Nonlinear Muscles, Passive Viscoelasticity and Body Taper Conspire To Create Neuromechanical Phase Lags in Anguilliform Swimmers

Abstract

Locomotion provides superb examples of cooperation among neuromuscular systems, environmental reaction forces, and sensory feedback. As part of a program to understand the neuromechanics of locomotion, here we construct a model of anguilliform (eel-like) swimming in slender fishes. Building on a continuum mechanical representation of the body as an viscoelastic rod, actuated by a traveling wave of preferred curvature and subject to hydrodynamic reaction forces, we incorporate a new version of a calcium release and muscle force model, fitted to data from the lamprey Ichthyomyzon unicuspis, that interactively generates the curvature wave. We use the model to investigate the source of the difference in speeds observed between electromyographic waves of muscle activation and mechanical waves of body curvature, concluding that it is due to a combination of passive viscoelastic and geometric properties of the body and active muscle properties. Moreover, we find that nonlinear force dependence on muscle length and shortening velocity may reduce the work done by the swimming muscles in steady swimming.

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 refined model had independent length and velocity factors, but the activation factor was dependent on the work done by the contractile element (the integral of the force-velocity product). It is interesting to note that the accuracy of the multiplicative model in McMillen et al was greatly improved when model parameters were fit directly to experimental data during sinusoidal movements, without increasing the model complexity (McMillen et al, 2008). We encountered a similar situation when modeling the passive (tonus) tension of leech muscles (Tian, 2008;Tian et al, 2007).…”

“…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). A model structure was fixed a priori, and least-square curve-fitting software was used to determine the model parameters.…”

“…Some coefficients of the ODEs were dependent on the length, and the tension was modeled as the product of two factors: a lengthdependent activation factor and a velocity factor. A Hill-type multiplicative model was developed (McMillen et al, 2008) by removing the length dependence of the ODE coefficients and multiplying a separate length factor in the previous model (Williams et al, 1998). 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.…”

“…Indeed, this expectation has been met in the case of periodic movements mimicking lamprey swimming (McMillen et al, 2008). Thus, direct modeling under functionally relevant conditions can be important for accurately characterizing the targeted behavior with least model complexity.…”