Fig 1.
Schematic diagram of the action potential-driven muscle-tendon model.
A. Muscle activation dynamics (A) in response to action potentials (ESs) from spinal motoneurons (dotted line box) and muscle mechanics (F) induced by contractile elements (CEs) and serial elastic elements (SEs). XCE and Xm show the length of the contractile element and entire muscle-tendon unit. C1 and 1/C2 indicate the constant concentration of calcium-binding troponin relative to the total troponin concentration (CaT/T0) for the half-maximal level of steady-state muscle activation () and the slope of the CaT/T0–
curve at C1. B. Relationship of steady-state C1 (C1∞) and C2 (C2∞) with the concentration of CaT/T0 (upper panel) and time constants (τC1 and τC2) for the C1 and C2 dynamics (bottom panel). C1n1 and C2n1 limit the saturation level, and C1n2 and C2n2 indicate a constant concentration of CaT/T0 for the midpoint between the minimum and maximum in the C1∞ and C2∞ curves.
Table 1.
Parameter values of fast muscle model for cat medial gastrocnemius muscles (CAT14 for model development and CAT12 for model validation).
Fig 2.
Tetani of the muscle-tendon model without changes in C1 and C2 in response to various input frequencies under isometric contraction at Xm,0.5 for CAT14.
A. Twitch (upper) and current stimulation (bottom). B. Unfused tetanus (upper) and current stimulation (10 Hz, bottom). C. Unfused tetanus (upper) and current stimulation (20 Hz, bottom). D. Unfused tetanus (upper) and current stimulation (40 Hz, bottom). E. Fused tetanus (upper) and current stimulation (100 Hz, bottom). F. Overall error value between simulated and experimental force data for the Panels A-E. Black and blue lines indicate the data obtained from the experiment and simulation.
Fig 3.
Tetani of the muscle-tendon model with changes only in C1 in response to various input frequencies under isometric contraction at Xm,0.5 for CAT14.
A. Twitch (upper), change in C1 (middle) and current stimulation (bottom). B. Unfused tetanus (upper), change in C1 (middle) and current stimulation (10 Hz, bottom). C. Unfused tetanus (upper), change in C1 (middle) and current stimulation (20 Hz, bottom). D. Unfused tetanus (upper), change in C1 (middle) and current stimulation (40 Hz, bottom). E. Fused tetanus (upper), change in C1 (middle) and current stimulation (100 Hz, bottom). F. Overall error value between simulated and experimental force data for the Panels A-E. Black and blue lines indicate the data obtained from the experiment and simulation.
Fig 4.
Tetani of the muscle-tendon model with changes in both C1 and C2 under isometric contraction at Xm,0.5 for CAT14.
A. Twitch (upper), change in C1 & C2 (middle) and current stimulation (bottom). B. Unfused tetanus (upper), change in C1 & C2 (middle) and current stimulation (10 Hz, bottom). C. Unfused tetanus (upper), change in C1 & C2 (middle) and current stimulation (20 Hz, bottom). D. Unfused tetanus (upper), change in C1 & C2 (middle) and current stimulation (40 Hz, bottom). E. Fused tetanus (upper), change in C1 & C2 (middle) and current stimulation (100 Hz, bottom). F. Simulation error with no variation in C1 & C2 (black), only variation in C1 (gray) and variation in both C1 & C2 (white) at the stimulation frequency of 1 Hz (A), 10 Hz (B), 20 Hz (C), 40 Hz (D) and 100 Hz (E). Black and blue lines in A-E indicate the data obtained from the experiment and simulation with variation in both C1 & C2. The comparison of experimental and simulated data for the case of no variation in C1 & C2 and only variation in C1 was presented in Figs 2 and 3.
Fig 5.
Variation in the calcium-force relationship during isometric force production at Xm,0.5 for CAT14.
A. Steady-state relationship of muscle activation () to calcium binding troponin relative to the total troponin concentration (CaT/T0) in the initial and maximally varied states at various stimulation frequencies. B. Transient relationship of muscle activation (A) to CaT/T0 at various levels of stimulation frequency. C. Transient relationship of muscle force (F) and sarcoplasmic calcium (Ca) at various levels of stimulation frequency. Insets indicate the transient relationship between calcium and activation (B) and calcium and force (C) on the relaxation phase of force production.
Fig 6.
Representative types of sag behavior at Xm,0.5 for CAT14.
A. Simple sag form at 20 Hz current stimulation (left) and changes in C1 and C2 (right) with default values of C2n2 and τC2. B-D. Complex sag forms at 20 Hz (B and C) and 30 Hz (D) stimulation frequencies (left) and changes in C1 and C2 (right) with variations in C2n2 and τC2. Arrows indicate the direction of force production after the initial peak force.
Fig 7.
Type map of sag behavior with changes in threshold (C2n2) and time constant (τC2) for the slope (C2) variation in the calcium-activation (CaT/T0–) relationship at Xm,0.5 for CAT14.
A. Type map under 20 Hz current stimulation. B. Type map under 30 Hz current stimulation.
Fig 8.
Length-force and velocity-force properties of the model CAT14 with changes in C1 and C2.
A. Force responses during lengthening (upper), changes in C1 and C2 (middle) and current stimulation (100 Hz, bottom). B. Force responses during shortening (upper), changes in C1 and C2 (middle) and current stimulation (100 Hz, bottom). C. Force responses during step lengthening (upper), changes in C1 and C2 (middle) and current stimulation (100 Hz, bottom). D. Profiles of the muscle-tendon length variation for A, B and C. Black and blue lines indicate the experimental and simulated data.
Fig 9.
Length dependence of the sag magnitude in the model of CAT14.
A1 & B1. Unfused tetanus (upper), changes in C1 and C2 (middle) and 20 Hz current stimulation (bottom) with no change in the slope of calcium-force relationship at the physiologically minimal (0 mm) and maximal (10 mm) muscle-tendon length. A2 & B2. Unfused tetanus (upper), changes in C1 and C2 (middle) and 20 Hz current stimulation (bottom) with slope variation in the calcium-force relationship at the physiologically minimal and maximal muscle-tendon length. Arrows indicate the degree of force decline after the initial peak force.