Improved Assessment of Orbital Stability of Rhythmic Motion with Noise

Mathematical techniques have provided tools to quantify the stability of rhythmic movements of humans and machines as well as mathematical models. One archetypal example is the use of Floquet multipliers: assuming periodic motion to be a limit-cycle of a nonlinear oscillator, local stability has been assessed by evaluating the rate of convergence to the limit-cycle. However, the accuracy of the assessment in experiments is questionable: Floquet multipliers provide a measure of orbital stability for deterministic systems, but various components of biological systems and machines involve inevitable noise. In this study, we show that the conventional estimate of orbital stability, which depends on regression, has bias in the presence of noise. We quantify the bias, and devise a new method to estimate orbital stability more accurately. Compared with previous methods, our method substantially reduces the bias, providing acceptable estimates of orbital stability with an order-of-magnitude fewer cycles.     

The validity of stability measures: A modelling approach

Measures calculated from unperturbed walking patterns, such as variability measures and maximum Floquet multipliers, are often used to study the stability of walking. However, it is unknown if, and to what extent, these measures correlate to the probability of falling.We studied whether in a simple model of human walking, i.e., a passive dynamic walker, the probability of falling could be predicted from maximum Floquet multipliers, kinematic state variability, and step time variability. We used an extended version of the basic passive dynamic walker with arced feet and a hip spring. The probability of falling was manipulated by varying the foot radius and hip spring stiffness, or varying these factors while co-varying the slope to keep step length constant.The simulation data indicated that Floquet multipliers and kinematic state variability correlated inconsistently with probability of falling. Step time variability correlated well with probability of falling, but a more consistent correlation with the probability of falling was found by calculating the variability of the log transform of the step time. Our findings speak against the use of maximum Floquet multipliers and suggest instead that variability of critical variables may be a good predictor of the probability to fall.     

The effects of sensory loss and walking speed on the orbital dynamic stability of human walking

Peripheral sensory feedback is believed to contribute significantly to maintaining walking stability. Patients with diabetic peripheral neuropathy have a greatly increased risk of falling. Previously, we demonstrated that slower walking speeds in neuropathic patients lead to improved local dynamic stability. However, all subjects exhibited significant local instability during walking, even though no subject fell or stumbled during testing. The present study was conducted to determine if and how significant changes in peripheral sensation and walking speed affect orbital stability during walking. Trunk and lower extremity kinematics were examined from two prior experiments that compared patients with significant neuropathy to healthy controls and walking at multiple different speeds in young healthy subjects. Maximum Floquet multipliers were computed for each time series to quantify the orbital stability of these movements. All subjects exhibited orbitally stable walking kinematics, even though these same kinematics were previously shown to be locally unstable. Differences in orbital stability between neuropathic and control subjects were small and, with the exception of knee joint movements (p=0.001), not statistically significant (0.380p0.946). Differences in knee orbital stability were not mediated by differences in walking speed. This was supported by our finding that although orbital stability improved slightly with slower walking speeds, the correlations between walking speed and orbital stability were generally weak (r(2)16.7%). Thus, neuropathic patients do not gain improved orbital stability as a result of slowing down and do not experience any loss of orbital stability because of their sensory deficits.     

Dynamic stability of human walking in visually and mechanically destabilizing environments

Understanding how humans remain stable during challenging locomotor activities is critical to developing effective tests to diagnose patients with increased fall risk. This study determined if different continuous low-amplitude perturbations would induce specific measureable changes in measures of dynamic stability during walking. We applied continuous pseudo-random oscillations of either the visual scene or support surface in either the anterior-posterior or mediolateral directions to subjects walking in a virtual environment with speed-matched optic flow. Floquet multipliers and short-term local divergence exponents both increased (indicating greater instability) during perturbed walking. These responses were generally much stronger for body movements occurring in the same directions as the applied perturbations. Likewise, subjects were more sensitive to both visual and mechanical perturbations applied in the mediolateral direction than to those applied in the anterior-posterior direction, consistent with previous experiments and theoretical predictions. These responses were likewise consistent with subjects' anecdotal perceptions of which perturbation conditions were most challenging. Contrary to the Floquet multipliers and short-term local divergence exponents, which both increased, long-term local divergence exponents decreased during perturbed walking. However, this was consistent with specific changes in the mean log divergence curves, which indicated that subjects' movements reached their maximum local divergence limits more quickly during perturbed walking. Overall, the Floquet multipliers were less sensitive, but reflected greater specificity in their responses to the different perturbation conditions. Conversely, the short-term local divergence exponents exhibited less specificity in their responses, but were more sensitive measures of instability in general.     

