On Path Integration of Grid Cells: Group Representation and Isotropic Scaling
Authors: Ruiqi Gao
Jianwen Xie
Xue-Xin Wei
Song-Chun Zhu
Ying Nian Wu
Venue: NeurIPS 2021 Poster (2021)
Keywords: grid cell
representation learning
path integration
place cell
recurrent network
matrix Lie group

Abstract: Understanding how grid cells perform path integration calculations remains a fundamental problem. In this paper, we conduct theoretical analysis of a general representation model of path integration by grid cells, where the 2D self-position is encoded as a higher dimensional vector, and the 2D self-motion is represented by a general transformation of the vector. We identify two conditions on the transformation. One is a group representation condition that is necessary for path integration. The other is an isotropic scaling condition that ensures locally conformal embedding, so that the error in the vector representation translates conformally to the error in the 2D self-position. Then we investigate the simplest transformation, i.e., the linear transformation, uncover its explicit algebraic and geometric structure as matrix Lie group of rotation, and explore the connection between the isotropic scaling condition and a special class of hexagon grid patterns. Finally, with our optimization-based approach, we manage to learn hexagon grid patterns that share similar properties of the grid cells in the rodent brain. The learned model is capable of accurate long distance path integration. Code is available at https://github.com/ruiqigao/grid-cell-path.

On Path Integration of Grid Cells: Group Representation and Isotropic Scaling
Ruiqi Gao ruiqigaoQucla. edu
Jianwen Xie
Xue-Xin Wei weixx@utexas.cm edu
jianvenducla, edu
Song-Chun Zhu 14,5
Ying Nian Wu! ywuêstat. ucla.cllleeeoooo edu
sczhuEstat. ucla. edu
"Cognitive Computing Lab, Baidu Research 3 Department of Neuroscience, UT Austin *Department Computer Science UCLA *Beijing Institute for General Artificial Intelligence (BIGAI)
Department of Statistics, UCLA
Abstract
Understanding how grid cells perform path integration calculations remains a fundamental problem. In this paper, we conduct theoretical analysis of general representation model path integration by grid cells, where the 2D self-position is encoded as a higher dimensional vector, and the 2D self-motion is represented by a general transformation of the vector. We identify two conditions on the transformation. One is group representation condition that is necessary for path integration. The other an isotropic scaling condition that ensures locally conformal embedding, so that the error in the vector representation translates conformally to the error in the 2D self-position. Then we investigate the simplest transformation, i.e., the linear transformation uncover its explicit algebraic and geometric structure as matrix Lie group of rotation, and explore the connection between the isotropic scaling condition and special class of hexagon grid patterns. Finally, with our optimization based approach, we manage to learn hexagon grid patterns that share similar properties of the grid cells in the rodent brain. The learned model capable accurate long distance path integration. Code is available htttt:::/ //gt::://tttttt com http//ruqqigoo/o/o prridceell-path
1
Introduction
Imagine walking in the darkness. Purely based on the sense of self-motion, one can gain sense of self- self-position by integrating the self motion a process often referred as path integration [10 14 [5] [27). While the exact neural underpinning of path integration remains unclear, it has been hypothesized that the grid cells 21
the mammalian medial entorhinal cortex The grid cells are named because individual neurons exhibit striking firing patterns that form hexagonal grids when agent (such as a rat) navigates in 2D open field [18] 21 [6]
24 23
12
(mEC) may be involved in this process 20
30
22
29 I. The grid cells also interact with the place cells in the hippocampus (28). Unlike a grid cell that fires at the vertices of a lattice, a place cell often fires at single (or few) locations.
6 34 5 7
The purpose of this paper understand how the grid cells may perform path integration calculations. We study a general optimization- based representational model in which the 2D self- self-position is
*The author is now a Research Scientist Google Brain team.
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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represented by higher dimensional vector and the 2D self- melf-motion is represented by transformation of the vector. The vector representation can be considered position encoding or position embedding. where the elements of the vector may be interpreted activities of population of grid cells. The transformation can be realized by recurrent network that acts on the vector. Our focus is to study the properties of the transformation.
Specifically, we identify two conditions for the transformation: group representation condition and an isotropic scaling condition, under which we demonstrate that the local neighborhood around each self-position in the 2D physical space embedded conformally as a 2D neighborhood around the vector representation of the self-position in the neural space.
We then investigate the simplest special case of the transformation, i.e., linear transformation, that forms matrix Lie group of rotation, under which case we show that the isotropic scaling condition connected special class of hexagonal grid patterns. Our numerical experiments demonstrate that our model learns clear hexagon grid patterns of multiple scales which share observed properties of the grid cells the rodent brain, by optimizing a simple loss function. The learned model is also capable of accurate long distance path integration.
Contributions. Our work contributes to understanding the grid cells from the perspective of represen- tation learning. We conduct theoretical analysis of (1) general transformation for path integration by identifying two key conditions and local conformal embedding property, (2) linear transformation by revealing the algebraic and geometric structure and connecting the isotropic scaling condition and special class of hexagon grid patterns, and (3) integration of linear transformation model and linear basis expansion model via unitary group representation theory. Experimentally we learn clear hexagon grid patterns that consistent with biological observations, and the learned model is capable of accurate path integration.
