A Simple State-Determined Model Reproduces Entrainment and Phase-Locking of Human Walking

Theoretical studies and robotic experiments have shown that asymptotically stable periodic walking may emerge from nonlinear limit-cycle oscillators in the neuro-mechanical periphery. We recently reported entrainment of human gait to periodic mechanical perturbations with two essential features: 1) entrainment occurred only when the perturbation period was close to the original (preferred) walking period, and 2) entrainment was always accompanied by phase locking so that the perturbation occurred at the end of the double-stance phase. In this study, we show that a highly-simplified state-determined walking model can reproduce several salient nonlinear limit-cycle behaviors of human walking: 1) periodic gait that is 2) asymptotically stable; 3) entrainment to periodic mechanical perturbations only when the perturbation period is close to the model''s unperturbed period; and 4) phase-locking to locate the perturbation at the end of double stance. Importantly, this model requires neither supra-spinal control nor an intrinsic self-sustaining neural oscillator such as a rhythmic central pattern generator. Our results suggest that several prominent limit-cycle features of human walking may stem from simple afferent feedback processes without significant involvement of supra-spinal control or a self-sustaining oscillatory neural network.     

Hexapodal gaits and coupled nonlinear oscillator models

The general, model-independent features of different networks of six symmetrically coupled nonlinear oscillators are investigated. These networks are considered as possible models for locomotor central pattern generators (CPGs) in insects. Numerical experiments with a specific oscillator network model are briefly described. It is shown that some generic phase-locked oscillation-patterns for various systems of six symmetrically coupled nonlinear oscillators correspond to the common forward-walking gaits adopted by insects. It is also demonstrated that transitions observed in insect gaits can be modelled as standard symmetry-breaking bifurcations occurring in such systems. The present analysis, which leads to a natural classification of hexapodal gaits by symmetry and to natural sequences of gait bifurcations, relates observed gaits to the overall organizational structure of the underlying CPG. The implications of the present results for the development of simplified control systems for hexapodal walking robots are discussed.     

The mathematical modeling of entrained biological oscillators

In this paper perturbation methods are used for the mathematical analysis of coupled relaxation oscillators. This study covers entrainment by an external periodic stimulus as well as mutual entrainment of coupled oscillators with different limit cycles. The oscillators are of a type one meets in the modeling of biological oscillators by chemical reactions and electronic circuits. Special attention is given to entrainment different from 1∶1. The results relate to phenomena occurring in physiological experiments, such as the periodic stimulation of neural and cardiac cells, and in the non-regular functioning of organs and organisms, such as the AV-block in the heart.     

Artificial neural network model for the generation of muscle activation patterns for human locomotion.

Skilled locomotor behaviour requires information from various levels within the central nervous system (CNS). Mathematical models have permitted researchers to simulate various mechanisms in order to understand the organization of the locomotor control system. While it is difficult to adequately characterize the numerous inputs to the locomotor control system, an alternative strategy may be to use a kinematic movement plan to represent the complex inputs to the locomotor control system based on the possibility that the CNS may plan movements at a kinematic level. We propose the use of artificial neural network (ANN) models to represent the transformation of a kinematic plan into the necessary motor patterns. Essentially, kinematic representation of the actual limb movement was used as the input to an ANN model which generated the EMG activity of 8 muscles of the lower limb and trunk. Data from a wide variety of gait conditions was necessary to develop a robust model that could accommodate various environmental conditions encountered during everyday activity. A total of 120 walking strides representing normal walking and ten conditions where the normal gait was modified in terms of cadence, stride length, stance width or required foot clearance. The final network was assessed on its ability to predict the EMG activity on individual walking trials as well as its ability to represent the general activation pattern of a particular gait condition. The predicted EMG patterns closely matched those recorded experimentally, exhibiting the appropriate magnitude and temporal phasing required for each modification. Only 2 of the 96 muscle/gait conditions had RMS errors above 0.10, only 5 muscle/gait conditions exhibited correlations below 0.80 (most were above 0.90) and only 25 muscle/gait conditions deviated outside the normal range of muscle activity for more than 25% of the gait cycle. These results indicate the ability of single network ANNs to represent the transformation between a kinematic movement plan and the necessary muscle activations for normal steady state locomotion but they were also able to generate muscle activation patterns for conditions requiring changes in walking speed, foot placement and foot clearance. The abilities of this type of network have implications towards both the fundamental understanding of the control of locomotion and practical realizations of artificial control systems for use in rehabilitation medicine.     

