HTM Mini-Columns into Hexagonal Grids!



So now I’ve read some Calvin.
I really like the way he writes, and he provides lots of interesting details and stories on each chapter. At the end of the day, however, I’m not convinced of the central thesis about grids.

I’m well aware that my reluctancy to accept it may be biased by the fact that such wave-based phenomenons, if intelligence turns out to be so inherently based on them, would be inconceivably more difficult to model that simple, almost-amenable-to-sequential layers and areas. Also, let me reiterate that I’m not an expert in any of this, so correct me if I’m misinterpreting something.

To begin with - but that may be something of a personal feeling - after reading “The Cerebral Code” and “How Brain Think” (another nice read, btw), I could not depart from the sensation that the darwinian process subtending his model was somehow shoehorned. Although quite intrigued at first with that proposition, by the end of this book, absolutely wanting to find all 6 darwinian-model-ingredients to the brain’s innerworkings looked like a hammer in search of a nail. A feeling even conveyed by the choice of the book’s very layout.

Inner-darwin aside, his reflexion upon the evolutionary side of the equation were insightful. And I see now why the idea of an evolving neocortex from a need of throwing accuracy matters to you, @bitking. Indeed he seems to have a point here, and the fact that his solution seems to involve his cortical grid concept tickles me.

To get this self-reinforcement mechanism turn out to produce waves interference patterns which he then sparse out as “high points”, would require, in my view :

  • First, [edit] a symmetry between connections to the proximal, “feedforward” part of the neighbouring neurons. Which does not seem to be the case as far as I can tell from my (limited) exposition to cortical columns flow diagrams. Most same-layer, lateral input from sensibly close columns seem to involve [edit, sorry] basal dendrites, which according to HTM is best described as resulting in a modulatory signal, effectively allowing cells that do have them to prevent cells that don’t from firing, all feedforward being equal.
  • Second, we’d need a super-fine regulatory system for it to make any sense (maybe what he means by “automatic gain control” ? I’m not sure). I mean, cells should fire “whenever” a faint input ask them to, but also should fire “only” as part of the grid when surrounded by most of their 6 neighbors also firing ?

Also, 100-step ceiling would seem to be hit quite soon if a pattern carrying semantic value has to :

  • settle from unsorted towards captured by an attractor
  • propagate along the grid
  • be detected as part of a spatiotemporal pattern

I’m also quite uncertain how the maths would solve out for the capacity side of the matter. I think I understand SDR statistical properties by now, but how lots of different attractors could share a common lattice I do not intuit well at all.

Besides, needing ridges and small passages and stuff for the diverging copies required for “speciation” left me a little puzzled, and long-range messaging stuff operating on similar copies also quite a bit. In other words, my curiosity about “H” was not really satisfied here.

All in all, thanks for those links, @bitking. As I said, even with such reserves there were still lots and lots of interesting details and things worth thinking about in those books.

Ah. Also. Something fun occurred to me here while reading the first chapter again :

One technological analogy is the hologram, but the brain seems unlikely to utilize phase information in the same way

or does it ? Grid cell modules and their “overlap” anyone ?



@gmirey, I see four main points in your assessment:

  1. Darwinism. Granted that when I first read Calvin I blew this off for much the same reasons you mentioned. I used to think that this was just a personal jihad that got mixed in with his otherwise good ideas.

Time has since whispered to me that my lack of understanding does not make the ideas wrong.

I think that the timescale of when the concept applies is an important consideration. I don’t think this applies to every presentation of a sensed stream. I do think it applies when competing grids are forming and the “boundary cells” are trying to learn what pattern they really are part of. So the correct time scale is during the exploration and learning phases. So - not all at once and not during every recognition event.

  1. Proximal vs. distal? I think that this is bigger than that. There is a large number of cell types in the cortex, each with different “reach” and different mixes of inputs and output, some excitatory, and some inhibitory. The grid-forming cells and related inhibitory cells are just two types in this mix. The Numenta temporal and it’s related intercolumn inhibitory cells are other cells in the mix.

I hope that you take away that I am only describing a part of the mix of cells in the cortical sheet. I have tried to be careful to distinguish that the layer 2 grid-forming cells are distinct from the standard temporal sensing cells central to the Numenta model. I see them working together as a system.

There are other cells that do not fit into either model such as the thalamocortical cells[1] in layer 4. I am sure that there are more stories to be told here.