General transformation
2.1 Position embedding
Consider an agent (e-g., a rat) navigating within 2D open field. Let (2) be the self- position of the agent. We assume that the self- position in the 2D physical space is repre- sented by the response activities of a popula- tion of d neurons (e.g., d = 200), which form vector v(x) (vi(x),i i= I.....dT in the d- dimensional "neural space", with each element vz(x) representing the firing rate of one neuron when the animal is at location x.
sôr
r ④ + 8x)
+ dx
v(x)
x
(a) physical space
(b) neural space
Figure 1: The local 2D polar system around self- position æ in the 2D physical space (a) is embedded conformally as 2D polar system around vector v in the d-dimensional neural space (h), with scaling factor (so that ör in the physical space becomes sor in the neural space while angle 0 preserved).
(xx) can be called position encoding or posi- tion embedding. Collectively, (w(x), Vz)) forms codebook of x E RR2 and (v(ar), Vox) is 2D manifold in the d-dimensional neural space, i.e., globally we embed 2² as a 2D manifold in the neural space. Locally, we identify two condi- tions under which the 2D local neighborhood around each æ is embedded conformally as 2D neighborhood around v(x) with a scaling factor. See Fig. As shown in Section 3.3 the conformal embedding connected the hexagon grid patterns.
2.2 Transformation and path integration
At self-position æ, the agent makes a self-motion Az = (Av,Ax2)) then it moves to æ + Az. Correspondingly, the vector representation v(x) is transformed v( Ax). The general form of the transformation can be formulated as:
u(x + Ax) = ((u(x),Az)
(I)
The transformation F(,,Az) can be considered representation of Ar. which forms 2D additive group. We call Eq. [I] the transformation model. It can be implemented by recurrent network to

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derive path integration model: if the agent starts from æ0. and makes sequence of moves (Az,,7
1.... T), then the vector is updated by 2, F AI,,), where vn v (z)), and I....
2.3 Group representation condition
The solution to the transformation model (Eq. ①T should satisfy the following condition.
Condition 1. (Group representation condition) (w(x:), Var) and (F(;,a)) VAx) form represenia- tion of the 2D additive Euclidean group I2 in the sense that
F(v(x)),0) v(x). Vx:
F(v(x))AA, + Az2) = ((F(v())) Ar2), VarArr Ari, Arr
(3)
(F(,, VAz) is Lie group of transformations acting on the codebook manifold (v(a)),
The reason for (2) is that if Ax = 0, then ((,,0) should be the identity transformation. Thus the codebook manifold (v(z), Vaz) consists of fixed points of the transformation F(.:0). If F(.,0) is
furthermore contraction around (vxx)), Vz), then (v Vx) are the attractor points.
The reason for (3) is that the agent then move by Azz. Both paths would end up at the same a: Ar1 + Ar2, which represented by the same v( + Aæy + Amz2)
in step by Ari + Az2, first move by Aæ and
The group representation condition a necessary self-consistent condition for the transformation model (Eq.
2.4 Egocentric self-motion
Self-motion Aæ can also be parametrized egocentrically as ()r where Ar the displacement along the direction O E [0,2x| that Aæ = (Ax) cosos cos 6, Ary Arsin 0). The egocentric self-motion may be more biologically plausible where 0 is encoded by head direction, and Ar can be interpreted as the speed along direction O. The transformation model then becomes
( + Ar) (r, 0),
(4)
where we continue to use F(-) for the transformation (with slight abuse of notation). (Ar, 0) form polar coordinate system around æ.
2.5 Infinitesimal self-motion and directional derivative
In this subsection, we derive the transformation model for infinitesimal self-motion. While we use Ax or Ar denote finite mon-infinitesimal) self-motion, we use dæ ôr denote infinitesimal self-motion. At self-posseee self-position æ. for an infinitesimal displacement òr along direction 0, 8a = (ōxi = òrcos 0, 8x2 = Örsin 0). See Fig. (a) for an illustration. Given that òr is infinitesimal, for any fixed 0, a first order Taylor expansion of F(w(x), 0) with respect to ör gives us
v(x + 6ω) F F(v(=), 8r,
(,
=( + fr
(5)
where F(v(z),0,0) v(x) according Condition D and fo(v())) ('(((x) 0, is the first derivative (v(x)) Ar, 0) with respect to Ar at 0. fo (o()( is the directional derivative of F(-) at self-position æ and direction 0.
For fixed (F,,,Ar 0), Vdr) forms a me-aarameter Lie group of transformations, and fo )) the generator of its Lie algebra.
2.6 Isotropic scaling condition
With the directional derivative, define the second condition follows, which leads locally conformal embedding and is connected to hexagon grid pattern.
Condition 2. (Isotropic scaling condition) For any fixed [[)e(v()))]] is constant over 0.
