Hill-type muscle model parameters determined from experiments on single muscles show large animal-to-animal variation

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. 2011a). 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.

Determining all parameters necessary to build Hill-type muscle models from experiments on single muscles

“…Assuming the muscle fibers are parallel to the muscle long axis therefore causes maximum muscle fiber length errors of less than 3.8 % (Guschlbauer et al 2007). We therefore used this approximation here and in Blümel et al (2012a,b), using a rest muscle fiber length of 1.41 mm (the mean reported in Guschlbauer et al 2007) and muscle fiber length changing 1:1 with changes in total muscle length (e.g., when the muscle was stretched 1 mm, muscle fiber length was taken to be 2.41 mm).…”

“…We measure method accuracy by giving the mean R 2 values of the various fits and the R 2 standard deviations for all 10 muscles. We present in the companion articles across-muscle comparisons and parameter correlations (Blümel et al 2012a) and the extent to which individual muscle-specific models improve model performance (Blümel et al 2012b). …”

Muscles: Non-linear Transformers of Motor Neuron Activity

“…Modeling extensor muscles correctly therefore requires measuring, in each individual's muscle, the values of all individual-specific parameters in the model equations (Blümel et al 2012c). Extensor across-animal parameter variation is large, ranging from 1.3 to 17-fold (Blümel et al 2012a), and modeling individual muscles with their own parameter values, as opposed to mean parameter values, halves simulation error (Blümel et al 2012b). A (non-exhaustive) review of the literature shows that large across-individual variation in muscle properties is widespread (spider: Siebert et al 2010;cockroach: Ahn and Full 2002;locust: Wilson et al 2010;rat: Bosboom et al 2001;Gilliver et al 2011;Grottel and Celichowski 1990;Hawkins and Bey 1997;cat: Brown and Loeb 2000;Herzog et al 1992;human: Bottinelli et al 1996;Gilliver et al 2009;Maughan et al 1983;Rubenson et al 2012;Wickiewicz et al 1984).…”

A Synthetic Nervous System Controls a Simulated Cockroach

“…This is an interesting observation, but also not that surprising. Animal experiments [52] and modeling studies alike [62] suggest that parameter values in the nervous system vary largely, even between individuals in the same species, despite indistinguishable performance. This suggests that laboring over the decision of how to tune redundant parameters is not important, as long as the resulting motion is useful.…”

“…Each muscle activation curve has four parameters that must be chosen: the amplitude, steepness, x-offset, and y-offset (see Equation (3)). We used data from insect muscles to pick the steepness and x-offset of the muscle activation curve such that it looked like the biologically observed data [51][52][53] (for an example, see Supplementary Materials (Section: Methods Explained)). The y-offset is constrained such that the muscle produces 0 N of active tension when the motor-neuron is at rest.…”