On Conformal Isometry of Grid Cells: Learning Distance-Preserving Position Embedding
Authors: Dehong Xu
Ruiqi Gao
Wenhao Zhang
Xue-Xin Wei
Ying Nian Wu
Venue: ICLR 2025 Oral (2025)
Keywords: grid cells
conformal isometry
distance-preserving
position embedding
representation learning
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We investigate the conformal isometry hypothesis that leads to the emergence of hexagon periodic patterns in grid cells, showing that learning a maximally distance-preserving position embedding naturally leads to these patterns.

Abstract: This paper investigates the conformal isometry hypothesis as a potential explanation for the hexagonal periodic patterns in grid cell response maps. We posit that grid cell activities form a high-dimensional vector in neural space, encoding the agent's position in 2D physical space. As the agent moves, this vector rotates within a 2D manifold in the neural space, driven by a recurrent neural network. The conformal hypothesis proposes that this neural manifold is a conformal isometric embedding of 2D physical space, where local physical distance is preserved by the embedding up to a scaling factor (or unit of metric). Such distance-preserving position embedding is indispensable for path planning in navigation, especially planning local straight path segments. We conduct numerical experiments to show that this hypothesis leads to the hexagonal grid firing patterns by learning maximally distance-preserving position embedding, agnostic to the choice of the recurrent neural network. Furthermore, we present a theoretical explanation of why hexagon periodic patterns emerge by minimizing our loss function by showing that hexagon flat torus is maximally distance preserving.

Published as a conference paper at ICLR 2025
ON CONFORMAL ISOMETRY OF GRID CELLS: LEARN- ING DUSTANCE-PRESERTEE
POSITION EMBEDDING
Dehong Xu', Ruiqi Gao', Wen-Hao Zhang', Xue-Xin Wei", Ying Nian Wu! UCLA PUT Southwestern Medical Center 'UT Austin xudehongl wwtpoonng999uucla..cmm ywuêstat ucla. edu
ABSTRACT
This paper investigates the conformal isometry hypothesis a potential explanation for the hexagonal periodic patterns in grid cell response maps. We posit that grid cell activities form high-dimensional vector in neural space, encoding the agent's s position in 2D physical space. As the agent moves, this vector rotates within a 2D manifold in the neural space, driven by recurrent neural network. The conformal hypothesis proposes that this neural manifold is a conformal isometric embedding of 2D physical space, where local physical distance is preserved by the embedding up to a scaling factor (or unit of metric). Such distance-preserving position embedding is indispensable for path planning in navigation, especially planning local straight path segments. We conduct numerical experiments to show that this hypothesis leads to the hexagonal grid firing patterns by learning maximally distance- ristance-preserving position embedding, agnostic to the choice of the recurrent neural network. Furthermore, we present a theoretical explanation of why hexagon periodic patterns emerge by minimizing our loss function by showing that hexagon flat torus is maximally distance preserving.
INTRODUCTION
The mammalian hippocampus formation encodes a "cognitive map" (Tolman, 1948; O'keefe & Nadel, 1979) of the animal's surrounding environment. In the 1970s, it was discovered that place cells in the rodent hippocampus fire at specific locations within the environment (O" Keefe & Dostrovsky, 1971). Later, another type of neurons, called grid cells, was identified in the medial entorhinal cortex (Hafting et al., 2005; Fyhn et al., 2008; Yartsev et al., 2011; 1; Killian et al., 2012; Jacobs et al., 2013; Doeller et al., 2010). Unlike place cells, each grid cell fires at multiple locations that form hexagonal grid pattern across the environment (Fyhn et al., 2004; Hafting et al., 2005; Fuhs & Touretzky, 2006; Burak Fiete, 2009; Sreenivasan & Fiete, 201 I; Blair et al., 2007: Couey et al., 2013; de Almeida et al., 2009; Pastoll al., 2013; Agmon & Burak, 2020). Grid cells are thought to interact with place cells and play a crucial role in path integration (Hafting et al., 2005; Ficte et al., 2008; McNaughton al., 2006; Gil et al., 2018: 8; Ridler et al., 2019; Homer al., 2016; Ginosar et al., 2023; Boccara et al., 2019), which calculates the agent's 's self-position by accumulating its self-motion. Thus, this has led to the view that grid cells form an internal Global Positioning System (GPS) in the brain (Moser & Moser, 2016). Although grid cells have primarily been studied in spatial contexts, recent research suggests that grid- rik--iie responses may also exist in more abstract, non-spatial cognitive spaces (Constantinescu et al., 2016; Bellmund et al., 2018).
Numerous computational models have been proposed to explain the characteristic firing patterns of grid cells. Early approaches focused on continuous attractor neural networks (CANN) (Amit, 1992; Burak & Fiete, 2009; Couey et al., 2013; Pastoll et al., 2013; Agmon & Burak, 2020). More recently, two papers (Cueva & Wei, 2018; Banino et al., 2018) learned recurrent neural networks (RNNs) on path integration tasks and demonstrated that grid patterns emerge in the learned networks. These results were further extended in subsequent studies (Gao et al., 2019; Sorscher et al., 2019; Cueva et al., 2020; Gao et al., 2021; Whittington et al., 2021; Dorrell et al., 2022; Xu et al., 2022; Sorscher et al., 2023). Besides RNN-based models, basis expansion models based on principal component
Project page: http://///tthiloooooooeeeeeeeeee

---

Published as a conference paper at ICLR 2025
analysis (PCA) with non-negativity constraints (Dordek et al., 2016; Sorscher et al., 2023; Stachenfeld et al., 2017) have been proposed to capture interactions between grid and place cells.
While prior models have advanced our understanding of grid cells, the underlying mathematical principles behind the emergence of hexagonal grid patterns remain elusive (Cueva & Wei, 2018; Sorscher et al., 2023; Gao et al., 2021; Nayebi et al., 2021; Schaeffer et al., 2022). Recently, the conformal isometry hypothesis has gained attention and has been studied various papers on grid cells. This hypothesis was formalized by Xu et al. (2022) with earlier explorations by Gao et al. (2021; 2019), and has been adapted and investigated in recent works (Schaeffer et al., 2023; Schoyen et al., 2024). The conformal isometry hypothesis posits that grid cell activities form a high sigh--imensional vector in neural space, encoding the agent's 's position in 2D physical space. As the agent moves, this vector rotates within 2D neural manifold, guided by a recurrent neural network (RNN). The hypothesis proposes that this manifold is a conformal isometric embedding of the 2D physical space, where local physical distance is preserved by the position embedding up scaling factor (or unit of metric). Such distance-preserringg position embedding is indispensable for path planning in navigation, including planning local straight-pan segments.
This paper investigates the conformal isometry hypothesis potential mathematical explanation for the formation of hexagonal periodic patterns in grid cell response maps. Unlike previous studies (Xu al., 2022; Gao et al., 2021) that examined conformal isometry within models combining both place cells and grid cells, we focus on a minimal setting: a single module of grid cells equipped with an explicit metric. By reducing the system its essentials, we bring the conformal isometry hypothesis to the forefront and study the grid cell system in isolation with fewer assumptions. This approach allows us gain sharper and deeper understanding of the hypothesis.
In this paper, we design numerical experiments in the minimalistic setting demonstrating that the conformal isometry hypothesis underlies the hexagonal periodic patterns in grid cell response maps. Specifically, we show that the hexagonal periodic patterns emerge by learning the maximally distance- preserving position embedding. To further validate this hypothesis, we conduct in-depth mathematical analysis to show that hexagon periodic patterns emerge by minimizing our loss function, due to the fact that the hexagon flat torus exhibits minimal deviation from local conformal isometry, i.e., the hexagon flat torus forms the maximally distance-preserviing position embedding.
Contributions. To summarize, our paper investigates the conformal isometry hypothesis as a possible mathematical principle that underlies the grid cell system. Our contributions are as follows. (1) We conduct a systematic numerical investigation of the conformal isometry hypothesis in a minimalistic setting with single module of grid cells. (2) We provide a general framework that is agnostic to specific forms of transformation models, grid scales, and the number of neurons. (3) We present theoretical analysis demonstrating that the hexagonal grid patterns emerge by minimizing our conformal and transformation loss function.
2 BACKGROUND
2.1 REPRESENTING SELF-POSITION
F(,)
F(,An)
v(xx)
u(x + Ax)
((rr
v(x + Ax)
Ar
x Ar
(a)
(b)
(c)
Figure 1: (a) The self-position x (211 in 2D Fuclidean space is represented by vector v(x) in the d-dimensional neural space. When the agent moves by An, the vector is transformed v(x Ar) F(n(()), Ar). (b) F(., Ar) is representation of the self-motion Ar. (c) (ы(=), x E D) a 2D
manifold in the neural space, and is an embedding of the 2D Euclidean domain D.
Suppose the agent is at the self celf-position x = (I1, .22 E R within a 2D Euclidean domain D. The activities of the population of d grid cells form a d-dimensional vector v(x) (v,( = 1, d). where (zxx) is the activity of the i-th grid cell at position æ. We call the d-dimensional vector space of the neural space, and embed x: of the 2D physical space vector (a::) of the d-dimensional
2