Let (x)) denote fo (x())) for 0 = 0, and (x:)( denote fe(((()) for x/2. Then we have the following theorem:
3

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Theorem 1. Assume group representation condition and isotropic scaling condition 2 At any fixed x. for the local motion öx (Örcos 0 Sisi 0) around æ. let =v= v v(2) be the
change of vector and x ( (1)) then we have |ov|| = s s|| Moreover,
do - (
- fo(v(())) 0 + (x)2(((a))))rriii 0 (
(6)
where fo(v(z)) and fx/2(v(z)) are two orthogonal basis vectors of equal norm s.
See Supplementary for a proof and Fig. Tb) for an illustration. Theorem T Iindicates that the local 2D polar system around self-position x in the 2D physical space embedded conformally as a 2D polar system around vector (x)) in the d-dimensional neural space, with scaling factor (our analysis is local for any fixed æz and s may depend on æ). Conformal embedding is a generalization of isometric embedding, where the metric can be changed by a scaling factor s. If s is globally constant for all æ, then the intrinsic geometry of the codebook manifold (v( Vz)) remains Euclidean, i.e., flat.
Why isotropic scaling and conformal embedding? The neurons are intrinsically noisy. During path integration, the errors may accumulate in u. Moreover, when inferring seff-position from visual image, it is possible that v is inferred first with error, and then æ is decoded from the inferred v. Due to isotropic scaling and conformal embedding, locally that the la error in v translates proportionally to the & error in æ, so that there will not be adversarial perturbations in (2)) that cause excessively big errors in æ. Specifically, we have the following theorem.
have [|8v|] s||sz||| which guarantees
Theorem 2. Assume the general transformation model (Eq. 4 and the isotropic scaling condition. For any fixed æ. let s s= fl (u(x))), which is independent of o. Suppose the neurons are noisy: v = v(x) + E, where E ~ (V (0, z Ia) and d the dimensionality of v. Suppose the agent infers its 2D position & from v by
arg min | - =0aa')1? )12, i.e,, v(+) the projection of o onto the 2D manifold formed by (v(a'), Ya'). Then we have
BI x 2= =2721 2
See Supplementary for proof.
Connection to continuous attractor neural network (CANN) defined on 2D torus. The group representation condition and the isotropic scaling condition appear be satisfied by the CANN that are typically hand-designed on 2D torus. See Supplementary for details.
models 2 6 29
3 Linear transformation
After studying the general transformation, we now investigate the linear transformation of v(z), for the following reasons. (1) It is the simplest transformation for which we can derive explicit algebraic and geometric results. (2) It enables to connect the isotropic scaling condition to special class of
hexagon grid patterns. (3) In Section 4 we integrate it with the basis expansion model, which is also linear in vxx), via unitary group representation theory.
For finite (non-ifiiniteimaa) self-motion, the linear transformation model is:
(8) where M (Ax) is a matrix. The group representation condition becomes M (Ax) + (x)) = M (Aæ²) M (Az) (Ax))<(x)) i.e., M (Ax) is matrix representation of self-motion Aæ, and M (Aæ) acts on the coding manifold (w( Væ)).. For egocentric parametrization of self-motion (Ar, 0). we can further write (Ax) Me Me(A) Ar) for Az = (Arcos 9 Arsin 0), and the linear model becomes v + Ax) = F(u(x) Ar, 8) = Ma (Ar))u(z))
+xx+Ax) = F(v(x), Ar) (Az)u(z))
3.1 Algebraic structure: matrix Lie algebra and Lie group
For the linear model (Eq. g ). the directional derivative is: fe(v(x)) = F'(o(z),0,0) =
M2 (0)v(z)== (z)) where B(0)= (O), which is the derivative of (A with respect to Ar at O. For infinitesimal self-motion, the transformation model in Eq. (5) becomes
va 6z) = + B (r))
where I is the identity matrix. It can be considered linear recurrent network where B()) is the learnable weight matrix. We have the following theorem for the algebraic structure of the linear transformation.

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Theorem 3. Assume the linear transformation model so that for infinitesimal self-motion (Ör, 0). the model is in the form of Eq. D. then for finite displacement år,
v + Aa) - (oo((a))(())=cxx((((())))))))
(10)
Proof. We can divide Ar into N steps, so that òr = Ar/N + 0 N and
v( + Az) = [I B(0) + (1 u(() exp ( (((O)))j(()))
(11)
as N co. The matrix exponential map is defined by exp(A)
The above math underlies the relationship between matrix Lie algebra and matrix Lie group in general [38]. For fixed 0, the set of My exp( Ar) for Ar E R forms a matrix Lie group, which is both group and manifold. The tangent space of Mg (Ar) at identity I is called matrix Lie algebra. (0) is the basis of this tangent space, and is often referred to as the generator matrix.
Path integration. If the agent starts from zo, and make sequence of moves ((Ar, 0,), I....T).. then the vector representation of self- self-position is updated by
1 exp (x))(()))Ar)))...
(12)
where vo v(zo), and I.....T.
