Kara has a charming habit of secreting herself behind some corner and landing a devastating swipe of her paw on the calves of passers-by. This would be very effective at crippling us, if only she had not been cruelly declawed at some point before we adopted her from the shelter. Today's post, however, will not be about the irreducible intensionality revealed by her desire to bring down prey much larger than herself, nor about the qualia of her phantom claws (which I am convinced she experiences). Rather, I want to talk about the mechanism which brings her paw — and with it her fore-arm and shoulder and the rest of her body — to move just so as to effectively ham-string us. This mechanism is accessible to experimental investigation, and, as it turns out, pretty nearly linear.
To elaborate a little on the experimental procedure: the cats were anesthetized with ketamine, which allowed for surgery to attach sensors measuring electrical activity to eight muscles, as well as brain surgery opening up the motor cortex. They also attached a sensor to the paw, tracking its position in space. Then, maintained on a steady flow of ketamine, the "body of the cat was laid on a cushion with its forelimbs hanging perpendicular to the ground and free to move in all directions against gravity". At this point, they began small-scale, low-current stimulation of the motor cortex, until they identified a number of points each of which, when stimulated, produced a repeatable and distinct motion of the paw and fore-limb. They recorded both the displacement of the paw, and the total activity, over the course of the movement, of each of the eight muscles. Thus, each point in the cortex corresponded to both a three-dimensional vector, in ordinary space, and an eight-dimensional vector, in muscle space.
In a linear system, if you add two inputs, you also add two outputs; this, and nothing more, is what "linear" means. Vectors add according to the parallelogram rule, so if the cat motor system is linear, if we stimulate two points simultaneously, the movement of the paw should be the superposition of the movements produced by either point on its own. I wish I could build some suspense at this point, by harping about how neurons are notoriously nonlinear devices, so it's madness to expect any sort of linearity here, but Ethier et al.'s title rather gives the show away. When they simultaneously stimulated pairs of points, they got paw motions which were almost perfect linear sums of the individual movements. This was true of both the paw-displacement vector, and the muscle-activation vectors. (See especially their figures 2 and 5, and accompanying text.) It didn't matter whether they electrically stimulated both points, or stimulated one point while chemically reducing the inhibition at the other. This is very nice, but what I like even more is the experiment summarized in their figure 7, where they took two cortical points, which produced nearly perpendicular movements on their own, and by varying the magnitude of the stimuli at each, got a sequence of movements which smoothly interpolated between them, exactly as one would hope for a linear control system.
Now, I should say that the linearity of response wasn't perfect. The largest systematic deviations from linearity occurred when summing the individual motions would have produced the largest displacements — in a word, the muscle response saturated. To avoid this effect, Ethier et al. first established an input-output relationship for each cortical point on its own, and kept the stimulus magnitude low enough to avoid saturation there. They suspect the place where the response became sub-linear was in the spinal cord, but they don't, that I can see, really establish it.
There are two different larger morals to be drawn from this story. One has to do with the functional anatomy of the motor cortex (a larger, long-running story nicely presented in an older paper of Capaday's). There is clearly a great deal of localization of function there — this point produces a swiping motion of the paw, that that one pulls it back towards the chest, neither makes the tail twitch — but of a peculiar sort. One might well imagine that each point in the motor cortex would correspond to a particular muscle or group of muscles; instead, at least in the part Ethier et al. worked with, they seem to correspond to motions involving many muscles to varying degrees, overlapping from one cortical point to another. Linearity means that a reasonably small set of motions could serve as a basis for a vast range of coordinated actions, without all of those having to be separately stored in the motor cortex.
The other moral has to do with the general principles of neural representation and computation. Neurons are, indeed, horribly nonlinear little things, so it would be entirely reasonable to suppose that neural codes are too; but that would be too quick. One of the few efforts in this area that is general, abstract and predictive enough that it seems to me to be worth calling a theory, the "neural engineering" advanced by Chris Eliasmith and Charles Anderson in a book of that title, takes as its first principle "nonlinear encoding and linear decoding". That is, while the mapping from input to output is hairy and nonlinear, for typical outputs you can recover the input, to high accuracy, using a linear rule. This is especially easy to arrange in neural systems where excitation and inhibition are nearly balanced, so Ethier et al.'s findings on dis-inhibition fit in nicely.
Nonlinear encoding and linear decoding is not just an assumption of Eliasmith and Anderson, but, e.g., features quite prominently in Spikes, and is implicit in the now-standard "reverse correlation" method. While I am not, usually, one to argue with scientific success, I have reservation about this. William James used to decry, as "the psychologist's fallacy", the "confusion of his own standpoint with that of the mental fact about which he is making his report" (Principles of Psychology, ch. 7). Something similar (the "computational neuroscientist's fallacy", perhaps?) seems to me involved here. Neural representations do not exist to be decoded by scientists, but to be used by other parts of the organism, and ultimately to produce adaptive actions. What is lacking, in most of these studies, is evidence that linear decodings of neural activity are in any way biologically relevant. (One of the nice things about Eliasmith and Anderson is they see at least part of this, since their second principle is that other parts of the brain use a neural representation by applying alternately-weighted linear transformations to it, i.e., biased linear decodings. But they present less evidence for this than for their first principle.)
In this case, however, I don't find much room for doubt: the points in the motor cortex represent actions, and those representations are, when the paw meets the calf, linearly decoded. It's still not clear to me that linear decodings and transformations are any easier for the brain to implement, but at least in this case that's what's going on, and it's an empirical fact we will have to incorporate in our models, or ideally explain in our theories.
And now, if you will excuse me, I'm being attacked.
Posted at June 23, 2006 23:40 | permanent link