---

Published as a conference paper at ICLR 2025
neural space. We may also call the vector v(x) the position embedding, following the deep learning terminoloyy ( (Vaswani et al., 2017). For each grid cell i, vy(x), as a function of æ, represents the response map of grid cell i.
2.2 REPRESENTING SELF-MOTION
At self-positioon (z1, x2), assume the agent makes movement Ax (Ar1, Az2) and moves to x + Av. Correspondingly, the vector v(x:) is transformed to v(x + Az). The general form of the transformation be formulated as:
v( Ax) F(v(x)) Ar),
(1)
where F can be parametrized by RNN. See Figure I(a). The input velocity A also be represented as (Ar, 8) in polar coordinates, where Ar = || is the displacement along the heading direction
[0, 2.7], so that Ar (Am1 Ar cos Azy Arsin 0). F(., Ax) is
a representation of the self-motion Aæ. See Figure (a-b). We can also write (()(x), Ax) =
F(v(w), Ar, 0) with slight overloading of notation F. The transformation model is necessary for path integration and path planning.
2.3 CONFORMAL ISOMETRY
For each æ E D. such as a Im x Im square environment, we embed æ as a vector v(x) in the d-dimensional neural space. Collectively, M (v(æ), Vo E D) a 2D manifold the neural space, and M is an embedding of D. See Figure I(c) (the shape of Mi in the figure is merely illustrative). As the agent moves Tr, ((x)) moves in M.
The conformal isometry hypothesis (Xu et al., 2022; Gao et al., 2021) proposes that M is a conformal embedding of ID. Specifically, at any x ID, for local Az, have
|| Ax) v=>)ll s Ax + o(]Aax))
(2)
where s constant scaling factor independent of and Av. As the agent moves in D by ||Aæ||| the vector v(x) moves in M by s| Az||.
Unit of metric. The scaling factor defines the metric unit for distance preserving. For example, ||a|| is measured in meters, and we want to work in centimeters, then 100.
Intrinsic and extrinsic curvatures. For constant s, the intrinsic geometry of the manifold M remains Euclidean or flat, i.e., M is a folding or bending of the flat D without distorting stretching or squeezing. The intrinsic curvature of M zero. But the extrinsic curvature of M non-zero, i.e., the o(]] [a]l) term in (equation 2) grows as || increases. We want to minimize this term for non-infinitesimal | so that distance maximally preserved within a ion-infinitesiaal local range.
A MINIMALISTIC SETTING
3.1 ASSUMPTIONS
In this section, we seek to study the grid cell system with minimal number of assumptions. Specifically, we make the following four assumptions:
x A +o{ Aall)) In the minimalistic setting, we specify s explicitly, in order to understand how s affects the learned hexagon patterns, and conversely what the hexagon patterns reveal about the underlying This will enable to gain deeper geometric understanding of the grid cell patterns. We shall discuss learning s in Appendix
Assumption 1. Conformal isometry: ||
Assumption 2. Transformation: v(x + Ax) F(e(2), Aæ), where F is recurrent neural network. We want to be agnostic about F, and our numerical experiments show that hexagon grid patterns emerge regardless of the form of F. In experiments, consider the following simple forms.
(1) Linear model: v(x + Ax) = v(x) + ((0))(z)Arr where Ar = |xx is the displacement, and 0 is the heading direction of Ar. (0) is a d square matrix.

---

Published as a conference paper at ICLR 2025
(2) Nonlinear model 1: v(x Ax) = (()((x)) + B(0)o(z)Ar + b). where R is elementwise nonlinearity such as ReLU, A and (0) are d x square matrices, and is bias vector.
(3) Nonlinear model 2: v( + Az) (x)v(()) ((0) + b), where (0)) is a a x vector, b is bias vector, and R is nonlinearity.
Assumption 3. Normalization: ||e( 1 for each D.
||(x)]] E, (x)) can be
interpreted as the total energy of the population of neurons in v at position z. This normalization assumption makes the conformal isometry assumption well-defined. Otherwise, we can multiply the vector v (a:) by an arbitrary constant c, so that the scaling factor 8 is changed to cs. Such undesirable arbitrariness is eliminated by the normalization assumption.
Assumption 4. Non-negativity: (a():) 2 for each i and æ. This the assumption studied by Dordek al. (2016); Sorscher et al. (2023); Stachenfeld al. (2017). This is obviously true for biological neurons. However, our ablation studies show that it is not necessary for the emergence of hexagon grid patterns. On the other hand, this assumption does enable more stable learning of clean patterns, because it greatly constrains the solution space.
The above assumptions form a minimalistic setting for studying grid cells, where place cells are not involved. This enables us to study the grid cells in isolation with an explicit metric
Scaling property. Under the above assumptions, for fixed constant c, ((x) e (cxz), then for Assumption 1, | + Az) - o(x))l
[[c(= + Ax)) - v(cz))l = cs||Ax| + o(1 Ar I]). For Assumption 2, 6( + Ar) v(er + cdr) = ((v(cz), cda) = F()(z), cáæ) (((()), Ar). Assumptions and continue to hold. Therefore, system with scaling factor s can be translated to system with scaling factor cs.
3.2 LEARNING METHOD
Let D be a Im x Im Euclidean continuous square environment. We overlay 40 x 40 regular lattice on D. We learn (xx) on the 40 x lattice points, but we treat as continuous, so that æ off the lattice points, we let v(x) be the bi-linear interpolation of the v(x) on the 4 nearest grid points.
The loss function consists of the following two terms:
L Ry,An v (z Ar) v(x))| ~||A>|| |)
La Es,A= |0 + Ar) F(v(x); Am))
where L1 ensures Assumption on conformal isometry, while Le satisfies Assumption 2 on transfor- mation. Since L2 is a one-step transformation loss, back-propaggtion through time is not required.
In L2, assume A to be infinitesimal for local motion (e.g.. | || < 0.075). In L1. however, we assume sAx to within a -on-infimitssiaal range, s >||Aæ| S D (c.g.. D = 1.25). In defining EAZ, assume Ar follows uniform distribution within the ranges specified above. In L1, can also assume a more general distribution p( (A)), so that I = Te [ [[ + Ax) - v(x)) ((A)))))AA)
We minimize L L1 XL2 over the set of v(x) the 40 x 40 lattice points well as the parameters in F, such as B(A) for the discrete set of directions 0 in the linear model. X > 0 is a hyper-parameter that balances Ta and L2. We use stochastic gradient descent to minimize L,, where in each iteration, the expectations in I. are replaced by the averages over Monte Carlo samples of (x, Ax).
After each iteration, we set any negative elements in v(x) to zero to enforce Assumption 4 on non-negativity. Then normalize [|=(())]] for each æ, ensuring Assumption 3 on normalization.
3.3 MAXIMALLY DISTANCE-PRESERVIIGG PRESERVING POSITION EMBEDDING
While L2 can be minimized to close to 0, L1 has a non-zero minimum due to the non-infinitesimal range D. For small s| | the deviation is small due to o[||a]]] term. In fact, our experiment in Section 3.5 shows that conformal isometry is nearly exact for s | << .8. Beyond that point, the deviation begins to increase due to the extrinsic curvature of the manifold. Minimizing L, amounts to finding the maximally distance-preserving position embedding a given scaling factor within
4

---

Published as a conference paper at ICLR 2025
the range D. The reason constrain the range of s||Aæ| is that under the scaling ú(x) v(ex). the deviation from conformal isometry in L1 satisfies: [0( + Ax) أ(z)) cs| Az = ||0(c cAx)
v(cx)]l s||cAæ||.
3.4 NUMERICAL EXPERIMENTS
We perform numerical experiments in the minimalistic setting, testing the generality of our method various RNN parameterizations. Additionally, varied the scaling factor 8 and the number of neurons. For A in 41, we constrained it locally, ensuring 8 s|| < 1.25. For Lg, || was restricted be smaller than 0.075.
The dimensions of v(x), representing the total number of grid cells, were nominally set to 24 for both the linear model and nonlinear model and 1000 for nonlinear model 2. Notably, similar results can be obtained with different numbers of cells, e.g. 500, for both the linear model and nonlinear model All the parameters are updated by Adam optimizer (Kingma Ba, 2014).
k
100
000
000
000
000
000
15
(a) Learned grid cells
(b) Toroidal analysis
Figure 2: Hexagonal periodic patterns learned linear models. (a) Learned patterns linear models with different scales. (b) Toroidal structure spectral analysis of the activities of grid cells.
Hexagonal patterns. We first trained the linear model with manually assigned scaling factor by minimizing L1 + AL2- Figure 2(a) shows the learned firing patterns of vax)) for linear models with the change of scaling factors which controlled the scale metric. In Figure 2(a), each image represents the response map for a grid cell, with each row displaying 6 randomly selected response maps. The emergence of hexagonal periodic patterns in these response maps is evident. Consistency in scale and orientation is observed within each scale, though variations in phases or spatial shifts are apparent.
We also trained nonlinear models with various activation functions. In Figures 3(a-d), hexagonal periodic patterns still emerge with nonlinear transformations, demonstrating that the grid-like patterns are stable and easily learned regardless of the transformation type, grid scales, or number of neurons.
To evaluate how closely the learned patterns align with regular hexagonal grids, we recruited the commonly used metric for quantifying grid cells, the gridness score, adopted by the neuroscience literature (Langston et al., 2010; Sargolini et al., 2006). The gridness scores and the valid rate were reported in Table 1. Compared to other existing learning based approaches, our models exhibit notably high gridness scores (f) and high percentage of valid grid cells (t).
Table 1: Gridness scores and validity rates of grid cells. The last two represent the results of our models.
Table 2: Scaling factor and estimated scale for learned patterns in single-module linear models.
MODEL
GRIDNESS(1)
VALID RATE(T)
BANINO ET AL. (2018)
0.18
25.2%
SCALING FACTOR
ESTIMATED SCALE
SORSCHER ET AI. (2023)
0.48
56.1%
0.82
GAO ET AL. (2021)
0.90
73.1%
OUR LINEAR
1.70
100.0%
10
0.41
OUR NONLINEAR
1.17
100.0%
15
0.27
5

