Numenta Numenta Journal Club, 16 Sep 2020

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From YouTube: "Accurate Representation for Spatial Cognition Using Grid Cells" Paper Review - September 16, 2020

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Karan Grewal reviews the paper ‘Accurate Representation for Spatial Cognition Using Grid Cells’ by Nicole Sandra-Yaffa Dumont & Chris Eliasmith. He first gives us an overview of semantic pointers and then discusses the use of grid cells in spatial representations.

Link to paper: https://cognitivesciencesociety.org/cogsci20/papers/0562/0562.pdf

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A

uh Today, I'm gonna talk about um this paper that I read um it's from a conference that I attended about a month ago. um There was another paper too that I also wanted to present. That was also on spatial representations, but I didn't, I ran out of time, didn't end up getting to that, but this one focuses more on grid cells, so I'll be talking about this um by the way I just want to say um this is my first time presenting on grid cells, so I did get a lot.

A

I did get the high level ideas out of this paper, but there are probably a lot of low level details that that I missed out on so just a disclaimer okay.

A

So, first before we talk about the grid cell aspect of spatial representations, there's a lot of background that might be worth going through.

A

So this research group, in particular, has spent a lot of time working on semantic pointers, and so just high-level idea of semantic pointers is that they they're all used to represent concepts like red square et, cetera um and, and it seems it seems, a lot like they're, there's they're somewhat similar to word to vect, or these neural representations that we get just because the some of the original papers that they cited on this was we're back from like going back to 2010, when we didn't really have things like where to back so some important things you can do with these semantic pointers.

A

Are you can bind them um like? uh You can bind two vectors and then they can be unbinded, so you can. We can, if you originally get? U, as v uh v, bind it with w. Then you can. You can get v back by doing this operation, um I'm not entirely sure if this is a sort of a one-to-one mapping that you that there's only one um thing that you can. They always get exactly vbac by unbinding, but um they claim that it seems to work like this.

A

So here you would use the inverse of w the um this is all pretty high level, but what's what's really going on here is they're using a they usually use a circular convolution to perform this operation, which is which I think is like a discrete fourier transform, um yeah and so uh here's a here's a bit more of a concrete example that I that I came across that might help explain things.

A

So if you have um a red square and a blue circle in a picture, then you could probably represent that as being as binding the red and square semantic pointers together and then add adding that to the blue and blue and circle semantic pointers also binded together.

A

And then, if you wanted to find out what the color of the square was in the picture, then you would apply the the the approximate inverse of the semantic pointer of square to to v and you'd get this. So the square inverse this distributes over addition, so you'd get so the square and the squares approximate inverse would cancel out so you'd get red here back and then and then to this term. They claim that you just get noise, so you'll be able to pretty much retrieve red.

A

I wasn't entirely sold on this, but I'll take it at face value. So this is the idea of a semantic pointer. It seems pretty similar to um like quarterback that we sort of have.

B

What was it you know? I read this paper, um but I didn't understand a lot of it because this semantic corner thing was it was, I think, I've run across it before, but I didn't really understand it and then and then, when they started talking about how how these operations actually work, but it wasn't clear there was: was this a neural model at all, or is this just a mathematical model for how to do these things? There wasn't a neural model. No, there was not. There was.

A

There was no like in in all the stuff they talked about on semantic pointers. There was nothing to do with neural networks. Okay,.

C

I'm just curious to see where this goes, because I have not looked at this paper, but this seems so much like vector, symbolic, architectures and we've. We've talked to people who work on those a little bit and drawn like a a uh a analog that displacement cells perform a sort of binding operation. And if you have a square square grid code and a red grid code, you can treat them as like a displacement. And then you can do all these same operations by setting up this binding operation as applying and removing a displacement.

C

So like it's just interesting just to see this, I'm wondering where this is going, but.

B

It also reminds me of the stuff that penny canerva did, but I forget what.

C

He calls is that the center, that that's it, that's it that's, okay, those vectors yeah.

