If Grid Cells are the Answer, What is the Question? A Review of Normative Grid Cell Theory

Reviewer #1 (Public review):

Summary:

The review by Dorrell and Whittington synthesizes the progress made over the past few years with respect to a normative theory of grid cells. The core question addressed by normative frameworks of grid cells is what primary computational function grid cells serve. The review discusses evidence from mechanistic models and experimental data that point to path integration as the computational function of grid cells, consistent with results from normative models. The main goal of the review is to clarify the normative grid cell theory literature. However, the current version of the article reads at times more like a perspective or opinion article in support of the path integration hypothesis rather than a critical review of normative frameworks in the grid cell literature that contrasts the benefits and limitations, as well as pitfalls and caveats, with other modelling approaches.

Some specific comments are as follows:

(1) Abstract: "The first question quickly attracted an answer: grid cells subserve path integration ..." - I am not sure if this statement is correct. The first grid cell paper by Hafting and Fyhn in 2005 suggested that grid cells are part of a path integration-based map, and the paper emphasizes the map part. It remained unclear, and is still debated, whether grid cells are part of a system performing path integration or whether grid maps reflect the output/result of a path integration process. Other theories about the function of grid cells were brought forward as well. Although the main competing theory is discussed in this review, this review article at times appears more as a perspective or opinion article with a clear bias toward the path integration hypothesis rather than objectively discussing the evidence.

(2) Grid cells may serve multiple functions. What would be the implications for our understanding of grid cells and for interpreting the results of normative models? In general, the review could discuss some pitfalls or caveats of normative models in more detail.

(3) A normative framework can be helpful in two ways: (a) Given sufficient details on biological constraints, a normative model can help identify the computational function of grid cells. If a computational function is given and - under the given simulated biological constraints - grid cells were part of the solution, the results of the model would support the hypothesis that grid cells serve the computational function in question. (b) If a computational function were identified beyond any doubt (e.g., assume experimental data demonstrated that grid cells are necessary and sufficient for path integration), a normative model would help identify biological parameters necessary to produce grid cell firing. Unfortunately, the review falls short in making this clear distinction between (a) and (b) and in discussing important caveats regarding mixing up these two ways. E.g., the neural network model approaches by Sorscher et al. and others have been criticized because they try to achieve two things at the same time: find support for the computational function of grid cells and identify optimal parameters that result in grid cells. But doing both at the same time provides a strong bias in tweaking the parameters in exactly the way you need for the model to produce grid cells as a solution (other solutions may be possible given other parameters), preventing strong conclusions regarding the computational function of grid cells and preventing conclusions about what the parameter choices mean for biological connectivity motifs. These caveats in setting up normative models and interpreting them could be discussed in greater detail.

(4) A common assumption underlying most grid cell models is that head direction is viewed as identical to movement direction. However, head direction can differ at times from movement direction, and entorhinal head direction cells code head direction rather than movement direction (Raudies et al., 2015; 10.1016/j.brainres.2014.10.053). This missing link in how movement direction signals reach and inform grid cells could be discussed.

(5) "Knowing that one neuron in a module is active and that you make a movement north uniquely determines which neuron in that module should be active next" - I agree that this rule follows from the fact that grid cells within one module differ in phase but share spacing and orientation. However, I am surprised that the authors do not also make the argument here for the value of a normative model. Rebecca R.G. et al. (10.7554/eLife.96627) use exactly the rule cited above as a normative function. They demonstrate that this rule begets grid cells. Isn't this a prime example of how a normative approach can contribute to scientific inquiry? First, a hypothesis about a computational function is derived from experimental data. And in turn, using a normative framework, the experimental data are derived from the computational function (under appropriate biological results). The paper is discussed later together with Nicolai Waniek's work (10.1162/neco_a_01255). However, in my opinion, their work seems to be somewhat misrepresented in that later paragraph. E.g., velocity is still required as an input to determine which neuron should be active next, neurons do not need to be binary units, and space is not discretized beyond the fact that space is encoded by neurons with spatial firing fields.

Reviewer #2 (Public review):

Summary:

This review by Dorrell and Whittington covers a number of aspects related to normative modeling of grid cells. They begin by discussing key experimental insights on grid cell phenomenology. Then, they discuss how grid cells can be used to perform path integration and how they size up as efficient codes of space. These two sections then lead the authors to discuss how combining path integration and efficient coding objectives leads to models of axis-aligned grid cells in a single module. Discussion on non-linear objectives leading to multi-modules is presented. The review ends with several outstanding questions and an optimistic outlook of how normative models (particularly, task-optimized RNNs) can be used as tools for advancing understanding in neuroscience.

Strengths:

(1) The review is timely and covers an area that has seen a lot of recent activity. This discussion around many of the different results (and kinds of models), I think, will be generally helpful for the field.

