Reviewer #1 (Public review):
Summary:
This work studies representations in a network with one recurrent layer and one output layer that needs to path-integrate so that its position can be accurately decoded from its output. To formalise this problem, the authors define a cost function consisting of the decoding error and a regularisation term. They specify a decoding procedure that, at a given time, averages the output unit center locations, weighted by the activity of the unit at that time. The network is initialised without position information, and only receives a velocity signal (and a context signal to index the environment) at each timestep, so to achieve low decoding error it needs to infer its position and keep it updated with respect to its velocity by path integration.
The authors take the trained network and let it explore a series of environments with different geometries while collecting unit activities to probe learned representations. They find localised responses in the output units (resembling place fields) and border responses in the recurrent units. Across environments, the output units show global remapping and the recurrent units show rate remapping. Stretching the environment generally produces stretched responses in output and recurrent units. Ratemaps remain stable within environments and stabilise after noise injection. Low-dimensional projections of the recurrent population activity forms environment-specific clusters that reflect the environment's geometry, which suggests independent rather than generalised representations. Finally, the authors discover that the centers of the output unit ratemaps cluster together on a triangular lattice (like the receptive fields of a single grid cell), and find significant clustering of place cell centers in empirical data as well.
The model setup and simulations are clearly described, and are an interesting exploration of the consequences of a particular set of training requirements - here: path integration and decodability. But it is not obvious to what extent the modelling choices are a realistic reflection of how the brain solves navigation. Therefore, it is not clear whether the results generalize beyond the specifics of the setup here.
Strengths:
The authors introduce a very minimal set of model requirements, assumptions, and constraints. In that sense, the model can function as a useful 'baseline', that shows how spatial representations and remapping properties can emerge from the requirement of path integration and decodability alone. Moreover, the authors use the same formalism to relate their setup to existing spatial navigation models, which is informative.
The global remapping that the authors show is convincing and well-supported by their analyses. The geometric manipulations and the resulting stretching of place responses, without additional training, are interesting. They seem to suggest that the recurrent network may scale the velocity input by the environment dimensions so that the exact same path integrator-output mappings remain valid (but maybe there are other mechanisms too that achieve the same).
The simulations and analyses in the appendices serve as insightful controls for the main results.
The clustering of place cell peaks on a triangular lattice is intriguing, given there is no grid cell input. It could have something to do with the fact that a triangular lattice provides optimal coverage of 2d space? The included comparison with empirical data is valuable as a first exploration, showing a promising example, but doesn't robustly support the modelling results.
Weaknesses:
The navigation problem that needs to be solved by the model is a bit of an odd one. Without any initial position information, the network needs to figure out where it is, and then path-integrate with respect to a velocity signal. As the authors remark in Methods 4.2, without additional input, the only way to infer location is from border interactions. It is like navigating in absolute darkness. Therefore, it seems likely that the salient wall representations found in the recurrent units are just a consequence of the specific navigation task here; it is unclear if the same would apply in natural navigation. In natural navigation, there are many more sensory cues that help inferring location, most importantly vision, but also smell and whiskers/touch (which provides a more direct wall interaction; here, wall interactions are indirect by constraining velocity vectors). There is a similar but weaker concern about whether the (place cell like) localised firing fields of the output units are a direct consequence of the decoding procedure that only considers activity center locations.
The conclusion that 'representations are attractive' (heading of section 2) is not entirely supported. The authors show 'attractor-like behaviour' within a single context, but there could be alternative explanations for the recovery of stable ratemaps after noise injection. For example, the noise injection could scramble the network's currently inferred position, so that it would need to re-infer its position from boundary interactions along the trajectory. In that case the stabilisation would be driven by the input, not just internal attractor dynamics. Indeed, the useful control analysis in Appendix D suggests such a mechanism: without a velocity signal, only for small noise injections the network returns to a high correlation state. Correlated representations are recovered for larger noise injections due to the same mechanism that allow the network to determine its position upon from an uninformative initial hidden state upon entering a new environment, i.e. boundary interactions.
The authors report empirical data that shows clustering of place cell centers like they find for their output units. They report that 'there appears to be a tendency for the clusters to arrange in hexagonal fashion, similar to our computational findings'. This is an interesting observation on the distribution of place field centres which seems justified based on the example animal shown, but not across the population of animals included.