A double-inverted pendulum model for studying the adaptability of postural control to frequency during human stepping in place

In order to analyze the influence of gravity and body characteristics on the control of center of mass (CM) oscillations in stepping in place, equations of motion in oscillating systems were developed using a double-inverted pendulum model which accounts for both the head-arms-trunk (HAT) segment and the two-legged system. The principal goal of this work is to propose an equivalent model which makes use of the usual anthropometric data for the human body, in order to study the ability of postural control to adapt to the step frequency in this particular paradigm of human gait. This model allows the computation of CM-to-CP amplitude ratios, when the center of foot pressure (CP) oscillates, as a parametric function of the stepping in place frequency, whose parameters are gravity and major body characteristics. Motion analysis from a force plate was used to test the model by comparing experimental and simulated values of variations of the CM-to-CP amplitude ratio in the frontal plane versus the frequency. With data from the literature, the model is used to calculate the intersegmental torque which stabilizes the HAT when the Leg segment is subjected to a harmonic torque with an imposed frequency. Received: 24 February 1997 / Accepted in revised form: 30 June 1998     

Pulse wave propagation in a model human arterial network: assessment of 1-D numerical simulations against in vitro measurements

A numerical model based on the nonlinear, one-dimensional (1-D) equations of pressure and flow wave propagation in conduit arteries is tested against a well-defined experimental 1:1 replica of the human arterial tree. The tree consists of 37 silicone branches representing the largest central systemic arteries in the human, including the aorta, carotid arteries and arteries that perfuse the upper and lower limbs and the main abdominal organs. The set-up is mounted horizontally and connected to a pulsatile pump delivering a periodic output similar to the aortic flow. Terminal branches end in simple resistance models, consisting of stiff capillary tubes leading to an overflow reservoir that reflects a constant venous pressure. The parameters required by the numerical algorithm are directly measured in the in vitro set-up and no data fitting is involved. Comparison of experimental and numerical pressure and flow waveforms shows the ability of the 1-D time-domain formulation to capture the main features of pulse wave propagation measured throughout the system test. As a consequence of the simple resistive boundary conditions used to reduce the uncertainty of the parameters involved in the simulation, the experimental set-up generates waveforms at terminal branches with additional non-physiological oscillations. The frequencies of these oscillations are well captured by the 1-D model, even though amplitudes are overestimated. Adding energy losses in bifurcations and including fluid inertia and compliance to the purely resistive terminal models does not reduce the underdamped effect, suggesting that wall visco-elasticity might play an important role in the experimental results. Nevertheless, average relative root-mean-square errors between simulations and experimental waveforms are smaller than 4% for pressure and 19% for the flow at all 70 locations studied.     

Dynamic reconstruction of the fish locomotor wave

Theoretical modeling substantiates the bionic solution of an optimal wave propulsor for water transport, which must have the amplitude and phase characteristics of the fish bending locomotor waves. Use is made of a computer model of multilink chain bending waves where arbitrary distributions can be set for amplitudes and phases of separate segments. With the linear increase in the amplitude of segmental oscillations, a numerical experiment determines the optimal phasing whereby each segment provides a maximal longitudinal component of the motive force. Two versions of hydrodynamic interaction of the segments with water are compared, implying (i) linear and (ii) quadratic drag: the optimal phasing of the transverse and longitudinal oscillations proves to be (i) orthogonal and (ii) nearly orthogonal. The computed bending wave shapes corresponding to dynamic optimization are consistent with the experimental data.     

Flutter in collapsible tubes: a theoretical model of wheezes

A mathematical analysis of flow through a flexible channel is examined as a model of flow-induced flutter oscillations that pertain to the production of wheezing breath sounds. The model provides predictions for the critical fluid speed that will initiate flutter waves of the wall, as well as their frequency and wavelength. The mathematical results are separated into linear theory (small oscillations) and nonlinear theory (larger oscillations). Linear theory determines the onset of the flutter, whereas nonlinear theory determines the relationships between the fluid speed and both the wave amplitudes and frequencies. The linear theory predictions correlate well with data taken at the onset of flutter and flow limitation during experiments of airflow in thick-walled collapsible tubes. The nonlinear theory predictions correlate well with data taken as these flows are forced to higher velocities while keeping the flow rate constant. Particular ranges of the parameters are selected to investigate and discuss the applications to airway flows. According to this theory, the mechanism of generation of wheezes is based in the interactions of fluid forces and friction and wall elastic-restoring forces and damping. In particular, a phase delay between the fluid pressure and wall motion is necessary. The wave speed theory of flow limitation is discussed with respect to the specific data and the flutter model.     

