Phase reset and dynamic stability during human gait

The human walking movement shows transient changes in response to single short-lived external perturbations, termed "stumbling reactions." During the stumbling reactions, the walking phase is reset. It has been considered that the reactions contribute to stabilizing the motion, but less evidence bridging between the rhythm reset and the dynamic stability of the gait has been provided. The present study tries to establish the relationship between them. To this end, we construct a simple dynamical system model of the human musculo-skeletal system interacting with the ground, whose joint kinematics during walking is constrained by a given periodic joint-angles-profile. We show first that the model can exhibit a stable limit cycle corresponding to the steady walking with no perturbations. The responses of the limit cycle oscillation are examined by applying a type of perturbations at various timings with various intensities, elucidating the stability of the model's walking when no phase reset is performed. We then observe that modifications of the periodic joint-angles-profile within a short time interval in response to the perturbation can alter the responses of the limit cycle oscillation and induce phase reset of the model's walking. It is shown that appropriate amounts of the phase reset can prevent the model from falling, even for the perturbation that induces falling in the case without the phase reset. This suggests that those phase resets can improve the dynamic stability of the gait. Moreover, the appropriate phase resets predicted by the model are compared with the experimentally observed phase resets during human stumbling reaction to show they share similar characteristics.     

Circadian rhythms in Neurospora: a new measurement, the reset zone

The authors define a new feature of a circadian rhythm, the reset zone, and point out its usefulness for predictions concerning oscillator behavior. The reset zone measures the responses of a circadian system to resetting pulses. It can be easily determined from a phase transition curve (PTC), which is simply a phase response curve (PRC) replotted as new phase versus old phase (Winfree's format). The reset zone is the range of new phases seen in such a plot and has two potentially useful characteristics: its size and its midpoint. A series of experiments with Neurospora involving temperature pulses indicated that the size of the reset zone changed in a nonlinear way in response to both the duration of 40 degrees C pulses and to the magnitude of temperature change for 3-h pulses. Other existing data are replotted to show how the reset zone size varies with growth temperature and with the period of different clock mutants. Employing exclusively reset zone data within the framework of a limit cycle displacement model, an equation is formulated that predicts the relative changes in the values of state variables of the oscillator for changes in any given environmental condition, such as temperature. Examples are also drawn from other organisms, such as hamsters, Gonyalaux, Kalanchoe, and Drosophila, illustrating the usefulness of the reset zone measurement. It can be used as a numerical scale for assessing the strength of a pulse, for comparing the relative effects of a given pulse applied to different organisms or mutants, for determining the directionality of the changes in state variables produced by various types of pulses, and possibly for measuring clock amplitude.     

Stumbling with optimal phase reset during gait can prevent a humanoid from falling

The human biped walking shows phase- dependent transient changes in gait trajectory in response to external brief force perturbations. Such responses, referred to as the stumbling reactions, are usually accompanied with phase reset of the walking rhythm. Our previous studies provided evidence, based on a human gait experiment and analyses of mathematical models of gait in the sagittal plane, that an appropriate amount of phase reset in response to a perturbation depended on the gait phase at the perturbation and could play an important role for preventing the walker from a fall, thus increasing gait stability. In this paper, we provide a further material that supports this evidence by a gait experiment on a biped humanoid. In the experiment, the impulsive force perturbations were applied using push-impacts by a pendulum-like hammer to the back of the robot during gait. The responses of the external perturbations were managed by resetting the gait phase with different delays or advancements. The results showed that appropriate amounts of phase resetting contributed to the avoidance of falling against the perturbation during the three-dimensional robot gait. A parallelism with human gait stumbling reactions was discussed.Electronic Supplementary Material Supplementary material is available for this article at and is accessible for authorized users.     

