Back to computation in the nullspace. How is it useful? Easy to see how its useful for preparatory motor activity: the null-potent space distinction allows for gating the downstream effect of neural activity, so that it can act in a preparatory manner. In other words, by living in the nullspace, preparatory activity is safe to plan the next movement without interfering with the current one.

But lets break this picture down a little further. What is a "movement"? Where is the distinction between the current movement and the next one currently being planned? Does the motor system really work this way? The simplest way to imagine what motor systems do is take some vector representing what you want achieved and then calculating and spitting out what set of muscle commands will achieve that. Where does preparatory come into play there? The idea in the literature seems to be that motor cortex is a "dynamical system", with the property that it behaves quite differently when put in different initial conditions. The preparatory activity's job is thus to choose the right initial conditions. But again - what does "initial" mean? In papers, the initiation of the movement is given to you on a platter with the delayed reaching task "Go" cue. But in the real world there are no Go cues. And there is no delayed period between target onset and go cue (except in really contrived situations, e.g ready,set,go!). So what does preparatory activity do then? It's a bit of a mystery to me. I should probably read more, but it sounds like a classic case of abstracting principles of neural activity from a highly contrived and artificial experimental task, i.e. modern calcium imaging era neuroscience.

One possibility is the existence of primitive motor movements. In this case, there are very distinct units called "movements", and each one can be produced by simply initializing the motor system in the right way and then letting it run (presumably with feedback). I'm pretty sure the idea of motor primitives has been around for a while, but I should look more into it. The idea is quite appealing from a learning perspective too, whereby simple tasks can be easily accomplished by a sequence of motor primitives (i.e. a sequence of different initial conditions) but harder tasks require learning new motor primitives, which might require some rewiring of e.g. motor cortex.

I'm actually currently working on this problem right now - although avoiding the issue of learning for now and just trying to see if we can hardwire this into a network. What I'm finding right now, in very preliminary stages (just trying to do this with a linear dynamical system - which could be harder to do than in a nonlinear network, but linear systems are easy to work with analytically), is that it's really quite hard to design a dynamical system with the desired properties. You want a dynamical system that (1) produces highly (meaningfully) distinct trajectories when starting at distinct initial conditions but (2) is robust to small perturbations of its initial condition. The brain is really noisy, so (2) is just as important as (1). It is worth noting that - at least in rat barrel cortex - cortical dynamics look to be pretty chaotic, i.e. condition (2) doesn't hold. But one could easily imagine motor cortex is wired up in a certain way for (2) to hold. But what I'm finding is that making (1) hold is pretty hard.

One caveat I that came to mind: when we speak about the initial conditions of the system, I think it's important to note that the system is a closed loop feedback system. One could imagine the feedback makes it easier to make sure (2) holds, while the wiring makes (1) hold.