Rapid and accurate estimation of release conditions in the javelin throw

We have developed a system to measure initial conditions in the javelin throw rapidly enough to be used by the thrower for feedback in performance improvement. The system consists of three subsystems whose main tasks are: (A) acquisition of automatically digitized high speed (200 Hz) video x, y position data for the first 0.1-0.2 s of the javelin flight after release (B) estimation of five javelin release conditions from the x, y position data and (C) graphical presentation to the thrower of these release conditions and a simulation of the subsequent flight together with optimal conditions and flight for the sam release velocity. The estimation scheme relies on a simulation model and is at least an order of magnitude more accurate than previously reported measurements of javelin release conditions. The system provides, for the first time ever in any throwing event, the ability to critique nearly instantly in a precise, quantitative manner the crucial factors in the throw which determine the range. This should be expected to much greater control and consistency of throwing variables by athletes who use system and could even lead to an evolution of new throwing techniques.     

Modeling complex neutrophil dynamics in the grey collie

We have developed a mathematical model for the peripheral regulation of neutrophil production mediated by granulocyte colony-stimulating factor. We have used that model to show that the pattern of neutrophil oscillations in nine grey collies is consistent with the hypothesis that cyclical neutropenia is due to an oscillatory stem cell input to the neutrophil regulatory system, and not due to autonomous oscillations in the peripheral neutrophil regulatory system. In the process of interfacing our model with the laboratory data, we have estimated parameters for the peripheral neutrophil control system consistent with higher than normal apoptotic cell loss within the recognizable neutrophil precursors. This is in agreement with other experimental data. Our estimated model parameters also predict that the peripheral neutrophil production system is globally stable in the grey collies we studied. This further supports our hypothesis that the origin of the oscillatory behavior in cyclical neutropenia is in the stem cell population, consistent with other clinical and experimental evidence.     

Diagnostics of plasma oscillations by the field of surface waves guided by a dielectric waveguide

The potentialities of the diagnostic method for determining the plasma parameters by recording the surface waves guided by a dielectric waveguide and scattered by plasma oscillations are discussed. The use of surface (slowed) waves makes it possible to improve both the sensitivity and spatial resolution of measurements. The scattering is the most intense near the waveguide cutoff, at which the dependence of the wave propagation constant on the plasma density is the steepest. It is shown experimentally that the method proposed makes it possible to determine the discharge plasma density and electron energy and to estimate the amplitude of the RF field of the plasma waves forming the discharge and the amplitude of plasma density oscillations in these waves. The data obtained from the measurements of the amplitudes of both high-and low-frequency plasma density oscillations by the proposed method agree satisfactorily with theoretical predictions. The experimental data on the plasma density are confirmed by other diagnostic measurements. The ways of reducing measurement errors are proposed.     

Model for energy oscillations in cells

Rapid periodic pulses have been observed in yeast cell walls and these pulsations must be accompanied by coherent oscillations of energy. Such energy oscillations are likely to be a common feature in cells and we explore other consequences, either known or unknown, that may originate from chemical oscillations of small amplitude. We do not consider specific mechanisms for the energy oscillations, but here we accept their existence as a fact following from the yeast experiments. Chemical oscillators are treated as generic quantum oscillators and this model predicts that observed frequencies should have a simple volume-dependence where smaller cells exhibit higher frequencies than their larger counterparts. An extension to multicellular organisms then affords a derivation of the celebrated Kleiber law that evaluates both the numerical coefficient and the 34 exponent of mass. Calculations of activation energies and efficiencies at experimental temperatures follow. Finally, the model is applied to derive established expressions for blood flow and pulse rate. We conclude that the model reflects some common metabolic process insofar as it agrees with diverse quantitative findings while using minimal input data and without introducing free parameters.     

High-frequency oscillations in a pulse wave signal and their relation to adaptation responses

High-frequency oscillations in a pulse wave signal in the range of 1-50 Hz and their relation to heart rhythms have been investigated. Informative parameters to estimate the adaptation reactions have been determined using the Kohonen maps, the results of simulating the pulse wave signal (single rheocycles), and the experimental data. The coefficient of harmonic variation was used as an informative parameter. It was shown that the space of signs in the frequency range of 1-12.5 Hz, formed on the basis of the coefficient of harmonic variation, possesses a significant degree of structuring relative to the distribution of data for different types of adaptation reactions.     

