Topological decoding of grid cell activity via path lifting to covering spaces

Iris H.R. Yoon; Yuxing Jared Yao

REVIEW 4 major objections 4 minor 1 cited by

Topological decoding of grid cell activity via path lifting to covering spaces

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The trajectory of a moving animal can be reconstructed from the activity of a single module of grid cells, without external position information or training, by lifting the population activity path from a torus to the plane; the reconstruct

desk verdict A clean, honest empirical paper showing that phase-unwrapping toroidal coordinates recovers local trajectories from a single grid module; the abstract overstates the affine claim in 2D, but the core result is worth engaging with seriously. read the letter →

arxiv 2510.16216 v2 pith:CWLA6LRL submitted 2025-10-17 q-bio.NC math.AT

Yuxing Jared Yao , Iris H.R. Yoon This is my paper · ORCID

classification q-bio.NCmath.AT

keywords gridcellspathintegrationtoroidalmanifoldpersistentcohomologyliftingcoveringspaceneuraldecodingtopologicaldataanalysis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that a single module of grid cells carries enough information to reconstruct the path an animal walks, even though each module's periodic code maps many physical locations to the same neural state. The key idea is to view grid-cell population activity as a path on a torus—the surface of a donut—and then 'unwrap' that torus into the plane using a standard topological operation called path lifting. No training, no external position data, and no knowledge of grid phases are needed; the recovered path matches the true trajectory up to a linear transformation (rotation, scaling, translation). Validation on both simulated and experimental recordings shows reliable reconstructions on short local segments and on one-dimensional tracks, suggesting that co-modular grid cells are sufficient for path integration. This is the first explicit demonstration that the toroidal organization of grid cells can be functionally leveraged for decoding.

What carries the argument

Path lifting to a covering space: the torus S¹ × S¹ is covered by the plane R² via the map p(x, y) = ((cos 2πx, sin 2πx), (cos 2πy, sin 2πy)), which tiles the plane into congruent 'tiles'—copies of the torus. A path on the torus lifts, from a chosen starting tile, to a unique path in R², provided each segment between consecutive samples crosses at most one tile boundary in each coordinate. The greedy lifting rule compares consecutive toroidal coordinates: if their difference is below a threshold ε, keep the same tile; otherwise move to the adjacent tile that minimizes the distance. The integer offsets (M_t, N_t) accumulate the tile choices and define the lifted coordinates (θ_x + 2πM_t, θ_y

What would settle it

Take a CAN-simulated trajectory in a 2D environment and deliberately downsample the recorded time points—or speed up the simulated motion—so that the consecutive toroidal angles skip past a tile boundary in at least one coordinate. If the reconstruction error then jumps to the level of random baseline pairs, the greedy lifting rule is the load-bearing assumption; if the error stays low, the method does not actually depend on the nearest-tile premise.

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Extended reading notes

Core claim

The paper introduces a two-stage decoding framework. First, persistent cohomology confirms that grid-cell population vectors lie on a torus and assigns each time point a pair of circular coordinates, producing a discrete path on the grid-cell torus. Second, this path is lifted to the plane via the covering map p: R² → S¹ × S¹, which tiles the plane into copies of the torus. Consecutive samples are placed in the same tile if their toroidal coordinates are close; otherwise they are placed in adjacent tiles, with the direction chosen to minimize the distance to the previous lifted point. The resulting lifted path is the reconstructed movement trajectory. In idealized noiseless settings the lift

Load-bearing premise

Between two consecutive time samples, the true path on the torus crosses at most one edge in each coordinate, so that the greedy nearest-tile rule never mistakes a fast crossing for noise or misses a needed tile change.