Differences between local and orbital dynamic stability during human walking

Currently there is no commonly accepted way to define, much less quantify, locomotor stability. In engineering, "orbital stability" is defined using Floquet multipliers that quantify how purely periodic systems respond to perturbations discretely from one cycle to the next. For aperiodic systems, "local stability" is defined by local divergence exponents that quantify how the system responds to very small perturbations continuously in real time. Triaxial trunk accelerations and lower extremity sagittal plane joint angles were recorded from ten young healthy subjects as they walked for 10 min over level ground and on a motorized treadmill at the same speed. Maximum Floquet multipliers (Max FM) were computed at each percent of the gait cycle (from 0% to 100%) for each time series to quantify the orbital stability of these movements. Analyses of variance comparing Max FM values between walking conditions and correlations between Max FM values and previously published local divergence exponent results were computed. All subjects exhibited orbitally stable walking kinematics (i.e., magnitudes of Max FM < 1.0), even though these same kinematics were previously found to be locally unstable. Variations in orbital stability across the gait cycle were generally small and exhibited no systematic patterns. Walking on the treadmill led to small, but statistically significant improvements in the orbital stability of mediolateral (p = 0.040) and vertical (p = 0.038) trunk accelerations and ankle joint kinematics (p = 0.002). However, these improvements were not exhibited by all subjects (p < or = 0.012 for subject x condition interaction effects). Correlations between Max FM values and previously published local divergence exponents were inconsistent and 11 of the 12 comparisons made were not statistically significant (r2 < or = 19.8%; p > or = 0.049). Thus, the variability inherent in human walking, which manifests itself as local instability, does not substantially adversely affect the orbital stability of walking. The results of this study will allow future efforts to gain a better understanding of where the boundaries lie between locally unstable movements that remain orbitally stable and those that lead to global instability (i.e., falling).     

Dynamics and stability of muscle activations during walking in healthy young and older adults

To facilitate stable walking, humans must generate appropriate motor patterns and effective corrective responses to perturbations. Yet most EMG analyses do not address the continuous nature of muscle activation dynamics over multiple strides. We compared muscle activation dynamics in young and older adults by defining a multivariate state space for muscle activity. Eighteen healthy older and 17 younger adults walked on a treadmill for 2 trials of 5 min each at each of 5 controlled speeds (80–120% of preferred). EMG linear envelopes of v. lateralis, b. femoris, gastrocnemius, and t. anterior of the left leg were obtained. Interstride variability, local dynamic stability (divergence exponents), and orbital stability (maximum Floquet multipliers; FM) were calculated. Both age groups exhibited similar preferred walking speeds (p=0.86). Amplitudes and variability of individual EMG linear envelopes increased with speed (p<0.01) in all muscles but gastrocnemius. Older adults also exhibited greater variability in b. femoris and t. anterior (p<0.004). When comparing continuous multivariate EMG dynamics, older adults demonstrated greater local and orbital instability of their EMG patterns (p<0.01). We also compared how muscle activation dynamics were manifested in kinematics. Local divergence exponents were strongly correlated between kinematics and EMG, independent of age and walking speed, while variability and max FM were not. These changes in EMG dynamics may be related to increased neuromotor noise associated with aging and may indicate subtle deterioration of gait function that could lead to future functional declines.     

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.     

Effects of perturbation magnitude on dynamic stability when walking in destabilizing environments

External perturbations applied to the walking surface or visual field can challenge an individual's ability to maintain stability during walking. Accurately quantifying and predicting changes in stability during walking will further our understanding of how individuals respond to challenges encountered during daily life and guide the development of assessments and rehabilitation interventions for individuals at increased risk of falling. This study is the first to determine how orbital and local dynamic stability metrics, including maximum Floquet multipliers and local divergence exponents, change in response to continuous mediolateral visual and surface perturbations of different amplitudes. Eleven healthy individuals walked in a fully immersive virtual environment. Participants completed two 3-min walking trials each under the following nine conditions: no perturbations, surface perturbations at each of 3 amplitudes, and visual perturbations at each of 5 amplitudes. All perturbations were applied as continuous pseudo-random oscillations. During both surface and visual perturbations, individuals were significantly more orbitally and locally unstable compared to un-perturbed walking. As walking surface perturbation amplitudes increased, individuals were more orbitally (but not locally) unstable. As visual perturbation amplitudes increased, individuals were more locally (but not orbitally) unstable between lower and higher amplitudes. Overall, these dynamic stability metrics were much less sensitive to changes in perturbation amplitudes than to differences between un-perturbed and perturbed walking, or to differences between mechanical and visual perturbations. This suggests that the type of perturbation(s) applied has a far greater impact than the magnitude of those perturbations in determining the response that will be elicited.     