The role of organ of Corti mass in passive cochlear tuning

The mechanism for passive cochlear tuning remains unsettled. Early models considered the organ of Corti complex (OCC) as a succession of spring-mass resonators. Later, traveling wave models showed that passive tuning could arise through the interaction of cochlear fluid mass and OCC stiffness without local resonators. However, including enough OCC mass to produce local resonance enhanced the tuning by slowing and thereby growing the traveling wave as it approached its resonant segment. To decide whether the OCC mass plays a role in tuning, the frequency variation of the wavenumber of the cochlear traveling wave was measured (in vivo, passive cochleae) and compared to theoretical predictions. The experimental wavenumber was found by taking the phase difference of basilar membrane motion between two longitudinally spaced locations and dividing by the distance between them. The theoretical wavenumber was a solution of the dispersion relation of a three-dimensional cochlear model with OCC mass and stiffness as the free parameters. The experimental data were only well fit by a model that included OCC mass. However, as the measurement position moved from a best-frequency place of 40 to 12 kHz, the role of mass was diminished. The notion of local resonance seems to only apply in the very high-frequency region of the cochlea.     

Parameter identification of the human lower limb under dynamic, transient torsional loading

The response of the lower limb to dynamic, transient torsional loading applied at the foot has been measured for a male test subject. The dynamic loading was provided by a computer controlled pneumatic system which applied single haversine (i.e. half cycle of a sine wave) axial moment pulses of variable amplitude (0-100 Nm) and duration (50-600 ms). Potentiometers measured the absolute rotations of the three leg segments. Test variables included rotation direction, weight bearing and joint flexion. Two approaches were explored for specifying parameters (i.e. inertia, damping, stiffness) of a three degree-of-freedom dynamic system model which best duplicated the measured response. One approach involved identification of linear parameters by means of optimization while the other approach entailed estimation. Parameter estimates, which included non-linear, asymmetric stiffness functions, were derived from the literature. The optimization was undertaken so as to identify parameter dependence on test variables. Results indicate that parameter values are influenced by test variables. Results also indicate that the non-linear, estimated model better approximates the experimental data than the linear, identified model. In addition to identifying parameters of a three degree-of-freedom model, parameters were also identified for a single degree-of-freedom model where the motion variable was intended to indicate the rotation of the in vivo knee. It is concluded that the simpler model offers good accuracy in predicting both magnitude and time of occurrence of peak knee axial rotations. Model motion fails to track the measured knee rotation subsequent to the peak, however.     

Computational Models Describing Possible Mechanisms for Generation of Excessive Beta Oscillations in Parkinson’s Disease

In Parkinson’s disease, an increase in beta oscillations within the basal ganglia nuclei has been shown to be associated with difficulty in movement initiation. An important role in the generation of these oscillations is thought to be played by the motor cortex and by a network composed of the subthalamic nucleus (STN) and the external segment of globus pallidus (GPe). Several alternative models have been proposed to describe the mechanisms for generation of the Parkinsonian beta oscillations. However, a recent experimental study of Tachibana and colleagues yielded results which are challenging for all published computational models of beta generation. That study investigated how the presence of beta oscillations in a primate model of Parkinson’s disease is affected by blocking different connections of the STN-GPe circuit. Due to a large number of experimental conditions, the study provides strong constraints that any mechanistic model of beta generation should satisfy. In this paper we present two models consistent with the data of Tachibana et al. The first model assumes that Parkinsonian beta oscillation are generated in the cortex and the STN-GPe circuits resonates at this frequency. The second model additionally assumes that the feedback from STN-GPe circuit to cortex is important for maintaining the oscillations in the network. Predictions are made about experimental evidence that is required to differentiate between the two models, both of which are able to reproduce firing rates, oscillation frequency and effects of lesions carried out by Tachibana and colleagues. Furthermore, an analysis of the models reveals how the amplitude and frequency of the generated oscillations depend on parameters.     

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.     