Editorial extensions

If this is right

The toroidal organization of grid-cell population activity is functionally relevant: it directly supports path integration without a learned decoder or external positional reference.
A single grid module—not multiple modules or phase information—is sufficient for local trajectory reconstruction, which narrows the requirements for neural decoding pipelines.
The method extends to realistic noise: moderate spontaneous firing preserves reconstructions, while noise that destroys the toroidal structure causes failure, setting a clear robustness boundary.
In one-dimensional environments, the pipeline recovers the topology and geometry of the track across hundreds of runs, suggesting it can serve as a model of how the brain solves the 'teleport' discontinuity problem in virtual navigation.
Local two-dimensional reconstructions are possible, but global reconstructions fail, indicating that error accumulation in the lift is the main limitation to be addressed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the greedy nearest-tile rule is the true mechanism, reconstruction error should grow monotonically with movement speed or with downsampling of the neural time series; this is a direct, testable prediction that the current data only partially address.
Because the reconstruction is only up to an affine transformation, the method recovers the shape of the environment but not absolute scale or orientation; anchoring the lift to a known landmark or combining two modules with incommensurate scales could fix these factors—an extension the paper mentions but does not explore.
The method's success on a one-dimensional virtual track and on short 2D segments suggests that interpolation or probabilistic unwrapping could push global 2D reconstructions from local to whole-session accuracy, which would make the framework a candidate for brain-machine interfaces that require continuous spatial awareness.
The path-lifting framework generalizes beyond grid cells: any neural population whose activity is organized on a torus or other compact manifold with a universal cover could use the same 'unwrap-and-lift' idea to decode internally represented states.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a solid empirical demonstration of something the reader should know: with the right toroidal coordinates, phase unwrapping (they call it path lifting) is enough to reconstruct short trajectories from a single grid module, without training. That is a useful result, and the 2D local results are real, even if the math itself is not new. The novelty is the packaging: persistent cohomology plus toroidal coordinates plus greedy unwrapping into one pipeline, validated on both simulations and real data.

What works well: the simulation study is clean. The reconstructions preserve the topology of the environment (holes included), and the z-scores against random baselines are strong. The noise robustness tables are systematic and convince me that moderate noise is tolerable. The 1D experimental result — median 3.6% error over 617 runs on the buildup track — is the best evidence in the paper and, for me, the main reason to take the work seriously. The authors also deserve credit for public code and data.

Now the soft spots, in proportion. The stress-test note is correct: the 'up to an affine transformation' claim in the abstract is overstated for the 2D experiment. The global reconstruction visibly fails, and the reported low errors come from fitting a separate 6-DOF affine map per 40-second segment. That is not a single affine equivalence; it is a shape-similarity test on short smooth segments. The mismatched-pair baseline partially controls for this, but it does not test whether a single affine relationship holds across segments. The fix is easy: qualify the claim as 'locally, up to a segment-wise affine transformation.' The abstract already says 'local trajectories' later, so the paper is internally inconsistent.

The other weaknesses are real but not fatal. The epsilon threshold is chosen from the data histogram and cannot distinguish a fast genuine crossing from noise; the paper's own footnote admits the key assumption that time points are dense enough. The 2D experimental evidence rests on one dataset, one module, one day. And the paper does not compare against standard phase unwrapping or other decoding baselines, which would clarify what the topological framing adds beyond the obvious. None of these sink the central claim of the paper, but they matter for interpretation.

Who should read this: anyone working on grid-cell decoding, neural manifolds, or spatial navigation. It provides a useful reference result and a public implementation. It deserves a serious referee. I would ask the authors to fix the abstract, add a held-out affine test for 2D, and compare with a baseline unwrapping method. Then I think the paper is publishable.

Referee Report

4 major / 4 minor

Summary. The paper proposes a training-free, topology-based method for reconstructing an animal's movement trajectory from the population activity of a single grid-cell module. The pipeline uses persistent cohomology to confirm toroidal structure and to assign toroidal coordinates to each population vector, then 'lifts' the resulting discrete path on the torus to the plane via a greedy nearest-tile rule. The authors validate the method on continuous-attractor-network simulations (with and without added spontaneous firing) and on two published experimental datasets: a 1D virtual linear track and a 2D open field. For simulations and the 1D recording, reconstruction errors are low and far below random baselines; for the 2D recording, only local 40-second segments are reconstructed accurately, while the global reconstruction fails. The central claim is that the reconstructed path differs from the original by an affine transformation, and that co-modular grid cells therefore carry sufficient information for path integration.