Evaluating functional roles of phase resetting in generation of adaptive human bipedal walking with a physiologically based model of the spinal pattern generator

The central pattern generators (CPGs) in the spinal cord strongly contribute to locomotor behavior. To achieve adaptive locomotion, locomotor rhythm generated by the CPGs is suggested to be functionally modulated by phase resetting based on sensory afferent or perturbations. Although phase resetting has been investigated during fictive locomotion in cats, its functional roles in actual locomotion have not been clarified. Recently, simulation studies have been conducted to examine the roles of phase resetting during human bipedal walking, assuming that locomotion is generated based on prescribed kinematics and feedback control. However, such kinematically based modeling cannot be used to fully elucidate the mechanisms of adaptation. In this article we proposed a more physiologically based mathematical model of the neural system for locomotion and investigated the functional roles of phase resetting. We constructed a locomotor CPG model based on a two-layered hierarchical network model of the rhythm generator (RG) and pattern formation (PF) networks. The RG model produces rhythm information using phase oscillators and regulates it by phase resetting based on foot-contact information. The PF model creates feedforward command signals based on rhythm information, which consists of the combination of five rectangular pulses based on previous analyses of muscle synergy. Simulation results showed that our model establishes adaptive walking against perturbing forces and variations in the environment, with phase resetting playing important roles in increasing the robustness of responses, suggesting that this mechanism of regulation may contribute to the generation of adaptive human bipedal locomotion.     

Dynamic stability of passive dynamic walking on an irregular surface

Falls that occur during walking are a significant health problem. One of the greatest impediments to solve this problem is that there is no single obviously "correct" way to quantify walking stability. While many people use variability as a proxy for stability, measures of variability do not quantify how the locomotor system responds to perturbations. The purpose of this study was to determine how changes in walking surface variability affect changes in both locomotor variability and stability. We modified an irreducibly simple model of walking to apply random perturbations that simulated walking over an irregular surface. Because the model's global basin of attraction remained fixed, increasing the amplitude of the applied perturbations directly increased the risk of falling in the model. We generated ten simulations of 300 consecutive strides of walking at each of six perturbation amplitudes ranging from zero (i.e., a smooth continuous surface) up to the maximum level the model could tolerate without falling over. Orbital stability defines how a system responds to small (i.e., "local") perturbations from one cycle to the next and was quantified by calculating the maximum Floquet multipliers for the model. Local stability defines how a system responds to similar perturbations in real time and was quantified by calculating short-term and long-term local exponential rates of divergence for the model. As perturbation amplitudes increased, no changes were seen in orbital stability (r(2)=2.43%; p=0.280) or long-term local instability (r(2)=1.0%; p=0.441). These measures essentially reflected the fact that the model never actually "fell" during any of our simulations. Conversely, the variability of the walker's kinematics increased exponentially (r(2)>or=99.6%; p<0.001) and short-term local instability increased linearly (r(2)=88.1%; p<0.001). These measures thus predicted the increased risk of falling exhibited by the model. For all simulated conditions, the walker remained orbitally stable, while exhibiting substantial local instability. This was because very small initial perturbations diverged away from the limit cycle, while larger initial perturbations converged toward the limit cycle. These results provide insight into how these different proposed measures of walking stability are related to each other and to risk of falling.     

MEG responses during rhythmic finger tapping in humans to phasic stimulation and their interpretation based on neural mechanisms

 The phase-resetting experiment was applied to human periodic finger tapping to understand how its rhythm is controlled by the internal neural clock that is assumed to exist. In the experiment, the right periodic tapping movement was disturbed transiently by a series of left finger taps in response to impulsive auditory cues presented randomly at various phases within the tapping cycle. After each left finger tap, the original periodic tapping was reestablished within several tapping cycles. Influences of the disturbance on the periodic right finger tapping varied depending on the phase of the periodic right finger tapping at which each left finger tap was made. It was confirmed that the periodic tapping was disturbed not by the auditory cues but by the left finger taps. Based on this fact, in this paper each single left tap was considered as the stimulus, and the phase of the periodic tapping of the right index finger when the left tap was executed as the phase of the stimulus. Responses of the neural activities (magnetoencephalography, MEG), the tapping movement, and the corresponding muscle activities (electromyography) were simultaneously measured. Phase-resetting curves (PRCs) representing the degree of phase reset as a function of the phase of the stimulus were obtained both for the left sensorimotor cortex MEG response and for the right index finger tapping response. The shapes of both PRCs were similar, suggesting that the phase reset of the left sensorimotor cortex activities and that of the finger tapping rhythm were the same. Four out of eight subjects showed type-0 reset in Winfree's definition, and the others showed type-1 reset. For general limit-cycle oscillators, type-0 reset is obtained for relatively strong perturbations and type 1 for weak perturbations. It was shown that the transient response of MEG to the single left tap stimuli in type-0 subjects, where the phase was progressively reset, were different from those in type-1 subjects. Based on detailed analysis of the differences, a neural network model for the phase reset of the tapping rhythm is proposed. Received: 10 February 2000 / Accepted in revised form: 15 January 2002     