---

Published as a conference paper at ICLR 2025
We also investigated the relationship between the manually assigned scaling factor and the estimated scale of the learned patterns following (Langston et al., 2010; Sargolini et al., 2006). As shown in Table 2, the estimated scale of the learned patterns is proportional to 1/s.
Topological analysis. As discussed in Section 4.1, the joint activities of grid cells should reside on torus-like manifold, and the positions the torus correspond to the physical locations of moving agent. To evaluate whether our empirically learned representations align with the topological properties of theoretical models, we employed nonlinear dimension reduction method (spectral embedding Saul et al. (2006)) to show that grid cell states fell on a toroidal manifold as depicted in Figure 2b(i). To further investigate periodicity and orientation within the same module, we conducted numerical simulations of pattern forming dynamics. In Figure 2b(ii), we applied 2D Fourier transforms of the learned maps, revealing that the Fourier power is hexagonally distributed along 3 principal directions k1, l2, and K3. Following Schøyen et al. (2022); Schaeffer et al. (2023), projecting the toroidal manifold onto the vectors, we can observe 3 rings in Figure 2b(ii). This indicates the manifold has 2D twisted torus topology.
(4) Nonlinear Model (Tanh) wlo v 0
(e) Linear model wio o >>
0 0
0
0
0
0.0
000
0000
0
000
Nonlinear Model (Tanh) 0>0
(f) Linear model w/o |v|
(c) Nonlineur Model 1 (ReLU) w/>0
(g) Lineur model
(d) Nonlinear Model 2 (ReLU)
(b) Linear model w/o
Figure 3: Left(a-d): Learned patterns for nonlinear models with different rectified functions. Right(e-h): Ablation for the linear model.
Ablation study. We show ablation results to investigate the empirical significance of each assumption for the emergence of hexagon grid patterns. First, we highlight the essential role of conformal isometry; in its absence, as shown in Figure 3(h), the response maps display non-hexagon patterns. Next, as shown in Figure 3(a) and (e), we also ablate the non-negative assumption. Without v(x) 2 0, the hexagonal pattern still emerges. For the transformation and normalization assumptions, Figure 3(f) and (g) indicate that they are necessary. See more results for normalization assumption in G.2.
3.5
NUMERICAL EVIDENCE FOR LOCAL CONFORMAL ISOMETRY
To further assess whether learned model achieves local conformal isometry, we examine the relationship between ||u( + Az) v(x)] and | our learned model. Specifi- cally, we aim to verify if the learned representation exhibits the expected local isometry for local displacements Ar and how deviations arise as the displacement increases. Here we use the linear model with the metric
1-10
Dusdrot Rt
[ewomed ieprosetiition
[[vix AM) V0011
10 for analysis.
As shown in Figure 4, we randomly sample Ax within the range (0, 0. 0.125), and plot || + Ar) v(x))] using the learned model. Then, we draw the linear reference line with slope s = 10, and also fit a quadratic curve to the data. The results show that the learned representation closely follows linear relationship for local Az, as expected from a conformally isometric embedding.
|AX|
Figure 4: The relationship between |o( + Ar) v())] and |Ar| in the learned representations.
As Ax increases, we observe a quadratic deviation from the linear trend, due to the extrinsic curvature of the manifold M. In our theoretical analysis, shall analyze this deviation and show that the

---

Published as a conference paper at ICLR 2025
hexagon flat torus achieves minimum deviation from local isometry, thus explaining the emergence of hexagonal grid patterns by minimizing our loss term on conformal isometry L1 in equation 3.
3.6 NEUROSCIENCE EVIDENCE FROM NEURAL RECORDING DATA
To further investigate conformal isometry. we perform analysis on real neural recordings using data from Gardner al. (2021), which contains simultaneous recording of hundreds of grid cells in rodents using Neuropixels probes. To test the conformal isometry hypothesis, we evaluate whether equation holds the recorded cell activations. Specifically, we calculate || v(x + locations z and for varying local shifts Az. We expect to observe a linear relationship.
Az) || for different
Neural data
Distribution of [[v]]
150
[|rix Da1 VOKII
Frequency
14
[|xx]]
[|]|]
(a) Conformal isometry in neural data
(b) Histogram of || neural data
Figure 5: Analysis of real neural data. (a) A clear linear relationship between ||v[ Ax) v(x) and ||
shown in the real neural data. The unit of Ar is meter. (h) Distribution of [[=[))]]] in neural data.
In the analysis, used data from one module, which includes 93 pure grid cells. As shown in Figure 5(a). despite the noise in the data, a clear linear relationship emerges between |[((ar)) v(x +
Ax)Il and | quadratic polynomial fit closely matches the linear curve, indicating minimal deviation from linearity for local | ||. Furthermore, we analyzed the distribution of [|v())]] in real neural data, finding a mean of 128.6 with a standard deviation of 15.1. When normalized to a mean of 1, the standard deviation becomes 0.12, indicating that the assumption of constant [[x(x)]] holds approximately.
Considering the noisy nature of neural responses and the fact that the data may only contain a subset of grid cells in the module, the results above are consistent with Assumption of local conformal isometry and Assumption 3 of constant ||æ( over æ. See Appendix J for more results.
THEORETICAL UNDERSTANDING
This section presents theoretical understanding the emergence of hexagon grid patterns when minimizing our loss function. We show that the hexagonal flat torus, which underlies grid cell patterns, achieves minimal deviation from local conformal isometry.
4.1 HEXAGON FLAAT TORUS IS MAXIMALLY DISTANCE PRESERVING
We start by showing that the manifold of the grid cell representation M = (n(æ:), Vo) forms a 2D torus.
Proposition 1. The transformations Ax), vλx) form a group acting on the manifold M (v(x), Yo), and the 2D manifold ML has torus topology.
Proof. The group (F(: Aw), VdΦ) representation of the 2D additive Euclidean group (IE2 +), ie.. F(v(#), Aæ +Ax2) = ((F(v(( A Aæ²) = F(F(v()), Az²), Ax1), Va, Ax, Are, and F(v(z)) 0) = v(x), Yo (Gao et al., 2021). See Figure for an illustration. Since (IR2, +) is abelian Lie group, (F(:, Ax), vAz) is also abelian Lie group. With Assumption 3 on normalization, ||=] = 1, the manifold (v(x), Yz) is compact, and (z), V Vλz) is a compact group. It is also connected because the 2D domain connected. According classical theorem
7

---

Published as a conference paper at ICLR 2025
in Lie group theory (Dwyer & Wilkerson, 1998), compact and connected abelian Lie group has topology of a torus. Therefore, the 2D manifold M is a 2D torus.
The torus topology is supported by neuroscience data (Gardner al., 2021) as well our numerical experiments. The torus manifold makes v v(x) a 2D periodic function of æ. See Appendix B for detailed explanation of the 2D periodic lattice.
Section 3.5 shows that our learned model achieves local isometry. Without loss of generality, assuming the conformal scaling factor s The 2D manifold M (v(x),, Vx) is thus a flat torus with local isometry, and it has zero intrinsic curvature, meaning it has a flat Euclidean geometry intrinsically.
So far we have shown that the geometry of the grid cell representation is a torus and flat locally. However, viewed from outside the manifold with bigger || A the extrinsic curvature of the torus within the ambient embedding space is non-zero, and our loss function E...A. [(] + Ar)
v(x) - ||=)|) [I) minimizes the deviation from the local flatness for bigger ||
Let us assume ||v( + Az) - v=))| = |A + of [|]2)||])) for infinitesimal ||a=]], enforced by the loss function and verified numerically in Section 3.5. For larger || we further analyze
[| + Ax) =(>)]l - Ax| using higher-order Taylor expansion, up to oxl |*).
Definition 2. We define notation for the derivatives of v()): v(2) 0v(x)/0x1, w(2) = (v(x)/000 /8x2.
(11) = pv(a))/ur) p(12) 8 v()(()) / 10,,00 w222) (()(a)/0xx) /Ong and their higher-order counterparts. The indices in the superscript indicate the variables with respect to which we take partial derivatives.
Definition 3. Assume [|v[( + Ax) - + o(llAar| Define high order deviation from local flatness as ||v( Ax) v(x) ||
Proposition 4. Assuming [|=( for every then the high order deviation
||( + Ar) 0(x)]12 ||
D(Ax)) o(1 Am:l))
(5)
12
where
D(Ar)=((p!))) =(1(1111) Azf + (2222 v) A + (1122) ,
Arj A Ar
(6)
+ 441(1112), v) Ao Aoy 160 v) D, Ar...
Proof.
With |u(x)]] 1, we have
||o + Ar) - = 2(v( + Ar), v)=)) (Aaf Aaa).
||
We can expand v(x + Ax) by 4th-order Taylor expansion, and calculate the inner products between the derivatives and vr (z). When the manifold is locally flat and isometric for infinitesimal Aæ, the first derivatives (1)) and ((2) form orthonormal basis, i.e., (v(1), u(?)) = 0 and (1)(0) 1 for i = 1, 2. By repeatedly differentiating these equations, we can derive a series of properties such as (1, v(z)) = (2) v(x)) = 0, (1(1), v(x)) = (2(22), v(x)) = -1, (1(12), v(x)) = 0. Moreover, the inner products between the third-order derivatives and v(x) all vanish. See Appendix C for detailed proof. Therefore, only the 4-th order terms D (())) remain.
This fourth-order term ((rr) captures deviation from local flatness or extrinsic curvature for larger ||a=|| (i.e., going beyond second-order A[] to up to o(I Ax]|+).]
We shall prove that among all flat tori (such as parallelogram, rectangle, square, and hexagon tori), the hexagonal torus minimizes the deviation derived above, due to its six-fold symmetry that evenly distributes the extrinsic curvature. We first establish the isotropy of hexagon flat torus.
Theorem 5. If the torus M (v(x), Væ) is a hexagon flat torus, then D(Ar) c||Aæ||* for constant coefficient e, i.e., D (Ax)) is isotropic.
Proof. A hexagon torus possesses 6-fold rotational symmetry, meaning it is invariant under 60-degree rotations. Under such a rotation, the coordinates transform as:
v3 2
1 Az + v3
Ar =
Ar
Arzi
Ar/2
AZZ.
(8)
2
2
2
8