Significance. If the central claim holds, the paper offers a conceptually novel decoding strategy that requires no external position information, no training, and no knowledge of grid phases, and it provides a concrete computational demonstration that a single grid module can support path integration. The simulation results are clean: the correct environment topology is recovered, and reconstruction errors are far below random baselines (z-scores between -5.6 and -8.6). The 1D experimental result (median error 3.6% over 617 runs) is a solid proof-of-concept. The paper ships code and uses public data, which are important strengths. However, the 2D experimental evidence is restricted to local segments and, as discussed below, the 'up to affine' claim is weakened by the per-segment affine fitting procedure.

major comments (4)

[Section 2.4 / Section 4.3.2] The abstract states that 'reconstructed paths differ from the original by an affine transformation,' but in the 2D experimental analysis the reported errors are computed after fitting a separate 6-DOF affine transformation for each 40-second local segment (Section 4.3.2). The global reconstruction visibly fails (Fig. 7B), so no single affine map is demonstrated to relate the reconstructed path to the true trajectory. For short, smooth segments an affine fit can absorb substantial shape error, so the low mean error (0.120) and the t-test against mismatched pairs (Fig. 7K) do not establish that the toroidal lift recovers metric information beyond a local shape similarity. Please test whether a single affine transformation, fitted on one segment (or on a training set of segments), generalizes to held-out segments; report the reconstruction error under that fixed map, and qualify the abstrac
[Section 2.1, Eq. (1), footnote 3] The greedy lifting rule assumes that between consecutive time samples the true path on the torus crosses at most one torus edge in each coordinate; otherwise the wrong tile is selected and errors accumulate. The footnote notes this assumption but does not validate it. This is load-bearing: the authors attribute the global 2D failure precisely to such lifting errors (Fig. 8, Section 2.4). Please quantify how often the nearest-tile rule makes a correct decision, using the CAN simulations where the ground-truth lift is known, and show how errors accumulate over time. Without this, it is unclear whether the 1D and local 2D successes reflect a generally valid property of grid-cell paths or favorable sampling conditions.
[Section 4.3.3] The proximity parameter ε controls every edge-crossing decision in the lift, yet it is selected by an ad-hoc procedure: restrict maximal coordinate differences to [2,2π], reverse their order, and take the value at which the CDF first exceeds α=0.99. No sensitivity analysis is reported for either ε or α. Since the method's ability to distinguish a genuine edge crossing from noise depends on this threshold, please report how reconstruction errors (or the success/failure of toroidal coordinate computation) vary with ε and α, at least for the simulated data. If the results are highly sensitive to these choices, the robustness claims in Section 2.2 need to be substantially qualified.
[Section 2.4] The 2D experimental analysis is based on a single dataset (rat R, module 1, day 2, OF). The claim that 'local trajectories can be reliably reconstructed' from 2D recordings would be much stronger if the method were applied to additional animals/modules/days from the same public dataset [15]. Given that the global reconstruction fails, the generality of the local result remains uncertain.

minor comments (4)

[Section 2.4] Typo: 'Accumulated errors of this type can can cause' should read 'can cause'.
[Section 2.1] The subheading 'F rom grid cell activity to path on a torus' contains a spacing error ('F rom').
[Reference list] Reference [1] is incomplete ('523 Neural Syst. 2012.' lacks the journal title and page/volume details).
[Section 2.2 / Table 1] In Table 1, the entries with N/A indicate failure to compute toroidal coordinates, but it is not stated whether these trials were excluded from the summary statistics or from the z-scores in Fig. 4E. Please clarify how failed trials are handled.

Circularity Check

2 steps flagged · score 4.0 of 10

Per-segment affine alignment makes the 'up to affine' claim in 2D partially fitted; the path-lifting pipeline itself is otherwise self-contained.

fitted input called prediction [Section 2.4; Section 4.3.1; Section 4.3.2]
"Using the estimateAffine2D function from the OpenCV package [3], we derived the optimal affine transformation matrix that maps the lifted coordinates to the original trajectory. This transformation is then applied to the entire lifted path, aligning it with the original space and allowing for subsequent calculation of the reconstruction error."

The abstract claims 'The reconstructed paths differ from the original by an affine transformation.' But the affine map used to demonstrate this is not predicted by the algorithm: it is fitted after the fact from the original trajectory, separately for each 40-second local segment in the 2D experimental analysis. After an optimal 6-DOF affine fit, the qualitative statement 'differs by an affine transformation' is true for essentially any segment by construction; only the residual error is informative. The mismatched-pair baseline in Fig. 7K partially controls for this, so the step is not fully circular, but it weakens the central claim as stated.
other [Section 4.3.3]
"To select ε, we first compute the maximal coordinate difference cd_t = max{|θ_t^x − θ_{t+1}^x|, |θ_t^y − θ_{t+1}^y|} for every time point t. We then observe the histogram of the maximal coordinate differences, which ranges from 0 to 2π."

