The biomechanics of an overarm throwing task: a simulation model examination of optimal timing of muscle activations.

A series of overarm throws, constrained to the parasagittal plane, were simulated using a muscle model actuated two-segment model representing the forearm and hand plus projectile. The parameters defining the modeled muscles and the anthropometry of the two-segment models were specific to the two young male subjects. All simulations commenced from a position of full elbow flexion and full wrist extension. The study was designed to elucidate the optimal inter-muscular coordination strategies for throwing projectiles to achieve maximum range, as well as maximum projectile kinetic energy for a variety of projectile masses. A proximal to distal (PD) sequence of muscle activations was seen in many of the simulated throws but not all. Under certain conditions moment reversal produced a longer throw and greater projectile energy, and deactivation of the muscles resulted in increased projectile energy. Therefore, simple timing of muscle activation does not fully describe the patterns of muscle recruitment which can produce optimal throws. The models of the two subjects required different timings of muscle activations, and for some of the tasks used different coordination patterns. Optimal strategies were found to vary with the mass of the projectile, the anthropometry and the muscle characteristics of the subjects modeled. The tasks examined were relatively simple, but basic rules for coordinating these tasks were not evident.     

Predicting spike timing of neocortical pyramidal neurons by simple threshold models

Neurons generate spikes reliably with millisecond precision if driven by a fluctuating current—is it then possible to predict the spike timing knowing the input? We determined parameters of an adapting threshold model using data recorded in vitro from 24 layer 5 pyramidal neurons from rat somatosensory cortex, stimulated intracellularly by a fluctuating current simulating synaptic bombardment in vivo. The model generates output spikes whenever the membrane voltage (a filtered version of the input current) reaches a dynamic threshold. We find that for input currents with large fluctuation amplitude, up to 75% of the spike times can be predicted with a precision of ±2 ms. Some of the intrinsic neuronal unreliability can be accounted for by a noisy threshold mechanism. Our results suggest that, under random current injection into the soma, (i) neuronal behavior in the subthreshold regime can be well approximated by a simple linear filter; and (ii) most of the nonlinearities are captured by a simple threshold process.     

Analysis of models for the activation and contraction of muscle

Huxley (1957) proposed a sliding filament model of muscular contraction to which Julian (1969) added equations for the activation produced by cations. Each parameter in the combined Huxley-Julian model has been varied systematically to determine its effect on the predicted twitches. The slower rate constant for Ca activation has a predominant effect on the relaxation phase of the twitch. The series elasticity and the rate constants for the making and the breaking of cross-bridges all strongly affect the contraction phase of the twitch. Further experimental work is required to determine which factor is rate limiting under a given set of conditions.Taking the Fourier transform of the twitch gives a prediction for the frequency response of the model. The predicted frequency response curves are well-fitted by those of a simple, second order system, in agreement with recent experiments (Mannard &; Stein, 1973). The parameters of the best-fitting, second-order, frequency response curves vary experimentally with mean stimulus rate. This variation probably results from a saturation at higher stimulus rates of the pXocesses for reuptake of Ca into the sarcoplasmic reticulum. The saturation of Ca reuptake, together with the saturation of the myofilaments by Ca at higher stimulus rates, can account qualitatively for the sigmoid rate-tension curves found experimentally.     

Effect of altering starting length and activation timing of muscle on fiber strain and muscle damage.

Muscle strain injuries are some of the most frequent injuries in sports and command a great deal of attention in an effort to understand their etiology. These injuries may be the culmination of a series of subcellular events accumulated through repetitive lengthening (eccentric) contractions during exercise, and they may be influenced by a variety of variables including fiber strain magnitude, peak joint torque, and starting muscle length. To assess the influence of these variables on muscle injury magnitude in vivo, we measured fiber dynamics and joint torque production during repeated stretch-shortening cycles in the rabbit tibialis anterior muscle, at short and long muscle lengths, while varying the timing of activation before muscle stretch. We found that a muscle subjected to repeated stretch-shortening cycles of constant muscle-tendon unit excursion exhibits significantly different joint torque and fiber strains when the timing of activation or starting muscle length is changed. In particular, measures of fiber strain and muscle injury were significantly increased by altering activation timing and increasing the starting length of the muscle. However, we observed differential effects on peak joint torque during the cyclic stretch-shortening exercise, as increasing the starting length of the muscle did not increase torque production. We conclude that altering activation timing and muscle length before stretch may influence muscle injury by significantly increasing fiber strain magnitude and that fiber dynamics is a more important variable than muscle-tendon unit dynamics and torque production in influencing the magnitude of muscle injury.     

