Energy efficient walking with central pattern generators: from passive dynamic walking to biologically inspired control

Like human walking, passive dynamic walking—i.e. walking down a slope with no actuation except gravity—is energy efficient by exploiting the natural dynamics. In the animal world, neural oscillators termed central pattern generators (CPGs) provide the basic rhythm for muscular activity in locomotion. We present a CPG model, which automatically tunes into the resonance frequency of the passive dynamics of a bipedal walker, i.e. the CPG model exhibits resonance tuning behavior. Each leg is coupled to its own CPG, controlling the hip moment of force. Resonance tuning above the endogenous frequency of the CPG—i.e. the CPG’s eigenfrequency—is achieved by feedback of both limb angles to their corresponding CPG, while integration of the limb angles provides resonance tuning at and below the endogenous frequency of the CPG. Feedback of the angular velocity of both limbs to their corresponding CPG compensates for the time delay in the loop coupling each limb to its CPG. The resonance tuning behavior of the CPG model allows the gait velocity to be controlled by a single parameter, while retaining the energy efficiency of passive dynamic walking.     

Perturbation analysis of 6DoF flight dynamics and passive dynamic stability of hovering fruit fly Drosophila melanogaster

Insects exhibit exquisite control of their flapping flight, capable of performing precise stability and steering maneuverability. Here we develop an integrated computational model to investigate flight dynamics of insect hovering based on coupling the equations of 6 degree of freedom (6DoF) motion with the Navier-Stokes (NS) equations. Unsteady aerodynamics is resolved by using a biology-inspired dynamic flight simulator that integrates models of realistic wing-body morphology and kinematics, and a NS solver. We further develop a dynamic model to solve the rigid body equations of 6DoF motion by using a 4th-order Runge-Kutta method. In this model, instantaneous forces and moments based on the NS-solutions are represented in terms of Fourier series. With this model, we perform a systematic simulation-based analysis on the passive dynamic stability of a hovering fruit fly, Drosophila melanogaster, with a specific focus on responses of state variables to six one-directional perturbation conditions during latency period. Our results reveal that the flight dynamics of fruit fly hovering does not have a straightforward dynamic stability in a conventional sense that perturbations damp out in a manner of monotonous convergence. However, it is found to exist a transient interval containing an initial converging response observed for all the six perturbation variables and a terminal instability that at least one state variable subsequently tends to diverge after several wing beat cycles. Furthermore, our results illustrate that a fruit fly does have sufficient time to apply some active mediation to sustain a steady hovering before losing body attitudes.     

Local dynamic stability of treadmill walking: Intrasession and week-to-week repeatability

Repetitive falls degrade the quality of life of elderly people and of patients suffering of various neurological disorders. In order to prevent falls while walking, one should rely on relevant early indicators of impaired dynamic balance. The local dynamic stability (LDS) represents the sensitivity of gait to small perturbations: divergence exponents (maximal Lyapunov exponents) assess how fast a dynamical system diverges from neighbor points. Although numerous findings attest the validity of LDS as a fall risk index, reliability results are still sparse. The present study explores the intrasession and intersession repeatability of gait LDS using intraclass correlation coefficients (ICC) and standard error of measurement (SEM). Ninety-five healthy individuals performed 5 min treadmill walking in two sessions separated by 9 days. Trunk acceleration was measured with a 3D accelerometer. Three time scales were used to estimate LDS: over 4–10 strides (λ_4–10), over one stride (λ₁) and over one step (λ_0.5). The intrasession repeatability was assessed from three repetitions of either 35 strides or 70 strides taken within the 5 min tests. The intersession repeatability compared the two sessions, which totalized 210 strides. The intrasession ICCs (70-strides estimates/35-strides estimates) were 0.52/0.18 for λ_4–10 and 0.84/0.77 for λ₁ and λ_0.5. The intersession ICCs were around 0.60. The SEM results revealed that λ_0.5 measured in medio-lateral direction exhibited the best reliability, sufficient to detect moderate changes at individual level (20%). However, due to the low intersession repeatability, one should average several measurements taken on different days in order to better approximate the true LDS.     