Taking your shoe-horn niggle to the Numenta model - they are trying to cram every bit of the cortical sheet into the temporal sensing model seemingly without trying to use these other layers & connections to form larger patterns and connections to distant structures.

All that aside: The grid-forming cells solve a problem that does not seem to get much attention in the Numenta model: binding. Even if I accept the Numenta model as doing everything that is claimed (and I don’t) we still have the problem that this finger says “cup” and that finger say “guard rail” and the palm says “gearshift knob.” Nothing ties them together into a whole and integrates the various sensed states into the whole. The grid-forming layer is in effect allowing the various sensations to vote on a learned thing over a spatial region of a map. Quickly, automatically, and in a biologically plausible way.

  1. The 100 step thing - excellent observation. First - see #1 above. Second - all the local cells are trying to recognize some pattern at the same time. If there is some overarching pattern that the local cell is part of it gets an extra “kick” from its grid-spaced neighbors helping it to become fully active. All of the grid-forming cells are trying to see the pattern at the same time so it is a relatively fast local process.

  2. Regulation. Good point - seldom discussed. Note that at the level of the Numenta model you almost ever hear anything about brain waves or tonic maintenance. I don’t have a ready answer but I think this deserves more attention. I did mention some things about this in an earlier post[1] but did not follow-up on it.

Discussion: As you may have noticed - I read a fair number of papers; I try to understand the ideas being presented and move on to the next one. Some central tendencies emerge and often I see that the work done in the paper is not all that helpful by itself but that unintentionally it does offer support to other work in other papers. The Calvin books fit in that space. When I first read them I was entertained and I did check out some of the related references. It all checked out but I could not see much use for what he was saying and I filed it away with the vast number of proposed models of “how the brain works.”
Then the Moser Grid findings hit the scene and I went back and looked at anything that I had seen related to grids and was struck with how nicely Calvin’s work anticipated this. Then I (finally) made the connection with the binding problem and got serious in looking at his work. Even if Calvin’s work turns out to be wrong it does such a good job of explaining the meta-behavior of grids that I think it is useful as a starting point to evaluate what grids are doing.

Note: I would be delighted to post some of the papers supporting Calvin’s work but these were done in a different time. All of them seem to be either behind a paywall or in a book. The ones that I did obtain through interlibrary loans did seem like good solid technical support for his main points.

[1] Cortical Oscillations: A topic seldom discussed in HTM circles:


3 posts were merged into an existing topic: Neuroscience newbie questions


Oh, I have :smiley:

About shoe-horning : Yeah well. As I said that’s more of a feeling than an argument that he’s wrong. I guess any top-down approach which also cares about the bottom-up side has to shoehorn things somewhere at edges before the whole picture is crystal clear. However since we’re both discussing things here on this forum, I assume we have a gut feeling that Numenta’s broad direction and/or methodology feels right. For me is trying to work from the initial insight that we’re predictive machines who’ll try to discover any structure in whatever input flow we get. I cannot prove it right, but it sure does not sound dissonant to my experience either, and it seems to point towards an explanation while trying to stay consistent with biology findings. An inner-darwin mechanism, on the other side, I have no clue whether it is a necessary requirement. This does not make this wrong either, but I wouldn’t bet my life on it.
So were he to see/feel/intuit/have proof of five over the six “ingredients”, then see a similarity with a darwin model, then try to get a grasp on the missing one, okay. Even if the sixth had required a little edge-rounding. But the book certainly reads like he saw the shiny hammer well before that 5/6 mark.

Granted, he has a life time exposition to these considerations, a bright, well-working brain, and this work shall not be dismissed on the basis that a novice like myself does not understand it. Very true. I’m not dismissing it per se. In fact the best I can do to try and understand, however rough this may sound, is to assault everywhere I do not understand until convinced that it holds.

From your linked post above, I’ve barely browsed ref. [8] and this seems well over my head already. However what comes after the burst does seem like a wave indeed.

pondering pondering

What I still do not get is the relationship you see between “the Moser Grid findings” and Calvin’s grid patterns. Other than the fact they both involve the hex lattice (which is in itself an optimal form for a lot of things, not necessarily intrinsic to brains or navigation). I mean in the very HTM school video you’re referring to, Matt makes it clear that those grid cells spike at fixed locations in the environment’s space, but this has nothing to do with their own layout in the cortex topology.