A comparison of models explaining muscle activation patterns for isometric contractions

One of the main problems in motor-control research is the muscle load sharing problem, which originates from the fact that the number of muscles spanning a joint exceeds the number of degrees of freedom of the joint. As a consequence, many different possibilities exist for the activation of muscles in order to produce a desired joint torque. Several models describing muscle activation have been hypothesized over the last few decades to solve this problem. This study presents theoretical analyses of the various models and compares the predictions of these models with new data on muscle activation patterns for isometric contractions in various directions. None of the existing models fitted the experimental data in all aspects. The best fit was obtained by models based on minimization of the squared sum of muscle forces (∑ _m φ² _m , which is almost equivalent to the Moore-Penrose pseudo-inverse solution), muscle stress σ (∑ _m σ _m ²) or muscle activation α (∑ _m α _m ²). Since muscle activation patterns are different for isometric contractions and for movements, it could well be that other models or optimization criteria are better suited to describe muscle activation patterns for movements. The results of our simulations demonstrate that the predicted muscle activation patterns do not depend critically on the parameters in the model. This may explain why muscle activation patterns are highly stereotyped for all subjects irrespective of differences between subjects in many neuro-anatomical aspects, such as, for example, in the physiological cross-sectional area of muscle. Received: 24 September 1998 / Accepted in revised form: 1 March 1999     

Time delay in simple chemostat models.

The models of Monod and Williams, for the growth of unicellular organisms in chemostats, give strongly damped transients in the biomass and cell number when the flow rate of the chemostat is changed. A simple trick is used to incorporate time delay in these models while still allowing a conventional stability analysis. For long enough time delays the equilibrium point is unstable and limit cycles can be compared. Results obtained using Williams' model, with weakly damped transients as a result of using moderately long time delay, are compared with his data in which cell numbers show weak damping but biomass shows strong damping.     

Deep insight from simple models of evolution.

On one hand, people admire the often strikingly efficient results of organic evolution. On the other hand, however, they decry mutation and selection to be a rather prodigal, inefficient trial-and-error strategy. Taking into account the parallel information processing in a heterogeneous population and sexual propagation with recombination, as well as the genetic control of the reproduction accuracy, computer simulated evolution reveals a couple of interesting, sometimes surprising, properties of nature's learning-by-doing algorithm. Survival of the fittest, Spencer's résumé of Darwin's view, turns out to be a bad advice if taken literally. Individual death, forgetting, and even regression show up to be necessary ingredients of the life game. Whether the process should be named gradualistic or punctualistic, is a matter of the observer's point of view.     

Filopodia: Complex models for simple rods

Filopodia are prominent cell surface projections filled with bundles of linear actin filaments that drive their protrusion. These structures are considered important sensory organelles, for instance in neuronal growth cones or during the fusion of sheets of epithelial tissues. In addition, they can serve a precursor function in adhesion site or stress fibre formation. Actin filament assembly is essential for filopodia formation and turnover, yet the precise molecular mechanisms of filament nucleation and/or elongation are controversial. Indeed, conflicting reports on the molecular requirements of filopodia initiation have prompted researchers to propose different types and/or alternative or redundant mechanisms mediating this process. However, recent data shed new light on these questions, and they indicate that the balance of a limited set of biochemical activities can determine the structural outcome of a given filopodium. Here we focus on discussing our current view of the relevance of these activities, and attempt to propose a molecular mechanism of filopodia assembly based on a single core machinery.     

Predicting optimal electrical stimulation for repetitive human muscle activation.

Functional electrical stimulation is the use of electrical currents to activate paralyzed muscles to produce functional movements. Muscle force output must meet or exceed the external load to maintain a posture or produce movements. A mathematical force-fatigue modeling system that predicts muscle force responses during repetitive electrical stimulation has been developed in our laboratory to help identify stimulation patterns that optimize force output for individual subjects. This study tests how well this model predicts the number of contractions that can be maintained above a required force level (successful contractions) during repetitive activation of a muscle. Healthy human quadriceps muscles were tested isometrically on 12 subjects. Data were first collected and used to parameterize the model. Next, the model was used to predict the number of successful contractions that were produced by trains with frequencies ranging from 5 to 100 Hz while the pulse durations and amplitudes were held constant. Finally, three clinically relevant stimulation frequencies were selected and tested to verify the model's predictions. Under these conditions, the model accurately predicted the number of successful contractions for clinically relevant stimulation frequencies. Furthermore, the model appears to have the potential to identify the stimulation frequency that maximizes muscle force output and minimizes fatigue for each subject.     