Effects of walking speed, strength and range of motion on gait stability in healthy older adults

Falls pose a tremendous risk to those over 65 and most falls occur during locomotion. Older adults commonly walk slower, which many believe helps improve walking stability. While increased gait variability predicts future fall risk, increased variability is also caused by walking slower. Thus, we need to better understand how differences in age and walking speed independently affect dynamic stability during walking. We investigated if older adults improved their dynamic stability by walking slower, and how leg strength and flexibility (passive range of motion (ROM)) affected this relationship. Eighteen active healthy older and 17 healthy younger adults walked on a treadmill for 5min each at each of 5 speeds (80-120% of preferred). Local divergence exponents and maximum Floquet multipliers (FM) were calculated to quantify each subject's inherent local dynamic stability. The older subjects walked with the same preferred walking speeds as the younger subjects (p=0.860). However, these older adults still exhibited greater local divergence exponents (p<0.0001) and higher maximum FM (p<0.007) than the younger adults at all walking speeds. These older adults remained more locally unstable (p<0.04) even after adjusting for declines in both strength and ROM. In both age groups, local divergence exponents decreased at slower speeds and increased at faster speeds (p<0.0001). Maximum FM showed similar changes with speed (p<0.02). Both younger and older adults exhibited decreased instability by walking slower, in spite of increased variability. These increases in dynamic instability might be more sensitive indicators of future fall risk than changes in gait variability.     

The forces and couples in the human trunk during level walking

The intersegmental force and couple exchanged between upper and lower body across a transverse section passing through the fourth lumbar vertebra were estimated during level walking on a straight line at speeds ranging from 0.99 to 2.23 ms-1. This was done using 3-D kinematic information relative to the head, upper limbs, and upper torso, obtained through a stereophotogrammetric technique, and the relevant inertial parameters obtained using anthropometric measurements and estimation techniques provided in the literature. Twenty walking cycles of five normal adult male subjects were analysed. The intersegmental force and couple components are presented as referenced to both a laboratory and pelvic set of axes. Using these results some considerations are made concerning the variations which the overall trunk muscles effort undergoes because of mean walking speed changes. The muscular action on the trunk is inferred from the intersegmental couple components. The various factors that contribute to the build-up of the intersegmental force and couple are analysed and their relative importance assessed.     

Floquet theory: a useful tool for understanding nonequilibrium dynamics

Many ecological systems experience periodic variability. Theoretical investigation of population and community dynamics in periodic environments has been hampered by the lack of mathematical tools relative to equilibrium systems. Here, I describe one such mathematical tool that has been rarely used in the ecological literature but has widespread use: Floquet theory. Floquet theory is the study of the stability of linear periodic systems in continuous time. Floquet exponents/multipliers are analogous to the eigenvalues of Jacobian matrices of equilibrium points. In this paper, I describe the general theory, then give examples to illustrate some of its uses: it defines fitness of structured populations, it can be used for invasion criteria in models of competition, and it can test the stability of limit cycle solutions. I also provide computer code to calculate Floquet exponents and multipliers. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Influence of simulated neuromuscular noise on the dynamic stability and fall risk of a 3D dynamic walking model

Measures that can predict risk of falling are essential for enrollment of older adults into fall prevention programs. Local and orbital stability directly quantify responses to very small perturbations and are therefore putative candidates for predicting fall risk. However, research to date is not conclusive on whether and how these measures relate to fall risk. Testing this empirically would be time consuming or may require high risk tripping experiments. Simulation studies therefore provide an important tool to initially explore potential measures to predict fall risk. This study performed simulations with a 3D dynamic walking model to explore if and how dynamic stability measures predict fall risk. The model incorporated a lateral step controller to maintain lateral stability. Neuronal noise of increasing amplitude was added to this controller to manipulate fall risk. Short-term (λ(S)(*)) local instability did predict fall risk, but long-term (λ(L)(*)) local instability and orbital stability (maxFM) did not. Additionally, λ(S)(*) was an early predictor for fall risk as it started increasing before fall risk increased. Therefore, λ(S)(*) could be a very useful tool to identify older adults whose fall risk is about to increase, so they can be enrolled in fall prevention programs before they actually fall.     