(2) Although I think the story could be a little more coherently made (see below), in general I enjoyed the author's flow from efficient coding -> efficient coding + path integration -> efficient coding + path integration + non-linear objective. This framing supports the specific conclusion the authors arrive at.

(3) I also really liked the message that the review made of how normative modeling, despite some of its challenges/limitations, can be used effectively in neuroscience. The discussion of cycling between "experimental" modeling (e.g., vanilla RNNs) and theoretically-grounded models was nice, and I think it helps demonstrate the value of this approach.

(4) Showing how the metric loss could be seen as a bandpass filter (Figure 3C) was nice and a contribution of the review.

(5) While the focus of P4 (conjunctive HD-grid cells) felt initially a little cast aside, the discussion around "brain and task-optimised RNNs with standard architectural choices use fundamentally different path-integration mechanism" was nice and I think helpful for steering the community to an interesting open problem.

(6) Identifying how "non-linear functionality" can lead to multi-modules was nice and not something that I have seen as clearly presented before.

Weaknesses:

(1) The authors view the experimental evidence for grid cells being linked to path integration as "specific and strong" and that the " key computational feature that defines entorhinal cortex [is] path-integration". I think experimentalists (at least the ones I work with) would push back on that. First, it's hard to isolate path integration in rodent experiments. So while Gil et al. (2018) did about as good a job as you could do, there are still other interpretations of the results that are not purely path integration dependent. And second, as the authors point out later in the review, there is experimental work finding that grid cells are disrupted in large environments and 3D. Path integration certainly happens (to some extent) in these spaces, which begs the question of how it is achieved with weakened grid coding. Thus, I think reducing the claims about how strongly grid cells are experimentally linked to path integration is called for.

(2) The authors introduce the idea of efficient coding of space and discuss how grid cells are not optimal. It is later clarified (Sec. 5.3) that multi-module codes can be efficient (even if not the most optimal). I was confused reading Section 3, because in Section 2 the multiple modules are discussed, but then in Section 3, they are dropped, and only a single module is being considered. Equation 2 was also a little confusing to me. Alpha is not defined, and I would have thought that it would be x^Tx' - g(x)^T g(x') and not x^Tx' g(x)^T g(x'). Given that there is no page limit here, I think a little more detail in Section 3 would be helpful.

(3) In Section 3, the authors make use of P2 (translation invariance within a module) to rule out (or, at least, question) certain models/approaches. While this is certainly a standard assumption made in theoretical work, it is not very well supported by experimental findings. In particular, Diehl et al. (2017), Ismakov et al. (2017), and Dunn et al. (2017) all found that individual grid fields systematically vary in their peak firing rate. In addition, Redman et al. (2025) found that, within a given module, there was a small but robust diversity of grid orientations and spacings. These suggest that grid cells within a single module may actually be able to encode properties of local space and give some support to normative models that find efficient space coding with grid cells by finding non-axis-aligned grid fields. I think this is all important to mention because: a) it provides more biological nuance to the question about spatial coding; b) it provides more ways in which to test models. For instance, in Redman et al. (2025), the Sorscher et al. (2022) model was shown to produce variability in grid properties that loosely matched what was found in real data. For tests like this (e.g., how much does a model reproduce variability in grid firing field peak rates), I think it is going to be important for continuing to evaluate models.

(4) The focus of the review, I know, is grid cells, but of course, grid cells are part of the MEC and the larger hippocampal network. I totally understand, at some level, you have to make a decision of what to model, but it seems that there are other functional classes of neurons (border cells, head direction cells) that all play an important role in path integration. And while the models the authors consider at the end of the review capture properties of grid cells really well, they do so at the cost of not modeling anything else. The authors mention this in the context of the models not capturing conjunctive grid-head direction cells, but I think the point is a deeper one, and more discussion of at what level it makes sense to consider grid cells only is important.

(5) As I mentioned in the Strengths section, I did enjoy the flow of the paper on how path integration + efficiency is needed to get grid single modules and path integration + efficiency + non-linearity is needed to get multiple grid modules. This creates the story that adding more of these theory-driven constraints helps lead to more "accurate" models of grid cells. But one alternative view is that, if path integration + efficiency is enough to get a single grid module (but only a single grid module), then maybe the utility (or need) of multiple grid modules comes from something else. That is, instead of saying "we need more constraints to get multiple modules", it could be evidence for "we need to re-think whether multiple modules might need a different theory to explain". While I understand this is a big picture question that maybe isn't entirely fair to ask of the authors, I think: 1) the authors do a nice job of positioning their review as a kind of discussion on what normative modeling can provide to neuroscience, so having this discussion on when the failure of a model to capture ALL aspects of the biological features motivates further constraints as opposed to a new approach, would be useful; 2) this question connects with the title of the paper, i.e. "what is the question?"

