Summary
This work reviews some of the issues with existing models of grid cells (cells in mammalian brains that fire when the animal is located at the vertices of a hexagonal grid) and suggests a new model based on recurrent neural networks (RNNs). The model is self-supervised, eliminating the worry that the structure of the readout in a supervised task could influence the outcome. Besides leading generically to grid-like firing, the model also exhibits multiple grid scales organized in modules such that the scale is the same within each module but the phase varies.
Strengths
*Originality:*
The paper suggests a way for grid cells to emerge from a self-supervised learning (SSL) paradigm, in contrast to previous work which works mostly in a supervised regime.
*Quality:*
The paper does a good job of surveying prior work and includes a fair amount of simulations.
*Clarity:*
The presentation is generally clear.
*Significance:*
Grid cells are of tremendous interest in neuroscience and there is a considerable volume of work attempting to explain their properties and the reasons behind their existence. This paper provides a novel model for how grid cell may emerge – as a self-supervised means of keeping track of an animal's location in space – and is thus of great interest to neuroscientists.
Weaknesses
1. The authors present this work as a significant advance over methods based on supervised learning because the latter depend on specific design choices. However, the same seems to be true in the new approach: for instance, while the separation and path-invariance loss are pretty natural, the capacity loss is counter-intuitive, as mentioned even by the authors in the Discussion. Excluding the capacity loss eliminates the multi-scale nature of the solution. Moreover, the emergence of grid cells is sensitive to the parameters used in the loss, as shown in Figure 7. It is thus not immediately obvious that the proposed method requires any less fine tuning to lead to grid cells than prior models.
2. Ideally the code used to run all the simulations would have been included with the supplementary material. It was promised for after acceptance, but this seems hard to justify, since the code exists already, and it could be useful for a thorough evaluation of the paper.
3. Most of the figure panels should be significantly larger – Figure 7 is a particularly bad example. I understand that space is a limiting factor, but some progress can be made by including fewer ratemaps (don't see the need for more than 3x3 or 4x2 examples of each kind). Also ensure that font sizes don't go below 7 or 8 – when they do, it may be better to just remove the text because it is very inconvenient to read.
Questions
1. The permutation invariance discussed around eq. (2) assumes Euclidean space, where the order of the steps does not change the outcome of a path. This seems overly restrictive since a typical animal's habitat is unlikely to feature a flat 2d plane, but instead is more likely to contain obstacles, hills, tunnels, etc. Can the authors comment on how their method might adapt to such cases?
2. The motivation for the capacity loss in eq. (9) seems a bit obscure. This was touched upon briefly in the Discussion, but it would be useful for the authors to indicate how they came up with this particular formulation. Also, were other capacity loss functions attempted, and how did they work?
Minor comments:
* line 127: $f$ is used here with a different meaning from $f$ in, e.g., eqns. (2) or (3), which is a bit confusing
* Figure 6, panel c: what non-linear dimensionality reduction technique was used?
Rating
6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
Confidence
4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Limitations
The authors have adequately discussed limitations of their work.