B

But I never felt that was a good neural model. I just couldn't never make it work, but you I I don't remember this. Could I I'm sorry if I got that I don't remember that.

C

Yeah, that was a little bit more me, marco and scott kind of okay. uh You know we're on a whiteboard about that, but the idea is that um that you can take. You can take what penti canerva has been talking about and use two concepts, displacement, cells and unions, and you get a full vector, symbolic architecture and what uh quran is shown here: red binded to square, that's a displacement vector, uh and then the plus edition is like a union uh like we we can.

C

We can translate all of this, not necessarily we should, but I'm just saying you can take these grid cell operations that we use um and and and do all this so anyway, I I've jumped in for long enough. I'm curious.

A

So marcus is interesting that you mentioned vsas, because in the in the papers where they describe these semantic pointers, they do mention that a lot. I don't have too much background on that area, so I sort of left it out just for simplicity, um yeah, okay, so those were semantic pointers um and then one of their recent papers from last year they showed how to build spatial semantic pointers which are basically the same concept but applied to spatial locations.

A

So here um what you would do is um for each uh for each say you have um x y, which is which gives you a point on a 2d grid right. If you want to represent that using semantic pointers, um they would you'd pick you'd pick um so big x and big y here are fixed semantic pointers and each of them. So these are semantic pointers that are um that represent the dimension. So so these are fixed they're, not they're, not variable they're they're fixed and they represent a certain dimension.

A

So one could be the x axis and the other one could be the y axis and then you would get the spatial semantic pointer by um basically getting. I guess semantic pointers for for each of the dimensions and then binding those together. So so here, what's going on, is continuous. Finding so see here, x's, x and y are small x and small y are real numbers they're, not um they're, not necessarily integers, so they using a fourier transform they just they define exactly what exactly how this is done.

A

uh The details aren't too important, um and so um so this is how they come up with their spatial semantic pointers, um and I guess the the neat thing here is that they they they aim to show that um when you have, when you have an uh a concept like h, so h can be uh just a regular semantic pointer for something like an object, landmark or person or something else.

A

Then you can actually bind that together with its um with its spatial with with its spatial semantic pointer, so that you're sort of binding the concept with the location. So you can say: if so, if certain landmark is at a certain position, you can bind that landmark with the position that it's in um and then and then since you and then, since it's derived like this, you can actually figure out the location of some object by taking the by taking like h, inverse and multiplying it um by this binding it together with this.

A

So this is just about a bunch of arithmetic going on here.

A

Okay, so that's the idea of spatial semantic pointers and these are going to be used for the um for to build the grid grid. Cell-Like representations.

A

So, by the way I just want to mention that it wasn't entirely clear to me right away what the advantage of having a spatial semantic pointer was over just a regular semantic pointer. That shows a uh that gives a particular location of an object, but I guess these from this probably has some properties that um regular semantic pointers don't satisfy.

A

Okay. So now we get to the um the modeling problem here so um so the goal here is to find grid cell representations to discover grid cell representations of the observed place cell responses, so the representations of locations that we've obtained from the spatial semantic pointers the ssps.

A

um Those are those are those are like our observations here and then we want to uncover grid cell representations for those, and so how this is done is.

A

Just through a minimization of okay, so this is sort of like a um this can be done in this can be a neural model or it can just be there's a regular regression problem.

B

I'm just making sure I understand you're, saying that they're they're trying to derive grid cells from place of representations is that right.

A

Yeah and then yeah so they're, it's it's almost like they're encoding it, so they want to encode these observations that they have, which are taken from their spatial semantic pointers which are supposed to represent place cell responses.

B

Under this kind of scheme, I mean to say: okay, we're trying to recreate some physiological um effect we see which are grid cells. Do they ascribe why they're doing that and what the grid cells are doing and or is it just like hey? We can get this this sort of response pattern. If we do this operation.

A

So in there so when they talked about their results, um they didn't really apply it to any particular task where grid cells would be particularly useful. But um I'm guessing that you could say you have some toy tasks where you would want to. uh You know you have some sort of spatial representations you could use. These grid cell representations that you derive that you learn from your play cell observations, instead of using the original ssps that they had before or any other sort of spatial representation.

