nonlinearity killed the cat

or: how our brain makes movement

When you move your arm, how does your brain make it end up where it should?  How can one play tennis so accurately? A musical instrument so quickly? The principle of movement generation in the human body looks very complex but is deceptively simple. In fact, it can be deduced quite straightforward by collecting and analysing a bunch of facts. Here are some key observations.

the premises

0. the brain is there to generate movement.  This may not be immediately apparent if you think of your own brain.  But look at it this way: every thought is aimed at optimising, in some way, a current or future movement, or learn from previous movements. Non-moving species don’t have brains (or neurons). And, as an exquisite example, the sea squirt shows: after it stops swimming around and settles on a rock, it digests its own brain (its cerebral ganglion, to be precise, explained below).

1. neural signals are slow. They travel through your nerves at speeds up to 100m/s.  This means that, a fast signal from your thumb muscles to your spine takes about 10ms. Sounds fast, doesn’t it? But it isn’t: in that time, your arm may have moved quite a bit. Look at how fast a professional pitcher moves his arm: the ball has speeds up to 170km/h or about 45m/s, meaning 45cm in 10ms. A fast move of only 20% of that would still mean 9cm of movement, before information arrives at the spinal cord.

Add to that the signal going back from your spine to your muscles plus the time to process the signal in the spinal cord, it at least doubles the delay. So we are talking more than 20ms. Now you still have to add the time for the muscles to react, ball park figure: 30ms. To cut a long story short: if you move fast, the time your neurons are able to react you will be quite a bit down the road. So, there is no way for your neurons to quickly correct movement errors it made—it must be good from the beginning.

2. muscle signals are noisy. Muscle movement is coded through two types of sensors: Golgi tendon organs (“GTO”) and muscle spindles. Muscle spindles give out signals related to the length of the muscle.  GTO signal the force in the tendon. Now, both of these are rather noisy signals. There is compelling evidence that, instead of these proprioceptive sensors, our body uses sensors in the skin to code the position of our joints. And you will have observed it: if you arm feels numb waking up from an uncomfortable position, you can’t move it without looking at it. Not because the muscles don’t work, but because your body is unable to tell where your arm is! Medically speaking, deafference is a very rare condition but a few cases of deafferented patients have been reported. In all cases, they lead to severe movement disabilities.

3. our movement is never unstable.  Chances are that you have never played with low-level control of a robot. I did, but I did not enjoy it. It is a lot of fiddling to get the parameters right; and if you don’t, the robot may well blow up in your face. Literally!  If your feedback controller, which amplifies an error signal with some factor, is wrong, that amplification may go the wrong way and magnify rather than dampen a vibration. The same can happen if you try to move a robot where it should not go: if it tries to reach a position but it cannot, because there’s a wall, it will increase its error over (short!) time and oscillate into disaster. This is one of the reasons why engineers never operate robots without emergency button—they know what can happen.

Animals don’t have emergency buttons. Sure, it would be silly to determine who presses whose; but really, in an intact specimen, movement will never become unstable. Why not? The solution is damping. Somehow, our musculoskeletal system is set up in such a way that everything is “soft” and yielding enough that we can easily absorb movement energy where necessary. How and how much precisely is a topic of research for biomechanics, and of a later blog. But this also means that the system is very forgiving for control errors. If our neurons give an imprecise command, our muscles and tendons will make sure it will be right enough.

4. our movements are precise enough. I mean fast movements, meaning: such movements which you do not constantly adapt and correct by looking, feeling, etc. For instance, returning a ball with a tennis racket. Or moving your legs for the next step. Moving your fingers to the next key. Hitting a golf ball. Catching, throwing, … All of them movements where a certain amount of precision is required, and with a bit of training we can do it.

You can turn this argumentation around and say, these tasks are all set up in such a way that our precision suffices. That is just as true. We could not type on a keyboard where the keys are 0.1mm apart. But the key observation is: we need not learn each and every possible movement that we make. Rather, we can interpolate very well, and thus learn many a task very rapidly. If you know how to do something at a point a, and you can do it at b, you are pretty well off doing it in the middle between a and b.

