Grid Cell related papers

Hi all

Are there papers detailing how grid cells work biologically and mathematically? I’m trying up understand more about GC and it’s math properties. But I did’t find good explanations on my own.


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Hey Marty,

there is not one single grid cell model, that can explain everything. A lot of models have been proposed, and each model has its own limitations and advantages. Additionally there are some combined models.

First it would be good to know why you are interested in grid cells.

  • Do you want to understand the anatomy of the entorhinal cortex (a structure that is highy interconnected with the hippocampus).
  • Do you want to understand the anatomy of a minicolumn in the neo-cortex (if grid cells even exist there…)?
  • Do you want to do path integration (e.g. for a robotic application)?

Depending on those questions, different research areas are more or less important.

I think entorhinal cortex and neo-cortex are both structures, that probably have grid cells or at least some cells that process location information (and even more abstract location information). But that does not mean, that they operate on the same principles. In particular the hippocampus has a special organization, that might affect the way, how grid cells emerge in the entorhinal cortex. It is possible that those grid cells work on a different principle than cortical grid cells.

Regarding mathematical models: There are a lot of models, that are out there and I think they can be grouped into 3 categories:

  • CAN (Continous Attractor Network): A special connectivity ensures that velocity information is integrated. The connectivity is rather unlikely to be found in biological systems and all the models that tried to produce the connectivity relied on the fact, that neurons are physically placed in a special way on the neural sheet. This however was not observed in the entorhinal cortex.
  • OI (Oscillatory Inference): In those models you have VCO cells (Velocity controlled oscillator cells) that project to the grid cell layer. Because of emerging interference patterns, grid cells form in that layer. Those models can explain theta precession, but require very stable oscillators, which is not very likely in biological systems.
  • Single Cell Models: In those models, a specific learning rule and some other parameters allow to obtain grid cells. However those model are not able to do path integration. “The Emergence of Grid Cells: Intelligent Design or Just Adaptation?” (

If you want a great introduction to the topic, I can recommend you this PhD Thesis:

And finally there are some models that explain, how path integration is done in bees:
However those models are very special (special neurons, special connectivity, etc.) and do not produce grid-cells. But still they are able to do path integration very reliable, even with noisy inputs.


This is a mathematical model of GCs.
I made a video presentation which explains how it works:


Thanks @dmac

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Ahh, that’s why I found nothing when searching. Thanks for the link! What should I look into if I want to understand the entorhinal cortex and the neucortex?

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Entorhinal cortex / Hippocampus:
Place cells and grid cells were discovered in this structure. However that does not imply that a widely accepted theory exists. OI and CAN models can describe how path integration might work. So you can look into them. There is also a paper, that combines them to get a even better model (, this was discussed in one of the Numenta research meetings). However it even is not clear, if path integration is a necessary component.

Numenta proposed that the neocortex also uses grid cells to compute the locations of objects relative to a reference frame (I dont know if any other research group ever proposed this, or they were the first?). But this is far from a widely accepted theory. So I can not recommend you a good introduction to this (Jeff however mentioned in one of the recent research meetings, that he thinks differently about the way grid cells work in the cortex, than what they described it in one of the papers[locations paper?]).

So in summary:
Because we to date lack a complete understanding of those systems, I cannot point you to a single theory. Rather I would ask you to look into all those models (OI, CAN, combined, single cell) yourself.

Currently I think, that the entorhinal cortex / hippocampus uses a combination of OI, CAN and single cell models. I come to that conclusion because every model has some benefits and can explain some part of the story.
I still understand the neocortex so little that I do not want to speculate if and how grid cells work there.


Thanks @adam! Your explanations are very helpful to structure my mental cognitive maps about grid cells modelling :slight_smile:

One of my favorite paper on grid cells is this one (about biology, not mathematical modelling):


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