---

Published as a conference paper at ICLR 2025
To maintain symmetry, we require that D(Ax') = D(Ax), which leads to the conditions:
( v) (v (2222), v) - - 3( 122), v) - (0(1112), v) - (0o (222)) v 0.
(9)
Thus, the deviation D D()x) becomes:
D(A=) el (frt + Art 2An/An3 = c(Arj Ax²)² = cl A
(10)
which is constant in all directions. This shows that the hexagonal torus is locally isotropic up to the fourth-order term. See Appendix D for a detailed treatment.
Define Ar = || Am = Ar cos A and Arς Ar sin A. D(Ax) is constant over direction A for fixed År. As shown in Figure Appendix, only the hexagon lattice has six-fold symmetry. Other flat tori do not have six-fold symmetry and thus do not have isotropic D (Ar).. For instance, a square torus has 4-fold symmetry, and its (Arr) is not isotropic. An example Clifford torus in 4D, v(x:) = (cos 31, sin 21, cos 22, sin 22), with ((Ar) x (cos 0 + sin 0) Ard, which is not isotropic.
Flat tori come with different sizes or surface areas, which are related to their average extrinsic curvatures. When comparing different tori, we should fix the size or average extrinsic curvature for fair comparison. Specifically, for any fixed Ar, we integrate over E [0, 2T] to get expectation with respect to 0.
Theorem 6. For any fixed average extrinsic curvature f D(Ax)dd, the overall deviation from local flatness
L(Ar) - + Az) - v(x)) Ax 13)2 = 12a / (D(
(11)
minimized if ((Ax) is constant over O.
Proof. This optimality is a result of Ep Ep(D(Az)²) (Es D(Ax))² + Vara (D(Ax)), which is minimized when (Drz()(Az)) 0. This is achieved by the hexagon torus which is isotropic. Moreover, for the same hexagon torus, we apply the above analysis and proves its optimality for all Ar. That is, the optimality of hexagon torus is independent of År.
inequality. can also be proved via the identity
This proves that the hexagon torus minimizes our loss function, i.e., the hexagon torus has local isometry up of ((²)) and it has the minimum deviation from local isometry up to o[] ()). To conclude, the hexagon flat torus forms the maximally distance-preserving preserving position embedding.
4.2 MULTIPLE MODULES CONFORMAL ISOMETRY
The minimalistic setting studied in the previous sections is about single module of grid cells with a fixed scaling factor s. Grid cells form multiple modules or blocks (Barry al., 2007: Stensola et al., 2012), and the response maps of grid cells within each module share the same scale and orientation. Thus, we can learn multiple modules, so that each module satisfies conform isometry with its own scaling factor
For small 8, the flat torus is big, and conformal isometry holds within big range, but the resolution is low, i.e., v( + Ax) is close to v()) in the presence of noises in neuron activities. On the contrary, for a big s, the flat torus is small, and conformal isometry holds within a small range, but the resolution is high.i.., v(x + Ax) is far from v(x) to resist noises in neuron activities.
The agent needs spatial awareness at different scales or resolutions, For instance, in playing golf, the first stroke and the final putt require very different s. Thus it needs multiple modules that satisfy conformal isometry at different scaling factors.
See Appendix H for conformal modulation scheme to ensure the transformation RNN satisfies conformal isometry automatically. See Appendix for learning multiple modules with such built-in conformal isometry by incorporating place cells.

---

Published as a conference paper at ICLR 2025
5 RELATION TO PAST WORKS
The study of grid cells using machine learning models (Cueva & Wei, 2018; Banino et al., 2018; Sorscher et al., 2019; Whittington et al., 2020; Nayebi al., 2021) is topic of interest both machine learning and computational neuroscience communities. Earlier papers (Cueva & Wei, 2018; Banino et al., 2018) adopted recurrent neural networks and used the objective of path integration to learn grid cells. Their models also involved interactions with place cells at the output layer, and the place cells are often modeled by predefined functions such as Gaussian tuning functions (Cueva Wei, 2018; Gao et al., 2021; Xu et al., 2022). However, hexagon grid patterns do not consistently emerge from the learned models (Schaeffer al., 2022). Thus extra assumptions or hypotheses are needed to explain the emergence of hexagon grid patterns.
One hypothesis is to assume the non-negativity of the grid cell responses coupled with Difference-of- Gaussian tuning functions for place cells, as in (Sorscher et al., 2019; Nayebi et al., 2021). However, the biological plausibility of center-surround Difference-of of-Gausian assumption is questioned (Scha- effer et al., 2022).
Another hypothesis is the conformal isometry hypothesis proposed in (Gao et al., 2021; Xu et al., 2022). However, these papers focused learning multiple modules of grid cells using specific transformation models. Additionally, they introduced extra assumptions regarding place cells and the interactions between grid cells and place cells.
Our paper is built on (Gao et al., 2021; Xu et al., 2022), and we have achieved scientific reductionism by isolating the grid cell system of a single module, without making any assumptions about the place cells as well as the interactions between grid cells and place cells. We have also achieved generality by being agonistic of the form of the transformation model. This enables us focus solely on the conformal isometry hypothesis and demonstrates its crucial role in the emergence of hexagon grid patterns. Moreover, we also provide neuroscience and theoretical evidences for this hypothesis. In our work, we study the conformal isometry hypothesis with explicit scaling factor In contrast, Gao et al. (2021); Xu et al. (2022) incorporate s implicitly within their learned models.
Schaeffer et al. (2023) proposed self-supervise learning framework to learn grid cells of multiple modules, where conformal isometry is incorporated.
DISCUSSION: CONFORMAL ISOMETRY FOR PATH PLANNING
Conformal isometry ensures that the spatial representation in the brain preserves the geometry the local environment, where distances are preserved locally. In terms of functional benefit, conformal isometry can be crucial for path planning. For instance, planning a straight path in an open field can be easily accomplished by following the steepest descent on the distance between neural representations of the target position and the current position on the path. This woometry-preservien facilitates the planning of more complex paths involving obstacles and landmarks. The scaling factor 8 allows path planning at different spatial scales or resolutions. This rationale suggests planning- centered representation learning in general, i.e., the learning of representations should serve the purpose of planning.
property also
7 CONCLUSION
This paper investigates the conformal isometry hypothesis as fundamental mathematical principle for the emergence of hexagonal periodic patterns in grid cells, both numerically and theoretically. The simplicity and naturalness of the conformal isometry hypothesis, along with its compelling geometric implications, make it powerful framework for understanding grid cell patterns. Moreover. the conformal isometry hypothesis is also general enough that it leads to hexagon grid patterns regardless of the specific forms of transformation models or other settings. We believe that this hypothesis will serve foundation for further development of normative models of grid cells and enhance our understanding of their interactions with other systems.
10

---

Published as a conference paper at ICLR 2025
ACKNOWLEDGMENT
We thank Minglu Zhao, Sirui Xie, and Deqian Kong for their insightful discussions and valuable suggestions on the experiments. Y.N. W. partially supported by NSF DMS-2015577, NSF DMS- 2415226, and gift fund from Amazon. X.X. W. is partially supported by NSF 2318065.
REFERENCES
Haggai Agmon and Yoram Burak. A theory of joint attractor dynamics in the hippocampus and the entorhinal cortex accounts for artificial remapping and grid cell field-to-fiedd variability. eLife, e56894, 2020.
Daniel J Amit. Modeling brain function: The world of attractor neural networks. Cambridge university press, 1992.
Neil W Ashcroft, N David Mermin, et al. Solid state physics, 1976.
Andrea Banino, Caswell Barry, Benigno Uria, Charles Blundell, Timothy Lillicrap, Piotr Mirowski, Alexander Pritzel, Martin J Chadwick, Thomas Degris, and Joseph Modayil. Vector-based navigation using grid-like representations in artificial agents. Nature, 557(7705):429, 2018.
Caswell Barry, Robin Hayman, Neil Burgess, and Kathryn J Jeffery. Experience-dependent rescaling of entorhinal grids. Nature neuroscience, 10(6):682-684, 2007.
Jacob LS Bellmund, Peter Gärdenfors, Edvard I Moser, and Christian F Doeller. Navigating cognition: Spatial codes for human thinking. Science, 362(6415) 2018.
Hugh Blair, Adam Welday, and Kechen Zhang. Scale-inaarian memory representations emerge from moire interference between grid fields that produce theta oscillations: computational model. Journal of Neuroscience, (1(1):331113222 2007.
Charlotte N Boccara, Michele Nardin, Federico Stella, Joseph O'Neill, and Jozsef Csicsvari. The entorhinal cognitive map attracted goals. Science, (633443)) 1447, 2019.
Yoram Burak and Ila Fiete. Accurate path integration in continuous attractor network models of grid cells. PLoS computational biology, 5(2)xc0000291, 2009.
Alexandra O Constantinescu, Jill X O' O'Reilly, and Timothy EJ Behrens. Organizing conceptual knowledge in humans with a gridlike code. Science, 36266292) 2016.
Jonathan Couey, Aree Witoelar, Sheng-Jia Zhang. Kang Zheng. Jing Ye, Benjamin Dunn, Rafal Czajkowski, May-Britt Moser, Edvard I Moser, Yasser Roudi, et al. Recurrent inhibitory circuitry as mechanism for grid formation. Nature neuroxcience, 16(3):318-333 2013.
Christopher J Cueva and Xue-Xin Wei. Emergence of grid-like representations by training recurrent neural networks to perform spatial localization. arXiv preprint arXiv:1803.07777, 2018.
Christopher J Cueva, Peter Y Wang, Matthew Chin, and Xue-Xin Wei. Emergence of functional and structural properties of the head direction system by optimization of recurrent neural networks. International Conferences on Learning Representations (ICLR), 2020.
Licurgo de Almeida, Marco Idiart, and John E Lisman. The input-output transformation of the hippocampal granule cells: from grid cells to place fields. Journal of Neuroscience, 29(23): 7504-7512 2009.
Christian F Doeller, Caswell Barry, and Neil Burgess, Evidence for grid cells human memory network. Nature, 463(7281):65)) 2010.
Yedidyah Dordek, Daniel Soudry, Ron Meir, and Dori Derdikman. Extracting grid cell characteristics from place cell inputs using non-negative principal component analysis. Elife, 5:e 10040 2016.
William Dorrell, Peter E Latham, Timothy EJ Behrens, and James CR Whittington. Actionable neural representations: Grid cells from minimal constraints. arXiv preprint arXiv:2209. 15563, 2022.
11