Of course, practice (= learning) improves accuracy. But still, after only a few tries we can deal with many, apparently challenging, tasks. Challenging for robots, for sure.

a bit on brain structure

the human brain

Brain structures seem to vary considerably between species. You will find many internet sources describing brain structure, but let me summarise as follows. Mammals have a cerebral cortex, the “grey matter”, of a few mm thick. It is folded in the brain, so there’s actually quite a lot of it. The most commonly known part of that is the neocortex, that part of the brain we relate to our thinking, behaviour, etc., and having large parts dedicated to motor (movement) and sensor (feeling, seeing, …) processing. Underneath the cortex is the white matter, which mostly acts as a communication centre, intelligently transmitting signals from one part of the cortex to the other.

The neocortex is only to be found in mammals. It was quite a surprise to me to learn that experiments with decorticated animals show that the cortex is not necessary to generate movement.  If you disable the cortex, a cat or dog can move almost just as well.  It is therefore rather likely that the motor cortex models and weighs movements, to subsequently make decisions based thereon. This has not been found in humans, by the way: with the cortex gone, there is little movement but not none.

In some experiments we did and reported in Nature, we were able to listen to neurons in a human motor cortex firing away as our participant, Cathy, observed a robot moving left and right, or when she tried to move it herself by thinking about doing the movement. Some neurons

simple brain dissectionThen there is the basal ganglia and the thalamus, located between the brain stem and the cortex.  The basal ganglia seems to be responsible for making movements, or filter out unwanted movements.  The effect of Parkinson’s disease (the inability to initiate movement) and Huntington’s disease (the inability to prevent unwanted movement) on the basal ganglia is well known and clearly indicates their function.  The Thalamus gates to and fro the brain stem, which then relays the movement to the spinal cord.  Through the spinal cord the muscle fibers are controlled.

The major role of the cerebellum seems to be supervised learning of motor patterns. But also decerebellation does not lead to complete movement loss. An individual with cerebellar lesions may be able to move the arm to successfully reach a target, and to successfully adjust the hand to the size of an object. However, the action cannot be made swiftly and accurately, and the ability to coordinate the timing of the two subactions is lacking. The behaviour will thus exhibit decomposition of movement—first the hand is moved until the thumb touches the object, and only then the hand is shaped appropriately to grasp the object.

putting them together

To best predict a movement which one has never done before—remember, all your sensory states play a role: what you see, what you feel, the weight of your clothes, the temperature, …  it’s never precisely the same—one needs a model.  This model tells you that, if my sensory state is a(t), and I want to go to sensory state a(t+1), I must use muscle activation f(a(t)).

Sounds simple.  But do not forget that, at the root, a(t) contains thousands or millions of signals, and f(a(t)) must activate thousands of motor units. On top of that, muscles are very nonlinear things: if you put in twice the activation, you do get an amount of force depending on where the muscle was, how stiff, what the other muscles do, etc. All in all, a very large collection of nonlinear things.  A model describing this must be very nonlinear, right?

Wrong.  The problem is inversion. The data you obtain describes how sensory state is changed by a movement. What you need is to find which movement to make to obtain a certain sensory state change. Meaning, you need to invert your model.  Inverting nonlinear models is difficult and, in many cases, impossible or unstable. Small errors in the model or data can lead to large errors in the behaviour—as in the robotics example above.

The only way out is to have a linear model. Because it allows for the following behaviour: if I have a model f1(a) for doing something from my sensor state a, and I have a model f2(b) for doing something from my sensor state b, then I can just use [f1(a)+f2(b)]/2 to get good behaviour in the middle. In fact, it’s the only method which generalises over all sensor states.

So this would mean that, if the result of a neuromuscular activation c is a movement x, then doubling that activation gives me 2x. Well, approximately at least. How can that be? After all, we are talking about several hundreds of control variables to activate a muscle, and many thousands to move an arm.

The answer lies in the structure of the muscles, in combination with how the spinal cord controls them. My answer is, I don’t know, precisely. But I do know that movements are, in principle, linear in the neuromuscular domain.

