Muscle Contraction Dynamics

In our electromyography data processing journey so far, we’ve gone from this:

Raw electromyography (EMG) signal from the sensor placed over the muscle group that extends the middle, ring and little finger. Read about the previous processing steps here and here.

to this:

Activation of the part of the Extensor Digitorum Communis muscle that controls the middle finger (EDCM)

Muscle activation is a number that tells us how active the muscle is, from 0 (not at all) to 1 (fully contracted).

But Activation Dynamics is only the first modelling step! Let’s jump straight into step 2.

Step 2: Muscle contraction

To estimate the force a muscle produces we use Hill’s three-element muscle model. This model describes muscles in terms of their force-producing capabilities as having three parts:

1. The contractile element, which represents the muscle fibres where force is actively produced by the formation of cross-bridges between the proteins actin and myosin.
2. The series elastic element, which represents the tendon. It cannot actively produce force itself, but when it is stretched, it tries to resist, just like a spring.
3. The parallel elastic element, which represents the various connective tissues that surround the muscle fibres. As you may have guessed, this element also does not produce active force, but produces passive force when it is stretched.

The Hill muscle model, from one of our articles. CE: contractile element, SEE: series elastic element (the tendon), PEE: parallel elastic element (connective tissues). LCE is the length of the contractile element, LM is the length of the entire muscle-tendon unit, and phi is the pennation angle, which is the angle at which the muscle fibres insert into the tendon.

What does all this mean in practice? The force of a muscle depends on its activation level, as this makes the contractile element produce more active force. But it also depends on its length, because if it is too short or too long, cross-bridges between actin and myosin cannot be formed. If it is too long, the passive elastic element is stretched, so even though there is no active force from the contractile element, there is passive force from the parallel elastic element.

On top of that, it’s not just the length, but also the velocity of a muscle that affects how much force it can produce: if it is shortening too quickly, there is no time for cross-bridges to form.

Finally, the muscle force depends on how big the muscle is. Imagine two muscles with exactly the same length, velocity and activation, but one is twice as big as the other (because its owner has been lifting weights instead of thinking they should start, which is how far I’ve got). The bigger the muscle, the larger the force.

We have mathematical equations that describe the Hill muscle model. They are very complex, but luckily, we don’t solve them by hand! (If you’d like to play with equations similar to these but a bit simplified, have a look here. You’ll need Matlab, but I’m hoping to convert to Python soon.)

In the case of the EDCM muscle, the activation we calculated from the EMG signal results in the following force:

This force depends on the activation, but also the length, velocity and size of the muscle. I’m not showing you these graphs because there must be a limit to how many graphs a post can include before all the readers run away screaming. I am worried I went past that three graphs ago.

Our computer hand model saga concludes in the next episode, when we will navigate the last couple of steps from muscle forces to joint angles. Along the way we’ll say hi to old favourites Archimedes and Newton, and have a narrow escape from the dangers of stiff differential equations. Don’t miss it :)