Curse of dimensionality (and why the brain avoids it)

45:59you can think let's go back to our analogy of the three neuron brain. Okay, you've got a three neuron brain. Neuron 1, neuron 2, neuron 3. But let's say neuron 3 is actually correlated to how neuron 1 and neuron 2 do. Okay. >> Okay. So, my dimensionality is actually not three dimensions because if I know what neuron 1 and neuron 2 are doing, I immediately know what neuron 3 is doing because these two, let's say, are input to them. >> And so, neuron 3's activity is on a plane, right? And so, instead of three dimensions, I'm really only talking about two dimensions, right? and my my brain states are confined to this

46:40manifold in a higher dimensional space. The manifold being this lower dimensional 2D structure. The idea is you can there because of the correlation that that you know exists you can abstract the complexity to a a simpler structure because if ball goes up like if neuron one goes up if you know when neuron one goes up neuron 2 goes right >> you you don't you can now use that as a >> I know that information >> information right and that becomes sort of like >> I can rule out all of this >> other stuff that otherwise would have been in the probabilities okay yeah >> exactly yeah you see what I'm saying so Now you're constraining your your brain states to a lowdimensional manifold in

47:21this very highdimensional space. So even though you have a thousand neurons that you're dealing with, maybe the brain actually only lives on a much lower dimensional surface in this thousand dimensional space just like the three brain the three neuron brain was only living in a 2D surface. M you could even imagine a one 1D surface where it's like a line, >> right? And so all three of them are connected in some way and there's no there's no independence. It's just like all you have to do is figure out where you are on the line rather than where you are in 3D space. >> Okay. So the central insight here is that if we have two different cognitive variables, you remember we're talking about the binding problem, right? That's what we're trying to solve. We had the

48:01red and we had the ball and we had the motion. These are three different things >> that we don't store in individual neurons. >> Exactly. Right. So, we've got to figure out a way for the brain as a state >> to store this information. >> Okay. So, the way we can do this is if we have a giant highdimensional space, but we can constrain ourselves to manifolds. What if there's two different manifolds for the two different variables that we're trying to store? >> Okay. >> Okay. >> Yes. That's the idea. The key idea is you can have in your let's let's go back to our three uh three neuron brain. You can have orthogonal planes. You can have

48:43planes that are perpendicular to each other. And I've got I've got a photo to show this. So, you know, we can have a color plane >> which is one plane. >> And then I can have a shape plane, >> let's say. >> And that's that's the second plane or a motion plane >> and that's the second plane. So now >> here's what here here's what we would do. >> Your brain is somewhere in 3D, right? In this 3D space. But >> what a downstream circuit can do is look at all of these neurons and then figure out what is the shadow of my brain state on one plane. That's going to tell me the color. >> Mhm. >> The shadow of my brain state on the other plane, that's going to tell me the >> shape.

49:23>> Shape. Yeah. >> You see? So I can keep track of a bunch of different information. by constraining myself along these >> the intersections of different manifolds. >> That's the idea. >> That's f because the idea is that like for for lack of a better way to put it, it's I'm still groing, but like the the you have a you have a high a very high dimensional space of potential potentialities. >> Yeah. But you have a very like acute space whether it's on a single manifold or at the intersection of the manifolds which is actually what you care about. And that surface area that that that the amount of information or surface is like

50:05small is fundamentally smaller and shorter to traverse than navigating through the entirety of that larger multi-dimensional. >> Yeah. Yeah. It's kind of like you can't you can't like fully have unstructured just like brain space. Right. >> Right. like your brain states have to have to adhere >> conform to some >> conform to some order in order to actually like learn and like like get ideas and like keep track of ideas, right? So what what you could do is you could have a brain state that's in this high dimensional space and then you can have another brain circuit that's somewhere else >> that is looking at this brain state and then what it does is calculate okay where along this plane am I? >> Yeah. And then another the color circuit

50:46is going to be like actually I care about this other plane. >> So where along this other plane am I? >> Right. And then and then the binding problem >> is kind of solved. >> Solved. Right. Right. Right. Because you don't you you you you now have an ability to um store those the the the color the shape and the motion >> along these with the same number of neurons. >> With the same number of neurons. I don't I don't need individual neurons for red for blue for this that right and I don't need I don't need an individual neuron for red ball red square red star blue ball square blue star right I I can now

51:26with the same number of neurons all I have to do is just calculate where I am on this plane and this plane >> yes >> you see what I'm saying >> yeah I do I do >> very cool >> it's yes please continue sorry but yes yes >> okay so it's cool it's cool to think about >> yes >> but um what what truly impressed me about this paper is how they actually went about proving this. Right. Exactly. >> Because this is this is quite >> like very abstract. >> Yeah. >> Right. So like how how are we actually doing this? >> Prove it. >> Yeah. Exactly. [laughter] Um so they used um two adult rius monkeys. The rius monkeys are chosen because the prefrontal cortex the the the neoortex of a rius monkey is very similar to a human um neoortex. Okay. So

52:07you know for it's better than for example a lot of the other model organisms that we use like rodents because this brain can be involved with rule switching abstract categorization. So it's very close that way and we can actually try to probe these higher order abstract ideas. Right? >> And the stimuli that the subjects were responding to were of two kinds. Okay? There's shape and then there's color. The shape that they were responding to is a bunny versus a T. Okay. And the bunny ears became a T. >> Okay. Got it. >> Okay. So, so that's the sort of shape axis

52:47>> that we're trying to discern. Right. And then on the other hand, we have the color axis which is green versus red. >> And one of thing, one of the things that was that they were very careful about is that these axes are circular. >> They don't have edges. >> Okay. So you smoothly go from a T to a bunny back to a T and from green to red back to green. Okay. The reason why this is important is because you don't have any edge effects, right? Like if you if you have like a square, let's say, then there might be some artifact from being in the corner here because there's nowhere to go over on this side. So your your brain state might be doing something weird, right? But here in this case, you've got a circle and then

53:27you've got another circle. There's no edges. >> So every point in the circle is equivalent. >> Yes. >> Right. topology of this manifold, right? It's a Clifford Taurus because it's a circle >> which is a circle like this way >> and then a circle around the donut and then like through the you know there's there's two circles in a donut. So So you're constraining your stimulus >> to a Taurus. >> Okay. So presumably the brain should also have some kind of representation of this Taurus. And the task that they're doing is the following. Okay. So you've got a stimulus. The subject looks at the

54:09stimulus and then he does a sakad which is the eyes go in one particular direction based on whether it's a tea or a bunny or if it's green or if it's red. Okay. So the subjects are um trained to categorize shape in which case it's a bunny or a T. And they respond in one axis. So the axis being like you know upper left, lower right.