The theory of sliding filament models for muscle contraction. II. Biochemically-based models of the contraction cycle

A seven-state sliding filament model is proposed which differs from the model of Eisenberg & Greene. It is based on a simplified version of the in-vitro contraction cycle of Stein et al., and also has some desirable dynamical features of the empirical three-state model of Nishiyama & Murase. Appropriate x-dependences for all reaction rates are derived from the transition-state theory. The seventh-state is assumed to be a high-tension intermediate of A.M.ATP, from which direct but x-dependent dissociation can occur. If the final A.M.ATP state has a sufficiently lower tension than that of A.M.ADP.Pi, then the dominant escape path from the intermediate state is shown to be direct dissociation of the actin-myosin bond. This leads to an approximate five-state model for active and relaxed muscle in which A.M and the final A.M.ATP state are omitted.     

Cooperative binding is not required for activation of muscle phosphorylase.

Muscle and liver glycogen phosphorylase isozymes differ in their responsiveness to the activating ligand AMP. The muscle enzyme, which supplies glucose in response to strenuous activity, binds AMP cooperatively, and its enzymatic activity becomes greatly enhanced. The liver isozyme regulates the level of blood glucose, and AMP is not the primary activator. In muscle glycogen phosphorylase, the residue proline 48 links two secondary structural elements that bind AMP. This amino acid residue is replaced with a threonine in the liver isozyme; unlike the muscle enzyme, liver binds AMP noncooperatively, and the enzymatic activity is not greatly increased. We have substituted proline 48 in the muscle enzyme with threonine, alanine, and glycine and characterized the recombinant enzymes kinetically and structurally to determine if proline at this position is critical for cooperative AMP binding and activation. Importantly, all of the engineered enzymes were fully activated by phosphorylation, indicating that enzymatic activity was not compromised. Only the mutant enzyme with alanine at position 48 responds like the wild-type enzyme to the presence of AMP, indicating that proline is not absolutely required for full cooperative activation. The substitution of either threonine or glycine at this position, however, creates enzymes that no longer bind AMP cooperatively. The enzyme with threonine at position 48 further mimics the liver enzyme, in that the maximal enzymatic activity is also reduced. Significantly, the glycine substitution caused the enzyme to be fully activated by AMP, although binding was not cooperative. The hyperactivation of the glycine mutant by AMP suggests that the total free energy of activation has decreased.(ABSTRACT TRUNCATED AT 250 WORDS)     

Some simple models of cannibalism

A sequence of mathematical models for a species which engages in cannibalism are investigated. The models treat the species as age-structured, and assume that adults consume the unhatched eggs of their own kind. The McKendrick or von Foerster partial differential equation model is first converted into a set of three coupled, nonlinear ordinary differential equations, and then adjusted to describe cannibalism. Some rather unusual dynamical effects are discovered. These include both a Hopf bifurcation and a catastrophic transition from an asymptotically stable equilibrium point to a stable limit cycle.     

Voluntary muscle activation varies with age and muscle group.

The consistency and the number of attempts required to achieve maximal voluntary muscle activation have not been documented and compared between young and old adults. Furthermore, few studies have contrasted activation between functional pairs of muscle groups, and no study has tested upper limb muscles. The purpose of this study was to measure and compare voluntary muscle activation of the elbow flexors and extensors in young and old men over two separate test sessions. With the method of twitch interpolation to measure activation, six young (24 +/- 1 yr) and six old (83 +/- 4 yr) men performed five maximal voluntary contractions (MVC) during each session for each muscle group. Elbow flexion and extension MVC was less (43 and 47%, respectively) in the old men, yet the best maximal voluntary muscle activation was similar between age groups. However, when all 10 attempts at MVC were compared, the mean activation scores were slightly less (approximately 5%) in the elbow extensors but were approximately 11% less (P < 0.001) in the elbow flexors of old men, compared with young men. During the second session, there was a significant improvement of 13% (P < 0.005) in mean elbow flexor activation in the old men. There were no session differences for either muscle group for the young men. The results indicate that, for aged men, elbow flexor maximal activation is achieved less frequently compared with elbow extensors, and thus mean activation for elbow flexors is less than for elbow extensors. However, if sufficient attempts are provided, the best effort for the old men is not different from that of the young men for either muscle group.     