Kinematic control of walking

The planar law of inter-segmental co-ordination we described may emerge from the coupling of neural oscillators between each other and with limb mechanical oscillators. Muscle contraction intervenes at variable times to re-excite the intrinsic oscillations of the system when energy is lost. The hypothesis that a law of coordinative control results from a minimal active tuning of the passive inertial and viscoelastic coupling among limb segments is congruent with the idea that movement has evolved according to minimum energy criteria (1, 8). It is known that multi-segment motion of mammals locomotion is controlled by a network of coupled oscillators (CPGs, see 18, 33, 37). Flexible combination of unit oscillators gives rise to different forms of locomotion. Inter-oscillator coupling can be modified by changing the synaptic strength (or polarity) of the relative spinal connections. As a result, unit oscillators can be coupled in phase, out of phase, or with a variable phase, giving rise to different behaviors, such as speed increments or reversal of gait direction (from forward to backward). Supra-spinal centers may drive or modulate functional sets of coordinating interneurons to generate different walking modes (or gaits). Although it is often assumed that CPGs control patterns of muscle activity, an equally plausible hypothesis is that they control patterns of limb segment motion instead (22). According to this kinematic view, each unit oscillator would directly control a limb segment, alternately generating forward and backward oscillations of the segment. Inter-segmental coordination would be achieved by coupling unit oscillators with a variable phase. Inter-segmental kinematic phase plays the role of global control variable previously postulated for the network of central oscillators. In fact, inter-segmental phase shifts systematically with increasing speed both in man (4) and cat (38). Because this phase-shift is correlated with the net mechanical power output over a gait cycle (3, 4), phase control could be used for limiting the overall energy expenditure with increasing speed (22). Adaptation to different walking conditions, such as changes in body posture, body weight unloading and backward walk, also involves inter-segmental phase tuning, as does the maturation of limb kinematics in toddlers.     

Learning to walk with a robotic ankle exoskeleton

We used a lower limb robotic exoskeleton controlled by the wearer's muscle activity to study human locomotor adaptation to disrupted muscular coordination. Ten healthy subjects walked while wearing a pneumatically powered ankle exoskeleton on one limb that effectively increased plantar flexor strength of the soleus muscle. Soleus electromyography amplitude controlled plantar flexion assistance from the exoskeleton in real time. We hypothesized that subjects' gait kinematics would be initially distorted by the added exoskeleton power, but that subjects would reduce soleus muscle recruitment with practice to return to gait kinematics more similar to normal. We also examined the ability of subjects to recall their adapted motor pattern for exoskeleton walking by testing subjects on two separate sessions, 3 days apart. The mechanical power added by the exoskeleton greatly perturbed ankle joint movements at first, causing subjects to walk with significantly increased plantar flexion during stance. With practice, subjects reduced soleus recruitment by approximately 35% and learned to use the exoskeleton to perform almost exclusively positive work about the ankle. Subjects demonstrated the ability to retain the adapted locomotor pattern between testing sessions as evidenced by similar muscle activity, kinematic and kinetic patterns between the end of the first test day and the beginning of the second. These results demonstrate that robotic exoskeletons controlled by muscle activity could be useful tools for testing neural mechanisms of human locomotor adaptation.     

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.     

Adaptative control of above knee electro-hydraulic prosthesis.

Conventional designs of an above-knee prosthesis are based on mechanisms with mechanical properties (such as friction, spring and damping coefficients) that remain constant during changing cadence. These designs are unable to replace natural legs due to the lack of active knee joint control. Since the nonlinear and time-varying dynamic coupling between the thigh and the prosthetic limb is high during swing phase, an adaptive control is employed to control the knee joint motion. Two dimensional simulation indicates that the adaptive controller can improve the appearance of gait pattern. It is adaptable to walking speed and can compensate for the variations of hip moment, hip trajectory and toe-off conditions.     

