Walking is a hard problem, and we’re not really sure how we do it. Like reaching, there are many muscles to coordinate in order to make a step forward. Unlike in arm reaching, these coordinated steps need to follow one another cyclically in such a way as to keep the body stable and upright while simultaneously moving it over terrain that might well be rough and uneven. Just think for a moment about how difficult that is, and what different processes might be involved in the control of such movements.

One question that remains unanswered is how we control

*variability*in walking. It’s a simple matter to control average position or velocity, but the variation in these parameters between steps is still unexplained. It is pretty well established that over the long-term people tend to try to minimize energy costs while walking – hence the gait we learn to adopt over the first few years of life. But there’s evidence that such a seemingly “optimal” strategy is not the whole story.

Consider walking on a treadmill. What’s the primary goal of continuous treadmill walking? Well, it’s to not fall off. The researchers in the article took that idea and reasoned that because the treadmill is moving at a constant speed, the best way not to fall off is to move at a constant speed yourself. That’s not the

*only*strategy of course – you could also do something a little more complicated like make some short, quick steps followed by some long, slow ones in sequence, which would also keep you on the treadmill.

To test how the parameters varied, the researchers used five different walking speeds. You can see this in the figure below (Figure 3 in the paper):

**Human treadmill walking data with speed as percentage of preferred walking speed (PWS)**

L is stride length, T is stride time and S is stride speed. So A-C in the figure show how these values change with the five different treadmill speeds – length increases, time decreases and speed increases. D-F show the

*variability*(σ) in these different parameters. G-I show something slightly more complex: a value called α that is defined as a measure of

*persistence*, i.e. how much or little the parameters were corrected on subsequent strides. Values of α > ½ mean that there was less correction, whereas values < ½ mean that there was more correction. So panels G-I show that variability in stride length and time were

*not*generally corrected quickly, but that variations in stride speed

*were*.

Read that last paragraph through again to make sure you get it. It will be important shortly!

So: now we have a measure of human walking parameters. The question is, how are these parameters produced by the motor control system? That is, what does the system care about when it initiates and monitors walking? Well, one thing we can get from the data here is that the system seems to care about stride speed, but doesn’t care about stride time and stride length individually. And if that’s the case, then as long as the coupled length and time lie on a line that defines the speed, the system should be happy. A line a bit like this (figure 2B in the paper):

**Human stride parameters lie along line of constant speed**

The figure shows the GEM (which stands for Goal Equivalent Manifold, essentially the line of constant speed) plotted against stride time and stride length. The red dots show some data. Right away you can see that the dots generally lie along the line. Ignore the green arrows, but do take note of the blue ones – they’re showing a measure of deviations tangent to (δ

_{T}) and perpendicular to (δ

_{P}) the line. Why is δ

_{T}so much bigger than δ

_{P}? Because perpendicular variations push you off the line and thus interfere with the goal, whereas tangential variations don’t. So the system is either not stepping off the line much in the first place or correcting heavily when it does.

Here’s one more figure (Figure 5C and D in the paper) showing the variability (σ) and persistence (α) for δ

_{T}and δ

_{P }:

**Variability and persistence of deviations**

You can see that δ

_{T}is much more variable than δ

_{P}, as you might expect from the shape of the data shown in the second figure. You can also see something else, however: the persistence for δ

_{P}is less than ½, whereas the persistence for δ

_{T}is greater than ½. Thus, the system cares very much about correcting not just stride speed but the combination of stride time and stride length that take the stride speed away from the goal speed.

Great, you may think, a lot of funny numbers to tell us that the system cares about maintaining a constant speed when it’s trying to maintain a constant speed! What do you scientists get paid for anyway? The cool thing about this paper is that the researchers are trying to figure out precisely how the brain produces these numbers. It turns out that if you just use an ‘optimal’ model that corrects for δ

_{P}while ignoring δ

_{T}, you don’t get the same numbers. So that can’t be it. How about if you specify in your model that you have to keep at a certain speed – say the same average speed as in the human data? That doesn’t work either. The numbers are better, but they’re not right.

The solution that seems to work best is when the deviations off the GEM line (i.e. δ

_{P}) are

*overcorrected*for. This controller is

*sub*-optimal, so basically efficiency is being sacrificed for tight control over this parameter. Thus, humans don’t appear to simply minimize energy loss – they also perform more complex corrections depending on the task goal.

I’ve covered in a previous post the inkling that this might be the case; while we do tend to minimize energy over the long term, in the short term the optimization process is much more centred around the particular goal, and people are very good at exploiting the inherent variability in the motor system to perform the task more easily. This paper does a great job of testing these hypotheses and providing models to explain how this might happen. What I’d be interested to see in the future is an explanation of why the system is set up to overcorrect like that in the first place – is it overall a more efficient way of producing movement than just a standard optimization over all parameters? Time, perhaps, will tell.

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Dingwell JB, John J, & Cusumano JP (2010). Do humans optimally exploit redundancy to control step variability in walking? PLoS computational biology, 6 (7) PMID: 20657664

Images copyright © 2010 Dingwell, John & Cusumano

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