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

Abstract

Characterizing muscle requires measuring such properties as force–length, force–activation, and force–velocity curves. These characterizations require large numbers of data points because both what type of function (e.g., linear, exponential, hyperbolic) best represents each property, and the values of the parameters in the relevant equations, need to be determined. Only a few properties are therefore generally measured in experiments on any one muscle, and complete characterizations are obtained by averaging data across a large number of muscles. Such averaging approaches can work well for muscles that are similar across individuals. However, considerable evidence indicates that large inter-individual variation exists, at least for some muscles. This variation poses difficulties for across-animal averaging approaches. Methods to fully describe all muscle’s characteristics in experiments on individual muscles would therefore be useful. Prior work in stick insect extensor muscle has identified what functions describe each of this muscle’s properties and shown that these equations apply across animals. Characterizing these muscles on an individual-by-individual basis therefore requires determining only the values of the parameters in these equations, not equation form. We present here techniques that allow determining all these parameter values in experiments on single muscles. This technique will allow us to compare parameter variation across individuals and to model muscles individually. Similar experiments can likely be performed on single muscles in other systems. This approach may thus provide a widely applicable method for characterizing and modeling muscles from single experiments.

Segment-specific and state-dependent targeting accuracy of the stick insect

“…Depending on how far distal in the leg the innervated muscle is, it takes an additional 1-5ms for the motoneuron spikes to travel to the neuromuscular end plate (Höltje and Hustert, 2003). Finally, the muscle needs a minimum of 20-40ms to build up the muscle tension needed for movement of the leg (Guschlbauer et al, 2007;Hooper et al, 2009;Blümel et al, 2012). It is unclear how many synapses and interneurons have to be crossed before the information reaches the motoneurons of the targeting leg, but both intersegmental and local interneurons have been described that could take part in the targeting process (Brunn and Dean, 1994).…”

Muscles: Non-linear Transformers of Motor Neuron Activity

“…The b2 panels of show the output of models constructed from experimental data. To avoid clutter we do not show the data in these panels; interested readers should see Blümel et al 2012c. Model output always well agreed with the data (R 2 values ≥ 0.93).…”

“…b2 In extensor muscle, F PE is an exponential function of L PE . (Modified from Blümel et al 2012c) power law to have finite values at t = 0), show that for extensor muscle this concern is unlikely, as at measurement times the b · ( t + c) d term is small. Another concern is modeling the power law decline.…”

“…b2 In stick insect extensor muscle F SE depends on L SE squared. (Modified from Blümel et al 2012c) 6 Muscles: Non-linear Transformers of Motor Neuron Activity and applied forces equal (inset; ΔL SE = L M2 − L M1 ). Performing these steps from multiple F1 to multiple F2 gives a large number of ΔF vs. ΔL M data points, from which the relationship between F SE and L SE can be determined (Blümel et al 2012c;Guschlbauer et al 2007).…”

A Synthetic Nervous System Controls a Simulated Cockroach

“…The parameters of interest for this section are muscle parameters k se , k pe , and b in Equation (1). Values for these parameters vary widely from muscle-to-muscle and even from organism-to-organism [50]. Guschlbauer …”