Learning Place Cell Representations and Context-Dependent Remapping
Authors: Markus Pettersen
Frederik Rogge
Mikkel Elle Lepperød
Venue: NeurIPS 2024 poster (2024)
Keywords: Place cells
remapping
AI
neuroAI
Primary area: neuroscience_and_cognitive_science
TL;DR: We train neural networks to minimize a similarity-based objective function to learn joint encodings of space and context, and observe place cell-like representations and remapping in network responses

Abstract: Hippocampal place cells are known for their spatially selective firing patterns, which has led to the suggestion that they encode an animal's location. However, place cells also respond to contextual cues, such as smell. Furthermore, they have the ability to remap, wherein the firing fields and rates of cells change in response to changes in the environment. How place cell responses emerge, and how these representations remap is not fully understood. In this work, we propose a similarity-based objective function that translates proximity in space, to proximity in representation. We show that a neural network trained to minimize the proposed objective learns place-like representations. We also show that the proposed objective is easily extended to include other sources of information, such as context information, in the same way. When trained to encode multiple contexts, networks learn distinct representations, exhibiting remapping behaviors between contexts. The proposed objective is invariant to orthogonal transformations. Such transformations of the original trained representation (e.g. rotations), therefore yield new representations distinct from the original, without explicit relearning, akin to remapping. Our findings shed new light on the formation and encoding properties of place cells, and also demonstrate an interesting case of representational reuse.

Learning Place Cell Representations and Context-Dependent Remapping R
Markus Pettersen
Department of Numerical Analysis and Scientific Computing
Simula Research Laboratory
Oslo, Kristian Augusts Gate 23
prrkuus@@simule
Frederik Rogge
Department of Biosciences University of Oslo Oslo, Blinderveien 31 frederik. uio. no
Mikkel Elle Lepperød
Department of Numerical Analysis and Scientific Computing
Simula Research Laboratory
Oslo, Kristian Augusts Gate 23 mikkel mikkel@simul... no
Abstract
Hippocampal place cells are known for their spatially selective firing patterns, which has led to the suggestion that they encode animal's location. However, place cells also respond to contextual cues, such as smell. Furthermore, they have the ability to remap, wherein the firing fields and rates of cells change in response to changes in the environment. How place cell responses emerge, and how these representations remap is not fully understood. In this work, propose similarity-basee objective function that translates proximity in space, to proximity in representation. We show that neural network trained to minimize the proposed objective learns place-like representations. We also show that the proposed ob- jective is easily extended include other sources of information, such context information, in the same way. When trained to encode multiple contexts, networks learn distinct representations, exhibiting remapping behaviors between contexts. The proposed objective is invariant to orthogonal transformations. Such trans- formations of the original trained representation (e.g. rotations), therefore yield new representations distinct from the original, without explicit relearning, akin to remapping. Our findings shed new light on the formation and encoding properties of place cells, and also demonstrate an interesting case of representational reuse.
1 Introduction
Animals and humans are capable of extraordinary feats of navigation, from birds migrating by following magnetic fields [Packmor al. [021]. rodents navigating mazes (O KeeKe 1976 [mmall 1901] and cab drivers memorizing and traversing nearly 26000 busy London streets Fernandez Velasco and Spiers 2023 In the brain, navigation ability is believed be supported by Hippocampal place cells Keefe and Dostrovsky 1971 O D'Keee and Nadel 1978). Place cells are known for
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