Approximation exponential map. For a finite but small Ar, exp(B(0)Ar) can be approximated by a second-order (or higher-order) Taylor expansion
exp(B())Ar)= = I+ B(0)Ar+ B(0))///2+0(AF)
(13)
3.2 Geometric structure: rotation, periodicity, metic and error correction
If we assume B0) -B)),, i.e.. skew-symmetric, then I + 8(0)8r in Eq. 9 is rotation matrix operating on v vxx), due to the fact that (I+ ((0)rr) B(0)rr) = =1+0(6 For finite Ar, exp B (0))r) is also rotation matrix,
equals to the product of N matrices I+ (rr))) (Eq. I1) ). The geometric interpretation is that, if the agent moves along the direction θ in the physical space, the vector vxx) rotated by the matrix (0) in the neural space, while the & norm [[=()))) remains fixed. We may interpret [[<[=)] Ed va(az)2 as the total energy of grid cells.
See Fig. (b).
The angle of rotation is given by [[())o( (0r( because [B)))(())) is the are length and [[)( is the radius. If we further assume the isotropic scaling condition, which becomes that ()) is constant over for the linear model, then the angle of rotation can be written as rr, where μ [ 0 yo (()) [elll/////e (==((( is independent of 0. Geometrically, μ tells us how fast the vector rotates in the neural space as the agent moves in the physical space. In practice, μ
(
can be much bigger than 1 for the learned model, thus the vector can rotate back to itself in short distance, causing the periodic patterns the elements of v(x). μl captures the notion of metric.
For μ >> 1, the conformal embedding in Fig. (b) magnifies the local motion in Fig. (a), and this enables error correction 34 More specifically, ly, we have the following result, which is based on Theorem
Proposition 1. Assume the linear transformation model (Eq. (9) and the isotropic scaling condition For any fixed x. let μ = B (O))(())( (()))///)))))( Suppose v(xx) + where E NO, (0, ra and 2
==<2(11 (1)((() (12/)////)) /d), so that o² measures the variance noise relative the average magnitude of (vi(x)2,i= i= d.....d. Suppose the agent infers its 2D position & from by & arg argming! (ee-v(x'))
Then we have
E a - =22/==20
(14)
See Supplementary for proof. By the above proposition, error correction of grid cells is due to two factors: (1) higher dimensionality d of v(x)) for encoding positions æ. and (2) a magnifying >>>>> 1 (our analysis is local for any fixed æ, and may depend æ).
5

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3.3 Hexagon grid patterns formed by mixing Fourier waves
In this subsection, we make connection between the isotropic scaling condition 2 and a special class of hexagon grid patterns created by linearly mixing three Fourier plane waves whose directions are 2x/3 apart. We show such linear mixing satisfies the linear transformation model and the isotropic scaling condition.
Let e(æ)) - (e)) equal norm, and the angle befween every puir of them is 2T 3. Let m(ar)= Ue(æ), where U is an arbitrary unitary matrix. Let (0) = U (*)(O)U, where D = diag(i (((((((a.)()))))))))))))) j=) 1.2.3), with
Theorem 4.
where (aj.j = 1,2,3) are three 2D vectors of
q(0) = (cos 0 sin iin Then (v (.) B))) satisfies the linear transformation model (Eq. and
the isotropic scaling condition Moreover, B(0) is skew-symmetric.
See Supplementary for proof. We would like to emphasize that the above theorem analyzes special case solution to our linear transformation model, but our optimization-base based learning method does not assume any superposition of Fourier basis functions as in the theorem. Our experimental results are learned purely by optimizing a loss function based on the simple assumptions of our model with generic vectors and matrices.
We leave it to future work to theoretically prove that the isotropic scaling condition leads to hexagon grid patterns in either the general transformation model the linear transformation model. The hexagon grid patterns are not limited to superpositions of three plane waves as in the above theorem.
3.4 Modules
Biologically, it is well established that grid cells are organized in discrete modules 4 37 or blocks.
We thus partition the vector (ax) into blocks, v(x) = (= I,..,,, Correspondingly the generator matrices B(Ø) diag( B₁ (0), k = I.....K K) are block diagonal, so that each sub-vector
(x) is rotated by sub-matrix (0). For the general transformation model, each sub-vector
is transformed by separate sub-network. By the same argument as in Section 3.2 let 14
(2))(())) (z/ [>/))) then μl is the metric of module
4 Interaction with place cells
4.1 Place cells
For each vzx), we need to uniquely decode x globally. This can be accomplished via interaction with place cells. Specifically, each place cell fires when the agent at a specific position. Let A(x,z') be the response map of the place cell associated with position x'. It measures the adjacency between x and x'. A commonly used form of A(x, x') is the Gaussian adjacency kernel A A(x x') = exp(- ||ar - x' I12
==1l2/(202)) /(262)). The set of Gaussian adjacency kernels serve as inputs to our optimization-
based method to learn grid cells.
4.2 Basis expansion
A popular model that connects place cells and grid cells is the following basis expansion model (or PCA-based
A(x, x')
model) 13:
A(x,x') (v(x), (o(x),u(x'))
Σ " Mi vi(x),
(15)
Ui
where v(x) = (v/(2), I..... d) and u(xx) ii=
Vy(x
d,...,d)T. T. Here (vi()), I, d) forms a of d basis functions (which are functions of x) for expanding A(x, (which function of æ for each place 2'), while uxx') is the read-out weight vector for place cell at x' and needs to learned. See Fig. 2 for an illustration. Experimental results on biological brains have shown that the connections from grid cells to place cells are excitatory 42 31. We thus assume that u 2 0 for all i and x'.