Cell Division Cycles and Orcadian Oscillators in Euglena

The algal flagellate Euglena grown photoautotrophically in L:D 3:3 displays a circadian rhythm of cell division. Oscillatory models for cell cycle (CDC) control (particularly those of the limit cycle variety) include the property of phase perturbation, or resetting. This prediction has been tested in synchronous cultures in which the free-running rhythm has been scanned by 3-hr light signals. A strong (Type 0) phase response curve (PRC), yielding both advances and delays as great as 15 hr, has been derived. A second prediction of the limit cycle model is that there exists a pulse of a critical intensity, which, if given at one specific phase of the rhythm (the singularity point), should result in a phaseless, motionless state in which the rhythmicity disappears. Such a point has been found in Euglena in the late subjective night for light pulses having an intensity ranging from 40 to 700 Ix. Finally, circadian oscillators typically display temperature-compensated period lengths within the physiological range of steady-state temperatures, although the length of the CDC is commonly thought to be highly temperature dependent. We have found that over a range of at least 10°C, the period of the division rhythm is only slightly affected, exhibiting a Q₁₀ of about 1.05-1.20. These observations, therefore, collectively implicate a circadian oscillator in the control of the CDC.     

A model of the neuro-musculo-skeletal system for human locomotion

The generation of human locomotion was examined by linking computational neuroscience with biomechanics from the perspective of nonlinear dynamical theory. We constructed a model of human locomotion, which includes a musculo-skeletal system with 8 segments and 20 muscles, a neural rhythm generator composed of 7 pairs of neural oscillators, and mechanisms for processing and transporting sensory and motor signals. Using a computer simulation, we found that locomotion emerged as a stable limit cycle that was generated by the global entrainment between the musculo-skeletal system, the neural system, and the environment. Moreover, the walking movements of the model could be compared quantitatively with those of experimental studies in humans.Part of this paper was presented to IVth International Symposium on Computer Simulation in Biomechanics, Paris, France, July 1, 1993     

Postural dynamics of walking in humans

The dynamics of postural control in human biped locomotion were studied using(1) a model, and(2) experimentally applied impulsive force disturbances. The model was planar, and contained five rigid segments, articulating at frictionless pin joints. The model was used to identify joint torque combinations which would successfully correct for an impulsive force disturbance applied at different points in the walking cycle. The simulation results suggested that(1) early responses (within 80ms) can be effective in compensating for impulsive disturbances,(2) the same strategies which successfully counteract similar disturbances during quiet standing are also effective in certain phases of the walking cycle,(3) modifications in the response strategies are needed to accomodate differences in the dynamics over the stride cycle, and(4) the swing leg is ineffective in compensating for disturbances in the short term. These model predictions were tested experimentally. Subject responses to an impulsive force disturbance applied during walking were studied. The electromyographic results generally support the model predictions.     

The Paramecium circadian behavioral rhythm: light phase response curves and entrainment

The population of a ciliate protozoan, Paramecium multimicronucleatum, exhibits a circadian rhythm as measured by the number of the cells traversing an observation point ("traverse frequency," or TF). The present study examined phase shifting of the TF rhythm by administering 2-hr light pulses at different phases of the circadian cycle to cultures free-running in constant darkness (DD). The results were summarized in a phase response curve (PRC), categorized as Type 1. This PRC indicated a relatively narrow phase zone insensitive to the light pulse ("dead zone"). Entrainment of the rhythm to light pulses repeated at 24-hr intervals was also examined, and it was found that the rhythm gradually reached a steady state, following several transient cycles, with the pulses falling at a phase corresponding to the narrow dead zone. Such a steady-state rhythm, with a minimum at approximately 3 hr after the pulse and a maximum at approximately 12 hr after the pulse, was mathematically simulated by superimposing a response function to the pulse on a sinusoidal function representative of the free-running rhythm in DD.     