---

their tendency to only fire at one, or few locations within a recording environment Park et al. 201T]. correlating with the position of the animal. Besides encoding allocentric location, place cells also respond to contextual cues, such as room identity, room geometry. odors or colors [Latuske et al. 2018 O' Keefe and Burgess] 1996] Leutgeb et al. 2005b] JJeffery] 201 In other words, place cells form conjunctive representations, merging spatial information with available contextual cues. It is also believed that place cells distinguish contextual information trough so-called remapping. In response to large changes in the environment, place cell responses modulate, and spatial representations can
become uncorrelated across contexts [Muller and Kubie 1987 Leutgeb et al. 2004 2005b].
How place cells obtain their striking behaviors remains matter of debate. Some argue that place cells inherit their firing fields from upstream cell types, such as grid cells [Hafting et al. [005 [Sosstad] et al.) [006 Jeffery (20TT] or border cells [Barry et al.] [006] Pettersen et al. [2024] or that place cell representations in different environments form distinct attractor states [Jeffery] 2011 Samsonovich and McNaughton, 1997]. However, exactly how place-like like representations emerge, and how they can be learned, remains poorly understood. recent related work, however, a range of normative models have demonstrated that neural networks trained to solve simple navigation tasks actually learn representations similar to biological spatial cells [Cueva and Wei 2018 Banino et al.
2018 Uria et al. 2022 Schaeffer
2020 Sorscher et al. 2022, Whittington et al. 2020 Xu et al. et al. 2023]. However, these models often feature complicated architectures and a range of different regularization strategies, making it difficult to discern why the observed representations actually arise. Furthermore, some of these works tend to focus on learning grid cell-like representations, placing less emphasis other emergent cell types, such as place cells.
2022 Dorrell et al.
In this work, we take inspiration from existing machine learning models (Cueva and Wei, 2018 Banino et al.] 2018 [Sorscher et al. 2022 Xu et al., [2022 Dorrell et al.) 2022 and propose a minimal, similarity- bbased objective. When a feedforward network is trained to minimize this objective, we find that it learns place-like spatial representations. We further show that the objective is easily extended to encompass joint representations of space and context. When trained in joint setting across multiple contexts, we find that the network learns uncorrelated representations when comparing different contexts, similar to Hippocampal global remapping. We further train a recurrent neural network to solve the same task, and find that band cell-like (Krupic et al.) 2012] representations emerge alongside the place code, showing that other cell types may be involved in path integration, and also that the objective extends to more naturalistic settings. Lastly, orthogonal transformations to learned spatial representations in order to generate new spatial maps while preserving the similarity structure. Thus, the proposed objective allows for switching between different maps without having to relearn them, offering an interesting perspective on Hippocampal remapping.
show that can apply
2 Results & discussion
2.1 A similarity-based objective for learning representations of space and context
We consider the problem of learning an encoding of some region of space (e.g. a square recording enclosure). We follow the example set by recent normative models (Dorrell et al.] [022]] Schaeffer et al. 2023), and argue that biologically plausible spatial representations can be obtained directly by specifying the properties of the network population vector. Considering properties that a spatial representation should have, we propose an objective where we demand that: 1) Points that are close in physical space should be represented by similar population vectors. 2) Points that are distant in physical space should be represented by dissimilar population vectors. 3) In the open field, point or direction is special, so the above propertics should be rotation- and translation invariant in space. 4) Unit activations are bounded. 5) Unit activations are non-negative.
To investigate representations with these properties, we train neural networks minimize spatial encoding objective. Consider a neural network with population vector p(z:), where each component ps (2.). = 1, 2, 3..,. np is the firing rate particular (output) unit of the network at a particular location, x, (where indexes e.g. time along a trajectory or sampled spatial locations). 2,, on the other hand, is the network's latent position estimate corresponding to x,- We impose non-negativity in the architecture of the neural network by selecting appropriate activation functions. With these