Figure 2: Illustration of basis expansion model A(æ,a²) | (;()), where v((z) is the response map of i-th grid cell, shown the bottom, which shows 5 dif- ferent A(x, x') ) is the response map of place cell associated with æ', shown at the top, which shows different x'. "e is the connection weight.

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4.3 From group representation to basis functions
The vector representation v(z) generated constrained) by the linear transformation model (Eq. (8) can serve as basis functions of the PCA-based basis expansion model (Eq. (15), due to the tundamental theorems of Schur [4T] and Peter-Weyy Weyl [38] which reveal the deep root of Fourier analysis and generalize it to general Lie groups. Specifically, if (Ax) is an irreducible unitary representation of Az that forms a compact Lie group, then the elements (M)(Az) form set of orthogonal basis functions of Ax. Let ( = (x)vv(()) (where we choose the origin O as the reference point). The elements of v(x). i.e., (vi(=), ,....d,, are linear mixings of the basis functions (M;( J. so that they themselves form new set of basis functions that serve to expand (A(x, x'), Va') that parametrizes the place cells. Thus group representation our path integration model is perfect match to the basis expansion model, in the sense that the basis functions are results of group representation.
The basis expansion model (or PCA-based model) (Eq. 15 assumee that the basis functions are orthogonal, whereas our work, we do not make the orthogonality assumption. Interest- ingly, the learned transformation model generates basis functions that are close to being orthogonal automatically. See Supplementary for more detailed explanation and experimental results.
4.4 Decoding and re-encoding
For a neural response vector v, such as v, in Eq. (12), the response of the place cell associated with location x' is (v, (v,")()))) We can decode the position & by examining which place cell has the maximal response, i.e.,
à = arg max u(x')).
(16)
After decoding a, we can re-eneode v + v(ē) for error correction. Decoding and re-encoding can also be done by directly projecting onto the manifold (v(z), Vx)), which gives similar results. See Supplementary for more analysis and experimental results.
Learning
We learn the model by optimizing loss function defined based on three model assumptions discussed
above: (1) the basis expansion model (Eq. (15)). (2) the linear transformation model (Eq. (10) and
(3) the isotropic scaling condition 2 The input is the of adjacency kernels A A(x, x²), Yx, x The
unknown parameters to be learned are (1) (v(x) ) = ( k=
l..k=,1.... I.....K K). Vx), (2) (u(x') (())), Vx') and K modules or blocks and B(0) is skew-symmetric, so
(3) (B(0), VO). We assume that there
I..... K). VO) and only the lower triangle parts of the matrices need to be learned. The loss function defined as a weighted sum of simple & loss terms constraining the three model assumptions: L = Lo + L + 22L2, where
that ((0) are parametrized as block-diagona matrices (Bx
Lo Eg [A(x, x') (m(x)), (=xx))())))))))))))))) ) (basis expansion)
(17)
L
E. Be 11 Ar) Bx (B) (transformation)
(18)
L2
Σ En, 0.40 (e 1=1
(O) (isotropic scaling)
(19)
In L1, Ax (Arcos A, 0). and d2 are chosen so that the three loss terms are of similar magnitudes. A A(xr, a.)) are given Gaussian adjacency kernels. For regularization, we add penalty on [lu(x')) [|2, and further assume u(x') > 0 so that the connections from grid cells to place cells are
excitatory 42 3T]. However, note that uxx') ) 2 0 is not necessary for the emergence of hexagon grid patterns as shown in the ablation studies.
Expectations in Lo, Lı and L2 approximated by Monte Carlo samples. is minimized by Adam (5 optimizer. See Supplementary for implementation details.
It is worth noting that, consistent with the experimental observations, we assume individual place field A(x, x²) to exhibit Gaussian shape, rather than a Mexican-hal pattern (with balanced excitatory center and inhibitory surround) as assumed in previous basis expansion models [1] 33 of grid cells.
7

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ReLU non-linearity. We also experiment with a non- non-linear transformation model where ReLU
activation is added. See Supplementary for details.
Experiments
OO
C
O
D
00
6.3
100
000
b00 00
80
o
5 8
50 000
o bo
0.0 00
O
Figure 3: Hexagonal grid firing patterns emerge in the learned network. Every response map shows the firing
pattern one neuron (i.c, one element of v) the 2D environment. Every row shows the firing patterns of neurons within the same block or modulc.
We conduct numerical experiments learn the representations described in Section 5 Specifically, we use a square environment with size 1m x Im, which is discretized into a 40 x 40 lattice. For direction, we discretize the circle [0, into 144 directions and use nearest neighbor linear interpola- tions for values in between. We use the second-order Taylor expansion (Eq. (13) to approximate the exponential map exp( (()(B)))) Ar). The displacement Ar sampled within small range, i.e., Ar is smaller than 3 grids on the lattice. For A Axx,a', a'), we use a Gaussian adjacency kernel with o 0.07. v (x) is of d 192 dimensions, which partitioned into K == 16 modules, each of which has 12 cells.