---

Published as a conference paper at ICLR 2025
William Gerard Dwyer and CW Wilkerson. The elementary geometric structure of compact lie groups. Bulletin of the London Mathematical Sociery, 30(4):337-365 1998.
lla Fiete. Yoram Burak, and Ted Brookings. What grid cells convey about location. Journal of Neuroscience, 28(27):6858 6871, 2008.
Mark C Fuhs and David S Touretzky. A spin glass model of path integration in rat medial entorhinal cortex. Journal of Neuroscience, 26( 26(16):4266-4276 2006.
Marianne Fyhn, Sturla Molden, Menno P Witter, Edvard I Moser. and May-Britt Moser. Spatial representation in the entorhinal cortex. Science, 325(5688) 2004.
Marianne Fyhn, Torkel Hafting, Menno P Witter, Edvard Moser, and May-Britt Moser. Grid cells in mice. Hippocampus, 18(12): 1238-1238, 2008.
Ruiqi Gao, Jianwen Xie, Song-Chin Zhu, and Ying Nian Wu. Learning grid cells as vector rep- resentation of self-position coupled with matrix representation of self-motion. In International Conference on Learning Representations, 2019.
Ruiqi Gao, Jianwen Xie, Xue-Xin Wei, Song- Chun Zhu, and Ying Nian Wu. On path integration of grid cells: group representation and isotropic scaling. In Neural Information Processing Systems, 2021.
Richard Gardner, Erik Hermansen, Marius Pachitariu, Yoram Burak, Nils A Baas, Benjamin J Dunn, May-Britt Moser, and Edvard Moser. Toroidal topology of population activity grid cells. bioRxiv, 2021.
Mariana Gil, Mihai Ancau, Magdalene Schlesiger, Angela Neitz, Kevin Allen, Rodrigo De Marco, and Hannah Monyer. Impaired path integration in mice with disrupted grid cell firing. Nature neuroscience, 21(1):81-91, 2018.
Gily Ginosar, Johnatan Aljadeff, Liora Las, Dori Derdikman, and Nachum Ulanovsky. Are grid cells used for navigation? on local metrics, subjective spaces, and black holes. Neuron, 111(12): 1858-1875, 2023.
Torkel Hafting, Marianne Fyhn, Sturla Molden, May-Britt Moser, and Edvard I Moser. Microstructure of a spatial map in the entorhinal cortex. Nature, 436(7052):801, 2005.
Aidan Horner, James A Bisby, Ewa Zotow, Daniel Bush, and Neil Burgess. Grid-like processing of imagined navigation. Current Biology, 26(6):842-84) 847, 2016.
Joshua Jacobs, Christoph T Weidemann, Jonathan F Miller, Alec Solway, John Burke, Xue-Xin Wei, Nanthia Suthana, Michael R Sperling, Ashwini D Sharan, Itzhak Fried, al. Direct recordings of grid-like neuronal activity in human spatial navigation. Nature neuroscience, 16(9):1188, 2013.
Nathaniel J Killian, Michael J Jutras, and Elizabeth A Buffalo. A map of visual space in the primate entorhinal cortex. Nature, 771,,726)761,,011
Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint 1412.6441 2014.
Rosamund F Langston, James A Ainge, Jonathan J Couey, Cathrin B Canto, Tale L Bjerknes, Menno P Witter, Edvard Moser, and May-Britt Moser. Development of the spatial representation system in the rat. Science, 1285558)):5576115 1580, 2010.
Bruce L McNaughton, Francesco P Battaglia, Ole Jensen, Edvard I Moser, and May-Brin Britt Moser. Path integration and the neural basis of the' cognitive map'. Nature Reviews Neuroscience, 7(8): 663, 2006.
May- Britt Moser and Edvard I Moser. Where am where am going? Scientific American, 314(1): 26-33, 2016.
12

---

Published as a conference paper at ICLR 2025
Aran Nayebi, Alexander Attinger, Malcolm Campbell, Kiah Hardcastle, Isabel Low, Caitlin S Mallory, Gabriel Mel, Ben Sorscher, Alex H Williams, Surya Ganguli, et al. Explaining heterogeneity in medial entorhinal cortex with task-driven neural networks. Advances in Neural Information Processing Systems, 34:12167-12179, 2021.
John O' Keefe and Jonathan Dostrovsky. The hippocampus spatial map: preliminary evidence from unit activity in the freely-moving rat. Brain research, 1971.
John O'keefe and Lynn Nadel. Précis of o' o'keefe & nadel's the hippocampus as cognitive map. Behavioral and Brain Sciences, 2(4):487-494, 1979.
Hugh Pastoll, Lukas Solanka, Mark CW van Rossum, and Matthew Nolan. Feedback inhibition enables theta-nested gamma oscillations and grid firing fields. Neuron, 77(1):141-154, 2013.
Thomas Ridler, Jonathan Witton, Keith Phillips, Andrew D Randall, and Jonathan T Brown. Impaired speed encoding is associated with reduced grid cell periodicity in mouse model of tauopathy, bioRxiv, pp. 595652, 2019.
David C Rowland, Horst A Obenhaus, Emilie R Skytøen, Qiangwei Zhang, Cliff G Kentros, Edvard I Moser, and May- Moser. Functional properties of stellate cells in medial entorhinal cortex layer ii Elife, 7:e36664, 2018.
Francesca Sargolini, Marianne Fyhn, Torkel Hafting, Bruce L McNaughton, Menno P Witter, May- Britt Moser, and Edvard Moser. Conjunctive representation of position, direction, and velocity in entorhinal cortex. Science, 312(5774):758-277 2006.
Lawrence K Saul, Kilian Q Weinberger, Fei Sha, Jihun Ham, and Daniel D Lee. Spectral methods for dimensionality reduction. Semi-supervised learning, 3, 2006.
Rylan Schaeffer, Mikail Khona, and Fiete. No free lunch from deep learning in neuroscience: case study through models of the aattorinna-hippocampppa circuit. Advances in Neural Information Processing Systems, 35: 10522-166166, 2022.
Rylan Schaeffer, Mikail Khona, Tzuhsuan Ma, Cristóbal Eyzaguirre, Sanmi Koyejo, and Ila Rani Fiete. Self-supervised learning of representations for space generates multi-modular grid cells. arXiv preprint arXiv:231 1..02331111116.... 2023.
Vemund Schøyen, Markus Borud Pettersen, Konstantin Holzhausen, Marianne Fyhn, Anders Malthe- Sørenssen, and Mikkel Elle Lepperød. Coherently remapping toroidal cells but not grid cells are responsible for path integration in virtual agents. bioRxiv, pp. 2022-08, 2022.
Vemund Sigmundson Schoyen, Kosio Beshkov, Markus Borud Pettersen, Erik Hermansen, Konstantin Holzhausen, Anders Malthe-Sorenssen Marianne Fyhn, and Mikkel Elle Lepperod. Hexagons all the way down: Grid cells as a conformal isometric map of space. bioRxiv, pp. 2024 02, 2024.
Ben Sorscher, Gabriel Mel, Surya Ganguli, and Samuel Ocko. A unified theory for the origin of grid cells through the lens of pattern formation. Advances neural information processing systems, 32, 2019.
Ben Sorscher, Gabriel C Mel, Samuel A Ocko, Lisa M Giocomo, and Surya Ganguli. A unified theory for the computational and mechanistic origins of grid cells. Neuron, 111(1):111-37 2023.
Sameet Sreenivasan and Ila Fiete. Grid cells generate analog error-correcting code for singularly precise neural computation. Nature neuroscience, 14(10):1330, 2011.
Kimberly I. Stachenfeld, Matthew M Botvinick, and Samuel J Gershman. The hippocampus predictive map. Nature neuroscience, 10611): 2017.
Martin Stemmler, Alexander Mathis, and Andreas VM Herz. Connecting multiple spatial scales to decode the population activity of grid cells. Science Advances, 1(11):e150081) 2015.
Hanne Stensola, Tor Stensola, Trygve Solstad, Kristian Frøland, May-Brit Britt Moser, and Edvard I Moser. The entorhinal grid map is discretized, Nature, 492(7427):72, 2012.
13

---

Published as a conference paper at ICLR 2025
Edward C Tolman. Cognitive maps in rats and men. Psychological review, 55(4):189, 1948.
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention all you need. Advances in neural information processing systems, 30, 2017.
Xue-Xin Wei, Jason Prentice, and Vijay Balasubramanian. A principle of economy predicts the functional architecture of grid cells. Elife, 4:c08362, 2015.
James CR Whittington, Timothy Muller, Shirley Mark, Guifen Chen, Caswell Barry, Neil Burgess, and Timothy EJ Behrens. The nolman-eichenbaum machine: unifying space and relational memory through generalization in the hippocampal formation. Cell, 183(5):1249-122) 2020.
James CR Whittington, Joseph Warren, and Timothy EJ Behrens. Relating transformers to models and neural representations of the hippocampal formation. arXiv preprint arXiv:21 2021.
Dehong Xu, Ruiqi Gao, Wen-Hao Zhang, Xue-Xin Wei, and Ying Nian Wu. Conformal isometry of lic group representation in recurrent network grid cells. arXiv preprint arXiv:2210.0264, 2022.
Michael M Yartsev, Menno P Witter, and Nachum Ulanovsky. Grid cells without theta oscillations in the entorhinal cortex of bats. Nature, 479(7371):103 2011.
Sheng-Jia Zhang, Jing Ye, Chenglin Miao, Albert Tsao, Ignas Cemiauskas, Debora Ledergerber, May- Britt Moser, and Edvard Moser. Optogenetic dissection of antotinal-hippocammpp functional connectivity. Science, 340(6128), 2013.
Minglu Zhao, Dehong Xu, Deqian Kong, Wen-Hao Zhang, and Ying Nian Wu. A minimalistic representation model for head direction system. arXiv preprint arXiv:2411.105%% 2024.
A MORE TRAINING DETAILS
All the models were trained on a single 2080 Ti GPU for 200, 000 iterations with a learning rate of 0.003. For batch size, we generated 4000 simulated data for each iteration. For single-module models, the training time is less than 15 minutes. For multi-module models, the training time is less than hour.
B 2D PERIODIC FUNCTION
Since and Ar 2D, the torus formed by (F(+,æ)) Ax), VAx) is 2D, then its topology is S1 x St, where each S is a circle. Thus we can find two 2D vectors A and A so that F(,, Az, Aaz2) F(; 0). As a result, vax) is a 2D periodic function that v(ar ki Aa; k2 (xx) for arbitrary integers k1 and liz. We assume Ar₁ and Ar. are the shortest vectors that characterize the above 2D periodicity. According the theory of 2D Bravais lattice (Ashcroft et al., 1976), any 2D periodic lattice can be defined by two primitive vectors (Aæ1, Azz).
If the scaling factor is constant, as the position æ of the agent moves from 0 Aæ; in the 2D space, v(x) traces a perfect circle in the neural space due to conformal isometry. i.e., the geometry of the trajectory of v(x) is a perfect circle up to bending or folding but without distortion by stretching. The same with movement from () to Arg.
Named after Auguste Bravais (181 1863, the theory of Bravais lattice was developed for the study of crystallography in solid-state physics.
In 2D, periodic lattice defined by two primitive vectors (Az1, Arz). and there are different types of periodic lattices, as shown in Figure 6.
For Fourier analysis, we need to find the primitive vectors the reciprocal space, (a1, .2), via the relation: (a:, An,) 2x0zy where d.j = 1 if i = j. and 6,j = 0 otherwise.
For 2D periodic function (xx on lattice whose primitive vectors are (a1,az) a2) in the reciprocal space, define Wy.- = klaj k2a2, where k (ki, k2) are a pair of integers (positive, negative, and zero), the Fourier expansion is f(x) Ee ((
14