---

prerequisites in mind, we explore the following objective function
2 C Er [B+ (1 (1- B)e 2 x, - -e- + + plz()))
(I)
where |-| denotes the L2-norm, while / is a lower bound on the target similarity, a is a hyperparameter that controls the scale of the learned similarity structure, and a hyperparameter governing an L.2 activity regularization term.
Intuitively, compares the (Gaussian) similarity of two points x; and X and asserts that the corresponding population vectors at those points (p(z+) and p(zz)) )) should exhibit the same similarity, In other words: Points that are close in physical space should be represented similarly, while distant points in physical space should be represented using dissimilar population vectors. This idea is illustrated in Fig. [Tp). 3, on the other hand. determines the similarity scale at which vectors are deemed dissimilar. This borrows from the concept of vectors being nearly orthogonal in hyperdimensional computing [Kanerva 2009]. which exploits that (random) vectors in higher dimensions tend to lie some intermediate distance from other vectors. For the higher-dimensions population vectors of neural networks, dissimilarity may therefore be meaningfully defined in terms of some intermediate, rather than zero similarity (and maximal population vector separation). See App. for more details on the influence of 3 on learned representations.
We also find that the objective in (T) is just special case of a more general encoding objective, where distinct sources of nonspatial information illustrates how contextual signals can be represented in similar manner to the spatial case. However, place cells do not encode contextual information exclusively, but rather respond to particular locations and contexts in conjunction. Again, the similarity objective can be extended to accomodate this. If we represent context information the simplest manner, as a scalar signal c, we can have the neural network encode spatial and contextual information jointly by training it to minimize
be encoded in a single population vector. Fig. [b)
C == By,e [B+ 3)e
20g (x. x,rl"
20g
(o)= - -e (p((,.))) (alkaaa))llleee))) plawow)) + Alp(2z, G
(2)
where, in general xx and og are spatial and contextual encoding scales, respectively. For this work, however, we set 0x de. As with the purely spatial and purely contextual case, we model distinct spatial/contextual combinations as similar or dissimilar population vectors. An illustration of this situation is provided in Fig. [p]. When trained minimize (T) or (2). both path integrating recurrent and position-encoding feedforward units learns place-like representations,
shown in Fig. [t). Due to the correspondence with recurrent representations, their simplicity and ease of training, we employ feedforward networks for most analyses in this work. However, we show that our findings extend to recurrent networks, and that these learn additional path integration. We also train recurrent networks in more biologically plausible manner, to demonstrate that our results hold in more naturalistic settings (see App. A
2.2 Feedforward networks learn place cell-like representations
Having shown that place-like representations emerge in networks trained to minimize ①. we now study the feedforward network and the influence of model hyperparameters in more detail. Importantly, networks are able to minimize the objective function for a large range of hyperparameter combinations (evidenced by saturated loss in Fig. [))),, indicating that models are capable of learning the desired similarity structure. This is also reflected in Fig. 2 which shows an example of the agreement between the desired spatial similarity, and the corresponding Iearned representational similarity at the center of the training environment.
Depending on the choice of model parameters, the learned representations exhibit several features observed in biological place cells. While most units display strong tuning to particular spatial locations (see Fig. 2b) for several examples), activity levels vary. with several units being completely
silent (Fig. 2)) c) and d)), similar to e.g. (Thompson and Best 1989 Alme et al. 2014]. For small
values of and nonzero 8, some units exhibit multiple place fields (e.g. Fig. (p), left), which is also
observed in biological cells (Park et al. 2011], especially for large spaces Harland et al. 2021]. We also observe that field locations cover the training arena, shown in Fig. enabling encoding of the entire environment.
3