6.1 Hexagon grid patterns
Fig. shows the learned firing patterns of v(x) (v,(*), I....,d over the x 40 lattice of æ. Every shows the learned units belonging to the same block module. Regular hexagon grid patterns emerge. Within each block or module, the scales and orientations are roughly the same, but with different phases spatial shifts. For the learned B B(0), each element shows regular sine/cosine tuning over 0. See Supplementary for more learned patterns.
Table 1: Summary of gridness scores of the patterns learned from different models. To determine valid grid cells, we apply the same threshold of gridness score as ( i.c., gridness score > 0.37. For our model, we run 5 trials and report the average and standard deviation.
Probab density
Model
Gridness score (1)
% of grid cells
0.35
0.40
Grid scale (m)
3 (LSTM)
0.18
25.20
Figure 4: Multi-moda distribution of grid
33 (RNN)
0.48
56.10
scales of the learned model grid cells. The scale ratios closely match real data 37
Ours
0.90 + 0.044
73.10 +: 1.33
We further investigate the characteristics of the learned firing patterns of v(x) using measures adopted from the literature of grid cells. Specifically, the hexagonal regularity, scale and orientation of grid-like patterns are quantified using the gridness score, grid scale and grid orientation [26] 32), which are determined by taking a circular sample of the autocorrelogram of the response map. Table I summarizes the results of gridness scores and comparisons with other optimization-base approaches 3 33 We apply the same threshold to determine whether learned neuron can be considered a grid cell as (i.e., gridness score 0.37). For our model, 73.10% of the learned neurons exhibit significant hexagonal periodicity in terms of the gridness score. Fig. shows the

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histogram of grid scales of the learned grid cell neurons (mean 0.33, range 0.21 to 0.49), which follows multi-modal distribution. The ratio between neighboring modes are roughly 1.52 and 1.51, which closely matches the theoretical predictions [3 36 and also the empirical results from rodent grid cells [7.. Collectively, these results reveal striking, quantitative correspondence between the properties of our model neurons and those of the grid cells in the brain.
Connection to continuous attractor neural network (CANN) defined on 2D torus. The fact that the learned response maps of each module are shifted versions of a common hexagon periodic pattern implies that the learned codebook manifold forms a 2D torus, and as the agent moves, the responses of the grid cells undergo cyclic permutation. This is consistent with the CANN models hand-crafee on 2D torus. See Supplementary for detailed discussion.
Ablation studies. We conduct ablation studies to examine whether certain model assumptions are empirically important for the emergence of hexagon grid patterns. The conclusions are highlighted as follows: (1) The loss term L2 (Eq. (19) constraining the isotropic scaling condition is necessary for learning hexagon grid patterns. (2) The constraint uxx) 0 is not necessary for learning hexagon patterns, but the activations can be either excitatory or inhibitory without the constraint. (3) The skew-symmetric assumption on (()) is not important for learning hexagon grid pattern. (4) Hexagon patterns always emerge regardless of the choice of block size and number of blocks. (5) Multiple blocks or modules are necessary for the emergence of hexagon grid patterns of multiple scales. See Fig. 5 for several learned patterns and Supplementary for the full studies.
Do
(a) without L
(b) without u u(x') 2 0
(c) without skew sem-syymetry
Figure 5: Learned response maps in ablation studies where a certain model assumption is removed. (a) Remove the loss term L2- (b) Remove the assumption u(x') 2 0. (c) Remove the
assumption on B(0).
6.2 Path integration
We then examine the ability of the learned model on performing multi-step path integration, which can be accomplished by recurrently updating v, (Eq. (12) and decoding v, to H; for r = 1...., (Eq. (6)). Re-encoding 2 < v(x,) after decoding is adopted. Fig. Gia) shows an example trajectory of accurate path integration for number of time steps T = 30. As shown in Fig. k(b), with re-encoding, the path integration error remains close to zero over a duration of 500 time steps (< 0.01 cm, averaged over 1,000 episodes), even if the model is trained with the single-time-stey transformation model (Eq. (18). Without re-encoding, the error goes slight higher but still remains small (ranging from 0.0 to 4.2 cm, mean 1.9 em in the Im x Im environment). Fig. ( summarizes the path integration performance by fixing the number of blocks and altering the block size. The performance of path integration would be improved the block size becomes larger, i.e., with more neurons in each module. When block size is larger than 16, path integration is very accurate for the time steps tested.
Error correction. See Supplementary for numerical experiments error correction, which show that the learn model is still capable of path integration when we apply Gaussian white noise errors or Bernoulli drop-out errors to t2.
6.3 Additional experiments on path planning and egocentric vision
We also conduct additional experiments on path planning and egocentric vision with our model. Path planning can be accomplished by steepest ascent on the adjacency to the target position. For egocentric vision, we learn an extra generator network that generates the visual image given the position encoding formed by the grid cells. See Supplementary for details.