A passive dynamic walking robot that has a deterministic nonlinear gait

There is a growing body of evidence that the step-to-step variations present in human walking are related to the biomechanics of the locomotive system. However, we still have limited understanding of what biomechanical variables influence the observed nonlinear gait variations. It is necessary to develop reliable models that closely resemble the nonlinear gait dynamics in order to advance our knowledge in this scientific field. Previously, Goswami et al. [1998. A study of the passive gait of a compass-like biped robot: symmetry and chaos. International Journal of Robotic Research 17(12)] and Garcia et al. [1998. The simplest walking model: stability, complexity, and scaling. Journal of Biomechanical Engineering 120(2), 281-288] have demonstrated that passive dynamic walking computer models can exhibit a cascade of bifurcations in their gait pattern that lead to a deterministic nonlinear gait pattern. These computer models suggest that the intrinsic mechanical dynamics may be at least partially responsible for the deterministic nonlinear gait pattern; however, this has not been shown for a physical walking robot. Here we use the largest Laypunov exponent and a surrogation analysis method to confirm and extend Garcia et al.'s and Goswami et al.'s original results to a physical passive dynamic walking robot. Experimental outcomes from our walking robot further support the notion that the deterministic nonlinear step-to-step variations present in gait may be partly governed by the intrinsic mechanical dynamics of the locomotive system. Furthermore the nonlinear analysis techniques used in this investigation offer novel methods for quantifying the nature of the step-to-step variations found in human and robotic gait.     

Coupled oscillators utilised as gait rhythm generators of a two-legged walking machine

The gait of current two-legged walking machines differs from that of humans, although the kinematic structures of these machines' legs frequently imitate human limbs. This paper presents a method of generating the trajectories of hip and knee joint angles resulting in a gait pattern similar to that of a human. For this purpose the solutions of coupled van der Pol oscillator equations are utilised. There is much evidence that these equations can be treated as a good model of the central pattern generator generating functional (also locomotional) rhythms in living creatures. The oscillator equations are solved by numerical integration. The method of changing the type of gait by changing appropriate parameter values in the oscillator equations is presented (change of velocity and trajectory of leg-ends). The results obtained enable enhanced control of twolegged walking systems by including gait pattern generators which will assume a similar role to that of biological generators.     

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.     

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.     

The circadian locomotor rhythm of hemideina thoracica (Orthoptera; Stenopelmatidae): A population of weakly coupled feedback oscillators as a model of the underlying pacemaker

A control systems model consisting of a population of weakly-coupled feedback oscillators has been developed to simulate the circadian locomotor rhythm of the insect, Hemideina thoracica (Orthoptera; Stenopelmatidae). The model is an extension of a previously published single oscillator feedback model (Gander and Lewis, 1979) which successfully simulates entrainment, phase response curves, temperature compensation and Aschoff's Rule for Hemideina activity rhythms. The population model described here has the additional properties of predicting some of the free-run period lability (Pavlidis, 1978a, b) observed in the Hemideina rhythm (Christensen and Lewis, 1982) which is unexplained by single oscillator systems. Model behaviour is compared with the experimental data derived from the insect activity rhythms.     

The estrous cycle in golden hamsters: a circadian pacemaker times preovulatory gonadotropin release

In a population of cycling female hamsters entrained to an LD 6:18 light cycle (lights 1000-1600 hours), preovulatory release of luteinizing hormone and follicle-stimulating hormone occurred in some animals at 1300-1400 hours and in others at 1900 hours. In every case peak release was phase-locked (2-3-hour positive phase angle) to the circadian rhythm of locomotor activity. The pattern of entrainment of gonadotropin release on LD 6:18 is fully explicable in terms of the hamster's phase response curve to light. We conclude that periodic gonadotropin release in cycling females is timed by a circadian oscillator (biological clock) that is probably the same oscillator driving the circadian rhythm of locomotor activity.     

An analysis of the nonlinear behaviour of the human thermal vasomotor control system.

Specific aspects of the activity of the human thermoregulatory system are analysed from the point of view of the associated biological rhythms. Fluctuations in digit blood flow were examined and analysis of them showed that they were consistent with a nonlinear model of the control system. Experiments were undertaken in which periodic thermal stimuli were applied which caused the entrainment of digit blood flow. A theoretical study was subsequently undertaken in which the experimental results were compared to the activity of a Van der Pol oscillator.     

Gait modification when decreasing double support percentage

Much is still unknown about walking stability, including which aspects of gait contribute to higher stability. Walking stability appears to be related to walking speed, although the exact relationship is unclear. As walking speed decreases, the double support (DS) period of gait increases both in time and as a percentage of the gait cycle. Because humans have more control over their center of mass movement during DS, increasing DS duration may alter stability. This study examined how human gait is affected by changing DS percentage independent of walking speed. Sixteen young, healthy adults walked on a treadmill at a single speed for six one-minute trials. These trials included normal gait as well as longer- and shorter-than-normal DS percentage gaits. Subjects were consistently able to decrease DS percentage but had difficulty increasing DS percentage. In some cases, subjects altered their cadence when changing DS percentage, particularly when attempting to increase DS percentage. The changes to gait when decreasing DS percentage were similar to changes when increasing walking speed but occurred mainly during the swing period. These changes include increased hip and knee flexion during the swing period, increased swing foot height, and larger magnitude peaks in ground reaction forces. The changes in gait when attempting to increase DS percentage trended toward changes when decreasing walking speed. Altering DS percentage induced gait changes that were similar to, yet clearly distinct from, gait changes due to walking speed. Further, the difficulty of increasing DS percentage when walking at a constant speed suggests that people walk more slowly when they want to increase time spent in DS.     

