Springs and Muscles, Part 1

[From Bruce Abbott (2015.02.11.1400 EST)]

I’ve spent a bit of time lately discussing equilibrium systems in general and springs in particular to set the stage for a discussion of “motor control,” or how we (and other animals) control our movements. (Now before anyone gets exercised over this term “motor control,” it doesn’t actually refer to control of output, but rather, to control of the perception of movement.) In this discussion I am going to assume that there is a real body with real muscles, bones, joints, etc. and that the perceptions generally align well with what these parts are actually doing. So, under normal circumstances, when I perceive my hand to be moving from an initial position toward the cup of coffee I’m reaching for, my perception is close enough to reality that my hand actually does reach the cup. In that sense I have controlled the movement of my hand, although I actually do so by controlling my perception of such.

So, how do we control the skeletal-muscular system to achieve such goals? At present we have only a small part of the picture, and even in that area there is still debate. I’m going to focus here only on the bottom level of what is actually a multi-level, highly complex system involving the spinal cord and brain. To make the problem somewhat tractable, I’ll focus on movement involving a single joint that can only move in a hinge-like fashion: an idealized version of the elbow joint and the muscles that act to change joint-angle.

Let’s assume that you are hanging your upper arm downward in a vertical position and your forearm at a right angle to that, parallel to the floor with your outstretched hand palm-up. To keep the forearm from being pulled downward by gravity, you are contracting your biceps muscle to a certain degree. The biceps is a flexor muscle because its contraction flexes the joint, causing it to close.

There is also an extensor muscle, the triceps, that acts on the opposite side of the joint to open it up. At the moment your triceps is relaxed, although it would be possible for you to contract it while still keeping the biceps contracted – so-called isometric contraction.

Without warning, I drop a book onto the surface of your outstretched. For a moment your forearm starts to sag, then rebounds so that your forearm is once again horizontal (or nearly so) without your conscious intervention. Your forearm is still in about the same position as before, but your biceps is now more strongly contracted in order to resist the stronger downward force being exerted by the mass of the book on your palm. How are we to explain this change?

As a starting point, imaging that the muscle with its tendons connecting to the bones of the forearm and upper arm acts like a simple damped spring attached to a mass. The weight of your forearm pulls the forearm downward. This force opens the elbow joint somewhat, stretching the spring, which develops a counterforce as given by Hook’s Law. The spring will stretch until the counterforce equals the force being exerted by the weight of the forearm on the spring at its attachment point on the forearm.

Now we drop the book, increasing the force acting to open the joint, causing the spring to stretch more until once again, the counterforce being generated by the spring equals the force acting to stretch the spring. The forearm will pivot at the joint and the joint will open, so that the forearm will now be sagging below the horizontal. The springiness of the muscle will produce a small rebound effect and the damping force generated in the muscle will bring the movement to a halt with the forearm lowered somewhat from its initial horizontal position. How much lower will depend entirely on the stiffness of the spring: high stiffness, little change; low stiffness, large change.

You can increase the apparent stiffness of the muscle “spring” by adding co-contraction of the opposing muscle (the triceps). To keep the joint from moving, the contraction of the biceps is increased as well, to oppose the increased pull in the opposite direction by the triceps. The increased contraction does not change the length of the muscle because the joint is kept from moving; instead it increases for force of the muscles pulling on their tendons and bones. This effectively increases the spring constant, and the added stiffness reduces the amount by which the forearm drops when the weight of the book is added.

Muscles, of course, are not exactly like simple springs, but they do have measurable spring-like properties – stiffness and a length at which they come to rest under the influence of forces acting to stretch them. In our scenario, the biceps was initially contracted enough to hold the forearm horizontal against the pull of gravity. As there was no tendency for the forearm to raise or lower, this was a position in which the forces acting on the muscle to stretch it and the counterforce opposing that stretch were in equilibrium.

Muscles are driven to contract by neural impulses that travel down alpha motor neuron axons to muscle end-plates. At the muscle end-plates, a neurotransmitter (acetylcholine) is released, binds to receptors on the surfaces of muscle cells, and thus initiates a length-wise contraction of those cells. The cell bodies of those alpha motor neurons are located in the spinal cord. By changing the rate at which those neural impulses are delivered to the muscles, one can increase or decrease the degree of contraction of the muscle.

Now, imagine that the alpha motor neurons are receiving impulses from neurons located higher in the nervous system, and that, everything else being equal, increasing the rate of these inputs increases the rate of firing of pools of alpha motor neurons whose axons connect to the biceps muscle. In this way, input from a higher level could determine the strength of contraction of the biceps, shortening the muscle sufficiently to bring the forearm to a horizontal position. In effect, we have created a spring whose zero-point is the current length of the muscle. If no other factors intervene, increasing the force tending to open the elbow joint (e.g., dropping that book onto the palm) will stretch the spring from this point, increasing the reaction-force, thus tending to resist the stretch produced by the increase in applied force. This reaction-force would be entirely passive, generated by the stretching of the spring. No change in the rate of neural impulses coming from the alpha motor neuron is required.

But this is only part of the story. Up next: Control systems. Before I continue, is everyone with me so far?