Abstract
Models built using mean data can represent only a very small percentage, or none, of the population being modeled, and produce different activity than any member of it. Overcoming this “averaging” pitfall requires measuring, in single individuals in single experiments, all of the system’s defining characteristics. We have developed protocols that allow all the parameters in the curves used in typical Hill-type models (passive and active force–length, series elasticity, force–activation, force–velocity) to be determined from experiments on individual stick insect muscles (Blümel et al. 2012a). A requirement for means to not well represent the population is that the population shows large variation in its defining characteristics. We therefore used these protocols to measure extensor muscle defining parameters in multiple animals. Across-animal variability in these parameters can be very large, ranging from 1.3- to 17-fold. This large variation is consistent with earlier data in which extensor muscle responses to identical motor neuron driving showed large animal-to-animal variability (Hooper et al. 2006), and suggests accurate modeling of extensor muscles requires modeling individual-by-individual. These complete characterizations of individual muscles also allowed us to test for parameter correlations. Two parameter pairs significantly co-varied, suggesting that a simpler model could as well reproduce muscle response.
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Abbreviations
- A, B :
-
Parameters in rest length force–activation curve (Eqs. 3, 7, 8) (unitless)
- A act :
-
Maximum amplitude of force–length curves (Eqs. 4, 5) (unitless)
- act:
-
Muscle activation (Eqs. 3, 6,7–9) (normalized to 200 Hz, therefore unitless)
- c neg, c pos :
-
Curvatures of Hill hyperbola for shortening (Eq. 7) and lengthening (Eq. 8) contractions, respectively (unitless)
- curv ω :
-
Curvature of hyperbola relating ω and act (Eq. 6) (mm0.5)
- F A :
-
Active force at rest length (Eq. 3) (mN)
- F L :
-
Active force at different muscle lengths (force–length curve) (Eq. 4) (normalized to F max, and therefore unitless)
- F max :
-
Maximum isometric force at rest length (Eq. 3) (mN)
- F P :
-
Steady-state passive force (parallel spring) (Eq. 1) (mN)
- F SE :
-
Series elastic spring force (Eq. 2) (mN)
- F V :
-
F L at different contraction velocities (force–velocity curve) (Eqs. 7, 8) (normalized to F max, and therefore unitless)
- k 1, k 2 :
-
Passive steady-state force length curve constants (Eq. 1) (k 1, mN; k 2, mm−1)
- k 3 :
-
Proportionality constant in quadratic force equation (Eq. 2) (mN/mm 2)
- L M :
-
Muscle length (Eqs. 1, 4) (mm)
- L SE :
-
Series elastic element length (Eq. 2) (mm)
- v :
-
Velocity of muscle length change (Eqs. 7, 8) (mm/s)
- v max neg, v max pos :
-
Maximum velocity of muscle length change for shortening (Eqs. 7, 9) and lengthening (Eq. 8) contractions, respectively (mm/s)
- v max(act=1) :
-
v max pos at an activation of 1 (Eq. 9) (mm/s)
- ω :
-
“Length” frequency of force–length curves (Eqs. 4–6) (mm−1)
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Blümel, M., Guschlbauer, C., Daun-Gruhn, S. et al. Hill-type muscle model parameters determined from experiments on single muscles show large animal-to-animal variation. Biol Cybern 106, 559–571 (2012). https://doi.org/10.1007/s00422-012-0530-6
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DOI: https://doi.org/10.1007/s00422-012-0530-6