The effect of walking speed on local dynamic stability is sensitive to calculation methods

Local dynamic stability has been assessed by the short-term local divergence exponent (λ_S), which quantifies the average rate of logarithmic divergence of infinitesimally close trajectories in state space. Both increased and decreased local dynamic stability at faster walking speeds have been reported. This might pertain to methodological differences in calculating λ_S. Therefore, the aim was to test if different calculation methods would induce different effects of walking speed on local dynamic stability. Ten young healthy participants walked on a treadmill at five speeds (60%, 80%, 100%, 120% and 140% of preferred walking speed) for 3 min each, while upper body accelerations in three directions were sampled. From these time-series, λ_S was calculated by three different methods using: (a) a fixed time interval and expressed as logarithmic divergence per stride-time (λ_S−a), (b) a fixed number of strides and expressed as logarithmic divergence per time (λ_S−b) and (c) a fixed number of strides and expressed as logarithmic divergence per stride-time (λ_S−c). Mean preferred walking speed was 1.16±0.09 m/s. There was only a minor effect of walking speed on λ_S−a. λ_S−b increased with increasing walking speed indicating decreased local dynamic stability at faster walking speeds, whereas λ_S−c decreased with increasing walking speed indicating increased local dynamic stability at faster walking speeds. Thus, the effect of walking speed on calculated local dynamic stability was significantly different between methods used to calculate local dynamic stability. Therefore, inferences and comparisons of studies employing λ_S should be made with careful consideration of the calculation method.     

Humans exploit the biomechanics of bipedal gait during visually guided walking over complex terrain

How do humans achieve such remarkable energetic efficiency when walking over complex terrain such as a rocky trail? Recent research in biomechanics suggests that the efficiency of human walking over flat, obstacle-free terrain derives from the ability to exploit the physical dynamics of our bodies. In this study, we investigated whether this principle also applies to visually guided walking over complex terrain. We found that when humans can see the immediate foreground as little as two step lengths ahead, they are able to choose footholds that allow them to exploit their biomechanical structure as efficiently as they can with unlimited visual information. We conclude that when humans walk over complex terrain, they use visual information from two step lengths ahead to choose footholds that allow them to approximate the energetic efficiency of walking in flat, obstacle-free environments.     

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.     

A model for insect locomotion in the horizontal plane: Feedforward activation of fast muscles, stability, and robustness

We develop a neuromechanical model for running insects that includes a simplified hexapedal leg geometry with agonist-antagonist muscle pairs actuating each leg joint. Restricting to dynamics in the horizontal plane and neglecting leg masses, we reduce the model to three degrees of freedom describing translational and yawing motions of the body. Muscles are driven by stylized action potentials characteristic of fast motoneurons, and modeled using an activation function and nonlinear length and shortening velocity dependence. Parameter values are based on measurements from depressor muscles and observations of kinematics and dynamics of the cockroach Blaberus discoidalis; in particular, motoneuronal inputs and muscle force levels are chosen to approximately achieve joint torques that are consistent with measured ground reaction forces. We show that the model has stable double-tripod gaits over the animal's speed range, that its dynamics at preferred speeds matches those observed, and that it maintains stable gaits, with low frequency yaw deviations, when subject to random perturbations in foot touchdown and lift-off timing and action potential input timing. We explain this in terms of the low-dimensional dynamics.     

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.     

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.     

Allele-specific PCR can improve the efficiency of experimental resolution of heterozygotes in resequencing studies

Resolution of the two haplotypes present in an individual that is heterozygous at a locus has been a difficult problem for nucleotide sequence-based population genetic studies. Here, we demonstrate a method in which allele-specific polymerase chain reaction (AS-PCR) and computational phasing are combined for relatively high-throughput, efficient resolution of phase in resequencing studies. Using data from multiple loci that were fully experimentally phased, we demonstrate that the popular computational tool PHASE can accurately phase heterozygous individuals with common SNPs (single nucleotide polymorphisms) and/or common haplotypes. However, we also demonstrate that experimental phasing with AS-PCR can efficiently supplement computational phasing, providing a rapid means to phase individuals with rare SNPs or haplotypes and with heterozygous insertion/deletion polymorphisms. By following simple stepwise procedures, AS-PCR can result in much more efficient and accurate experimental phasing of haplotypes than is possible with traditional methods such as cloning.     

Autophagy can repair endoplasmic reticulum stress damage of the passive Heymann nephritis model as revealed by proteomics analysis

Membranous nephropathy is a common cause of nephrotic syndrome in adults. Although many mechanisms have been proposed, whole proteomic research is still lacking. We analyzed the passive Heymann nephritis animal model using label-free quantitative proteome technology. Results showed 160 differential proteins between control and PHN model groups at days 14 and 21. The expression level of endoplasmic reticulum stress (ERS)-associated protein GRP78 and GRP94 protein was up-regulated on day 14 or 21, which was confirmed by Western blotting. The results also showed that the autophagy marker LC3 was up-regulated in the models. Furthermore, we used tunicamycin to induce ERS of podocytes in vitro to investigate the mechanism. Results of Western blotting revealed that the expression of GRP78, GRP94, and LC3 was up-regulated, while that of the cytoskeletal protein tubulin-β was down-regulated, and immunofluorescence displayed disordered distribution of tubulin-β. These suggest that ERS plays an important role in podocyte damage. Autophagy can repair the cytoskeleton damage caused by ERS as a protective mechanism. This provides an important basis for a thorough understanding of the mechanism of podocyte damage and the pathogenesis of membranous nephropathy.     