Sorry to go heavy on that same question twice, when you already took the time to try and answer me, but I’m really confused here.



Me again, sorry.
I’m reading “Network mechanism of grid cells”. Haven’t finished, but wanted to let you know. Should have read it before, it seems. Here too they’re proposing that same correlation between environmental-grid-responses and internal-layout-“gridness”. I’m well open to the possibility that there is such a correlation, although I still cannot infer why.


The parts of Calvin’s book that are central for me are just -

  1. the neural mechanism for grid formation.
  2. A simple, biologically plausible, solution for the binding problem.
  3. the level of thinking once you realize that grids do exist.
    This is a big one or me. We go from a particular behavior (grid forming) to cooperative/competitive learning and what happens when competing population are perceiving some pattern. It fills a conceptual space somewhere between individual cells and whole maps. This is a new level of representation that we can discuss - It may not be Darwinism but how does it work?

My takeaway: There is a higher level of organization that can be tested and may have some new predictive power.

When I first read Calvin it was just an interesting idea: a sort of alphabet and grammar for coding. Please keep in mind that when Calvin proposed this nobody have ever seen this - it was just a prediction. When I read about the Moser work this suddenly got real - some actual real-world information coded just as Calvin predicted it would be.

I put this at about the same level as finding that DNA and the 4 letter alphabet is how genes code data.


I’m erring through papers. Currently jumped from Continuous attractor network models of grid cell firing based on excitatory–inhibitory interactions to its referenced Noise promotes independent control of gamma oscillations and grid firing within recurrent attractor networks.
What I find interesting in those two is the mention that “not all” of the cells of the precise type within the precise layer in which they find grid patterns, do actually arrange into grids. And that some bright people start to find models where indeed some cells do, and some don’t.

Each paper I read actually does a good job at hinting how much further knowledge I lack.

So I don’t know about an “all-in” for cortical grid patterns. But those grids are definitely something worth exploring.

Thanks again @bitking


As you wade through the sea of papers please keep in mind that not all areas of the brain are grid-forming.
I see that there is a progression:

  1. Early sensory areas where “edges” or transitions are sensed and sharpened
  2. Intermediate areas where the sensed streams are rearranged to distribute them over much wider topographical areas.
  3. Hub areas where the “abstracted” contents of the associated lobe are communicated to other areas. This is where I see grid-forming cells. The grid-forming cells are also the level II/III cells that mainly project to other cortical areas. Other cells in the area may have other important functions but these are local functions.

The genetically defined parameters of individual cells control the relative lengths and targets of dendrites and axons to define the behavior of individual cells. This, in turn, results in the “processing characteristics” of the area.


Things have not clicked into place in my mind yet… far from it, but I’m beginning to perceive part of what drives your insights.

good catch all this


Just following up on your note:


Papers offering support for William Calvin’s grid theory.


A post was split to a new topic: Posters summary for the Grid cells meeting in UCL

Grid Cell Spacing and Receptive Field Spacing

I did a clean-up pass to add new material and clarify the relationship between the hex-grid-forming aspect vs. Moser spatial grid representations.


Hi there,

I’m currently reading The cerebral code from Calvin and see that hexagonal lattices might tend to spatially propagate with different inhibition mechanisms that seems to allow a better consistency of the representation. This propagation/inhibition “rules” made me think about Conway’s life game, would there be any related article about this similarity, if ever there is one?



Conway’s Life? That is an interesting insight.

I am not aware of any connection.

On the other hand, I have never thought of looking either.


Cellular automata, of which Conway’s thing is an instance, capture some essential feature of computation in an interconnected environment. Even with a 1D automaton, some very simple update rules are unexpectedly able to produce “complexity”… chaos, randomness ; and some are provably turing-complete. This has been studied by Stephen Wolfram (author of Mathematica, Wolfram alpha…) and explained in his “A new kind of science” book.

(I own a hard cover at home, this is cool stuff…)

Why isn't HTM mainstream yet
Allocentric Grid Fields on a Moving Sensory Surface

Hmmm interesting, and currently half way through Calvins book.

Is any one interested in implementing such an approach on top of what is present in NuPic or would that prove to difficult?


The Calvin cells implemented in L2/3 would work well to do the spatial sparsing function. This could be dropped into the standard HTM model with very little work. The impact on binding/tiling is a rather pleasant side effect at that point.