The effects of gait strategy on metabolic rate and indicators of stability during downhill walking

When walking at a given speed, humans often appear to prefer gait patterns that minimize metabolic rate, thereby maximizing metabolic economy. However, recent experiments have demonstrated that humans do not maximize economy when walking downhill. The purpose of this study was to investigate whether this non-metabolically optimal behavior is the result of a trade-off between metabolic economy and gait stability. We hypothesized that humans have the ability to modulate their gait strategy to increase either metabolic economy or stability, but that increase in one measure will be accompanied by decrease in the other. Subjects walked downhill using gait strategies ranging from risky to conservative, which were either prescribed by verbal instructions or induced by the threat of perturbations. We quantified spatiotemporal gait characteristics, metabolic rate and several indicators of stability previously associated with fall risk: stride period variability; step width variability; Lyapunov exponents; Floquet multipliers; and stride period fractal index. When subjects walked using conservative gait strategies, stride periods and lengths decreased, metabolic rate increased, and anteroposterior maximum Lyapunov exponents increased, which has previously been interpreted as an indicator of decreased stability. These results do not provide clear support for the proposed trade-off between economy and stability, particularly when stability is approximated using complex metrics. However, several gait pattern changes previously linked to increased fall risk were observed when our healthy subjects walked with a conservative strategy, suggesting that these changes may be a response to, rather than a cause of, increased fall risk.     

Can stability really predict an impending slip-related fall among older adults?

The primary purpose of this study was to systematically evaluate and compare the predictive power of falls for a battery of stability indices, obtained during normal walking among community-dwelling older adults. One hundred and eighty seven community-dwelling older adults participated in the study. After walking regularly for 20 strides on a walkway, participants were subjected to an unannounced slip during gait under the protection of a safety harness. Full body kinematics and kinetics were monitored during walking using a motion capture system synchronized with force plates. Stability variables, including feasible-stability-region measurement, margin of stability, the maximum Floquet multiplier, the Lyapunov exponents (short- and long-term), and the variability of gait parameters (including the step length, step width, and step time), were calculated for each subject. Sensitivity of predicting slip outcome (fall vs. recovery) was examined for each stability variable using logistic regression. Results showed that the feasible-stability-region measurement predicted fall incidence among these subjects with the highest sensitivity (68.4%). Except for the step width (with an sensitivity of 60.2%), no other stability variables could differentiate fallers from those who did not fall for the sample included in this study. The findings from the present study could provide guidance to identify individuals at increased risk of falling using the feasible-stability-region measurement or variability of the step width.     

Reliability and Minimum Detectable Change of Temporal-Spatial,Kinematic, and Dynamic Stability Measures during Perturbed Gait

Temporal-spatial, kinematic variability, and dynamic stability measures collected during perturbation-based assessment paradigms are often used to identify dysfunction associated with gait instability. However, it remains unclear which measures are most reliable for detecting and tracking responses to perturbations. This study systematically determined the between-session reliability and minimum detectable change values of temporal-spatial, kinematic variability, and dynamic stability measures during three types of perturbed gait. Twenty young healthy adults completed two identical testing sessions two weeks apart, comprised of an unperturbed and three perturbed (cognitive, physical, and visual) walking conditions in a virtual reality environment. Within each session, perturbation responses were compared to unperturbed walking using paired t-tests. Between-session reliability and minimum detectable change values were also calculated for each measure and condition. All temporal-spatial, kinematic variability and dynamic stability measures demonstrated fair to excellent between-session reliability. Minimal detectable change values, normalized to mean values ranged from 1–50%. Step width mean and variability measures demonstrated the greatest response to perturbations with excellent between-session reliability and low minimum detectable change values. Orbital stability measures demonstrated specificity to perturbation direction and sensitivity with excellent between-session reliability and low minimum detectable change values. We observed substantially greater between-session reliability and lower minimum detectable change values for local stability measures than previously described which may be the result of averaging across trials within a session and using velocity versus acceleration data for reconstruction of state spaces. Across all perturbation types, temporal-spatial, orbital and local measures were the most reliable measures with the lowest minimum detectable change values, supporting their use for tracking changes over multiple testing sessions. The between-session reliability and minimum detectable change values reported here provide an objective means for interpreting changes in temporal-spatial, kinematic variability, and dynamic stability measures during perturbed walking which may assist in identifying instability.     