B

I mean the bigger thing about grid cells, there's several very important things one is they do this? This path, integration based on movement.

D

B

Yeah, so so, just just deriving them from placeholders doesn't give you that at all yeah and and then you have these other things, they have the the whole um data procession um effect, which is very critical and so again that wouldn't come it's a time-based thing, a movement-based thing. So I just didn't know how deep they went here. You know some people just say: hey, look. We can get cells to respond like x, but that's not really what we need.

B

You know we need something much richer than that, so I don't know if they it sounds like they didn't. Go there too much.

A

They did note that um it right now the grid cell representations that they uncovered. They don't satisfy the path, integration, property that.

B

Seems to be like a huge hole.

C

So it's like question for clarity, the so right now you're showing uh this this other stuff from this other paper. This pattern formation, the one from the kangali group uh is this- is this? Did the semantic pointer paper also do these techniques did they cite this paper? I'm curious! What made you jump in and yeah.

A

So I think so this task here of um constructing of basically encoding from your observations to grid cell, like representations and back to your observations. This was a problem or this was framed in the ganguly paper and then there. This is what they're applying their ssbs to okay, yeah. um So yeah.

C

So like to jump in just a little bit about this paper uh that their group decided to take a different look at grid cells for for just to entertain the idea of, can you explain good cells from the length through the lens of pattern formation? It was it's like the terminology they they use, so they they de-emphasize path. Integration in this paper, but these same authors will uh will include path: integration in other papers, so they're just exploring this angle.

A

Okay, so you're so you're, familiar with um with this work marcus is that right.

C

Somewhat uh yeah, and so michelangelo has also looked at this some but uh and then plus syria ganguly is a professor at stanford. This this group is a stanford group.

A

Okay cool, so when they um wanted to come up with these uh representations for uh in this in this layer g here um they had one they had a constraint, and that was that the the activation values of these sort of units here would would take on this form. So what you? What we have here is um is basically you have we're summing over three terms, and so there are these three, what they call plane cells.

A

It wasn't exactly clear how these plane cells are derived, but it, but they have three different plane cells, and then it also takes as input the the spatial, the ssp, the spatial representation, which is the observation here and ultimately, what what?

A

What this is is um just a sum of these different sort of cosine, curves and so, and so, but they mentioned that it's each cell or unit that um that has this activation like this, but they didn't mention whether there's a relationship between the different units as to whether, like you know um as to whether they're related or whether they're all independent of each other, because if they were related, that would sort of make sense.

A

Because you have this sort of periodic pattern going on, but they didn't make any mention of that, but otherwise um they did constrain their sort of representations here to sort of follow this pattern, and so that's how they, I guess, got things that look like this.

E

That's a great question: I understand the plane reference, but they have k there as a vector which means that the three components say at three different frequencies could actually be heading in different directions. They don't necessarily have to be aligned with each other. Is that something that I mean? That's that's. What I see with from that equation is that something they were talking about.

A

Yeah, so they are in the case where they have three of these um they're all exactly 120 degrees apart, and they all sum up to the zero vector in the space that they live in.

E

Okay, so it's not arbitrary, they actually, uh those those k's are discrete.

A

Exactly and they're all the same, they're all the same length just pointing in different directions.

E

Oh they're, the same length yeah. I see three different frequencies being summed here: spatial frequencies being sum. So when you say the same length, what does that mean.

A

The norm of the k, vectors.

E

Oh okay, got it got it got it yeah. Okay,.

A

Yeah, okay, so um so one important, um I guess, engineering aspect here that they had to look into was um how they're exact, how they're going to pick their um pick.

A

These these semantic pointers the big x and the big y, and so they refer to these as bases, um because how you how you pick these um basically determines what your grid cell representation looks like here, and so they picked certain ones um so that they would get uh to the so that you would get this sort of activation over here and- um and so they showed they showed in these pictures here that um when they had when they had this, these specific encoding, so when they, when they picked their x and y um semantic pointers, so that you would get that sort of activation, that you'd get you'd, get these sort of firing patterns.

