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 km ≈ 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.
Table 1.
Parameters that were used throughout the simulations.
The muscle properties were selected to be representative of average muscle properties for lumbar spine extensors (erector spinae).
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).