---

Neuron 3
Dissimlar
Dissimilar Similar
Similar
P1
Ps
P
Neuron
Neuron
Neuron
d)
Neuron
Nouron
Po
Dissimiarr
P
Neuron 2
Neuron
Figure 1: Overview of models and objective. a) Illustration of the spatial objective: locations that are close should be encoded by similar population vectors, distant locations by dissimilar population vectors. b) Similar to a); similar context signals should be represented by similar population vectors, dissimilar contexts by dissimilar population vectors. c) Similar to a) and b), but for joint encoding of space and context. d) Ratemaps of randomly selected units in networks trained to minimize the spatial objective function. Shown are learned representations for a recurrent network performing simultaneous path integration, and feedforward network performing spatial encoding.
We find that the properties of the learned representations follow readily from model hyperparameters. For example, Fig. 2p), d) and e) demonstrates that unit field size increases with increasing scale parameter a, and that unit is strongly linked to the similarity lower bound 33 (see App. B for more on the influence of different hyperparameters)
rates increase accordingly. Also, the number of place fields for given
Next, we investigated whether the learned representations in our model constitute useful spatial representations, in the sense of being decodable. First, we decoded the position using weighted mean of peak locations based on (5). varying the number of units included in the analysis. The results, shown in Fig. [h) indicate that the decoding performance (measured by the mean Euclidean distance between the actual and decoded positions) differs based on the number of units utilized, and model parameters, and demonstrate that the peak locations of learned representations can be used for accurate position decoding (achieving best-case mean error of few percentage points relative to the arena size). Fig. and j) show comparison of all three methods. Interestingly, simple linear decoder performs best in terms of mean error, while the population decoding using uniformly distributed memories exhibits the lowest maximum error. Moreover, the errors are uniformly distributed over space, while for the linear decoder and the Top n decoding we find areas of higher errors, especially the edges (see Fig. k)). Together, these findings show that the learned representations can be used to decode position efficiently using fairly simple decoding schemes.
2.3 Feedforward networks learn global-type remapping
When trained to encode multiple contexts in conjunction with spatial location, feedforward units exhibit dramatic firing field shifts, when comparing across contexts. This effect shown in Fig. 3h), where ratemaps of randomly selected units are tracked while varying the context signal.
Notably, firing fields exhibit several place-like behaviors, such as multiple firing fields (Park et al. 2011] (c.g. row seven), and field shifts between contexts, i.c. remapping [Muller and Kubic 1987]. In our case, the observed remapping is not sudden, attractor-like shift, which has been observed in place cells [Wills et al.) 2005]. but rather gradual transition between representations, which has also been seen during Hippocampal remapping [Leutgeb et al.) 2005a]. However, we find that spatial representations are uncorrelated between dissimilar contexts, an example of which is shown in Fig. (p). In fact, for sufficiently different contexts, the median spatial correlation between active units tends to zero, which aligns with global remapping in biological place cells [Leutgeb et al. [2044 (see Fig. (3b) and d)). Spatial correlations Fig. (3b) also support the observation that units remap gradually for smooth context transitions, correlations decay away from the diagonal.

---

a)
b)
10
e-- pi-
10"11
awas Aw
85 8 13
100
and Beous
g50.35 BH0.
CCO. Bro.
100
-00
h)
k)
Leae devkl
Liane drooos
Predecio deckki
488
62
My Aabi dooll
1/C
Spa
0.0
Figure 2: Feedforward network results. Training loss for different parameter combinations. Line shows the mean of 10 models and error bands show the min and max across models. Note that training data is generated continuously. b) Example ratemaps of randomly selected active units for models with different scale parameters a. Color scale is relative to individual unit activity. c) Distributions of the proportion of active units (mean rate 0) for different parameter combinations across models. d) Distribution of mean rate of units for each parameter combination (shown for example model each). e) Field sizes in pixels for each parameter combination (shown for one example model each). f) Left: Example target similarity structure relative to center. Middle: corresponding similarity for the learned representations of model with 0.25 and 0.5. Right: difference between target and learned similarity. g) Peak locations of all units for different parameter combinations (shown for one example model each). h) Mean position decoding error as function of the number of units used for Top decoding. Dashed line shows the naive case where every decoded position is at the center. i) and Mean and max decoding error for different decoding methods for trained 10 models, each with = 0.25 and 3 0.5. k) Example decoding error maps for different decoding methods (a 0.25 and 5 0.5).
Soaal Ocr.
Mediaa
Connedt
Coutext
Contest
Context
Figure 3: Feedforward network remapping results. Ratemaps function of context, for random selection 10 units. Each row corresponds to one unit and each column to a particular context value. b) Example distribution of spatial correlations for ratemaps corresponding to two distinct contexts (context =-0.9. context = 1.2). c) Median spatial correlations when comparing across all contexts. d) Number of units included (units active in both contexts) in the analysis in c).