UVA‐induced reset of hydroxyl radical ultradian rhythm improves temporal lipid production in Chlorella vulgaris

We report for the first time that the endogenous, pseudo‐steady‐state, specific intracellular levels of the hydroxyl radical (si‐OH) oscillate in an ultradian fashion (model system: the microalga, Chlorella vulgaris), and also characterize the various rhythm parameters. The ultradian rhythm in the endogenous levels of the si‐OH occurred with an approximately 6 h period in the daily cycle of light and darkness. Further, we expected that the rhythm reset to a shorter period could rapidly switch the cellular redox states that could favor lipid accumulation. We reset the endogenous rhythm through entrainment with UVA radiation, and generated two new ultradian rhythms with periods of approximately 2.97 h and 3.8 h in the light phase and dark phase, respectively. The reset increased the window of maximum lipid accumulation from 6 h to 12 h concomitant with the onset of the ultradian rhythms. Further, the saturated fatty acid content increased approximately to 80% of total lipid content, corresponding to the peak maxima of the hydroxyl radical levels in the reset rhythm. © 2014 American Institute of Chemical Engineers Biotechnol. Prog., 30:673–680, 2014     

Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment

A new principle of sensorimotor control of legged locomotion in an unpredictable environment is proposed on the basis of neurophysiological knowledge and a theory of nonlinear dynamics. Stable and flexible locomotion is realized as a global limit cycle generated by a global entrainment between the rhythmic activities of a nervous system composed of coupled neural oscillators and the rhythmic movements of a musculo-skeletal system including interaction with its environment. Coordinated movements are generated not by slaving to an explicit representation of the precise trajectories of the movement of each part but by dynamic interactions among the nervous system, the musculo-skeletal system and the environment. The performance of a bipedal model based on the above principle was investigated by computer simulation. Walking movements stable to mechanical perturbations and to environmental changes were obtained. Moreover, the model generated not only the walking movement but also the running movement by changing a single parameter nonspecific to the movement. The transitions between the gait patterns occurred with hysteresis.     

Probing the circadian pacemaker of a mouse using two light pulses

In two separate sets of experiments, the phases of the locomotor activity rhythm of the nocturnal field mouse Mus booduga were probed using two light pulses (LPs). In the first set of experiments, the circadian pacemaker underlying the locomotor activity rhythm was perturbed at circadian time 14 (CT 14) using a resetting light pulse LP1 of 1000 lux intensity and 15 min duration. The phases of the resetting pacemaker were then probed at all even CTs between CT 16 and CT 14 using a PRC probing light pulse LP2 of equal strength. The "LP2 PRC" thus obtained was then compared with the single light pulse PRC in terms of the area under delay (D) and advance (A) zones of the PRCs. The time course and waveform of the two LP PRCs suggest that the LP2 PRC resembled the single LP PRC, displaced by 2 h toward the right. The LP1 PRC had smaller D compared to the single LP PRC (p = 0.007), whereas both the PRCs had A of equal magnitude (p = 0.23). This suggests that the pacemaker phase shifts rapidly after LP perturbations. In the second set of experiments, the LP1 was administered at CT 14. The phase of the pacemaker was then perturbed on day 1 (next cycle after LP1) either 2 h after activity onset (at ca. CT 14 of the transient cycle) or 8 h after activity onset (at ca. CT 20 of the transient cycle) using an LP2 of equal strength. It was observed that the steady-state phase shifts evoked by positioning an LP2, 2 h after activity onset, were positively correlated with the phase shifts observed on day 1. The steady-state phase shifts observed, when the LP2 was positioned, 8 h after activity onset, were negatively correlated with the phase shifts observed on day 1. These results suggest that the transient cycles do not mirror the state of the pacemaker oscillator.     

