Muscle short-range stiffness behaves like a maxwell element, not a spring: Implications for joint stability

Abstract

Introduction: Muscles play a critical role in supporting joints during activities of daily living, owing, in part, to the phenomenon of short-range stiffness. Briefly, when an active muscle is lengthened, bound cross-bridges are stretched, yielding forces greater than what is predicted from the force length relationship. For this reason, short-range stiffness has been proposed as an attractive mechanism for providing joint stability. However, there has yet to be a forward dynamic simulation employing a cross-bridge model, that demonstrates this stabilizing role. Therefore, the purpose of this investigation was to test whether Huxley-type muscle elements, which exhibit short-range stiffness, can stabilize a joint while at constant activation.

Methods: We analyzed the stability of an inverted pendulum (moment of inertia: 2.7 kg m2) supported by Huxley-type muscle models that reproduce the short-range stiffness phenomenon. We calculated the muscle forces that would provide sufficient short-range stiffness to stabilize the system based in minimizing the potential energy. Simulations consisted of a 50 ms long, 5 Nm square-wave perturbation, with numerical simulations carried out in ArtiSynth.

Results: Despite the initial analysis predicting shared activity of antagonist and agonist muscles to maintain stable equilibrium, the inverted pendulum model was not stable, and did not maintain an upright posture even with fully activated muscles.

Discussion & conclusion: Our simulations suggested that short-range stiffness cannot be solely responsible for joint stability, even for modest perturbations. We argue that short-range stiffness cannot achieve stability because its dynamics do not behave like a typical spring. Instead, an alternative conceptual model for short-range stiffness is that of a Maxwell element (spring and damper in series), which can be obtained as a first-order approximation to the Huxley model. We postulate that the damping that results from short-range stiffness slows down the mechanical response and allows the central nervous system time to react and stabilize the joint. We speculate that other mechanisms, like reflexes or residual force enhancement/depression, may also play a role in joint stability. Joint stability is due to a combination of factors, and further research is needed to fully understand this complex system.

Fig 1. The active force-Length curve demonstrating the difference between transient and static stiffness.

If a muscle at point A is rapidly lengthened or shortened, then the force produced follows the curve labelled k + k′ (i.e. with stiffness k + k′) rather than k (i.e. with stiffness k).

Fig 2. Simplified inverted pendulum model for this analysis.

The pendulum has a length ℓ, mass m, and is supported by two spring-like elements of length b on either side of the pin-joint. They have moment arms ±a and are aligned with the pendulum’s long axis in the upright position. On the right is the pendulum during the perturbation, with its angle relative to vertical indicated by θ, and muscle lengths by x₁ and x₂.

Fig 3. An overview of the two-state Huxley model used in this investigation.

The myosin heads can exist in two states: (A) unbound, or (B) bound, to actin. Once bound, the model tracks the displacements among the myosin molecules, which have an associated stiffness of k_m ≈ 0.2 to 5.0 pN/nm. The rates between these two states are characterized by an attachment rate, parameterized by f(s), and a detachment rate, parameterized by g(s), both of which are graphically depicted in (C). A hypothetical displacement distribution function (D), where the area under this graph between displacements s₁ and s₂ is approximately the proportion of myosin heads that are bound with displacements between s₁ and s₂.

Fig 4. Angle time-histories before, during, and after the 50 ms, 5 Nm perturbation (shaded region).

Constant muscle activation, even with short-range stiffness, was unable to stabilize the pendulum as evidenced by its eventual loss of equilibrium. The dashed linear spring represents muscles as springs whose stiffness matched the 100% activation short-range stiffness. For comparison is a damped oscillation which is considered asymptotically stable; one second after the perturbation even with high activations, the Huxley models did not return to the upright configuration.

Fig 5. Longer-time simulations showing that the high activations (50% and 100%) eventually fell over in finite time.

For comparison there is the stable case from the springs, and an asymptotically stable case from springs with a dashpot included for comparison.

Fig 6. Muscle force (left column), fibre lengths (middle column) and pendulum angles (right column) for perturbations of varying muscle activations (rows) 250 ms after the initiation of the perturbation.

At 50% or 100% activation, the pendulum oscillates, and the muscles alternate between concentric and eccentric loading. When each muscle is lengthening, its force is amplified, and diminished when shortening.

Fig 7. Approximation of the inverted pendulum in this analysis (left) with one supported by standard viscoelastic solid models (right).