of gait frequency. Second, we discovered the speed-dependent time-evolving behaviours of the phase by estimating the eigenfunctions via our approach on the conventional low-dimensional structures. We also verified our approach using the double pendulum and walking model simulation data. Our results of locomotion analysis suggest that our approach can be useful to analyse biological periodic phenomena from the perspective of nonlinear dynamical systems.

Adaptive hindlimb split-belt treadmill walking in rats by controlling basic muscle activation patterns via phase resetting

To investigate the adaptive locomotion mechanism in animals, a split-belt treadmill has been used, which has two parallel belts to produce left–right symmetric and asymmetric environments for walking. Spinal cats walking on the treadmill have suggested the contribution of the spinal cord and associated peripheral nervous system to the adaptive locomotion. Physiological studies have shown that phase resetting of locomotor commands involving a phase shift occurs depending on the types of sensory nerves and stimulation timing, and that muscle activation patterns during walking are represented by a linear combination of a few numbers of basic temporal patterns despite the complexity of the activation patterns. Our working hypothesis was that resetting the onset timings of basic temporal patterns based on the sensory information from the leg, especially extension of hip flexors, contributes to adaptive locomotion on the split-belt treadmill. Our hypothesis was examined by conducting forward dynamic simulations using a neuromusculoskeletal model of a rat walking on a split-belt treadmill with its hindlimbs and by comparing the simulated motions with the measured motions of rats.

Critical Points and Traveling Wave in Locomotion: Experimental Evidence and Some Theoretical Considerations

The central pattern generator (CPG) architecture for rhythm generation remains partly elusive. We compare cat and frog locomotion results, where the component unrelated to pattern formation appears as a temporal grid, and traveling wave respectively. Frog spinal cord microstimulation with N-methyl-D-Aspartate (NMDA), a CPG activator, produced a limited set of force directions, sometimes tonic, but more often alternating between directions similar to the tonic forces. The tonic forces were topographically organized, and sites evoking rhythms with different force subsets were located close to the constituent tonic force regions. Thus CPGs consist of topographically organized modules. Modularity was also identified as a limited set of muscle synergies whose combinations reconstructed the EMGs. The cat CPG was investigated using proprioceptive inputs during fictive locomotion. Critical points identified both as abrupt transitions in the effect of phasic perturbations, and burst shape transitions, had biomechanical correlates in intact locomotion. During tonic proprioceptive perturbations, discrete shifts between these critical points explained the burst durations changes, and amplitude changes occurred at one of these points. Besides confirming CPG modularity, these results suggest a fixed temporal grid of anchoring points, to shift modules onsets and offsets. Frog locomotion, reconstructed with the NMDA synergies, showed a partially overlapping synergy activation sequence. Using the early synergy output evoked by NMDA at different spinal sites, revealed a rostrocaudal topographic organization, where each synergy is preferentially evoked from a few, albeit overlapping, cord regions. Comparing the locomotor synergy sequence with this topography suggests that a rostrocaudal traveling wave would activate the synergies in the proper sequence for locomotion. This output was reproduced in a two-layer model using this topography and a traveling wave. Together our results suggest two CPG components: modules, i.e., synergies; and temporal patterning, seen as a temporal grid in the cat, and a traveling wave in the frog. Animal and limb navigation have similarities. Research relating grid cells to the theta rhythm and on segmentation during navigation may relate to our temporal grid and traveling wave results. Winfree's mathematical work, combining critical phases and a traveling wave, also appears important. We conclude suggesting tracing, and imaging experiments to investigate our CPG model.

Adaptive Control Strategies for Interlimb Coordination in Legged Robots: A Review

Walking animals produce adaptive interlimb coordination during locomotion in accordance with their situation. Interlimb coordination is generated through the dynamic interactions of the neural system, the musculoskeletal system, and the environment, although the underlying mechanisms remain unclear. Recently, investigations of the adaptation mechanisms of living beings have attracted attention, and bio-inspired control systems based on neurophysiological findings regarding sensorimotor interactions are being developed for legged robots. In this review, we introduce adaptive interlimb coordination for legged robots induced by various factors (locomotion speed, environmental situation, body properties, and task). In addition, we show characteristic properties of adaptive interlimb coordination, such as gait hysteresis and different time-scale adaptations. We also discuss the underlying mechanisms and control strategies to achieve adaptive interlimb coordination and the design principle for the control system of legged robots.

A Fuzzy Controller for Movement Stabilization Using Afferent Control: Controller Synthesis and Simulation

Stimulation of spinal sensorimotor circuits can improve motor control in animal models and humans with spinal cord injury (SCI). More recent evidence suggests that the stimulation increases the level of excitability in the spinal circuits, activates central pattern generators, and it is also able to recruit distinctive afferent pathways connected to specific sensorimotor circuits. In addition, the stimulation generates well-defined responses in leg muscles after each pulse. The problem is that in most of the neuromodulation devices, electrical stimulation parameters are regulated manually and stay constant during movement. Such a technique is likely suboptimal to intercede maximum therapeutic effects in patients. Therefore, in this article, a fuzzy controller has been designed to control limb kinematics during locomotion using the afferent control in a neuromechanical model without supraspinal drive simulating post-SCI situation. The proposed controller automatically tunes the weights of group Ia afferent inputs of the spinal cord to reset the phase appropriately during the reaction to an external perturbation. The kinematic motion data and weights of group Ia afferent inputs were the input and output of the controller, respectively. Simulation results showed the acceptable performance of the controller to establish adaptive locomotion against the perturbing forces based on the phase resetting of the walking rhythm.