Dynamic Stability of Passive Bipedal Walking on Rough Terrain:A Preliminary Simulation Study

A simplified 2D passive dynamic model was simulated to walk down on a rough slope surface defined by deterministic profiles to investigate how the walking stability changes with increasing surface roughness.Our results show that the passive walker can walk on rough surfaces subject to surface roughness up to approximately 0.1％ of its leg length.This indicates that bipedal walkers based on passive dynamics may possess some intrinsic stability to adapt to rough terrains although the maximum roughness they can tolerate is small.Orbital stability method was used to quantify the walking stability before the walker started to fall over.It was found that the average maximum Floquet multiplier increases with surface roughness in a non-linear form.Although the passive walker remained orbitally stable for all the simulation cases,the results suggest that the possibility of the bipedal model moving away from its limit cycle increases with the surface roughness if subjected to additional perturbations.The number of consecutive steps before falling was used to measure the walking stability after the passive walker started to fall over.The results show that the number of steps before falling decreases exponentially with the increase in surface roughness.When the roughness magnitude approached to 0.73％ of the walker's leg length,it fell down to the ground as soon as it entered into the uneven terrain.It was also found that shifting the phase angle of the surface profile has apparent affect on the system stability.This is probably because point contact was used to simulate the heel strikes and the resulted variations in system states at heel strikes may have pronounced impact on the passive gaits,which have narrow basins of attraction.These results would provide insight into how the dynamic stability of passive bipedal walkers evolves with increasing surface roughness.     

Isochron-based phase response analysis of circadian rhythms

Circadian rhythms possess the ability to robustly entrain to the environmental cycles. This ability relies on the phase synchronization of circadian rhythm gene regulation to different environmental cues, of which light is the most obvious and important. The elucidation of the mechanism of circadian entrainment requires an understanding of circadian phase behavior. This article presents two phase analyses of oscillatory systems for infinitesimal and finite perturbations based on isochrons as a phase metric of a limit cycle. The phase response curve of circadian rhythm can be computed from the results of the analyses. The application to a mechanistic Drosophila circadian rhythm model gives experimentally testable hypotheses for the control mechanisms of circadian phase responses and evidence for the role of phase and period modulations in circadian photic entrainment.     

Light-Induced resetting of the circadian pacemaker: quantitative analysis of transient versus steady-state phase shifts.

The suprachiasmatic nuclei of the hypothalamus contain the major circadian pacemaker in mammals, driving circadian rhythms in behavioral and physiological functions. This circadian pacemaker's responsiveness to light allows synchronization to the light-dark cycle. Phase shifting by light often involves several transient cycles in which the behavioral activity rhythm gradually shifts to its steady-state position. In this article, the authors investigate in Syrian hamsters whether a phase-advancing light pulse results in immediate shifts of the PRC at the next circadian cycle. In a first series of experiments, the authors aimed a light pulse at CT 19 to induce a phase advance. It appeared that the steady-state phase advances were highly correlated with activity onset in the first and second transient cycle. This enabled them to make a reliable estimate of the steady-state phase shift induced by a phase-advancing light pulse on the basis of activity onset in the first transient cycle. In the next series of experiments, they presented a light pulse at CT 19, which was followed by a second light pulse aimed at the delay zone of the PRC on the next circadian cycle. The immediate and steady-state phase delays induced by the second light pulse were compared with data from a third experiment in which animals received a phase-delaying light pulse only. The authors observed that the waveform of the phase-delay part of the PRC (CT 12-16) obtained in Experiment 2 was virtually identical to the phase-delay part of the PRC for a single light pulse (obtained in Experiment 3). This finding allowed for a quantitative assessment of the data. The analysis indicates that the delay part of the PRC-between CT 12 and CT 16-is rapidly reset following a light pulse at CT 19. These findings complement earlier findings in the hamster showing that after a light pulse at CT 19, the phase-advancing part of the PRC is immediately shifted. Together, the data indicate that the basis for phase advancing involves rapid resetting of both advance and delay components of the PRC.     

