A new model for force generation by skeletal muscle, incorporating work-dependent deactivation

Abstract

SUMMARYA model is developed to predict the force generated by active skeletal muscle when subjected to imposed patterns of lengthening and shortening, such as those that occur during normal movements. The model is based on data from isolated lamprey muscle and can predict the forces developed during swimming. The model consists of a set of ordinary differential equations, which are solved numerically. The model's first part is a simplified description of the kinetics of Ca 2+ release from sarcoplasmic reticulum and binding to muscle protein filaments, in response to neural activation. The second part is based on A. V. Hill's mechanical model of muscle, consisting of elastic and contractile elements in series, the latter obeying known physiological properties. The parameters of the model are determined by fitting the appropriate mathematical solutions to data recorded from isolated lamprey muscle activated under conditions of constant length or rate of change of length. The model is then used to predict the forces developed under conditions of applied sinusoidal length changes, and the results compared with corresponding data. The most significant advance of this model is the incorporation of work-dependent deactivation, whereby a muscle that has been shortening under load generates less force after the shortening ceases than otherwise expected. In addition, the stiffness in this model is not constant but increases with increasing activation. The model yields a closer prediction to data than has been obtained before, and can thus prove an important component of investigations of the neural-mechanical-environmental interactions that occur during natural movements.

Optimal shape and motion of undulatory swimming organisms

“…Mathematical models of muscle behaviour during swimming have mostly been developed to study the muscle response to a given neural activation [9][10][11][12], with model parameters being often fine-tuned for particular species [12][13][14]. The actuation-response relationship varies widely among species [15][16][17], making these models unsuitable for the optimization of morphological traits of a generic organism.…”

Mechanisms underlying rhythmic locomotion: dynamics of muscle activation

“…The modeling work described above (Williams et al, 1998;Williams, 2010) was based on tetanic isometric/isovelocity experiments, and experimental tension data under sinusoidal movements and intermittent tetanic activation were used for validation purpose only. McMillen et al are among the few researchers who have performed modeling directly based on tension data under sinusoidal movements (although this part of their modeling was not the focus of this reference) (McMillen et al, 2008).…”

“…The method first validates the multiplicative model structure, and then determines the time courses of activation and length factors from tension data for developing component models. The novel dualsinusoid experiments were conducted on nerve-cord and body-wall preparations, where the muscle was activated through sinusoidal current injection into a MN to induce rhythmic bursts at realistic impulse frequencies, rather than tetanic electrical stimulation directly applied to muscle intermittently (Williams et al, 1998;Williams, 2010). The experiments were designed for the physiological conditions during swimming -rhythmicity and the phase relationship between muscle strain and activation -thus were specifically tuned for locomotion study.…”

“…With the parameters of this model determined from isometric/isovelocity experiments, the multiplicative model was able to predict the tension under rhythmic activation or movements with a reasonable accuracy. This model was further refined to improve its accuracy (Williams, 2010) by incorporating the effect of work-dependent deactivation, which reduces tension following shortening more than would otherwise be expected. By having some of the calcium-ionbinding constants dependent on the velocity of muscle shortening, the refined model successfully predicted the tension observed during sinusoidal changes in length.…”

“…A common cause of the failure is the assumption that the activation, length and velocity factors in the multiplicative structure are independent of each other. In reality, the activation factor may depend on the velocity (Williams, 2010), and the velocity factor may depend on the activation level (Otten, 1987;Winters and Stark, 1985). The tension developed under such coupling effects may not be well explained by a model with the decoupled multiplicative structure.…”