The threshold ε decides exactly which toroidal-coordinate jumps count as edge crossings and trigger a tile change in the lift. It is selected from the same coordinate-difference distribution to which it is then applied, with no independent ground-truth labels of true edge crossings. In the 2D experimental case, where global reconstruction fails and only local segments are reported, this data-derived threshold is part of what makes the local lifts look consistent. This is a fitting/hyperparameter concern rather than a derivation by definition, and the reconstruction errors are still nontrivial, so it does not by itself make the whole pipeline circular.

full rationale

The paper's core pipeline is not circular in the strongest sense: toroidal coordinates come from external persistent-cohomology tools (DREiMac, circular-coordinate algorithms from de Silva et al. and Gardner et al.), and the discrete path lift is deterministic given those coordinates and ε, with no position information or training used to construct the lifted path. The experimental datasets are external, and there is no load-bearing chain of self-citations: the cited toroidal-coordinate and CAN-model work is by other groups, not the present authors. The two genuinely concerning steps are (1) the 'reconstructed paths differ by an affine transformation' claim, which is evaluated only after fitting an optimal affine map from the original trajectory to each reconstructed local segment, especially in 2D where the paper admits 'Globally, the reconstructed path differed in overall shape from the true trajectory'; and (2) the data-dependent selection of ε from the same coordinate differences it is used to threshold. Both reduce part of the reported success to post hoc fitting, but the residual errors after affine alignment and the mismatched-pair baseline still provide independent evidence that local shape is recovered. The central novelty is therefore not circular by construction, but the headline affine claim is overstated as a prediction. Overall score 4: some post-fit and data-fitted elements around the main claim, while the underlying lift approach retains independent content.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central 'prediction' is the lifted path; it is not fit to the original trajectory. However, the path depends on ε/α, which are selected from the same dataset's toroidal coordinates. The evaluation metric additionally fits a 6-DOF affine transformation to the original path, so reported reconstruction error is in-sample after removing scale/rotation/translation. The substantive assumptions are topological (torus detection and faithful coordinates) and sampling-density (no tile skipping).

free parameters (5)

epsilon (ε) = data-dependent (selected via CDF threshold α=0.99)
Proximity threshold in Eq. (1) for deciding whether consecutive toroidal coordinates are similar; chosen from the histogram of coordinate differences, Section 4.3.3.
alpha (α) = 0.99
Hand-chosen CDF threshold used to select ε from the order-reversed coordinate-difference distribution, Section 4.3.3.
histogram lower bound (2 radians) = 2
Arbitrary cutoff for the range of 'large' coordinate differences considered when selecting ε, Section 4.3.3.
persistent cohomology filtration/generator parameters = chosen by DREiMac or [15] cohomological decoding
The toroidal coordinates depend on a Vietoris-Rips filtration threshold and a cohomology generator; not independently justified in this paper, Section 4.2.
affine transformation parameters (6 DOF) = fitted per dataset/path
The reconstruction error is computed after an optimal affine alignment to the original trajectory, Section 4.3.1; this is an in-sample evaluation fit.

assumptions (5)

domain assumption Grid cell population activity from a single module is organized on a torus and persistent cohomology reliably detects this torus.
Used throughout; validated by persistence diagrams, but for noisy or short recordings the torus may not be detected (see Table 1 N/A entries).
domain assumption The toroidal coordinates produced by DREiMac/cohomological decoding are a faithful, smoothly varying parameterization of physical space modulo a lattice, and the two coordinates are decorrelated so they can be treated independently.
Section 4.2.3; relies on correctness of [29,30,15]. If coordinates are noisy or correlated, the lift is wrong.
standard math The covering map from R^2 to the torus and the path-lifting lemma (Lemma 54.2 [24]) apply; any continuous path on the torus has a unique lift once a starting tile is chosen.
SI Section 1.1; standard topology, assumed without proof.
ad hoc to paper Consecutive time samples are dense enough that the underlying path crosses at most one torus edge in each coordinate between samples; otherwise the greedy lift picks the wrong tile.
Footnote 3 in Section 2.1; explicitly assumed, acknowledged to fail for fast movement (Fig. 8).
domain assumption The true path does not revisit the same torus point in a way that makes the nearest-tile rule ambiguous (no self-intersection within one lattice period that confuses the lift).
The unwrapping is local; if the path loops ambiguously on the torus, the greedy rule may accumulate errors. Not explicitly discussed.