An artificial neural network that utilizes hip joint actuations to control bifurcations and chaos in a passive dynamic bipedal walking model

Chaos is a central feature of human locomotion and has been suggested to be a window to the control mechanisms of locomotion. In this investigation, we explored how the principles of chaos can be used to control locomotion with a passive dynamic bipedal walking model that has a chaotic gait pattern. Our control scheme was based on the scientific evidence that slight perturbations to the unstable manifolds of points in a chaotic system will promote the transition to new stable behaviors embedded in the rich chaotic attractor. Here we demonstrate that hip joint actuations during the swing phase can provide such perturbations for the control of bifurcations and chaos in a locomotive pattern. Our simulations indicated that systematic alterations of the hip joint actuations resulted in rapid transitions to any stable locomotive pattern available in the chaotic locomotive attractor. Based on these insights, we further explored the benefits of having a chaotic gait with a biologically inspired artificial neural network (ANN) that employed this chaotic control scheme. Remarkably, the ANN was quite robust and capable of selecting a hip joint actuation that rapidly transitioned the passive dynamic bipedal model to a stable gait embedded in the chaotic attractor. Additionally, the ANN was capable of using hip joint actuations to accommodate unstable environments and to overcome unforeseen perturbations. Our simulations provide insight on the advantage of having a chaotic locomotive system and provide evidence as to how chaos can be used as an advantageous control scheme for the nervous system.     

A weakly nonlinear analysis of a model of avascular solid tumour growth

 In this paper we study a mathematical model that describes the growth of an avascular solid tumour. Our analysis concentrates on the stability of steady, radially-symmetric model solutions with respect to perturbations taken from the class of spherical harmonics. Using weakly nonlinear analysis, previous results are extended to show how the amplitudes of the asymmetric modes interact. Attention focuses on a special case for which the model equations simplify. Analysis of the simplified model equations leads to the identification of a two-parameter family of asymmetric steady solutions, the dimensions of whose stable and unstable manifolds depend on the system parameters. The asymmetric steady solutions limit the basin of attraction of the radially-symmetric steady state when it is linearly stable. On the basis of these numerical and analytical results we postulate the existence of fully nonlinear steady solutions which are stable with respect to time-dependent perturbations. Received: 25 October 1998 / Revised version: 20 June 1998     

A collisional model of the energetic cost of support work qualitatively explains leg sequencing in walking and galloping, pseudo-elastic leg behavior in running and the walk-to-run transition

Terrestrial legged locomotion requires repeated support forces to redirect the body's vertical velocity component from down to up. We assume that the redirection is accomplished by impulsive leg forces that cause small-angle glancing collisions of a point-mass model of the animal. We estimate the energetic costs of these collisions by assuming a metabolic cost proportional to positive muscle work involved in generating the impulses. The cost of bipedal running estimated from this collisional model becomes less than that of walking at a Froude number (v2/gl) of about 0.7. Two strategies to reduce locomotion costs associated with the motion redirection are: (1) having legs simulate purely elastic springs, as is observed in human running; and (2) sequencing the leg forces during the redirection phase; examples of this sequencing are the ba-da-dump pattern of a horse gallop and having push-off followed by heel-strike in human walking.     

Rapid Phase Resetting of a Mammalian Orcadian Rhythm by Brief Light Pulses

The phase-shift (Δψ) responses of the circadian rhythm in the field mouse Mus booduga to brief light pulses (LPs) of 15 minutes duration and 1000 lux intensity were measured in 90 experiments. In each experiment, a resetting light pulse LP₁ was administered at CT14 (CT, circadian time), and a scanning light pulse LP₂ was then variously administered in separate experiments at CT16, CT20, and CT22 in the same and in the next circadian cycle. The Δψ obtained in all these two-pulse experiments did not differ significantly from theoretical values computed on the assumption that LP₁ reset the phase response curve (PRC) rapidly. In each case, the steady-state Δψ observed after LP₁ and LP₂ differed significantly from the Δψ obtained at the same CT in determination of the single-pulse PRC (control) and also differed significantly from the values on the assumption of no Δψ in the PRC following LP₁. These results indicate that the circadian pacemaker of M. booduga, as measured by its PRC, is substantially reset within 2h after a light pulse at CT14. (Chronobiology International 14(6), 537–548, 1997)