And now one can also hypothesise the role of the cerebellum. These movement models, between which we interpolate our movement, are stored in the cerebellum. Learning a movement more accurately means, that a larger number of cerebellar microzones are allocated to that particular movement, and that the interpolation between these become finer.  After all, we are not completely linear.

I am talking about afferent, i.e., feedback, nerves. The fastest nerves, the Ia and Ib afferent nerves, feed signals from the Golgi tendon organs (GTO) cq. muscle spindles back to the spinal cord at 70–100m/s. Type II feedback signals from the muscle spindles to the spinal cord, reacting on length change are a bit slower: about 35–70m/s. Finally, skin signals are transmitted through type III afferents at 10–30m/s. See E. Kandel, J. Schwartz, T. Jessell, Principles of neural science, McGraw-Hill, 2000.

In fact, we measured delays of approximately 25ms between the activation of a wrist muscle spindle and the activation of the corresponding muscle.

Please quantify! I hear you say. That is not easy. This paper says something about that, but it does not try to find a signal-to-noise ratio. Just believe me, these sensors are too noisy. And on top of that: please remember that there is a lot of flexible tissue between the sensors and the joint itself; they do not measure where the movement is. That is another source of error, and even a highly accurate technical sensor could not solve that.

Published in the following papers:

  1. Edin BB, Quantitative analyses of dynamic strain sensitivity in human skin mechanoreceptors. J Neurophysiol 92(6):3233–3243, 2004
  2. Johnson KO, Closing in on the neural mechanisms of finger joint angle sense. Focus on “quantitative analysis of 
dynamic strain sensitivity in human skin mechanoreceptors”. J Neurophysiol 92(6):3167–3168, 2004
  3. Edin BB, Cutaneous afferents provide information about knee joint movements in humans, J Physiology 531(1): 289–297, 2001

There will probably be more papers, but I happen to know these:

  1. Ghez C, Gordon J, Ghilardi MF (1995) Impairments of reaching movements in patients without proprioception. II. Effects of visual information on accuracy. J Neurophysiol 73:361-372
  2. Gordon J, Ghilardi MF, Ghez C (1995) Impairments of reaching movements in patients without proprioception. I. Spatial errors. J Neurophysiol 73:347-360
  3. Rothwell JC, Traub MM, Day BL, Obeso JA, Thomas PK, Marsden CD (1982) Manual motor performance in a deafferented man. Brain 105 (Pt 3): 515-542

Yes, that includes humans, of course. We’re in the Animalia kingdom, our phylum is Chordata, class Mammalia, order: Primates.

Nasty business, really, to deactivate the cortex of such an animal, but there it is. These two, and some more recent, papers describe that these animals exhibit little change in their movement—except that males cannot mate anymore. Think of that.

  1. Culler E, Mettler FA (1934) Conditioned behavior in a decorticate dog. Journal of Comparative Psychology 18:291-303
  2. Bjursten LM, Norrsell K, Norrsell U (1976) Behavioural repertory of cats without cerebral cortex from infancy. Experimental Brain Research 25:115-130

Doya K (2000). “Complementary roles of basal ganglia and cerebellum in learning and motor control”. Curr. Op. Neurobiology 10 (6): 732–739. doi:10.1016/S0959-4388(00)00153-7. PMID 11240282.

The scientifically interesting paper Holmes, G (1939). The cerebellum of man. Brain, 62, 1–30 describes a few clinical cases related to shot wounds.

One counts in the order of 250.000 muscle fibers in the biceps muscle, for instance. See this publication, which you can read online, or others which unfortunately are behind a paywall.  But then, one motor unit—i.e., one alpha motor neuron and the muscle fibers it controls—has between 10 (eye) and 1000 (thigh) muscle fibers.  So, you can count a few hundred motor units, therefore a few hundred motor neurons, per large muscle.

This is, in fact, very much but not precisely in line with the old models described in Marr DA (1969), A theory of cerebellar cortex, J. Physiol. 202, 437–470 and Albus JS (1971), A theory of cerebellar function, Math. Biosci. 10, 25–61.