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Cite this review

Pith. "Pith review of Topological decoding of grid cell activity via path lifting to covering spaces." pith.science (2026). https://pith.science/paper/CWLA6LRL

@misc{pith2026251016216,
  author       = {Pith},
  title        = {Pith review of: Topological decoding of grid cell activity via path lifting to covering spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CWLA6LRL}},
  note         = {Machine review of arXiv:2510.16216}
}

read the original abstract

High-dimensional neural activity often reside in a low-dimensional subspace, referred to as neural manifolds. Grid cells in the medial entorhinal cortex provide a periodic spatial code that are organized near a toroidal manifold, independent of the spatial environment. Due to the periodic nature of its code, it is unclear how the brain utilizes the toroidal manifold to understand its state in a spatial environment. We introduce a novel framework that decodes spatial information from grid cell activity using topology. Our approach uses topological data analysis to extract toroidal coordinates from grid cell population activity and employs path-lifting to reconstruct trajectories in physical space. The reconstructed paths differ from the original by an affine transformation. We validated the method on both continuous attractor network simulations and experimental recordings of grid cells, demonstrating that local trajectories can be reliably reconstructed from a single grid cell module without external position information or training data. These results suggest that co-modular grid cells contain sufficient information for path integration and suggest a potential computational mechanism for spatial navigation.

Figures

Figures reproduced from arXiv: 2510.16216 by the authors.

**Figure 1.** Constructing an internal representation of space from grid cell activity. A. The input data is grid cell activity collected while the mouse moves in an environment. Grid cell population activity is represented as a population vector P(t) evolving over time. B. Persistent cohomology indicates that the population vectors are organized on a torus. C. Each population vector P(t) is assigned toroidal coordinates (θ t x ,… view at source ↗

**Figure 2.** Lifting a discrete path Θ on the torus to a path Θ in ˜ R 2 . A. Similar toroidal coordinates Θ(t) and Θ(t + 1) are lifted to the same tile (no edge crossings), and dissimilar toroidal coordinates are lifted to adjacent tiles (crossing one or more torus edges). B. An example lifting of dissimilar toroidal coordinates Θ(2) and Θ(3), shown in circle and star in bottom panel. The first possibility is to assume that the… view at source ↗

**Figure 3.** Illustration of path lifting on a simulated path. A. A simulated movement path, with a highlighted segment. B. Enlarged view of the highlighted segment. The color indicates that the simulated mouse moves from dark to light. C. Toroidal coordinates for each location on the map. The repeated values indicate that the map is large enough to require nontrivial lifting during path reconstruction. D. The toroidal coordinat… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Path lifting on simulated grid cell activity reconstructs the original movement path. A. Simulated movement trajectories in environments with 0, 1, and 2 holes. B. The persistence diagrams computed on the reconstructed path recover the correct number of holes in the en…

**Figure 5.** Figure 5: Example simulated grid cell firing rates with spontaneous firings that lead to low path reconstruction errors. (Top) An example simulated grid cell firing rates. (Center) Firing rates with additional spontaneous firings, generated with h = 0.4, p = 0.1% and σ = 50. The…

**Figure 6.** Figure 6: Path reconstruction recovers one-dimensional environment from grid cell activity. A. Mouse position in a 320 cm virtual build-up track; the mouse is teleported to the start upon reaching the end. B. The persistence diagram confirms that grid cells are organized on a to…

**Figure 7.** Figure 7: Reconstruction of local paths from two-dimensional experimental data [15]. A. The original trajectory of a rat exploring a 1.5m × 1.5m open-field arena. B. The reconstructed global trajectory, which differs in overall shape from the original path. C. The persistence di…

**Figure 8.** Figure 8: Two possible errors in path reconstruction arising from sparsity of time points. A. The first type of error occurs when two consecutive toroidal coordinates are lifted to two distinct tiles when they should be lifted to a single tile. (Left) Original movement path. Cir…

**Figure 9.** Figure 9: Toroidal coordinates parametrizes points on the torus via (θx, θy), where θx, θy ∈ [0, 2π). adapts the original circular coordinates algorithm. Instead of computing the toroidal coordinates for every population vector, the method computes the toroidal coordinates for a…

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