A

And then, when you picked random, encodings you'd get um you get very sparse patterns that look like this.

A

And finally, they showed some results. uh They didn't really apply this to any tasks, but they did. um I guess reconstruction was the only real task they did here where they tried to go from their.

A

They try to go from the original spatial semantic pointer to the representation level and then back to the um back to their observation, and they show that, if they're using, um if they're, using if they encode it such that it it looks like uh it looks like those the some of those the sinusoidal curves you're, getting you're you're, reconstructing it a lot better. Otherwise random is just a random encoding which doesn't necessarily give you grid cell-like representations at all and that's giving you a higher verbenius norm.

A

And then this is kind of similar. They talk about place cell centers being able to reconstruct the center of your observation or how far the center was from your input to your output. Yeah. So that's it.

A

But I guess I guess the high level idea is trying to get they sort of got something that looks, looks like grid cells but, as jeffy pointed out, um there's a lot of properties that are that are still missing and they made note of that um in their work too.

D

This reminds me a lot of the. um If you look at the, if you show your graphs, it might be a lot of the paper where, uh if you start with place cell risk responses, and then you do non-negative principle component analysis, you get you get responses that look like grid cells and, of course, pca will automatically be trying to minimize the reconstruction error, and so um it seems, like you, kind of have to get this graph almost like it's. You know, you don't have a choice.

D

B

Great is that suggesting it's it's less interesting, trivial.

D

Well, it's I'm just saying this. This. These graphs are not surprising, given the previous result that pca on place cell responses will give you grid cell-like responses. That implies that grid cell responses will minimize the reconstruction error or place cell.

D

Mathematical equivalence- almost the only trick, is that they they had to do this thing called non-negative principle component analysis. So that's the only piece but.

D

A

In that sense,.

D

It's not surprising, I guess.

A

Yeah, I would have liked to seen them use their um grid cell, like representations applied to some some sort of navigation task, which they didn't really do, um but I think that would have that. I think, if that, if it were actually able to perform well, that would have made it a lot more interesting.

D

Yeah, but I think, as you mentioned, you know they you have to do the uh you know: path: integration, property, yeah, yeah. You have to have that now. You need to really address that.

B

You know no one really understands how grid sales and place cells will you know really work. I mean in terms of um there's a lot of uncertainty about how they, how we navigate using them and um and so there's a lot of there's a series of literature, and I would include this as part of it, which is saying.

A

Okay, we don't really understand.

B

The bigger picture of how all this stuff works- let's see if we can just recreate this physiology or explain how these things come about, uh what might be some of the mechanisms that lead to these things, which is valuable, but I I've seen, I would be very especially surprised if they tried to actually do some more of an end analysis on this.

E

Frank, can I ask you, go to the previous slide again.

A

E

Yeah, so uh that equation you have in the upper left hand corner is that a convolution operation.

A

uh This is a um circular convolution, okay,.

E

A

E

Okay, I just want to make sure, because the sxy seemed to be in the in the spatial mean rather than the frequency domain, so so the x and y are basically uh uh some of these. These plane equations is that what it is they're they but they're in the spatial domain, when they're, when they're doing these operations, are you.

A

Talking about the big x and big y yeah, um so those are so those are from what I understood. Those are semantic pointers representing um one representing the x dimension, one representing the y dimension um they're in they're in the they're in the same space as each other.

D

A

So so this so this operation doesn't change the dimensionality of.

E

No, I just want to make you you made references to 4a transform. So that's why I was. I was trying to make sure I understood what operation which, which domain you were actually in when you're doing this.

D

E

Okay, so so, basically you're you're convolving convolving, plane waves with other plane waves in different dimensions. So all right, I can see how you're going to get these these patterns, I'm just it's a it's a it's just a uh interesting uh okay! Thank you.

A

uh In fact, if you wanna, if you're interested in the details, it looked, um they had it here somewhere. So it looked something like this, so I think f here is the fourier transform. um This is exactly how they're doing it.