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Figure 6: The learned model can perform accurate path integration. (a) Black: example trajectory. Red: inferred irajectory. (h) Path integration error over numher of time steps, for procedures with re-encoding and without re-encoding. (c) Path integration error with fixed number of blocks and different black sizes, for and 100 time steps. The eroor band in (h) and error bar in (c) are standard deviations computed over 1,000 episodes.
7 Related work
Our work is related to several lines of previous research modeling grid cells. First, RNN models have been used to model grid cells and path integration. The traditional approach uses simulation-basee models with hand-crafted connectivity, known continuous attractor neural network (CANN) 6 6 7 29, M. On the other hand, more recently two pioneering papers [ 3 developed optimization-
RNN approaches learn the path integration model and discovered that grid-like response patterns can emerge in the optimized networks. These results are further substantiated in 33 8. Our work analyzes the properties of the general recurrent model for path integration, and these properties seem to be satisfied by the hand-crafted CANN models. Our method belongs to the scheme optimization- based approaches, and the learned response maps share similar properties assumed by the CANN models.
in that, unlike PCA. we make no assumption about the orthogonality between the basis functions, and the basis functions are generated by the transformation model. Furthermore, in previous basis expansion models [13 33).
Second, our work differs from the PCA-based basis expansion models 13 33 35
place fields with Mexican-hat patterns (with balanced excitatory center and inhibitory surround) had be assumed in order to obtain hexagonal grid firing patterns. However, experimentally measured place fields in biological brains were instead well characterized by Gaussian functions. Crucially, in our model, hexagonal grids emerge from learning with Gaussian place fields, and there is no need to assume any additional surround mechanisms or difference of Gaussians kernels.
In another related paper, [19] proposed matrix representation of 2D self-motion while work analyzes general transformations. Our investigation the special case of linear transformation model reveals the matrix Lie group and the matrix Lie algebra of rotation group. Our work also connects the linear transformation model to the basis expansion model via unitary group representation theory.
8 Conclusion
This paper analyzes the recurrent model for path integration calculations by grid cells. We identify group representation condition and an isotropic scaling condition that give rise to locally conformal embedding of the self-motion. We study linear prototype model that reveals the matrix Lie group of rotation, and explore the connection between the isotropic scaling condition and hexagon grid patterns. In addition these theoretical investigations, our numerical experiments demonstrate that our model can learn hexagon grid patterns for the response maps of grid cells, and the learned model is capable of accurate long distance path integration.
In this work, the numerical experiments are mostly limited to the linear transformation model, with the exception of an experiment with ReLU non-linearity. We will conduct experiments on the other non-linear transformation models, especially the forms assumed by the hand-crafted continuous attractor neural networks. Moreover, we assume that the agent navigates within square open-field environment without obstacles or rewards. is worthwhile to explore more complicated environments, including 3D environment.
10

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Acknowledgments and Disclosure of Funding
The work was supported by NSF DMS-2015577, ONR MURI project N00014-16-1-2007 DARPA XAI project N66001-11-2-4022, and XSEDE grant ASC170063. We thank Yaxuan Zhu from UCLA Department of Statistics for his help with experiments on egocentric vision. We thank Dr. Wenhao Zhang for sharing his knowledge and insights on continuous attractor neural networks. We thank Sirui Xie for discussions. We thank the three reviewers for their constructive comments.
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13


Reviews:
[{"review_id": "wfBGhszfU4H", "reviewer": "Reviewer_Lump", "summary": "The paper introduces a general framework for modeling 2D navigation via grid cells. The main idea is to embed a 2D position x into a higher-dimensional positional encoding v(x) and model 2D translations via group transformations in this embedding space. A specific case are linear transformations via rotation matrices that act on the embedding v(x). Under this assumption, it can be shown that hexagonal patterns which are characteristic for neural grid cells naturally emerge as an optimal encoding. ", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "1. Presentation: According to the submission history, the ICML AC had concerns about the exposition and presentation of the paper. The presentation of the paper in its current form is very clear. I find the structure of the methods part adequate (Sec. 2 - Sec. 5) and easy to follow. \n\n2. Quality of theoretical contribution/mathematical validity: Overall, the theory sections are mathematically sound and the stated assumptions are reasonable (l. 82, l. 107), save for a number of minor flaws. It should be straightforward to address these issues in a minor revision:\n- The defined set of transformations F are intimately related to the standard mathematic concept of a \"flow\". Curiously, the authors do not mention this connection. This would also provide a stronger justification for Condition 1 (l. 82).\n- Why does the angle Phi not depend on positional differential dx. In Eq. (5), the expansion of dv would envolve both dr and dPhi. If this is an implicit requirement, it should be stated as an explicit assumption.\n- What is the motivation for restricting the approach to 2D environments? One could easily conceive of a formulation for more general Euclidean domains, like R^3. Most of the theory should be analogous. \n\n3. Results:\n- It is fascinating that the hexagonal patterns emerge from the optimization framework. It's also a strong justification for the validity of the proposed method and assumptions. Furthermore, this effect is validated quantitatively in Table 1.