---

f)
Figure 4: Recurrent network results with and without context. a) Ratemap examples of randomly selected units a recurrent network without context. b) Example trajectories used for training. c) Training loss for recurrent networks with and without context (10 models each, error bands show min and max). d) Histogram of peak values of a recurrent network without context and example ratemaps of units of different parts of the distribution. e) Histogram of mean rates of a recurrent network without context. f) Similarity structure in the center location of the learned representations of recurrent network without context (left) and the objective (center), as well as the difference between the two (right). g) Example trajectories decoded from network representations h) Comparison of the mean decoding error using linear decoder or population decoding across trajectories for 10 different models each. i) 2D UMAP projection of spatial representations for different contexts.
2.4 Recurrent networks learn place- and band-like representations and path integration
When we train a recurrent network to solve the proposed objective functions I and (2) while path integrating along simulated trajectories, we find that its units learn place-like and band-like Krupic et al. 2012) spatial tuning. Example ratemaps and trajectories are shown in Fig. 4h) and b), respectively. We also find that the recurrent network performs on par with the feedforward network. achieving similar loss minima, both for spatial and joint encoding (see Fig. 4c), suggesting that
the network has learned path integrate and minimize the objective. As band-like representations emerge only in the recurrent network, we speculate that these representations may be involved in path integration, which has also been found in other neural network models (Schøyen et al. 2023). Also worth noting is that while mean rates are similar, peak rates are markedly different between unit types (Fig. 4t and e)), further hinting at different functional roles. As with the feedforward model, recurrent responses accurately capture the desired similarity structure (Fig. 4D).
To verify that the RNN is path integrating, we use both population and línear decoding schemes to extract position estimates from the network representation (Fig. 4g) and h)). We find that positions can be accurately decoded using both methods for sequences longer than the network training trajectory length (see Fig. 4g) for example decoded trajectories). However, the trainable linear decoder is more performant during carly timesteps, but exhibits larger error over trajectory time, suggesting that the population decoding scheme can decode locations more robustly.
Besides simply being able to encode locations and contexts, find that the RNN learns low- dimensional structures representing distinct contexts. After applying UMAP to recurrent representa- tions in different contexts, we observe that low-dimensional projections capture both the geometry of the square enclosure, and the identity of the context (Fig. 4 )). It therefore appears that the network has learned veritable cognitive maps Keeeeeee and Nadel 1978 of different contexts, accessible from the representations themselves. See App. D for recurrent units representations across contexts.
2.5 Reuse by orthogonal transformation
Having demonstrated that networks learn distinet representations of different contexts, we turn to an interesting feature of the similarity objective. Assuming that we have trained (feedforward) network, with representations p(x.) (for all x, in the domain), the loss function only depends on the norm of, and distance between population vectors (and data). Thus, the objective is invariant

