Originality: This is a highly original manuscript which goes beyond using known algorithms to study technically challenging problems and instead attempts to explain mechanisms of fundamental neuro-systems in the MEC and HCF. Quality: The claims, while profound, appear to be well built upon the 3 main figures in the article. The authors acknowledge that further study could be done along the lines of quantitatively how grid cell solutions change when the environment is deformed, rewards or obstacles are incorporated, or place cells are lesioned, although few limitations are actually noted and overall strengths are highlighted in the article. Clarity: The article is well written and easy to understand even supposing the readers have little knowledge of place/grid cells. Significance: The article attempts to advance science in a theoretical sense which could have impact on future work both theoretical and applied given the current neuromodulation attempts to restore/improve human memory (and subservient to those navigation) systems

-The paper is clearly written. The figures are well-thought-out and nicely complement the explanations provided in the text. I find the presentation of some of the material in the SI slightly odd. Some of them are standard results in linear algebra (or follow directly from standard results), but they read like they are "new results". I assume the intent is to make the paper as self-contained as possible but please make this clear. -The findings appear to be technically sound. The analytical results are supported by appropriate simulations. - While the normative framework utilized in the work is not new, the analysis that establishes the conditions under which hexagonal grid cells emerge is original and interesting and I believe relevant to NeurIPS. -In the introduction, it is claimed that there is no theoretical clarity on when and why grid cells emerge (l42-43). This is not quite true. [a] (and relatedly reference [17] ) has shown that hexagonal grid cells are optimal because they maximize spatial resolution. Please discuss. In your experiments, do you find that hexagonal lattices provide better reconstruction (e.g relative to square)? Relevant citations: [a] Mathis, Stemmler, Herz, eLife, 2015, Probable nature of higher-dimensional symmetries underlying mammalian grid-cell activity patterns, [b] Whittington et al, NeurIPS, 2018, another related normative explanation of grid cells as basis functions for encoding transition statistics. ___________ Post author's feedback update: Thanks for your response. Please include relevant discussion from your response in the final manuscript. I've increased my score.

The work addresses the issue of how grid-patterned spatial activity in medial entorhinal cortex might arise as a function of learning using hippocampal place cell inputs in simulated random-walk experiments. This is not the first work to do so, but does take such efforts significantly forward in comparing and contrasting multiple approaches and providing a key means by which such networks can be generated to produce hexagonal versus square grid patterns. The work is presented quite clearly and in detail. In terms of significance, it would seem that the entire field of researchers working on this problem fail to consider that grid-patterning in medial entorhinal cortex is not "learned" during random explorations of the environment, but develops in the absence of any significant such behavior. Furthermore, even if such patterning were "learned", it would almost certainly involved many more sources of information (e.g., head direction cell input and egocentric border cell input). While my comments call into question the value of the work in this field to neuroscience, it seems that the authors have done an admirable job of addressing the questions as they stand and the work may prove useful to non-neuroscientific applications.