The authors train an RNN to perform deduced reckoning (ded reckoning) for spatial navigation,  and then study the responses of the model neurons in the RNN.  They find many properties reminiscent of neurons in the mammalian entorhinal cortex (EC): grid cells, border cells, etc.  When regularization of the network is not used during training, the trained RNNs no longer resemble the EC.  This suggests that those constraints (lower overall connectivity strengths, and lower metabolic costs) might play a role in the EC's navigation function.  The paper is overall quite interesting and the study is pretty thorough:  no major cons come to mind.  Some suggestions / criticisms are given below. 1) The findings seem conceptually similar to the older sparse coding ideas from the visual cortex.  That connection might be worth discussing  because removing the regularizing (i.e., metabolic cost) constraint from your RNNS makes them learn representations that differ from the ones seen in EC.  The sparse coding models see something similar:  without sparsity constraints, the image representations do not resemble those seen in V1,  but with sparsity, the learned representations match V1 quite well.  That the same observation is made in such disparate brain areas (V1, EC) suggests that sparsity / efficiency might be quite universal constraints on the neural code. 2) The finding that regularizing the RNN makes it more closely match the neural code is also foreshadowed somewhat by the 2015 Nature Neuro paper by Susillo et al.  That could be worthy of some (brief) discussion. Sussillo, D., Churchland, M. M., Kaufman, M. T., & Shenoy, K. V. (2015). A neural network that finds a naturalistic solution for the production of muscle activity. Nature neuroscience, 18(7), 1025-1033. 3) Why the different initializations for the recurrent weights for the hexagonal vs other environments?  I'm guessing it's because the RNNs don't "work" in all environments with the same initialization (i.e., they either don't look like EC, or they don't obtain small errors in the navigation task).  That seems important to explain more thoroughly than is done in the current text. 4) What happens with ongoing training?  Animals presumably continue to learn throughout their lives.  With on-going (continous) training, do the RNN neurons' spatial tuning remain stable, or do they continue to "drift" (so that border cells turn into grid cells turn into irregular cells, or some such)?  That result could make some predictions for experiment,  that would be testable with chronic methods (like Ca2+ imaging) that can record from the same neurons over multiple experimental sessions. 5) It would be nice to more quantitatively map out the relation between speed tuning, direction tuning, and spatial tuning (illustrated in Fig. 3).  Specifically, I would quantify the cells' direction tuning using the circular variance methods that people use for studying retinal direction selective neurons.  And I would quantify speed tuning via something like the slope of the firing rate vs speed curves.  And quantify spatial tuning somehow (a natural method would be to use the sparsity measures sometimes applied to neural data to quantify how selective the spatial profile is to one or a few specific locations).  Then make scatter plots of these quantities against each other.  Basically, I'd love to see the trends for how these types of tuning relate to each other over the whole populations:  those trends could then be tested against experimental data (possibly in a future study).