E

Okay, so but they're using a hadamard product, which would be what you would do in the in the uh in the uh frequency domain, so the fourier transform and the frequency, okay, so yeah. So that's there is they're, essentially doing a convolution. This is just a cheap form of doing the convolution.

A

E

And it has to be circular because, if they're using ffts, it's inherently circular, it's not clear that that grid cells and whatever they're trying to represent you know, are inherently circular.

A

Right right, right.

E

Okay, thank you.

C

One other just side note the paper subatai mentioned about negative pca, um so just to throw in a couple of details there like it, was first found that principal component analysis uh will lead to something kinda grid like but they're like more square grids than hexagonal, and then that paper that subatai mentioned noted that that, if you do non-negative pca, some reason is hexagonal.

C

um This other the paper that quran was showing from um sarcher and ganguly and all um that jumped in and gave a theoretical explanation for why the non-negativity constraint caused it to be hexagonal rather than a square grid. So that was one of their contributions um in general. There's this like space of questions of.

C

Why does principal component analysis or or some people talk about it in terms of eigenvectors? Why does that lead to these grids and what? uh How do we make sense of that? And does that have anything to do with grid cells in the and then rhino cortex, and they they end up causing questions for each other like, even even if it doesn't have anything to do with internal cortex? It still raises a bunch of interesting analytical questions like why did it become hexagonal, so people then write papers about that uh and anyway, that's yeah.

D

That's one way to think.

C

D

I think the tie-in is very intriguing, because hebbian learning will do principal components extraction so there's just like there seems to be like all these elements that seem to somehow point to grid cells being a natural representation in in this scenario,.

C

You could also look at it as a as like a warning or as a thing. You need to be cautious about that. um If you see grid cells in your network, are you seeing actual grid cells, or are you just seeing principal components that are kind of a red herring? Are they the the the trap you can fall into? Are they actually? Is it actually like a siren song of principal components as yeah, I'm not saying an opinion on this, I'm being cautious like? Is it maybe it's a siren song? Maybe it's the real thing.

B

Yeah, I have a question about the figure that's on on the screen right now, so there's uh three sets of grid cell uh modules at different spatial resolutions. There does.

A

D

B

The bottom ones: no, the upper left here, the top center under the g.

B

um So so you see, there's like a the first four is one resolution. The next is small and extra smaller. Does this produce all three of those at the same time, or is this like I'm just curious? Why why the different results, the resolutions there is.

A

I think I think that's just for um diagram purposes. um This is so. This image was actually taken from the from from this paper from the sorcerer paper that marcos was talking about, but as far as I understood it in their setup, g and p were just are just matrices where, um where the first axis just corresponds to the the example uh like the the input example and then the other axis actually corresponds to the different, the different cells in that unit. In that layer.

B

All right, so I guess my answer is I'm not sure I got the answer, but it seems like you're saying it's there for illustration purposes like this is what grid cells look like, as opposed to this set of particular p's going under this operation, will leave to us a series of grid cell modules at different resolutions.

A

Yeah, I don't think I don't think it's the lateral one.

B

Okay, all right they're, still just leaving looking at that. You know. There's one other possibility, the one one you know we just talked about how good cells may just be a sort of a artifact of pca. I guess is what you're saying, um but you know one thing that I've been working on and I've talked about a bunch of times- is that there's a real dealing in two-dimensional.

B

uh Good cell arrays is really problematic in a lot of ways, um and so I've been keep finding myself going down the road of wanting to do path, integration in one dimensions and representing movements in one dimensions.

B

And of course you combine movements, multiple movement, vectors, you end up with a grid, cell-like representation, but also that it could be just that we need one of the things that grid cells have to do. Is they have to maintain their activity? In the absence of sensory input, um they have to be stable. You know like you're, standing in a room, and you close your eyes. You still know where you are.

