Grid Cell Spacing and Receptive Field Spacing


I was trying to write a critical review on the binding problem and offering grid cells as a potential solution. I realized that for this to work, the spacing of actual grid cells should have a pattern (am I wrong?). However, the studies I know point to the fact that the receptive fields of these cells adhere the hex pattern not the grid cell activation. I just checked the HTM Columns into Hex Grids thread by @Bitking and to my surprise there was also a reference to the binding problem there. That thread offers the pattern of grid cells as a solution to the binding problem not the pattern of the receptive fields.

The Stable Enforced Pooling thread by @gmirey is also built on top of the idea that grid cells form a pattern not their receptive fields.

The HTM experiment by @dmac presents that the HTM SP can somewhat produce the said spacing on receptive fields. The actual grid cell activation at any given time is just random without any spatial pattern, without even local inhibition.

So here is where I got stuck; how do we know that “grid cell spacing” exists in the sense what we theorize? The sources point onto the spacing of the receptive fields. Yet including me, most of us are trying to thinker with an idea of hex activation patterns of grid cells. I should be missing something here.

HTM Hackers' Hangout - Jul 6, 2018

Maybe we can sort things out.

@Bitking tried to convince me that what I’d call ‘cell grids’ (Calvin-inspired stuff, of grids between firing cells forming over a cortical sheet) and what is called ‘grid cells’ (Nobel 2014, of cells firing at given regular positions when subject is evolving in an environment) were either fundamentally same, or shared some ‘one-implements-the-other’ relationship. It seems there are even papers hypothesizing in that sense, but they’re over my head.

So, to this day, I can’t help with the discussion, if this is what you also perceive yourself (although I’m not against another shot here, at understanding what is it, that you both see similar in it, apart from using the hex lattice). And I was not personally referring to ‘grid cells’ in that thread you quoted, but to ‘cell grids’ :grin:

If to the contrary, what you’re struggling with is, in essence, same kind of eye-frowing than me at that very question, then welcome on board :dark_sunglasses:


You can say that.

I understand that and my question still stands. Can you point me towards some sort of biological evidence or a hint that cell grids may exist? Is there a neural patch that has a somewhat global pattern to it? If that pattern is too complicated in the sense that we cannot observe, then can the pattern really be a cell grid?


Calvin’s whole Cerebral-Code thesis may not qualify as ‘evidence’ but at least is a hint.
Currently we would find many papers about all kinds of hexy things but I’m still struggling with same questions myself, distinguishing ‘cell grids’ from ‘grid cells’. Some ‘models’ seem to refer to the cortical topology indeed… or at least I believe they do… I don’t wan’t to add to the confusion by providing misinformed links here.

Maybe @Bitking knows of actual in-vivo recordings about cortical patches specifically. If we ask him nicely to sort one from the other ?


It is unfortunate that the hexagonal patterns formed by “grid cells” is also a grid.

In all cases where I refer to grid cells, I am referring to the emergent organization of these cells to make regular hexagonal arrays of activation in response to whatever it is the cells are representing.

This pattern of activation is local form tiles as they are responding to whatever it is that these cells hit on. I see these patterns a ranging from a little island to covering large sections of a given map. It is entirely possible that more than one of these islands can be present at the same time on the same map.

I like @gmirey convention of referring to the hex pattern as “cell grids” to differentiate from the Moser work of cells that respond to an external pattern that happens to be a grid. I shall endeavor to use this going forward.

I have no clue as to how the sense outside world is organized and presented to cell grids as feed-stock to make the hex pattern that they form but it is clear that they do in fact make this hex pattern in response. My intuition is that there is something very close to the “butterfly” operations used in FFT calculations but I don’t have a shred of evidence to back this up.

I do think that the cell grids are an elegant solution to the binding problem at the local map level and that the reciprocal connections between maps extend this binding solution through higher dimensional space.

One of my ongoing research antennas is on the possibility that the multi-dimensional maps that form with map-to-map connections pass through an island of activation on a given map at the same time as a different multi-dimensional map pass through a different island on the same map. Saying it a different way - two different multi-dimensional multi-map activation patterns forming at the same time. I see this a a possible key to generalization and the flow of activation from one concept to another. If you encounter any links to this please pass them on.


This paper indirectly explores some of the strings that hold the “cell grid” puppet up:

González-Burgos, G. et al. (2000) Horizontal synaptic connections in monkey prefrontal cortex: an in vitro
electrophysiological study. Cereb. Cortex 10, 82–92

The paper is a hard read if you are not into cell recording and is heavy on lab technique.
Some tantalizing tidbits from this paper:

Given that the stripes revealed by a given tracer injection in the PFC appear to be reciprocally connected (Pucak et al., 1996), it is reasonable to hypothesize that other pyramidal neurons in the superficial layers are the principal synaptic targets of these connections (Melchitzky et al., 1998).

They do confirm that the target is “300–500 μm away” The inhibitory circuit responses are evident in the results but it does not look like they choose to interpret them this way.

" initial experiments showed that the PSCs typically had mixed excitatory and inhibitory components (EPSCs and IPSCs, respectively …"

In fact - they considered these inhibitory signal a nuance and tried to separate them out. This is unfortunate as I see the inhibitory surround as a key part of the grid-forming behavior. They did not consider the possibility that it may take a trio of cells to start a strong response.