---

Published as a conference paper at ICLR 2025
Oblique
Rectangular
Centered rectangular
Square
Hexagonal
Figure 6: 2D periodic lattice is defined by two primitive vectors.
C
DERIVATIVE PROPERTIES FOR FLAT TORI
Given:
|e(x))| for all
The metric tensor Pij (v))), r11)) 6ij
Let us derive the properties:
Lemma 7. (w(1), vxx)) = (v(2), v(x)) 0
Proof. Differentiate ||0(z)]13 with respect to æ1:
(w(z), v(z)) = 2( v(x)) =
dz
Therefore, v(x)) = O. Similarly for Ø2-
Lemma 8, (11)), v(x)) = (m(22), v(x)) -1
Proof. Differentiate (v(1), v(x)) = 0 with respect #1:
(w v(x)) (v (())
But (,(1), o()) = (from the metric tensor), so (111), v(x)) = -1. Similarly for (22).
Lemma 9, (1(12), v(w)) = 0
Proof. Differentiate (v v(w)) = 0 with respect to x2: (1(12), v(x)) (v o(2))
But (v(2), ((2) 0 (from the metric tensor), (1(12), v(z)) 0.
Now, let's derive the third-order terms:
Theorem 10. (1111), v(x)) (222), v(x)) (1(112) v(x)) (1(112)) v(x)) 0
Proof. 1) For (I!1))) v(x)); Differentiate (v(11) v(x)) - with respect to or
(I11)),, v(x)) + (1)))))))))))) = 0
But (1(11), (()) = 0 (differentiate (1),, v(1) = 1), so (1(111)) (((11)))n(()))) = 0.
2) Similarly, (2(2))) 2 (xx)) = 0.
3) For (112), v(x))): Differentiate (1)),) v(x)) = -1 with respect to 22!
(1(1))) v(x)) + (11) (()) 0
But (1(21), x(2)) = 0, which we can show by differentiating (1) (2)))
(1(11) 22)) + (122) = 0
(v(), (12)) O, which can be derived by differentiating (w(1), ())) with respect x2. Therefore, (112)) v(x))
4) Similarly, (122) v(z)) 0.
15

---

Published as a conference paper at ICLR 2025
These derivations show that all third-order inner products (w(())) v(x)) are for flat torus with the given conditions.
D HEXAGONAL SYMMETRY IN 4TH-ORDER POLYNOMIALS
Theorem 11. If a 4th-order polynomial f(z, y) = axx + bx y + cx²y² + day" + ey² has hexagonal symmetry (invariant under 60 rotation). then it must be of the form f(z,8) y) a(x² + y²?
This result can be applied to our analysis of hexagon flat torus in the main text, where the 4th-order terms the Taylor expansion isotropic under hexagonal symmetry.
Proof. Let rotate our coordinate system (I, y) by an angle 0 to align with the reflection symmetry axes the hexagon. Let (a', 3²) be the new coordinate system, so that f(z, y) = g(a', which is 4th-order polynomial of 2' and y'. In this new (æ', y/) coordinate system, we have reflection symmetries:
x' x' 3// y
x x. d' y
Due to these reflection symmetries, in the (x', (x', system, our polynomial must take the form:
g(ø', v/) = Ax t + Bx? Cyd
(12)
Without loss of generality, assume A = 1. Thus:
g( y²) xt + yx C2a²4
(13)
Due to 60" rotational symmetry, we must have:
0(1 v(1/2,
9(0,1) = g0(30/g233/3///
These conditions give equations:
( 2//22 CC(33/2)
1 = 1/16 + 3B16+3//1+99//
(14)
= (V3/2)* (V/32 (1/2)
C = 9/16 33/--B 1/16
Solving these equations simultaneously yields:
B 3 2 and C
(15)
Therefore, in the rotated coordinate system:
g(x' v') = 1 2x 12 + 1 = (222 + 1/222
(16)
Since (x² + y²) is invariant under rotation,
conclude that in the original coordinate system:
f(x, y) = a(=² 22
(17)
where we have reintroduced the coefficient a that we normalized to earlier.
Thus, we have proved that if 4th-order polynomial has hexagonal symmetry, must be of the form [a, y) = alx² + 2))2
16

---

Published as a conference paper at ICLR 2025
E PERMUTATION GROUP AND RELATION TO ATTRACTOR NETWORK
The learned response maps of the grid cells in the same module are shifted versions of each other, i.e., there a discrete set of displacements { {Ax), such as for each Aæ in this set, we have 0,(2 + Ax) (2)), where j = o (i, Aæ), and is a mapping from i to j that depends on Az. In other words, Aa causes permutation of the indices of the elements v(x), and F(, Aa) a(: Ar), that is, the transformation group for the discrete set of A is equivalent This is consistent with aanddddigned CANN (Amit, 1992; Burak & Fiete, 2009; Couey et al., 2013; Pastoll et al., 2013; Agmon & Burak, 2020). A CANN places grid cells on 2D "neuron sheet" with periodic boundary condition, i.e., a 2D torus, and lets the movement of the "bump" formed by the activities of grid cells mirror the movement of the agent in conformal way, and the movement of the "bump" amounts to cyclic permutation of the neurons. Our model does not assume such an a priori 2D torus neuron sheet, and is much simpler and more generic.
subgroup of the permutation group.
F EIGEN ANALYSIS OF TRANSFORMATION
For the general transformation, a( + Ar) = ((u(x)) Ar), we have
v v(x) F(v(œ)), 0),
(18)
v x + Ax) F'v(x Az).
(19)
thus
Av v(x + Ax) v(z) ((( +
(20)
where
(1())= = 0s F(u+ + A,0) -==
(21)
Thus Av is in the 2D eigen subspace of F (z)))) with eigenvalue 1. At the same time,
v(x + Ax) = F(v(x), Ax) = v(x) + F'sm Ax.
(22)
where
(A)( DAx
F(v, Aa) Ia
(23)
Thus
Av ())
(24)
that is, the two columns of FA (v(x)) are the vectors that span the eigen subspace of F", (v(
with eigenvalue 1. If we further assume conformal isometry, then the two column vectors of F (v())) are orthogonal and of equal length, that Av is conformal to Az.
The above analysis is about vxx) on the manifold. We want the remaining eigenvalues of F (vx))) to be less than 1, so that, off the manifold. F(v, 0) will bring u closer to the manifold, iec. the
manifold consists of attractor points of F, and F is an attractor network.
G MORE EXPERIMENT RESULTS ON MINIMAL SETTING
G.1 MORE ABLATION STUDIES ON ACTIVATION FUNCTIONS
To further examine the impact of the transformation model F, we conducted additional ablation studies using different activation functions in nonlinear models. As illustrated in Figure 7, we tested Leaky ReLU and Swish activations. Both successfully produced hexagonal patterns, demonstrating that the specific form F does not dictate the emergence of grid patterns. This highlights the robustness of our approach to variations in the transformation model.
17

---

Published as a conference paper at ICLR 2025
O
(a) Nonlinear Model w/ Leaky ReLU
(b) Nonlinear Model I w/ Swish
Figure 7: More activation functions: (a) Leaky ReLU and (b) Swish.
G.2 INFLUENCE OF NORMALIZATION ASSUMPTION
Our normalization assumption, || = 1, critical for the emergence of hexagonal grid patterns, as shown in our ablation study where removing normalization prevents their formation. To further explore this, we conducted experiments with imperfect normalization by introducing noise: 1 1-6 S [[]()))]| I 1 + c. During training, in the normalization process, allow ||w|| to randomly vary within the range (1 - e, 1 c). This enables us to assess the maximum allowable e that still preserves hexagonal patterns.
As shown in Figure 8, we tested four different noise levels. The results indicate that hexagonal patterns robust to small deviations in normalization but collapse becomes large, highlighting the importance of precise normalization for pattern stability.
e 0.05
(b) 8
(c) 0.2
(d) = 0.4
Figure 8: Imperfect normalization with different noise levels.
G.3 INFLUENCE OF THE RANGE OF DISPLACHMENT
As mentioned in the experiments, for L1, enforce the constraint s|| s||Aæ|| 1.25 ensure local conformal isometry. To further explore the influence of the range of Ax on the patterns, we conducted additional experiments by increasing and decreasing the range. As shown in Figure perturbed the range and found that the emergence of hexagonal patterns remains stable, demonstrating the robustness of the framework to variations in this syyper-arametee
| 1 1.5
b00
s 1 I 1.25
1
Figure 9: Varying the range of ||Az|| conformal training.
H CONFORMAL MODULATION FOR BUILT-IN CONFORMAL ISOMETRY
So far have used a conformal isometry loss to constrain the conformal isometry property (equation 3). Conformal isometry can be directly baked into the transformation model via a mechanism we call conformal modulation of input velocity that conformal isometry is automatically satisfied.
18

---

Published as a conference paper at ICLR 2025
H.1 CONFORMAL MODULATION
Consider the general recurrent transformation o(æ Aæ) - F(v(x), Aœ) - F(v(x), Ar, 0), where
Ax (Ar) Ar cos Azg Arsin 0), is the heading direction, and Ar is the displacement, the directional derivative F at (v, 0) is defined as
a da
f(u, 8)
F(v, a, la=0
(25)
With the above definition, the first order Taylor expansion of the recurrent transformation Ar gives us
(a Ar) (ax)) [(v(())) o(Ar).
(26)
The conformal modulation of Ar is defined as
Ar [f(v(z),)) Ar,
(27)
where || || is the & norm, and is learnable parameter. With conformal modulation, the transfor-
mation is changed to
v(x + Az) = F(v(()), 0) F(u(2),Ar,0)
(28)
Then the first order Taylor expansion gives
v(x Ax) v(x) f(v(x), 0)Ar ((rr) v(x:) ( (()((x)))))))) o(Ar),
(29)
where f(v, f(v, ()t,))/l[((r 0)ll is unit vector with [1ʃ(u, 0) 1, which leads ||00
Ar) - (=))] s s|Aa ( A=l)), i.e., conformal isometry. Thus F F(n((x), Ar, 0) F(v(#), Ar)
satisfies conformal isometry automatically.
H.2 IMPLEMENTATION OF CONFORMAL MODULATION
For the linear model na + Ar) = v(x) + (()))())Ar; the directional derivative f(v,0) (f)) The conformal modulation is
SAr
(30)
B(O))())]]]
The linear transformation model then becomes
B ((0))(z) B [()))())|
v(x Ax) v(x) +
Ar.
(31)
The above model is similar to the "add + layer norm" operations the Transformer model (Vaswani al., 2017).
For the nonlinear model v(x + Az) = R(Av(z) + B(f)u(z)Ar+ + b), where is a learnable matrix, and R(-) is element-wise nonlinear rectification, the directional derivative is f(v, 0) R'' (Av) @ B(0), where I²'()) is calculated element-wise, and @ is element-wise multiplication. Thus, the nonlinear transformation model becomes
B(0)v(x) (a + Ax) R(Av(z)))+)))) + (AR) @ B(0)al Ar 6).
(32)
While the linear model is defined for (xx) on the manifold M, the nonlinear model further constrains v(x) = R Av(x)) for v(x) MI, where Ar O. If Av) is furthermore contraction for that are off Mi, then Mi consists of the attractors of R(Av) for v around M. The nonlinear model then becomes continuous attractor neural network (CANN) (Amit, 1992; Burak & Fiete, 2009; Couey et al., 2013; Pastoll et al., 2013; Agmon & Burak, 2020).
In Appendix I, learn multiple modules of grid cells within combined model with both place cells and grid cells, where each module undergoes its own transformation with conformal modulation. and each module has learnable scaling factor s.
19