‘Initial state’ coordinations reproduce the instant flexibility for human walking

An important feature of human locomotor control is the instant adaptability to unpredictable changes of conditions surrounding the locomotion. Humans, for example, can seamlessly adapt their walking gait following a sudden ankle impairment (e.g., as a result of an injury). In this paper, we propose a theoretical study of the mechanisms underlying flexible locomotor control. We hypothesize that flexibility is achieved by modulating the posture at the beginning of the stance phase—the initial state. Using a walking model, we validate our hypothesis through computer simulations     

Circadian organization in Aplysia californica.

Substantial progress has been made in unraveling the organization of the circadian system of Aplysia californica. There are at least three circadian pacemakers in Aplysia. One has been localized in each eye and a third lies outside the eyes. Removal of the eyes disrupts the free-running locomotor activity rhythm; however, an extraocular oscillator can mediate a free-running rhythm in some eyeless animals. Although photoreceptors sufficient for entrainment of the ocular oscillator have been localized in the retina, photoreceptors outside the eyes are capable of "driving" a diurnal rhythm of locomotor activity and may also influence entrainment of ocular pacemakers. Finally, attention has been focused on the optic nerve as a coupling pathway between various parts of the system. The evidence suggests that information transmitted in the optic nerves is involved in entrainment of the ocular pacemaker by light, and in ocular control of the locomotor activity rhythm.     

Phase plane description of crayfish swimmeret oscillator

Crayfish swimmeret system shows rhythmic, coordinated behavior when the command fibers are stimulated chronically by electrical pulses, and the oscillating frequency becomes faster with increasing stimulus frequency. This behavior is organized by the distributed neural oscillators in the abdominal ganglia. We investigated the dynamics of the neural oscillators which are controlled by command fibers. Phase resetting experiment technique was used for this purpose; a temporary cessation of commanding pulses, which was regarded as suppressive perturbation for the neural oscillator, was applied to the chronically stimulated oscillator, and phase transition curves (PTCs) were measured. For the short cessation of command pulses, type 1 PTCs were obtained. With increasing cessation length, the PTC shifted downward, and finally changed into type 0. We also measured PTCs for temporarily increased stimulus frequency, which was an excitatory perturbation for the neural oscillator and increased the frequency of the oscillation transiently. For the short excitatory perturbation, the PTCs were also type 1 and shifted upward. PTCs changed their shapes from type 1 into type 0, as increasing the perturbation length. These shapes of the PTCs contain important information about the properties of the neural oscillator. Analyzing these PTCs, we present a phase plane diagram which describes the character of the command control of the neural oscillator.     

Joint kinetic response during unexpectedly reduced plantar flexor torque provided by a robotic ankle exoskeleton during walking

During human walking, plantar flexor activation in late stance helps to generate a stable and economical gait pattern. Because plantar flexor activation is highly mediated by proprioceptive feedback, the nervous system must modulate reflex pathways to meet the mechanical requirements of gait. The purpose of this study was to quantify ankle joint mechanical output of the plantar flexor stretch reflex response during a novel unexpected gait perturbation. We used a robotic ankle exoskeleton to mechanically amplify the ankle torque output resulting from soleus muscle activation. We recorded lower-body kinematics, ground reaction forces, and electromyography during steady-state walking and during randomly perturbed steps when the exoskeleton assistance was unexpectedly turned off. We also measured soleus Hoffmann- (H-) reflexes at late stance during the two conditions. Subjects reacted to the unexpectedly decreased exoskeleton assistance by greatly increasing soleus muscle activity about 60 ms after ankle angle deviated from the control condition (p<0.001). There were large differences in ankle kinematic and electromyography patterns for the perturbed and control steps, but the total ankle moment was almost identical for the two conditions (p=0.13). The ratio of soleus H-reflex amplitude to background electromyography was not significantly different between the two conditions (p=0.4). This is the first study to show that the nervous system chooses reflex responses during human walking such that invariant ankle joint moment patterns are maintained during perturbations. Our findings are particularly useful for the development of neuromusculoskeletal computer simulations of human walking that need to adjust reflex gains appropriately for biomechanical analyses.