Reviewer #3 (Public review):

Summary:

The authors present an extensive review of the literature on normative grid cell theory, asking what kind of cost function might be minimized by the entorhinal grid cell code. The authors show which of the main features of grid cells emerge from combinations of terms in a cost function that optimizes for spatial fidelity, biological plausibility, and path integration. They conclude by outlining potential future directions for the field.

Strengths:

The structure of the review makes it particularly useful for researchers who are familiar with grid cells but not necessarily with normative models. Equations are kept to a minimum and are usually explained conceptually.

Weaknesses:

I identified one main weakness, related to the fact that the introduction to experimental results around grid cells and what they allow us to conclude is less nuanced than the rest of the review. However, since this is not the main focus of the manuscript, I consider this a secondary limitation.

The review organizes the current literature on the subject within a coherent conceptual framework, helping to define possible paths forward for the field.

Author response:

We thank the reviewers for their time and attention which will significantly improve the paper. Further, we are grateful for their appreciation of our goals and work. In sum, the reviewers point to our overstated discussion of experimental evidence which we will tone down, some slightly confusing points of argumentation which we will clarify, and some discussion points on the role of normative theories that we will add text to address. We believe this will improve the paper significantly and hope you agree!

Major Concern: Experimental Support for Path-Integration is not as strong as suggested

The major point raised by all reviewers (reviewer 1 comment 1, reviewer 2 comment 1, reviewer 3’s only weakness) was that our presentation of the experimental perturbation evidence for path-integration is stronger than the reality. On reflection, we agree with this evaluation. We thank the reviewers for raising it; we will moderate our writing and include the sensible caveats raised. In sum, we still think that the convergence of evidence points to path-integration: first, disruptions to grid cells lead to path-integration problems, though these perturbations admittedly aren’t perfectly precise; second, normative theories of path-integration lead to grid cells and predict grid cell behaviour; third, mechanistic models of path-integration match grid cell behaviour and predict connectivity subsequently measured in entorhinal cortex. However, the evidence is not as all-encompassing as we suggested.

That said, we’d like to further comment on one point. It is argued (reviewer 1, comment 1) that there are other theories of grid cell function, and that we discuss these theories. We discuss efficient-coding only models of grid cells and emphasise strongly why we reject them. We also briefly discuss oscillatory-interference models of path-integration and our reasons for not pursuing them further. As such, the reviewer is correct that our reading of literature strongly points us towards path-integration rather than other theories. We will slightly change the framing of the paper to make it clear that we are making a case. However, we are not aware of other theories the reviewer might be referring to. If the reviewer can point us to the other suggested theories that we do not address we would be happy to evaluate and include them.

We now turn to the remaining comments, and how we plan to address them.

Reviewer 1, Comment 2 – There could be multiple roles for grid cells

The reviewer is indeed right that grid cells might perform multiple functions. This could just mean that the same computational motif (e.g. path-integration) is reused across different computations though that introduces no changes to the required normative theory. A stronger claim would be that grid cells perform both path-integration and some other function. This, according to a normative perspective, would most likely change how grid cells were optimally structured. We use the fact that large parts of the grid cell code can be captured with only path-integration as an argument against additional roles for grid cells. That said, there exist properties of grid cells not well-captured by path-integration which could well be smoking guns for additional roles of grid cells. The review already discusses both discrepancies between grid cells in three and two dimensions, and inhomogeneities in the grid in complex environments, and we will add two more (heading direction and peak-to-peak/angular variability, discussed below) that we are grateful to the reviewers for raising, and we discuss each of these in detail below.

That said, whether these are necessarily arguments against purely path-integration or a reflection of interesting mappings of the core path-integration mechanism to the measurements we make remains to be seen. We would argue that both 3D grid cells (as explained below: there appear to be 2D slices in which grid cells behave as you’d expect) and spatial inhomogeneities (as explained in the paper: mappings of torus to world can introduce warping) can be explained without reference to additional computational roles of grid cells, which remain to us the most parsimonious explanation. We discuss next the slight update to path-integration only that the heading direction story suggest. But in sum, our view is that these discrepancies are likely not fatal for our path-integration-centric view of grid cells, but may well suggest some very interesting clarifications.

Reviewer 1, Comment 4 – The system has two heading signals: true & internal, why?

The reviewer is right to point to the puzzle over true vs. purely internal heading direction and which drives grid cells. We believe recent work from Abraham Vollan has effectively solved this puzzle: there appear to be two parallel circuits, one theta-modulated and following internal heading direction, another theta-unmodulated and aligning more with true heading direction. We will make sure to include discussion of this exciting work in our revised submission. This serves as a good example of an update we concede to the most austere version of the path-integration only view. Rather, it seems there are two parallel path-integrators working with different heading signals. The reasons for this remain unclear, but seem to be related to attention and planning (Vollan et al. 2026).