---

Published as a conference paper at ICLR 2025
1 MULTI-MODULE SETTING WITH PLACE CELLS
We can extend the multimodal setting defined in Section 4.2 to further incorporate the interaction with place cells. Define the adjacency kernel A(x, æ²) exp(-Ila || == ==112/20²) /2a²) for certain scale parameter a. A )') is a model of the response map (as function of æ) of place cell for the place w'. We can then use v(x) decode A via A(æ, x') = (w(x'), v(x)), where w(x') is the decoding vector associated with place cell w'. Below we demonstrate the learning of such a system.
1.1 MOLTIE SETUP
The conformal modulation follows the definition in equation 31 and equation 32. For multi-modules in our model, the vector representations v (xz) are divided into sub-vectors analogous to modules. Accordingly, we assume that the transformation is also module-wise by assuming (0) block- diagonal where each block corresponds to module.
1.2 LEARNING MULTIPLE MODULES
We assume multiple modules (sub-vectors) in vx+), each has its own transformation with conformal modulation. The loss consists of two terms:
Lu Eg,w [(Axx x²) (w(xx) [x
(33)
L2
[ Ax) F(v(2), Az)
(34)
where F() is the transformation after conformal modulation so that we do not need conformal isometry loss term L1. We assume w(x') 2 0 because the connections from grid cells to place cells
are excitatory (Zhang et al., 2013; Rowland et al., 2018). We do not enforce the non-negativity of v (x=). We continue to enforce [[=(())]|
In the numerical experiments, Axx x²) is the given input, and we jointly learn the position embedding (a), read-out weights wax'), and the transformation model F by minimizing the total loss: L = Lo + L2.
I.3 NUMERICAL EXPERIMENTS
We conducted numerical experiments to train multi-module models with linear and nonlinear trans- formations. Each module consisted of 24 cells, with 15 modules for the linear model and 8 for the nonlinear model. For the response maps of the place cells, we used a Gaussian adjacency kernel with 0.07. Regarding the transformation, the one-step displacement Ar restricted to be smaller than 3 grids. Additionally, the scaling factor s was treated as a learnable parameter for each module.
As shown in Figure 10(ac),, each row shows the units belonging to the same module and randomly present 3 cells in each module. The empirical results illustrate that multi-scale hexagonal grid firing patterns emerge in the learned vxx) across all models with conformal modulation. Figures 11 and 12 show more patterns.
Additionally, Figure 10(d) shows the histogram of grid scales of the learned grid cell neurons, which follows a multi-moda distribution. The distribution is best fitted by a mixture of 3 Gaussians with means 0.57, 0.77, and 1.01. The ratios between successive modes are 1.37 and 1.31. The scale rela- tionships in our learned grid patterns across modules align with both theoretical predictions (Stemmler al., 2015; Wei et al., 2015) and empirical observations from rodent grid cells (Stensola et al., 2012).
1.4 PATH INTEGRATION
Suppose the agent starts from æo. with vector representation Vn v(zo). If the agent makes sequence of moves (Aw,, t = 1, T"), then the vector v is updated by v, = F(V-11 Ax;). ). At time the self-position of the agent can be decoded by â arg max (v,, w(x')): i.e., the place x' that is the most adjacent to the self- self-position represented by vy- This enables the agent to infer and keep even in darkness.
track of its position based on its
Following Gao et al. (2021): Xu et al. (2022); Zhao et al. (2024), we assess the ability of the learned model to execute accurate path integration in two different scenarios. First of all, for path integration
20

---

Published as a conference paper at ICLR 2025
0.77
1.0L
00
Pooobbiiiyy
06 0.6 03
1.2
Grdd scale im)
(a)
Linear
(b) Nonlinear
(c)
Nonlinear(Tanh)
(d) Grid scale distribution
Figure 10: Results of linear and nonlinear models with GELU and Tanh activations and grid scale distribution.
D0 00
880
pod
200 98
00 06
0 00
200
100
100
P0c
200
100
20g
200
200
000
000
00
00
100
200
DOX
POX
Do
0
Figure 11: In the lincar model, hexagon grid firing patterns observed in the learned v(x). Each row displays the firing patterns of all the cells within single module, with cach module comprising 24 cells. The units illustrate the neuron activity throughout the entire 2D square environment. The figure presents patterns from randomly chosen modules.
with re-encoding, we decode v - á: to the 2D physical space via á = arg max (1,, un(x))) )) and then encode v < v(Φ) back to the neuron space intermittently. This approach aids in rectifying the errors accumulated in the neural space throughout the transformation. Conversely, in scenarios excluding re-encoding, the transformation is applied exclusively using the neuron vector v. In Figure 13(), the model adeptly handles path integration up to 30 steps (short distance) without the need for re-encoding. The figure illustrates trajectories with a fixed step size of three grids, enhancing the visibility of discrepancies between predicted and actual paths. is important note that the physical space was not discretized in our experiments, allowing the agent to choose any step size flexibly. For path integration with longer distances. we evaluate the learned model for 100 steps over 300 trajectories. As shown in Figure (BB,, with re-encoding, the path integration eгrоr for the last step is as small as 0.003, while the average error over the 100-step trajectory is 0.002. Without re-encoding, the error is relatively larger, where the average error over the entire trajectory is approximately 0.024, and it reaches 0.037 for the last step.
CONFORMAL ISOMETRY IN NEURAL DATA
J.1 MORE DETAILS AND RESULTS FOR NEURAL DATA
In Figure 5, the real neural recording that we used is rat 'R' day 1, module from Gardner et al. (2021). To provide more results for real neural data, here we choose another module of grid cells (rat 'R' day 1, module 3) to further evaluate conformal isometry.
similar results in the main paper: linear relationship As shown in Figure 14(a), || and || || and a very small deviation from the linear relationship in between (x)) vxx + Ax) local A ||. Also, similar distribution shape of ||l|| is shown in Figure 14(b). This provides more empirical evidence of the conformal isometry hypothesis.
J.2 QUANTITATIVE COMPARISON BETWEEN NEURAL DATA AND NUMERICAL SIMULATION
In this section, we present additional quantitative analysis to further compare real neural data with numerical simulations in the context of conformal isometry.
21

---

Published as a conference paper at ICLR 2025
0o
OO
(a) Tanh activation
GELU activation
Figure 12: Results of the non-lincar models. We randomly chose modules and showed the firing patterns with different rectification functions.
A
B
Start
Real Path
W/D re encode
Inferred Path
Srid
0.03
1.00
Figure 13: Results for path integration. (A) Path integration for 30 steps without re-encoding. The black line represents the real trajectory and the red the predicted trajectory by the learned model. (B) Results for long-distance (100-step) path integration error with and without re-cncoding over time by the non-linear model.
To align with the settings in our simulation, first normalize the side length of the square domain (1.5 m) in the real neural data to a unit length of 1, as is the case with our numerical simulation. Based on the scale of the grid module (rat R, module 1, day reported in Gardner et al. (2021), the spatial grid spacing is 0.5 m, which corresponds to 0.33 spatial grid spacing in our numerical simulation after normalization (0.5m/1.55) We also normalize || in the neural data, noting that this pre-processing step was not applied in the original results shown in Figure 5.
To quantitatively examine the scaling factor in the normalized neural data, first fit linear relationship between vo + Ar) v(x) (after normalization) and | AAx| with the intercept fixed O. As shown in Figure 15, the fitted scaling factor is 13.4. In Table 2, we show that the estimated spatial grid spacing of the learned patterns is inversely proportional to s. Specifically, the product of the scaling factor and the estimated spacing in the real neural data (13.4 x 0.33 4.42) comparable to our numerical simulation result (10 x 0.41 = 4.1).
To further examine the local conformal isometry, we fit quadratic curve to the data, similar to the approach in Section 3.5. The deviation arises as || A is larger than 0.08, and this result is in agreement with the result in our numerical simulation.
In summary, the real neural data is in agreement with the results in our numerical simulation in terms of the relationship between the scaling factor a and the estimated grid spacing, as well as the range of conformal isometry observed.
22

---

Published as a conference paper at ICLR 2025
Neural data
Distribution of ||v]|
|| + (X) vbo0l
Frequency
30c
Quadratic Rit
Neural data
|
||v|l
(a) Conformal isometry in neural data
(b) Histogram for ||a|| in neural data
Figure 14: Results for different module of grid cells.
Neural data
175
v(X1|
1.25
||v(x + Ax)
Linear Fit
Quadratic c Fit
Neural data
0.0
0.58
6/2
||XX||
Figure 15: Conformal isometry in normalized neural data.
23