Non-linear transmission and simple models for bovine tuberculosis

1.  A new model is presented for a possum–tuberculosis (TB) system ( Trichosurus vulpecula – Mycobacterium bovis ) that is both realistic and parsimonious. The model includes a phenomenological treatment of heterogeneity of risk for susceptible hosts, similar to that used in insect host–parasitoid systems.
2.  Parameter values for the model reflect current knowledge and differ significantly from those in other recent models of this system. Associated with these structural and parametric changes are substantially different predictions for the dynamics and control of TB in possums.
3.  The model predictions include (i) only limited host suppression due to the disease (< 10%, cf. several earlier simple models for TB in both possums and badgers); (ii) asymptotically stable disease dynamics (cf. homogeneous-mixing models that predict either extremely weak stability such that disease fails to recover when host density is temporarily reduced, or oscillatory behaviour and potential elimination of disease following such a perturbation); (iii) TB that is harder to control than in the homogeneous-mixing model equivalents, in line with practical experience; and (iv) a threshold host density for disease elimination that differs substantially from the host equilibrium density in the presence of disease.
4.  Homogeneous-mixing models are unable to reproduce this behaviour, whatever parameter values are chosen. Heterogeneous-mixing models with non-linear transmission may therefore be worth consideration in other endemic wildlife disease systems, as is now commonplace for insect–parasitoid and insect–pathogen ones.     

From simple to detailed models for cell polarization

Many mathematical models have been proposed for the process of cell polarization. Some of these are ‘functional models’ that capture a class of dynamical behaviour, whereas others are derived from features of signalling molecules. Some mechanistic models are detailed, and therefore complex, whereas others are simplified. Each type contributes to our understanding of cell polarization. However, the huge variety at different levels of detail makes comparisons challenging. Here, we provide examples of both elementary and more detailed models for polarization. We also display how a recent mathematical method, local perturbation analysis, can provide an appropriate tool for such comparisons. This technique simplifies and speeds up the model development process by revealing the effect of model extensions, parameter variations and in silico manipulations such as knock-out or over-expression of key molecules. Finally, simulations in both one dimension and two dimensions, and particularly in deforming two-dimensional ‘cells’, can highlight behaviour not captured by traditional simulation methods.     

A simple framework for selection of water quality models

Water quality modeling is no longer just the preserve of specialists who seek to describe water quality processes but also for use by non specialists in everyday water quality management issues. With so many models already developed, it becomes prudent to adapt them to a situation than to develop a completely new model that would probably do the same simulations. The question is: which is the most appropriate model to apply to a situation? The specialist can always draw on past experiences to make a decision. However, this is not the case for the non specialist. A lot of different criteria can be used to decide which model to use for a particular situation based on some important factors. The objectives of modeling exercises differ and each water body is unique so there cannot be hard and fast rules on which is the best criteria for selecting the appropriate model. Furthermore, there is usually hardly any time on the project work plan allocated for model selection. Therefore there is need for a simple procedure to select the appropriate model. The objective of this paper was to develop a simple framework for selecting water quality models to aid the non specialist. The framework was then applied to a case study in order to evaluate its usefulness. The results from the case study show that after a thorough literature review, models can be evaluated against chosen criteria and the most appropriate model singled out. It was concluded that the framework is only effective if the research objective is adequately defined and the models are reviewed thoroughly, but it saves time for the actual modeling exercise.     

A simple method for testing two-locus models of inheritance

A graphic method for testing simple two-locus models of inheritance is developed. The model assumes two alleles at each locus where both loci exhibit dominant, both exhibit recessive, or one locus exhibits dominant and one locus exhibits recessive inheritance. Examples of applying the graphs using published data on three diseases are given.     

Anatomic consequences of intrinsic tongue muscle activation.

We recently showed respiratory-related coactivation of both extrinsic and intrinsic tongue muscles in the rat. Here, we test the hypothesis that intrinsic tongue muscles contribute importantly to changes in velopharyngeal airway volume. Spontaneously breathing anesthetized rats were placed in a MRI scanner. A catheter was placed in the hypopharynx and connected to a pressure source. Axial and sagittal images of the velopharyngeal airway were obtained, and the volume of each image was computed at airway pressures ranging from +5.0 to -5.0 cm H2O. We obtained images in the hypoglossal intact animal (i.e., coactivation of intrinsic and extrinsic tongue muscles) and after selective denervation of the intrinsic tongue muscles, with and without electrical stimulation. Denervation of the intrinsic tongue muscles reduced velopharyngeal airway volume at atmospheric and positive airway pressures. Electrical stimulation of the intact hypoglossal nerve increased velopharyngeal airway volume; however, when stimulation was repeated after selective denervation of the intrinsic tongue muscles, the increase in velopharyngeal airway volume was significantly attenuated. These findings support our working hypothesis that intrinsic tongue muscles play a critical role in modulating upper airway patency.