Kinematic perturbation in the flexion-extension axis for two lumbar rigs during a high impact jump task

The measurement of lumbar spine motion is an important step for injury prevention research during complex and high impact activities, such as cricket fast bowling or javelin throwing. This study examined the performance of two designs of a lumbar rig, previously used in gait research, during a controlled high impact bench jump task. An 8-camera retro-reflective motion analysis system was used to track the lumbar rig. Eleven athletes completed the task wearing the two different lumbar rig designs. Flexion extension data were analyzed using a fast Fourier transformation to assess the signal power of these data during the impact phase of the jump. The lumbar rig featuring an increased and pliable base of support recorded moderately less signal power through the 0-60 Hz spectrum, with statistically less magnitudes at the 0-5 Hz (p = .039), 5-10 Hz (p = .005) and 10-20 Hz (p = .006) frequency bins. A lumbar rig of this design would seem likely to provide less noisy lumbar motion data during high impact tasks.     

A rigid body model of the forearm

In this article the forearm, with its complex, continuous motion of masses during pronation/supination, was approximated by a rigid body model consisting of a radial segment rotating around an ulnar segment. The method used to obtain the model parameters is based on three-dimensional voxel data that include velocity information. We propose a criterion that allows the voxels to be attributed to either of the two segments. It is based on the notion that the rotational kinetic energy determined from the voxel data equals the kinetic energy of the rigid body model. To obtain a three-dimensional smoothing we further propose a parameterization of the shape of both segments. These shapes can then be used to determine the dynamic integrals of the segments, i.e. mass, center of mass, and inertia. Using this approach we determined all model parameters for a human forearm from three series of MRI scans in a supinated, a pronated, and an intermediate position. In the appendix, a procedure is described that allows the dynamic quantities to be scaled homogeneously without recalculation of the integrals. Thus, this article provides all essential parameters required for three-dimensional dynamic simulations of general movements of the forearm.     

Model of calcium oscillations due to negative feedback in olfactory cilia

We present a mathematical model for calcium oscillations in the cilia of olfactory sensory neurons. The underlying mechanism is based on direct negative regulation of cyclic nucleotide-gated channels by calcium/calmodulin and does not require any autocatalysis such as calcium-induced calcium release. The model is in quantitative agreement with available experimental data, both with respect to oscillations and to fast adaptation. We give predictions for the ranges of parameters in which oscillations should be observable. Relevance of the model to calcium oscillations in other systems is discussed.     

Modeling of control and learning in a stepping motion

In a previous study (Beuter et al. 1986) the authors modeled a stepping motion using a three-body linkage with four degrees of freedom. Stepping was simulated by using three task parameters (i.e., step height, length, and duration) and sinusoidal joint angular velocity profiles. The results supported the concept of a hierarchical control structure with open-loop control during normal operation. In this study we refine the dynamic model and improve the simulation technique by incorporating the dynamics of the leg after landing, adding a foot segment to the model, and preprogramming the complete step motion using cycloids. The equations of the forces and torques developed on the ground by the foot during the landing phase are derived using the Lagrangian method. Simulation results are compared to experimental data collected on a subject stepping four times over an obstacle using a Selspot motion analysis system. A hierarchical control model that incorporates a learning process is proposed. The model allows an efficient combination of open and closed loop control strategies and involves hardwired movement segments. We also test the hypothesis of cycloidal velocity profiles in the joint programs against experimental data using a novel curve-fitting procedure based on analytical rather than numerical differentiation. The results suggest multiob-jective optimization of the joint's motion. The control and learning model proposed here will help the understanding of the mechanisms responsible for assembling selected movement segments into goaldirected movement sequences in humans.     

Modeling of weak blast wave propagation in the lung

Blast injuries of the lung are the most life-threatening after an explosion. The choice of physical parameters responsible for trauma is important to understand its mechanism. We developed a one-dimensional linear model of an elastic wave propagation in foam-like pulmonary parenchyma to identify the possible cause of edema due to the impact load. The model demonstrates different injury localizations for free and rigid boundary conditions. The following parameters were considered: strain, velocity, pressure in the medium and stresses in structural elements, energy dissipation, parameter of viscous criterion. Maximum underpressure is the most suitable wave parameter to be the criterion for edema formation in a rabbit lung. We supposed that observed scattering of experimental data on edema severity is induced by the physiological variety of rabbit lungs. The criterion and the model explain this scattering. The model outlines the demands for experimental data to make an unambiguous choice of physical parameters responsible for lung trauma due to impact load.