Reviewer #3 (Public Review):

Summary:

The paper proposes an alternative to the attractor hypothesis, as an explanation for the fact that grid cell population activity patterns (within a module) span a toroidal manifold. The proposal is based on a class of models that were extensively studied in the past, in which grid cells are driven by synaptic inputs from place cells in the hippocampus. The synapses are updated according to a Hebbian plasticity rule. Combined with an adaptation mechanism, this leads to patterning of the inputs from place cells to grid cells such that the spatial activity patterns are organized as an array of localized firing fields with hexagonal order. I refer to these models below as feedforward models.

It has already been shown by Si, Kropff, and Treves in 2012 that recurrent connections between grid cells can lead to alignment of their spatial response patterns. This idea was revisited by Urdapilleta, Si, and Treves in 2017. Thus, it should already be clear that in such models, the population activity pattern spans a manifold with toroidal topology. The main new contributions in the present paper are (i) in considering some forms of recurrent connectivity that were not directly addressed before (but see comments below). (ii) in applying topological analysis to simulations of the model. (iii) in interpreting the results as a potential explanation for the observations of Gardner et al.

Strengths:

The exploration of learning in a feedforward model, when recurrent connectivity in the grid cell layer is structured in a ring topology, is interesting. The insight that this not only aligns the grid cells in a common direction but also creates a correspondence between their intrinsic coordinate (in terms of the ring-like recurrent connectivity) and their tuning on the torus is interesting as well, and the paper as a whole may influence future theoretical thinking on the mechanisms giving rise to the properties of grid cells.

Weaknesses:

1. It is not clear to me that the proposal here is fundamentally new. In Si, Kropff and Treves (2012) recurrent connectivity was dependent on the head direction tuning and thus had a ring structure. Urdapilleta, Si, and Treves considered connectivity that depends on the distance on a 2d plane.

2. The paper refers to the connectivity within the grid cell layer as an attractor. However, would this connectivity, on its own, indeed sustain persistent attractor states? This is not examined in the paper. Furthermore, is this even necessary to obtain the results in the model? Perhaps weak connections that do not produce an attractor would be sufficient to align the spatial response patterns during the learning of feedforward weights, and reproduce the results? In general, there is no exploration of how the strength of collateral interactions affects the outcome.

3. I did not understand what is learned from the local topology analysis. Given that all the grid cells are driven by an input from place cells that spans a 2d manifold, and that the activity in the grid cell network settles on a steady state that depends only on the inputs, isn't it quite obvious that the manifold of activity in the grid cell layer would have, locally, a 2d structure?

4. The modeling is all done in planar 2d environments, where the feedforward learning mechanism promotes the emergence of a hexagonal pattern in the single neuron tuning curve. This, combined with the fact that all neurons develop spatial patterns with the same spacing and orientation, implies even without any topological analysis that the emerging topology of the population activity is a torus.

However, the toroidal topology of grid cells in reality has been observed by Gardner et al also in the wagon wheel environment and in sleep, and there is substantial evidence based on pairwise correlations that it persists also in various other situations, in which the spatial response pattern is not a hexagonal firing pattern. It is not clear that the mechanism proposed in the present paper would generate toroidal topology of the population activity in more complex environments. In fact, it seems likely that it will not do so.

5. Moreover, the recent work of Gardner et al. demonstrated much more than the preservation of the topology in the different environments and in sleep: the toroidal tuning curves of individual neurons remained the same in different environments. Previous works, that analyzed pairwise correlations under hippocampal inactivation and various other manipulations, also pointed towards the same conclusion. Thus, the same population activity patterns are expressed in many different conditions. In the present model, the results of Figure 6 suggest that even across distinct rectangular environments, toroidal tuning curves will not be preserved, because there are multiple possible arrangements of the phases on the torus which emerge in different simulations.

6. In real grid cells, there is a dense and fairly uniform representation of all phases (see the toroidal tuning of grid cells measured by Gardner et al). Here the distribution of phases is not shown, but Figure 7 suggests that phases are non uniformly represented, with significant clustering around a few discrete phases. This, I believe, is also the origin for the difficulty in identifying the toroidal topology based on the transpose of the matrix M: vectors representing the spatial response patterns of individual neurons are localized near the clusters, and there are only a few of them that represent other phases. Therefore, there is no dense coverage of the toroidal manifold that would exist if all phases were represented equally. This is not just a technical issue, however: there appears to be a mismatch between the results of the model and the experimental reality, in terms of the phase coverage.

7. The manuscript makes several strong claims that incorrectly represent the relation between experimental data and attractor models, on one hand, and the present model on the other hand. For the latter, see the comments above. For the former, I provide a detailed list in the recommendations to the authors, but in short: the paper claims that attractor models induce rigidness in the neural activity which is incompatible with distortions seen in the spatial response patterns of grid cells. However, this claim seems to confuse distortions in the spatial response pattern, which are fully compatible with the attractor model, with distortions in the population activity patterns, which would be incompatible with the attractor model. The attractor model has withstood numerous tests showing that the population activity manifold is rigidly preserved across conditions - a strong prediction (which is not made, as far as I can see, by feedforward models). I am not aware of any data set where distortions of the population activity manifold have been identified, and the preservation has been demonstrated in many examples where the spatial response pattern is disrupted. This is the main point of two papers cited in the present manuscript: by Yoon et al, and Gardner et al.

8. There is also some weakness in the mathematical description of the dynamics. Mathematical equations are formulated in discrete time steps, without a clear interpretation in terms of biophysically relevant time scales. It appears that there are no terms in the dynamics associated with an intrinsic time scale of the neurons or the synapses, and this introduces a difficulty in interpreting synaptic weights as being weak or strong. As mentioned above, the nature of the recurrent dynamics within the grid cell network (whether it exhibits continuous attractor behavior) is not sufficiently clear.

In my view, the weaknesses discussed above limit the ability of the model, as it stands, to offer a compelling explanation for the toroidal topology of grid cell population activity patterns, and especially the rigidity of the manifold across environments and behavioral states. Still, the work offers an interesting way of thinking on how the toroidal topology might emerge. Perhaps with certain additional elements this may motivate new theoretical insights.

Figures and data

Attractors with a 2D or 1D architecture align grid maps.

Quantification of the alignment and contraction of grid maps by different attractor architectures.

The population activity in all attractor conditions is locally 2-dimensional, with no boundary or singularities.

For 1D and 2D architectures, the population activity is an orientable manifold with the homology of torus.

Features of the attractor architecture observed in the persistent homology of neural activity.

Multiple configurations for the alignment of grid maps by one-dimensional attractors.

Two different visualizations of the twisted torus obtained with Isomap.

Methods in topological data analysis and examples.

Persistent homology and Betti numbers for condition 1DL.

Persistent homology and Betti numbers for condition 1D_L.