Conclusion offered: The Majority of Layer 3 Neurons Are Postsynaptic Targets of Long-distance Horizontal Monosynaptic Connections


A large proportion of the excitatory input to cortical pyramidal neurons is generally assumed to be provided by short-distance, local axon collaterals of neighboring pyramidal cells (Douglas et al. , 1995; Markram, 1997). However, in many of our experiments, monosynaptic EPSCs elicited from distal stimulation sites had a similar or larger amplitude than EPSCs elicited from more proximal sites
Therefore, this finding suggests that long-distance, horizontal, intrinsic projections are a relatively strong source of excitatory input to layer 3 pyramidal neurons.


What Proportion of Layer 3 Pyramidal Cells Receive Long-distance, Excitatory, Monosynaptic Inputs?
Our findings also suggest that most pyramidal neurons in layer 3 are targets of long-distance, horizontal projections. Specifically, low-intensity stimulation at long distances from the recorded layer 3 pyramidal cell evoked monosynaptic EPSCs in the majority (77%) of these neurons. However, this proportion is likely to be an underestimate since some long-distance axon collaterals were probably severed by slicing of the tissue blocks.

Here is a bit for @gmirey and the V1 studies:
" Interestingly, the range of horizontal distances (135–810 μm) between the stimulation sites that evoked peaks in EPSC amplitude was very similar to the range of distances (200–1200 μm) between centers of stripe-like clusters in anatomical studies of monkey PFC (Levittet al.,1993). In fact, previous studies in other cortical regions suggest that clustered connections are the basis of a peaked distribution in EPSC amplitudes. For example, using whole-cell recordings and multisite electrical stimulation in visual cortex slices, Weliky and colleagues (Weliky and Katz, 1994; Weliky et al.,1995) observed EPSC peaks when stimulation was applied at the locations of clusters of neurons that shared orientation selectivity with the recorded cell. These iso-orientation clusters of visual cortex neurons are known to be linked by intrinsic horizontal axons (Gilbert and Wiesel, 1989)"
So - griddy cells are possible useful when they are not forming grids.


Grid cells no doubt solve the binding problem when they respond to large & contiguous areas of the input space. Grid cells appear to be a special case of the binding problem: they bind themselves to locations, and the locations are special because of their size, shape and periodic nature. I suspect the brain uses some of the same methods for both grid cells and other areas which solve the binding problem (L2/3?).

My hypothesis is that the cortex ‘binds’ sensory inputs into a single representation on the basis of how often the inputs are seen together (adjacent in either space or time).

I’m sceptical that the physical locations of grid cell is important.

  • Can ‘cell grids’ function with a single macro-columns worth of grid cells? Or is a single macro column in isolation broken. Imagine a mouse with a single whisker…
  • Can ‘cell grids’ represent spaces which are not 2D like the cortex, such as the volume of space which is within arms reach?


I would like to inject a point of order here.
@gmirey has made the point that cells that react to “whatever” by forming regular grid structures are not necessarily the same thing as cells that are coding some spatial aspect of the surrounding environment.

These grid-forming cells combine a sparse collection of columns into a larger hexagonal array.

These cells (thought to be in Layer 2/3) that form a regular array can offer a separate and highly useful tool in the neural network toolbox. They can act to take a local response from a column responding to some locally sensed condition and join it to other local columns that are also responding to what they see of their part of some locally sensed pattern. This is a form of binding.

The Moser/grid cells use these hex-array forming cells to code the external sensed environment so we have a hex-grid signaling an external grid.

I have not been careful to separate these in my writing and I fear that is has been a source of considerable confusion.

My bad / mea-culpa.


Thanks for the clarification.

I’ve hesitated replying to this with maybe counter-arguments to my own position, risking muddying the waters in the process… but I feel it deserves some modulation.

I was insisting on the distinction between the two, as they are two different concepts in my mind, but I’m not refusing the hypothesis that they are correlated.
Now, anyone proposing such correlation in the hope to be understood would need, in my view, to first recognize the original distinction… and subsequently point out where the two concept could be joined back.

And the following statement

Would still deserve an explanation.

Yet I’m aware that the question ‘what would be the source of an environmental 60° phasing’ could maybe be brought back to an underlying cortical-grid topology, in which those concerns could be backed-in (Although I still don’t know how).

I’m also aware that the following paper:

essentially seems to propose that the correlation exits. A few others do

I personally can’t make sense of it at this point, though.

Anyway. These are models. Not recorded patterns as you asked for, @sunguralikaan, I guess.


@Bitking and @gmirey, thanks for the sources hinting at cell grids. I was hoping that there was a shortcut to internalize cell grids since I am tinkering a lot on that pooling layer. I concede… Started reading Cerebral Code.


I think it is fair to say that we both found the emphasis on evolution in the Calvin work - mey.
Please don’t let this put you off from the core of the tile mechanism which is excellent.

Also a personal note - he keeps representing the tiles as all showing the same pattern - say a banana or a note. This takes away from what should be a more correct idea of “that relative part of a banana.”
I think that this arises from his evolution bias. If you substitute this as “part of a pattern” as you read you should find that it more closely explains many of the details of pattern binding.


Hi Bitking. Pls quit your day job and go work at Numenta. From my layman’s perspective I think you can speed up the theoretical integration of a whole lotta existing brain research stuff.


Thanks for the kind words.
I’ve been reading this stuff for a long time. Perhaps it’s time to write down some of the juicier bits.
I’ve seen how a few things seem to fit together so I write about them - sort of a conspiracy story thing.
Looking forward to seeing you around here.