Reviewer 2, Comment 3: Real Grid Cells have peak-to-peak variability & Angular variability

The reviewer is right to point to the discrepancy in peak-to-peak firing rate and angles within a module that we did not adequately address. First, it is Sorscher’s RNN models, not nonnegative PCA that can generate a distribution of grid angles (Redman et al. 2025), which suggests that path-integration and such variability are compatible. We emphasise this point because the non-path-integration results from nonnegative PCA produce grid cells oriented at 30 degree offsets, something not measured even when you’re careful as in Redman et al. 2025. Thus, this becomes an interesting target for future work: perhaps using theories of path-integration up to an error threshold (rather than perfect) such angular diversity would be recovered. We will include this in our discussion. Further, we will include discussion of peak-to-peak variability that, as yet, has no obvious role.

Reviewer 2, Comment 1: grid cells are inhomogeneous in 3D or complex environments, doesn’t that break the theory?

Disrupted grid coding in extended or 3D environments indeed deserve more discussion, which we will add. In particular, we will add recent evidence that grid cells in 3D can be understood via the correct sequence of 2D projections(Qi & Yartsev, 2026). These two phenomena seem, to us, consistent with a path-integration only view of grid cells, as discussed above, and we hope to make this position clearer.

Reviewer 2, Comment 5: Couldn’t there be other reasons for multiple modules?

We have suggested a consistent normative framework in which multiple modules are explained through their role in non-linear coding. We think this elegant, and the most parsimonious current theory. We could, of course, be wrong. The discrepancies pointed to above might be good clues to follow to work out what else these modules might be doing, but currently these alternative explanations seem not to exist. We will text to clarify this.

Reviewer 1, Comment 3: The review confuses computational and parameter parts of normative theory

We disagree with the reviewer’s dichotomisation of normative theory. We view a normative theory as the complete procedure that produces the predictions. Almost all such theories have parameters and hence fitting a theory to data comprises both elements (a) [computational role] and (b) [specific parameters] identified by the reviewer. Occasionally theories have no parameters in the traditional sense, e.g. Rebecca et al.; instead they have heavy assumptions that play an equivalent role. It is true that, as the reviewer says, Sorscher et al.’s work was criticised for producing grid cells only for specific parameter values. We never found this as damning as Schaeffer et al. argued: simply it says that that theory is only correct within the given parameter range. Rather, arbitrating between models, parameters, or assumptions seems the same basic process: see what they predict and keep working with models while they remain useful ways to understand measured phenomena. If a model with very specific parameter values remains useful, that seems okay. In fact, we argued extensively why we think the nonnegative PCA model is not a useful model, but this was for completely different reasons. To us this story just reinforces the importance of hygiene in normative research: perform parameter sweeps and clarify how they constrain the claims you are making, carefully arbitrate what models can capture. Indeed, that is the whole goal of this review. We might be misunderstanding and, if so, we welcome correction.

Reviewer 2, Comment 4: Normative Models of Cells Beyond Grid Cells

The reviewer is right that extending these models to other cell types is an interesting area for further work, and that other cell types do seem to be involved in aspects of navigational computations both in RNNs and the brain. We will include a discussion to this effect in the revised manuscript. That said, we think the modularity of grid cells and their tight-linking to path-integration calculations should also be appreciated as a win!

Reviewer 2, Comment 2: Multi-modularity is not cleanly explained

We thank the reviewer for the comments, we agree. We will clarify the story regarding multiple modules, and will explain the equation further.

Reviewer 1, Comment 5: the early introduction of phase-shifted Grid Cells seem the perfect place to normatively argue for Path-integration!

We agree with the reviewer that this point can be made both normatively (‘oh look! If I try to do this optimally, I get translations!’) or, as we did early in the paper, mechanistically (‘oh look! With these cells I can do this!’). Indeed, a large part of the point of our paper is that path-integration is what is required to normatively derive phase-shifted grid modules, something discussed by Rebecca et al., our earlier work, and RNN studies, and appreciated for two decades. The earlier part of the paper does not discuss these papers as that section is aimed at giving intuition for the solution (mechanism). Later sections then heavily discuss the normative angle. We hope that division of labour makes sense.

Finally, we will refine our summary of Rebecca et al. The reviewer is right that neurons don’t have to be discrete, we apologise for that error, but our understanding is that the only meaningful role of a neuron in Rebecca et al.’s work is the region in which is active, effectively making every neuron a binary unit, which seems dubious. We will clarify that by “predict velocity from each current and next encoding” we mean that the normative constraint they enforce is axiom 1: sequential activity of sets of neurons i then j can be uniquely interpreted as a trajectory, i.e. a step or velocity. Their work is elegant, and we will try to do more justice to it in the revision.

To conclude, we thank the reviewers for their extensive comments, and look forward to releasing a version that addresses their concerns.