Reviews:
[{"review_id": "Sr8ICyLWIO", "reviewer": "Reviewer_hz9p", "summary": "The paper studies the conformal isometry hypothesis for grid cells. They studied this in a minimal setting, in the isolation of modeling place cells. They show analysis that the hypothesis underlies the hexagonal periodic patterns that the grid cells exhibit. Moreover, they mathematically show that the hexagon patterns are minimal deviations from the local conformal isometry, which can emerge by minimizing the discussed loss function. They perform topological analysis and show the similarities of the arose patterns with their mathematical model to the neural data.", "questions": "- It would be nice to know the consequence of their model assumptions on what we know about neural representations in the brain; their model seems to have removed some assumptions that the literature seems to think are needed for grid cells to arise.\n\n- Can the authors also discuss what parameter or assumption in the model may break the emergence of hexagonal patterns? For example, can the authors conduct an ablation study where an assumption is violated or added, resulting in no longer optimal hexagons? This study may bring much value. For example, the authors emphasize the importance of the normalization step, which plays a crucial role in finding the correlation between terms in the theoretical analysis. Possible to see numerical analysis in the absence of such constraint?\n\n- It was reported briefly how this work differs from Gao et al. 's. (2021) and Xu et al. (2022) on their study on conformal isometry, please discuss the importance of the difference. Why the proposed approach of only modeling the grid cells is better than the previously studied combined model?\n\n- Could the authors please explain the choice of 1.25 and 3 in the experiment design in Section 3.3.\n\n- Three models are chosen, showing that the grid pattern still emerges. Can the authors clarify if their results are fully independent of the choice of such models?\n\n- More clarifications are needed for Figure 2. How exactly are Figure 2b(ii) and (iii) constructed?\n\n- Is the finding on the proportion of the scale of patterns and the s parameter unique and new from this paper?\n\n- What is the impact of \\lambda on the presented analysis and results?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "This is of interest to the community. It studies how grid cells encode spatial navigation and how these representations arise in the brain. The paper is written well, and concepts are clearly explained. I enjoyed reading the paper, and the math is thorough.\n\nThe proposed framework is novel compared to prior work; it reduces the assumptions required for the emerging hexagonal patterns.", "weaknesses": "I am not an expert in grid cell research; hence, it is a bit hard to fully appreciate and evaluate the significance of their minimalistic assumptions to model grid cells and study the conformal isometry hypothesis. This suggests a more in-depth discussion on why this proposal is better than prior works. Table 1 briefly shows that the framework offers better grid patterns for learning. Moreover, the paper discusses that their assumptions are less strict than prior works. These comparisons should be included in-depth in the main paper (in this version, only the intro discusses). Moreover, the first half of the paper is organized well; however, the theoretical analysis can be organized better. I recommend combining Sections 4 and 5.\n\n\nI have comments about additional experiments and discussions that can help to improve the paper (see my questions).\n\n\nminor \n\n- Some sentences in the abstract are exactly repeated in the intro. I suggest to revise.\n- Line 76: Please clarify the \"our loss function\". Same for line 86.\n- Line 126: no need to re-state \"D is the 2d Euclidean domain\". It has already been stated at the end of page 2.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "TIhiWSMoWg", "reviewer": "Reviewer_982F", "summary": "This paper explores the conformal isometry hypothesis as an explanation for the hexagonal patterns observed in grid cell responses. The authors conduct numerical experiments to show that this hypothesis leads to the hexagon periodic patterns of grid cells, agnostic to the choice of transformation models. Also, a theoretical understanding that hexagon patterns emerge by minimizing the loss function is presented.", "questions": "- It would be valuable, however, to see a comparison between the proposed conformal model and other standard models of grid cell activity, as this would provide more context on the advantages or limitations of the proposed approach.\n\n- Could the authors expand on how this hypothesis might be experimentally tested in biological systems or how it could inspire new architectures in machine learning?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The authors conduct numerical experiments to show that this hypothesis leads to the hexagon periodic patterns of grid cells, agnostic to the choice of transformation models.\n\n- The use of RNN serves as a foundation for further development of normative models of grid cells.\n\n- Some theoretical analysis is presented showing that the hexagonal grid patterns emerge by minimizing the loss function.", "weaknesses": "- A primary limitation of this paper is that the explored neural networks are restricted to ReLU, GELU, and Tanh activation functions. Could the authors consider extending their framework to support a broader range of activation functions?\n\n- Could the authors clarify the structure of neural network studied in this paper? There is no mention about the number of layers.Additionally, the reader could benefit from explaining why the authors selected the studied activation functions.\n\n- The paper would benefit from a deeper discussion on how the conformal isometry hypothesis compares to alternative theories of grid cell patterns which are missing in the current submission.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "J0g2j5Llfb", "reviewer": "Reviewer_vT6s", "summary": "The authors present numerical and theoretical evidence that the grid cell tuning to 2D position (which follows a hexagonal lattice) would arise whenever the neural representation is (approximately) a conformally isometric embedding of space. This follows up on a select number of previous works that posited the \"conformal hypothesis\" of grid cell tuning. This paper aims to present a simplified analysis that is stripped down to only the most necessary details.", "questions": "* As mentioned above, I would like to know whether there is a normative explanation or theory behind the conformal hypothesis of grid cell coding. Is there a potential benefit (e.g. in terms of energy efficiency or decoding accuracy) for this kind of representation?\n\n* Put differently, is there a justification for why the objective function defined in equations 3 and 4 is a good proxy for the constraints faced by real biological neural systems?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "* The paper is generally well written and self contained. I had not heard of the \"conformal hypothesis\" before reading this paper, but was able to follow the essential message without reading the prior cited works in detail.\n\n* The paper studies a very simple, minimal model that is nonetheless sufficient to produce striking grid cell firing patterns\n\n* The numerical experiments are convincing (although some additional details on how the simulation is set up would be helpful). I particularly like the ablation experiments in Fig 3e-h.\n\n* The core theoretical insight (which I view as being shown in sections 4.1) is quite interesting. The rest of the theoretical analysis in sections 4.3, 4.4, and 5.2 seems technically sound. However, the contribution of these additional sections is less clear from the presentation.", "weaknesses": "* As far as I can tell, there is no normative explanation or hypothesis that would explain *why* the neural manifold encoding 2D space *ought* to be a isometric embedding of physical space. What are the potential benefits of this coding scheme? Absent any functional benefit, it is hard to interpret the significance of these results. Is it just a coincidence that this model gives the same grid coding scheme as real networks? Or is there a deeper reason why biological and artificial networks will converge upon a conformal map of space?\n\n* Sections 4.2 - 4.4 are confusing. Because there are three sections, it feels as though the authors are trying to list three different results, but I think all three of these sections are really just supporting the same result (i.e. that hexagonal flat torus results in minimal deviation from local isometry). I suggest that the authors merge these three subsections into one. Furthermore, it would be easier to read if the main result were formally presented as a proposition and, potentially, if the result in section 4.3 were formally presented as a lemma. Overall, I would suggest re-writing this portion of the paper to make the motivation more clear.\n\n* It would be useful to investigate how important assumption 3 (normalization) is to the results. Figure 3f shows that the hexagonal grids do not form when normalization is removed entirely. Real biological networks are likely normalized imperfectly. Is imperfect normalization sufficient to get hexagonal grids? That is, what if $1 - \\epsilon \\leq \\Vert \\mathbf{v}(\\mathbf{x}) \\Vert \\leq 1 + \\epsilon$ for some $\\epsilon > 0$? How big can $\\epsilon$ get before the grids fail to form?\n\n* There is a question about whether ICLR is the optimal venue for this kind of work, which I think will be primarily interesting to experimental and theoretical neuroscientists. I think this work will have little to no cross-over appeal to machine learning or artificial intelligence communities.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oE9IlHow1p", "reviewer": "Reviewer_aXqp", "summary": "This work explores the conformal isometry hypothesis (i.e., that grid cells in MEC encode a conformal isometry that may be useful for spatial navigation) in artificial neural networks. The authors take a minimalistic approach, showing that hexagonal grids emerge robustly when optimizing a conformal isometry loss function. They show evidence for conformal isometry in real neural recordings. They analytically probe why hexagonal grid cells would emerge from their optimization, finding them to be optimal among flat tori. Finally, the authors demonstrate that modifying their loss function can lead to the emergence of multiple grid modules.", "questions": "1. Why is the hexagonal torus optimal (as compared to other flat tori)? \n\n2. How does having multiple grid modules affect the conformal isometry? \n\n3. Why might the neural data satisfy the conformal isometry over a larger scale than the simulations?\n\n4. Which modules was analyzed in Fig. 5 and Fig. 11, and was any pre-processing performed on them?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This paper thoroughly studies different aspects of the conformal isometry hypothesis. The authors' results on: 1) the robustness of grid cells emerging from optimizing a conformal isometry loss function; 2) the connection of the grid scale to the scalar $s$ in the conformal isometry; 3) the analytical results showing why a hexagonal torus is optimal; 4) the demonstration of real neural data satisfying the conformal isometry; 5) the expansion of the authors' results to multiple modules, all are nice additions to the computational study of grid cells.", "weaknesses": "I think there are two points that could be improved upon to increase the quality of this work:\n\n1. While the authors made it clear why a hexagonal torus is local isotropic, it was not clear to me when/if the authors ever showed it was optimal because it \"achieve the least deviation from local flatness because it distributes the extrinsic curvature evenly over all directions\" (Sec. 4.4). Providing more discussion on this (and explaining why other flat tori do not distribute extrinsic curvature evenly) would make their point of optimality more clear. \n\n2. While it was interesting to see that the framework developed by the authors could be expanded to include the generation of multiple modules (that were organized approximately $\\sqrt(2)$ apart in grid spacing), how exactly the multiple modules tied in to the idea of conformal isometry was lost for me. In an otherwise well structured paper, it felt a little like it was tacked on at the end and the thread was lost. Providing more discussion on how having multiple modules, in addition to the need to decode place responses, affects the kinds of representations that lead to conformal isometry, would be helpful. \n\nMINOR POINTS: \n\n1. What is meant by \"$\\Delta x$ is constrained to be smaller than 3 grids?\" By this do you mean 3 grids of the underlying space discretization? Maybe using a different word would be helpful to keep the reader from confusing this with grid cells. \n\n2. What is valid rate that is reported in Table 1? \n\n3. The real neural data analyzed by the authors satisfies conformal isometry (Fig. 5 - up to $||\\Delta x|| = 0.2$) for a much larger scale than the grid cells generated by the authors' models (Fig. 4 - up to $||\\Delta x|| = 0.08$). Why do the authors think this is? Could this be due to the Gardner et al. data having variability in grid properties (Redman et al., 2024) not present in the authors' model?\n\n4. Which module from the Garnder et al. data set was analyzed? If I remember correctly, none of the modules had 93 grid cells. Did the authors perform some screening on which grid cells to include?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Oral)