Modeling the Role of Diffusion Coefficients on Turing Instability in a Reaction-diffusion Prey-predator System

The paper is concerned with the effect of variable dispersal rates on Turing instability of a non-Lotka-Volterra reaction-diffusion system. In ecological applications, the dispersal rates of different species tends to oscillate in time. This oscillation is modeled by temporal variation in the diffusion coefficient with large as well as small periodicity. The case of large periodicity is analyzed using the theory of Floquet multipliers and that of the small periodicity by using Hill's equation. The effect of such variation on the resulting Turing space is studied. A comparative analysis of the Turing spaces with constant diffusivity and variable diffusivities is performed. Numerical simulations are carried out to support analytical findings.     

Dynamic stability of signal transduction networks depending on downstream and upstream specificity of protein kinases

Mathematical methods are used for explaining the structural design of signal transduction networks, e.g. MAP kinase cascades, which control cell proliferation, differentation or apoptosis. Taking into account protein kinases and phosphatases the interrelation between the topology of signaling networks and the stability of their ground state are analysed. It is shown that the stability is closely related to the system's dimension and to the number of cycles within the network. Systems with a higher number of kinases and/or cycles tend to be more unstable. In contrast to that increasing phosphatase activity stabilises the ground state.     

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.     

Effect of noise and filtering on largest Lyapunov exponent of time series associated with human walking

This study aimed to determine the effect of added noise, filtering and time series length on the largest Lyapunov exponent (LyE) value calculated for time series obtained from a passive dynamic walker. The simplest passive dynamic walker model comprising of two massless legs connected by a frictionless hinge joint at the hip was adopted to generate walking time series. The generated time series was used to construct a state space with the embedding dimension of 3 and time delay of 100 samples. The LyE was calculated as the exponential rate of divergence of neighboring trajectories of the state space using Rosenstein’s algorithm. To determine the effect of noise on LyE values, seven levels of Gaussian white noise (SNR = 55–25 dB with 5 dB steps) were added to the time series. In addition, the filtering was performed using a range of cutoff frequencies from 3 Hz to 19 Hz with 2 Hz steps. The LyE was calculated for both noise-free and noisy time series with different lengths of 6, 50, 100 and 150 strides. Results demonstrated a high percent error in the presence of noise for LyE. Therefore, these observations suggest that Rosenstein’s algorithm might not perform well in the presence of added experimental noise. Furthermore, findings indicated that at least 50 walking strides are required to calculate LyE to account for the effect of noise. Finally, observations support that a conservative filtering of the time series with a high cutoff frequency might be more appropriate prior to calculating LyE.     

Musculo-skeletal loading conditions at the hip during walking and stair climbing.

Musculo-skeletal loading plays an important role in the primary stability of joint replacements and in the biological processes involved in fracture healing. However, current knowledge of musculo-skeletal loading is still limited. In the past, a number of musculo-skeletal models have been developed to estimate loading conditions at the hip. So far, a cycle-to-cycle validation of predicted musculo-skeletal loading by in vivo measurements has not been possible. The aim of this study was to determine the musculo-skeletal loading conditions during walking and climbing stairs for a number of patients and compare these findings to in vivo data.Following total hip arthroplasty, four patients underwent gait analysis during walking and stair climbing. An instrumented femoral prosthesis enabled simultaneous measurement of in vivo hip contact forces. On the basis of CT and X-ray data, individual musculo-skeletal models of the lower extremity were developed for each patient. Muscle and joint contact forces were calculated using an optimization algorithm. The calculated peak hip contact forces both over- and under-estimated the measured forces. They differed by a mean of 12% during walking and 14% during stair climbing.For the first time, a cycle-to-cycle validation of predicted musculo-skeletal loading was possible for walking and climbing stairs in several patients. In all cases, the comparison of in vivo measured and calculated hip contact forces showed good agreement.Thus, the authors consider the presented approach as a useful means to determine valid conditions for the analysis of prosthesis loading, bone modeling or remodeling processes around implants and fracture stability following internal fixation.