\n- It is a little disappointing that all evaluations focus on linear transformation models. One of the motivations of the proposed framework is that it allows for more general group transformations. This should be shown empirically as a proof of concept.\n- The path integration experiments lack baseline comparisons which makes it hard to put these results into context. I would like to see comparisons to recurrent [3] and PCA based [33] models in Figure 6.\n- The presented ablation study (l. 289) is useful to assess under which circumstances the hexagonal patterns emerge.\n\nMinor remarks:\n- I believe that the condition of F(v, 0, Phi) = v (l. 101) is central to the proposed framework and should be stated as a part of \"Condition 1\" (l. 82).\n- The embeddings v(x) lie on the Lie group manifold characterized by rotation matrices. This means that they are contained in an elliptic space with positive curvature. This seems counterintuitive, since the goal is to model 2D trajectory prediction in Euclidean space. E.g. there might be group transformations, such that 3 consecutive steps with 90 degree turns between them yield a closed loop. Is this a phenomenon you encountered or did you find that this effect is relevant in practice?", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "There is a small discussion on limitations in Sec. 8 but no dedicated section or paragraph.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3h."}, "extra_text": {}}, {"review_id": "qKBt0NMuJiL", "reviewer": "Reviewer_G4HX", "summary": "This paper focuses on the find that how the grid cells perform path integration. Speciafically, this paper introduces the isotropic scaling condition to produce the hexagon firing pattern for grid cells. The optimization experiments show that the proposed formulation of place and grid cells demonstrated this.", "questions": "", "limitations": "", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "Although I appreciated that the authors provide an interesting formulation for the relationship of place and grid cells, I have the following concerns:\n\n1.\tThe authors claim that they don’t make any orthogonality assumption and the isotropic scaling guarantees the l_2 error in L122. However, in the proof of Theorem 3 of this error bound, explicit orthogonal assumptions are made. I expect the authors to provide, in the rebuttal, a more rigorous proof for this error bound without any orthogonal assumptions.\n\n2.\tMany literatures have shown that cos/sin functions can produce hexagon patterns, e.g., [a, b]. Can you provide more connections to those results, because I highly suspect that the three conditions introduced in this paper will result in the cosine functions (Maybe the formulation (e.g., the isotropic scaling) in this paper is mathematically equivalent to the cosine functions?), especially after seeing the Fig 7 in Appendix. If this conjecture is correct, then I see no contribution for this paper, since we can simply use the predefined functions as in [a] to do both path integration and generate hexagon. Then what’s the purpose for the optimization?\n3.\tThe authors didn’t show that the linear model with isotropic condition has a hexagon grid pattern in the proof, thus the tittle of Sec 3.3 is misleading, “hexagon grid pattern” in the section title should be removed. \n4.\tTheoretic guarantee for generating the hexagon pattern by the introduced formulation will strengthen this paper a lot.\n5.\tIt’s not clear how to derive the third item from the second item in Eq (9).\n\n[a] Burgess N, Barry C, O'keefe J. An oscillatory interference model of grid cell firing. Hippocampus. 2007 Sep;17(9):801-12.\n\n[b] Fuhs MC, Touretzky DS. A spin glass model of path integration in rat medial entorhinal cortex. Journal of Neuroscience. 2006 Apr 19;26(16):4266-76.\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "It seems that the only contribution is to introduce the isotropic scaling to generate the hexagon pattern during path integration. However, why should we do this? What ‘s the necessity to connect hexagon pattern and path integration via optimization, if predefined cosine functions can generate hexagon pattern very well?", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "6"}, "extra_text": {}}, {"review_id": "U0zc4Lwoc30", "reviewer": "Reviewer_Wn1K", "summary": "The paper derives a model of grid cells from first principles, starting with the basic assumption that position is represented as a high-D vector and that it is updated as a function of direction and speed.  The authors stipulate an isotropic scaling condition in order to make the representation robust to neural noise.  The elements of the resulting high-D embedding resemble grid cells, and importantly, the ability for error correction during path integration.\n", "questions": "", "limitations": "", "rating": 10, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "I find this to be a super interesting and exciting paper.  There are by now many papers that claim to show how grid cells emerge as a result of some kind of learning rule, but in my view this is the most insightful analysis I've seen to date.  It is a sensible and mathematically grounded approach.  Although they show how the grid cells emerge from learning, the resulting solution corroborates the theory - and so we understand why this is happening.  An important part of the story is robustness to noise, which is clearly linked to the isotropic condition and hence hexagonal grids.  This is high impact work that I believe will of interest to both the NeurIPS and computational neuroscience communities.\n\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "yes", "review_text": "", "extra_scores": {"ethical_concerns": "none", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1.5 hour"}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers are overall positive. The presented approach is mathematically sound and shows that grid cells can emerge from an optimization framework with minimal assumptions. The main criticism raised is whether the approach is useful beyond already existing models. The authors should clarify potential applications more explicitly in the revision, as well as include the additional experiments they cited in the rebuttal.

Decision: Accept (Poster)