Yank: the time derivative of force is an important biomechanical variable in sensorimotor systems

Abstract

The derivative of force with respect to time does not have a standard term in physics. As a consequence, the quantity has been given a variety of names, the most closely related being 'rate of force development'. The lack of a proper name has made it difficult to understand how different structures and processes within the sensorimotor system respond to and shape the dynamics of force generation, which is critical for survival in many species. We advocate that ∂[Formula: see text]/∂t be termed 'yank', a term that has previously been informally used and never formally defined. Our aim in this Commentary is to establish the significance of yank in how biological motor systems are organized, evolve and adapt. Further, by defining the quantity in mathematical terms, several measurement variables that are commonly reported can be clarified and unified. In this Commentary, we first detail the many types of motor function that are affected by the magnitude of yank generation, especially those related to time-constrained activities. These activities include escape, prey capture and postural responses to perturbations. Next, we describe the multi-scale structures and processes of the musculoskeletal system that influence yank and can be modified to increase yank generation. Lastly, we highlight recent studies showing that yank is represented in the sensory feedback system, and discuss how this information is used to enhance postural stability and facilitate recovery from postural perturbations. Overall, we promote an increased consideration of yank in studying biological motor and sensory systems.

Keywords: Biomechanics; Muscle; Spindle.

Fig. 1.

Yank can be derived from time recordings of ground reaction force (GRF) during motor behaviors. (A) In our previous study, kangaroo rats jumped over a vertical barrier from a standing posture (Schwaner et al., 2018). The GRF vector (red arrow) was measured using a force plate. The mass of the animal shown is 0.12 kg, corresponding to a GRF of 1.2 N at the initiation of the jump. (B) Yank is calculated from the measured GRFs (only the vertical component is shown). Only the first portion of negative yank values before the feet left the ground is shown (dashed line) so that the positive yank generated can be appropriately scaled on the plot.

Fig. 2.

A simple model showing how changes in yank magnitude change behavioral outcomes during a jump. See main text for details. (A) The center of mass of an animal is accelerated upwards by the vertical GRF until the takeoff point, X_o. Gravity and non-vertical forces are neglected for simplicity. (B) The GRF is assumed to be exponential (see Eqn 1) and the resulting yank, position and velocity are shown. The two cases shown are with nominal values (solid line) and with a 20% decrease in the exponential time constant (dashed line). The time of takeoff is indicated by a cross. Note the time to takeoff decreases and jump velocity increases with an increase in yank. Model parameters (Eqn 1) are: A=0.2 N, τ=0.025 s and body mass=0.1 kg.

Fig. 3.

The multi-scale anatomical structures and processes that determine the magnitude of yank. The structures/processes are indicated in red. The plasticity within each structure which influences yank is represented by changes in specific variables, indicated in blue. Yank can be calculated using the measured variables from in vivo, in situ or in vitro experiments, indicated in green. t, time; [Ca²⁺], intracellular calcium concentration; F_fiber(t), single fiber force (measured in vitro); Y_fiber(t), yank of fiber force; L(t) and F(t), muscle length and force; F_MT(t), musculotendon force; Y_MT(t), yank of musculotendon force; M(t), joint moment (measured in vivo); Y_M(t), yank of joint moment; PCSA, physiological cross-sectional area; SERCA, sarco/endoplasmic reticulum; and k_tr, time constant of force recovery.

Fig. 4.

Proposed role of yank in the sensorimotor feedback loop. The initial rise in yank caused by the postural perturbation is carried through the feedback loop and leads to faster reactive muscle force responses to counteract the perturbation. When a postural perturbation occurs, muscle fibers are stretched and have initial force and yank responses that drive muscle spindle Ia afferent sensory feedback. This sensory feedback, along with that from other receptors, drives motoneuron potential changes with initial bursts of activity. These motoneuronal potential changes lead to an initial burst in electrical activation of muscles resembling the yank in the stretched muscle, which in turn causes rapid muscle contraction, facilitated by the catch-like property of the muscle, resulting in a high initial yank of the balance-correcting muscle force.

Fig. 5.

Yank is encoded in the dynamic response of muscle spindle primary afferents. (A) A ramp–hold–release stretch protocol (bottom panel) is applied to the isolated, relaxed triceps surae muscle–tendon unit (MTU) of a cat, which causes a force and yank response in the MTU (middle panels), and a spiking response of the primary afferent neuron (top panel; IFR, instantaneous firing rate). (B) Enlargement of the boxed regions in A. The afferent firing rate closely resembles yank during the positive velocity ramp phase of the stretch and more closely resembles the force during the plateau phase of stretch.