# Walking in the groove

While I surfing the web doing a bit of background reading for last week’s post I came across this graph.

Ralston HJ (1958) Energy-speed relation and optimal speed during level walking. Int Z angew. Physiol. einschl. Arbeitsphysiol. 17 (8): 273-288.

It’s another of the classic outputs of Verne Inman’s group, from Henry Ralston, and shows data for a healthy subject to support his hypothesis that we select our walking speed to minimise the energy cost of walking (the energy used to travel a certain distance). The hypothesis is so plausible that it has been almost universally accepted.

What interests me is that despite being so widely accepted I’ve never seen any suggestion of the mechanism through which we might achieve this. It’s a fairly basic principle of control theory that if we want to minimise any particular variable (such as distance walked for a given amount of energy) we need some way of measuring it. Thus it is very difficult to drive a car fuel efficiently if you just have a speedometer and a standard fuel gauge. If you add a readout to the dashboard telling you how much fuel you are using per kilometre travelled and the task becomes trivial. They should be compulsory in a fuel challenged world!

I’m not aware of any proprioceptive mechanism that would allow the brain to “know” how much energy it is using per unit distance walked. I can see that there are complex mechanisms regulating cardiac and pulmonary rate based primarily on carbon dioxide concentration in the blood which might allow us to sense how much energy we are using per unit time, but how can we possible sense how much energy we are using per unit distance. I’m not saying it’s impossible – the brain is a marvellous organ and it is possible that it integrates such a measure of energy rate (per unit time) with information about cadence and proprioception of joint angle and in order to derive a measure of energy cost (per unit distance). This is a complex mechanism however and certainly suggests that, as with so much in biology, whilst the basic hypothesis is extremely simple the mechanisms required to achieve this is far more complex than we might have imagined. As Ralston himself put it, “one of the most interesting problems in physiology is to elucidate the built in mechanism by which a person tends to adopt an optimum walking velocity such that energy expenditure per unit distance is a minimum”.

But this also makes me want to question the underlying hypothesis. Going back to the original paper (which you can read here), Ralston only produces data from one healthy subject and one amputee to support his hypothesis. I’m not aware of many others having explored the hypothesis on an individual level (the conclusion that the self-selected walking speed is close to speed of minimum energy cost for a sample does not mean that the relationship holds for individuals within that sample). I’d be interested to hear from readers of papers that have investigated this relationship in more detail.

The other point that Ralston made which is almost always overlooked is that the curve is “almost flat”. The curve only looks so steep because it has been plotted over such a wide range of values (from 0 through to 150m/s). Just looking at the data plotted I’d suggest that the speed can range from about  56 to 84 m/min whilst the energy cost remains within 5% of the minimum energy cost value. This is almost certainly within the range of measurement error for a variable such as energy cost. In other words the really remarkable thing about the energy curve is that it allows us to walk over quite a range of speeds without having a measureable effect on our energy cost. It is interesting that Ralston managed to make this point and suggest that we select walking speed to minimise energy cost in the same paper!

1. Mike says:

Richard,
I think that this effect has actually been at least evaluated if not confirmed a number of times. Holt and Hamill Predicting the minimal energy cost of human walking; Zarrugh and Radcliffe, Predicting the Metabolic Cost of Level Walking; Minetti and Capelli, Effects of stride frequency on mechanical power and energy expenditure of walking, and these papers call out others.

I have to admit that I have never critically thought about the mechanism behind the regulation, but, rather fell into the trap of simply thinking that it made sense. However, I think that the paper by Holt and Hamill points to a compelling and plausible mechanism. Essentially, we walk most efficiently at a rate near the resonant frequency of our limbs when acting as damped pendulums. At speeds below this frequency, the subject is essentially actively damping the oscillation, an action that will take more immediate effort than matching the resonant frequency. At speeds above this frequency, the subject must exert more effort to increase the energy in the system with each step.

Muscular effort is generally perceived as a non linear, but monotonic exponential function which would provide a fairly simple, step by step perceptual feedback mechanism. Depending on the particular exponent for walking, this could be a pretty sensitive signal.

I suspect that the key is that the body is not performing some kind of complex temporal integration of energy across time, it simply needs to notice that speeds above and below the resonant frequency require more immediate effort on a step by step basis. I will admit that there is some hand waving here. However, the paper by Hold and Hamill (n=8) presents data that the predicted and actual stride periods, of an admittedly adjusted calculation, were within about 10% of each other, pretty good if you consider all of the non linearities and damping effects added into the system by the knee and soft tissues. Further, the paper by Minetti and Capelli (n=6) provides some compelling evidence that when walking at a given speed, we can actually pick a stride frequency that optimizes efficiency by modulating our stride length. Again, I suspect that the key is that the feedback loop is simply immediate effort.

As to the flatness of the curve, I suspect that if you subtracted off a metabolic baseline for standing still to get you closer to the actual effort due to traveling, you would have a doubling of the percentage changes. Mixing this with an exponential psychophysical perception of effort, and I think the mechanism is at least plausible.

Thoughts?

Mike

2. Thanks, Mike, It’s great to get such a detailed reply. I’ve had a chance to look at these papers and have a few comments. (I only seem to be able to access the Introduction and Methods for Zarrugh and Ralston but I’ve been more successful with the other two)

I think you’ll find that all three papers you mention are looking at a slightly different question to Ralston’s in two respects. The first is that they all report energy rate (per time) rather than cost (per distance). The other is that they focus on the effects of varying cadence at constant speed. I agree that they present quite strong evidence that if we are constrained to walk at a constant speed we will choose to do so at a cadence that minimises energy rate.

They also both show that oxygen rate increases monotonically with speed. It would be really interesting to see these data re-analysed to present the variation of cost with speed rather than rate (of course the relationship with cadence at fixed speed will be the same because rate=cost/speed).

I would also add that Figure 1 in the paper from Minetti et al. (sorry, WordPress doesn’t allow pictures in comments) emphasises the flatness of these minima. The corresponding graph from Holt, Hamil and Andres crops and scales the axes in such a way that the minimum looks much more pronounced. It’s amazing how different this essentially similar relationship appears in the two papers just because the graphs are plotted differently.

So thanks for the comment – its been stimulating to look at this stuff. I’m still not convinced, however, that these papers answers the question of whether and how we choose walking speed to minimise energy cost!