B

You know it doesn't go away for a long period of time, and so there has to be stable representations and the general belief is that sort of a continuous, attractive networks will produce that kind of situation. So, if you're trying to produce a stable representation in a two-dimensional sheet of cells, um using sort of uh general surround inhibition expectation principles, you'll end up with something like this.

B

It's like just what you end up with so another possibility that I find very intriguing is that the whole two-dimensional representation of grid cells is really just an artifact of of a needing at some point in time to form a stable representation, um but that's not really the operating um uh view of the world. If that makes any sense.

C

So I'll uh in my next presentation, which is probably monday I'll, show some stuff that kind of supports that idea.

C

It really kind of connects a lot of what's what's drawn here to uh to that. It's funny seeing this the the connections you'll see. uh Actually I have a figure out that I could show real, quick well.

B

I'd be really happy because you know it's it's kind of going out on a limb and saying hey all this two-dimensional good stuff. Grit cell stuff is not really true at all. It's really a bunch of one-dimensional grid cells, and that would be you know, really contrary to almost all conventional views, and so, if there's any, but I keep finding myself deciding that must be the case, so I can't figure out how to get a right away from that. So, if there's a parallel evidence for other evidence, that'd be exciting. For me,.

C

There is, I, let's see, there's probably there's a mix of empirical evidence, some of it and there's also other theorists who uh who have the same desires as you uh like. I.

C

I would say that, um like neil burgess, tim barron's uh that um and um this other paper I'll present some of their um some of their insights, um they also embrace the same idea, and so you know the you know like velocity, controlled, oscillators, vcs yeah, the people working on that that's conceptually the same thing as like a 1d grid cell, except it's the phases in time.

B

Yeah now interesting, I think when um uh we had a presentation um where we were talking about different ways that those could come about, one of the velocity control, oscillators, definitely sort of like these are one-dimensional things you combine them, but then there was also the idea that an individual neuron could have different velocity control oscillators on each of its dendrite branches.

B

um Remember that, um like there was inherent oscillations in the dendrites, um is that like familiar.

C

Yeah I mean I remember that there are some oscillatory models that depend on that.

B

Yeah I mean so that would be one where, like you don't go, you don't actually do. There would be no separate, one-dimensional uh grit cells that somehow that they go right to 2-d, because the cells do it themselves, but anyway, all right, well, good, so yeah you're right. um I guess you could think of the vcos and supporting that idea. You're right, yeah and that's compatible with what I've been thinking so great.

C

Yeah, if I, if I can share my screen real quick I'll, just give you a preview of some figures that I'll I'll be showing soon, uh and it's just funny to see. So, um let's see here's this and by the way this this mirror uh notebook. uh Whiteboard thing is cool, I'm gonna be using it more.

B

Is that the thing that someone sent to me.

C

Yeah, I think what will warren sent to and I've been using it, and it's good, like you can draw in on on an ipad. You can then write on it on your computer, and it also is just syncing live and it works really nicely. um So the the figures up here up top are a little bit reminiscent of what of what quran was just showing, except this is the more square version of them.

C

I went through a similar technique to generate these and an insight that I'm going to show is that okay, so I'm going to describe all this in terms of these are eigenvectors of of space. Essentially, but these down here, you can kind of rotate this set of four to get these instead rotation in the sense like a high dimensional rotation. uh So these are also eigenvectors of space, uh and these are gonna.

C

I'm gonna draw a connection from here to uh vcos to velocity, controlled oscillators into rings, and, and these are the different scales uh and um the different scales of the grid cell modules and by the way like what I would have described here. Other people have pointed out that the paper from burgess barons etc generates these types of figures, uh but I will make a nice like observation of how these are actually the same as these just rotated they're in the same eigen space quote unquote, but I'll talk about that on monday.

C

One thing one reason I wanted to show this is uh these actually do come out in sets of four sort of like how quran was showing these sets of four uh these. These sets of four cells at one scale sense of four uh fields at another scale that actually might not be a pure visualization trick. There's actually a technical reason why these come out in fours like this, it's like the sine and cosine sine and cosine, going in two different.

C

Directions so anyway, that's that's more of a preview.