Impacts of dispersal on rapid adaptation and dynamic stability of Daphnia in fluctuating environments

Prior ecological research has shown that spatial processes can enhance the temporal stability of populations in fluctuating environments. Less explored is the effect of dispersal on rapid adaptation and its concomitant impact on population dynamics. For asexually reproducing populations, theory predicts that dispersal in fluctuating environments can facilitate asynchrony among clones and enhance stability by reducing temporal variability of total population abundance. This effect is predicted when clones exhibit heritable variation in environmental optima and when fluctuations occur asynchronously among patches. We tested this in the field using artificial ponds and metapopulations composed of a diverse assemblage of Daphnia pulex clones. We directly manipulated dispersal presence/absence and environmental fluctuations in the form of nutrient pulses. Consistent with predictions, dispersal enhanced temporal asynchrony among clones in the presence of nutrient pulses; this in turn stabilized population dynamics. This effect only emerged when patches experienced spatially asynchronous nutrient pulses (dispersal had no effect when patches were synchronously pulsed). Clonal asynchrony was driven by strong positive selection for a single clone that exhibited a performance advantage under conditions of low resource availability. Our work highlights the importance of dispersal as a driver of eco-evolutionary dynamics and population stability in variable environments.     

Mate-search efficiency can determine the evolution of separate sexes and the stability of hermaphroditism in animals

Limited availability of mating partners has been proposed as an explanation for the occurrence of simultaneous hermaphroditism in animals with pair mating. When low population density or low mobility of a species limits the number of potential mates, simultaneous hermaphrodites may have a selective advantage because, first, they are able to adjust the allocation of resources between male and female functions in order to maximize fitness; second, in a hermaphroditic population the likelihood of meeting a partner is higher because all individuals are potential mates; and, third, in the absence of mating partners, many simultaneously hermaphroditic animals have the option of reproducing through self-fertilization. Recognizing that mate availability is central to the existing theory of hermaphroditism in animals, it is important to examine the effects of mate search on predictions of the stability of hermaphroditism. Many hermaphroditic animals can increase the number of potential mates they contact by active searching. However, since mate search has costs in terms of time and energy, the increased number of potential mates will be traded off against the amount of resources that can be allocated to the production of gametes. We explore the consequences of this trade-off to the evolution of mating strategies and to the selective advantage of self-fertilization. We show that in low and moderate population densities, poor mate-search efficiency and high costs of searching stabilize hermaphroditism and bias sex allocation toward female function. In addition, in very low population densities, there is strong selective advantage for self-fertilization, but this advantage decreases considerably in species with high mate-search efficiency. Most important, however, we present a novel evolutionary prediction: when mate search is efficient, disruptive frequency-dependent selection on time allocation to mate search leads to the evolution of searching and nonsearching phenotypes and, ultimately, to the evolution of males and females.     

Mathematical modeling of the hypothalamic-pituitary-adrenal system activity

Mathematical modeling has proven to be valuable in understanding of the complex biological systems dynamics. In the present report we have developed an initial model of the hypothalamic-pituitary-adrenal system self-regulatory activity. A four-dimensional non-linear differential equation model of the hormone secretion was formulated and used to analyze plasma cortisol levels in humans. The aim of this work was to explore in greater detail the role of this system in normal, homeostatic, conditions, since it is the first and unavoidable step in further understanding of the role of this complex neuroendocrine system in pathophysiological conditions. Neither the underlying mechanisms nor the physiological significance of this system are fully understood yet.     

