Modeled grid cells aligned by a flexible attractor

Reviewer #1 (Public Review):

This study investigates the impact of recurrent connections on grid fields generated in networks trained by adjusting the strength of feedforward spatial inputs. The main result is that if the recurrent connections in the network are given a 1D continuous attractor architecture, then aligned grid firing patterns emerge in the network following training. Detailed analyses of the low dimensional dynamics of the resulting networks are then presented. The simulations and analyses appear carefully carried out.

The feedforward model investigated by the authors (previously introduced by Kropff & Treves, 2008) is an interesting and important alternative to models that generate grid firing patterns through 2-dimensional continuous attractor network (CAN) dynamics. However, while both classes of model generate grid fields, in making comparisons the manuscript is insufficiently clear about their differences. In particular, in the CAN models grid firing is a direct result of their 2-D architecture, either a torus structure with a single activity bump (e.g. Guanella et al. 2007, Pastoll et al. 2013), or sheet with multiple local activity bumps (Fuhs & Touretzky, Burak & Fiete, 2009). In these models, spatial input can anchor the grid representations but is not necessary for grid firing. By contrast, in the feedforward models neurons transform existing spatial inputs into a grid representation. Thus, the two classes of model implement different computations; CANs path integrate, while the feedforward models transform spatial representations. A demonstration that a 1D CAN generates coordinated 2D grid fields would be surprising and important, but it's less clear why coordination between grids generated by the feedforward mechanism would be surprising. As written, it's unclear which of these claims the study is trying to make. If the former, then the conclusion doesn't appear well supported by the data as presented, if the latter then the results are perhaps not so unexpected, and the imposed attractor dynamics may still not be relevant.

Whichever claim is being made, it could be helpful to more carefully evaluate the model dynamics given predictions expected for the different classes of model. Key questions that are not answered by the manuscript include:

- At what point is the 1D attractor architecture playing a role in the models presented here? Is it important specifically for training or is it also contributing to computation in the fully trained network?

- Is an attractor architecture required at all for emergence of population alignment and gridness? Key controls missing from Figure 2 include training on networks with other architectures. For example, one might consider various architectures with randomly structured connectivity (e.g. drawing weights from exponential or Gaussian distributions).

- In the trained models do the recurrent connections substantially influence activity in the test conditions? Or after training are the 1D dynamics drowned out by feedforward inputs?

- What is the low dimensional structure of the input to the network? Can the apparent discrepancy between dimensionality of architecture and representation be resolved by considering structure of the inputs, e.g. if the input is a 2 dimensional representation of location then is it surprising that the output is too?

- What happens to representations in the trained networks presented when place cells remap? Is the 1D manifold maintained as expected for CAN models, or does it reorganise?

Reviewer #2 (Public Review):

Summary:
The authors proposed that grid cells may be aligned by simpler, 1D attractors, and they showed that the structure and the representational space of an attractor network can be two different topological objects.

Strengths:
It is very interesting that the toroidal topology of the population activity (the representational space) and the structure of the attractor network do not necessarily to be the same. The authors carried out extensive computational modeling to support such evidence. The results presented by the authors in this study could have an impact in the grid cell field, which will motivate future experimental studies to examine the detailed structure of the grid cell population.

Weaknesses:
The authors mentioned that "the recurrent collateral structure defines the geometry of the manifold..." and pointed out that this assumption is wrong. I am afraid this claim is too strong. The Gardner Torus paper showed evidence of the 2D CAN exists in the EC as a possible substrate of the grid pattern. Do the authors mean here that even the population activity in the grid cells show the torus structure, it does not necessarily mean that the grid cells form a 2D CAN? I understand that from the computational modeling view, it is doable to find counter-examples (like the 1D attractor network) in which the representational space is a torus but the structure is different. However, from the experimental view, do you expect that the grid cell network is a low-dimensional attractor network? To prove this, is there any evidence from the experimental data?

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.