Symmetry and reproducibility of the components of dynamic stability in young adults at different walking velocities on the treadmill

In the literature, analysis of dynamic gait stability using the extrapolated center of mass concept is often an objective that assumes reproducible and symmetrical data. Here, we examined the validity of this assumption by analyzing subjects walking at different velocities. Eleven healthy young subjects walked on a treadmill at six different velocities (1.0-2.0m·s(-1)). Dynamic stability at touchdown of the left and right foot (10 gait trials for each body side) was investigated by using the margin of stability, determined as the difference between base of support and extrapolated center of mass. Dynamic stability parameters showed no significant differences (P>0.05) between gait trials, with a root mean square difference in margin of stability of less than 1.62cm. Correlation coefficients between trials were above 0.70 for all parameters, demonstrating that two gait trials are sufficient to obtain reproducible data. In more than 90% of the cases, the absolute symmetry index was below 8% with no relevant functional differences between body sides. We concluded that analyzing two gait trials for one body side is sufficient to determine representative characteristics of the components of dynamic stability in healthy young adults while walking on the treadmill at a wide range of velocities.     

A reliable Taylor series-based computational method for the calculation of dynamic sensitivities in large-scale metabolic reaction systems: Algorithm and software evaluation

Dynamic sensitivity analysis has become an important tool to successfully characterize all sorts of biological systems. However, when the analysis is carried out on large scale systems, it becomes imperative to employ a highly accurate computational method in order to obtain reliable values. Furthermore, the preliminary laborious mathematical operations required by current software before the computation of dynamic sensitivities makes it inconvenient for a significant number of unacquainted users. To satisfy these needs, the present work investigates a newly developed algorithm consisting of a combination of Taylor series method that can directly execute Taylor expansions for simultaneous non-linear-differential equations and a simple but highly-accurate numerical differentiation method based on finite-difference formulas. Applications to three examples of biochemical systems indicate that the proposed method makes it possible to compute the dynamic sensitivity values with highly-reliable accuracies and also allows to readily compute them by setting up only the differential equations for metabolite concentrations in the computer program. Also, it is found that the Padé approximation introduced in the Taylor series method shortens the computation time greatly because it stabilizes the computation so that it allows us to use larger stepsizes in the numerical integration. Consequently, the calculated results suggest that the proposed computational method, in addition to being user-friendly, makes it possible to perform dynamic sensitivity analysis in large-scale metabolic reaction systems both efficiently and reliably.     

The 3' UTR of the human CTLA4 mRNA can regulate mRNA stability and translational efficiency

T cell activation results from the integration of signals generated through the T cell antigen receptor-CD3 complex with those from additional positive and negative regulatory pathways mainly mediated by the engagement of costimulatory receptors on T cells. Disruption of this balance leads to a defective immune response or alternative over-activation of the immune system. CTLA-4 plays a critical role in downregulating T cell responses. Autoimmune diseases have shown genetic linkage to the CTLA4 locus. In this report we demonstrate that the 3' UTR of CTLA4 regulates firefly luciferase reporter gene expression, can confer instability to CTLA4 mRNA and can influence its translation efficiency in vitro.     

Leg muscles that mediate stability: mechanics and control of two distal extensor muscles during obstacle negotiation in the guinea fowl

Here, we used an obstacle treadmill experiment to investigate the neuromuscular control of locomotion in uneven terrain. We measured in vivo function of two distal muscles of the guinea fowl, lateral gastrocnemius (LG) and digital flexor-IV (DF), during level running, and two uneven terrains, with 5 and 7 cm obstacles. Uneven terrain required one step onto an obstacle every four to five strides. We compared both perturbed and unperturbed strides in uneven terrain to level terrain. When the bird stepped onto an obstacle, the leg became crouched, both muscles acted at longer lengths and produced greater work, and body height increased. Muscle activation increased on obstacle strides in the LG, but not the DF, suggesting a greater reflex contribution to LG. In unperturbed strides in uneven terrain, swing pre-activation of DF increased by 5 per cent compared with level terrain, suggesting feed-forward tuning of leg impedance. Across conditions, the neuromechanical factors in work output differed between the two muscles, probably due to differences in muscle-tendon architecture. LG work depended primarily on fascicle length, whereas DF work depended on both length and velocity during loading. These distal muscles appear to play a critical role in stability by rapidly sensing and responding to altered leg-ground interaction.     

Computational centrosomics: An approach to understand the dynamic behaviour of centrosome

Centrosomes are the key regulating element of cell cycle progression. Aberrations in their functional mechanism leads to several cancer related disorders. Although genomic studies in the field of centrosome have been extensively carried out, with the lack of structural conformation, the proteomic analysis of pathological genetic mutation is still a challenging task. Several computational algorithms and high range force fields are used to design the 3D structure conformation of proteins, which has now become the leading platform for in-silico drug discovery approaches. Application of these highly efficient platforms in centrosomics studies will be a novel approach to develop an efficient drug therapy for the treatment of their dysfunction disorders.