---

0%
Dilfarence
Learned
Permuration
Reflection
Rotation
1.0
0.33
0.00
100
0.35
Random orthogonal
Orthogonal
-0
220
43000
4,005
43005
4 X00
1.000
0,200
0LSC
1.00
00000
1000
80000
2.9
300
0.5
100p
Figure 5: Remapping by orthogonal transformations. a) Random global orthogonal transforma- tions (reflection, rotation, and permuation) applied to a trained representation (top) all preserve the similarity objective (bottom left), while producing spatially decorrelated representations (bottom right). b) Best-fit orthogonal transformations applied to learned representations of a feedforward network across two contexts. Inset is the original representation. the orthogonally transformed repre- sentation, and the secondary representation alongside the difference between the two for example units. c) Jitter plot of Pearson correlation between ratemaps across contexts for the transformation in b); shading indicates mean unit activity. d) Mean squared error between transformed and orig- inal representations for random and best-fit orthogonal transforms across all learned contexts. e) Ratemaps of units where a learned representation (left of dashed line) is extended by a continous orthogonal representation into novel representation (right of dashed line) without learning. Inset is the corresponding similarity structure, measured from the center of the enlarged environment.
to an orthogonal transformation Q the representation (and its inputs). In other words, we can transform the entire set of population vectors (using global orthogonal transform), and still have representation that minimizes (2). We demonstrate this in Fig. 54) which shows how different global (and random) orthogonal transformations the similarity structure but exhibit low spatial correlation with the original representation, similar to (global) remapping. In particular, observe that permutations induce more strongly decorrelated representations than reflections or rotations.
be used to produce new representations, that preserve
Furthermore, we find that global orthogonal transformations can be used to explain some of the representational changes learned by the network through training. Specifically, we computed global orthogonal transformations to match spatial representations across different learned contexts (see 4.3 for a description). As an example, Fig. 55)) shows unit ratemaps in two dissimilar contexts, alongside ratemaps (and ratemap differences) for orthogonal transformation taking context I into context 2. Notably, learned and transformed ratemaps are highly aligned. This is also shown in Fig. 5b), which demonstrates that transformed and learned representations are highly correlated. Furthermore, Fig. (5)) shows that best-fit orthogonal transformations achieve low errors (substantially lower than random orthogonal baseline) across all learned contexts, suggesting that the orthogonal transformations can account for much the learned remapping behaviors of the network.
Finally, we demonstrate that continuously applied orthogonal transformations can even be used to extend existing learned representations into novel ones. An example this shown in Fig. (5)). where population vectors of trained representation is transformed along the horizontal axis into

---

previously unseen region without explicit training. Notably, extended representations appear to maintain their place-like tuning, and approximately adhere to the original similarity structure. Note that the extended representations are no longer necessarily non-negative (see App. C for details).
As result, orthogonal transformations could prove viable way of modeling Hippocampal remapping. and possibly field formation itself. Our findings could also be extended to study what kind of upstream representations are needed induce orthogonal transformations in Hippocampal representations. Doing so could conceivably shed light on interactions between place cells and other cell types during remapping. in which other spatial cells such as grid cells are often implicated [Latuske et al. 2018
2.6 Limitations
While our model provides a fresh perspective on the formation and remapping of place cells, there several factors that limit its scope. For one, consider models where label locations and pre- processed context signals are available during training time, which could be biologically implausible (but see App. A for a recurrent model with more plausible inputs). A related concern is the use of scalar context signal. However, our model could accommodate more complex context signals, such as spatially bound contexts, contexts with multiple features. Extending the context signal in this way could prove to be an interesting avenue of research. Another limitation is the use of rate coding; is somewhat unclear how the proposed objective could be extended to spiking networks. A third limitation is the choice of the similarity measure and its lower bound, i.c. Gaussian similarity with a particular 3. However, this could conceivably be addressed by exploring representational similarity in experimental data in the future.
3 Conclusion
This work introduces a similarity- objective to explain the functional characteristics of place cells. Using this minimal self-supervised objective, we are able to directly demonstrate how place cell-like representations can be learned, and how they can be understood as translating similarity in location similarity in representation. Furthermore, a consequence of joint encoding of space and contexts. Finally, we demonstrate that remapping may be enacted through orthogonal transformations without explicit relearning. By demonstrating how place-like representations can be constructed to encode both space and contexts our findings contribute deeper understanding of the neural basis of navigation and memory.
observe emergent global remapping as
4 Methods
4.1
Models and training details
We trained two distinct networks to minimize the proposed objective functions ( and feedforward network and recurrent network, as illustrated in Fig. [Td). and trained using the Pytorch library [Paszke et al.] 2019]. The feedforward network featured two densely connected layers with 64 and 128 hidden units, followed by an output layer containing n, = 256 units. Every layer was equipped with the ReLU activation function, ensuring non-negativity. Notably, the output units of the network together form the representation that is used to compute the loss, i.e. p. The weights of the feedforward network were all initialized according an input size-dependent uniform distribution, following the PyTorch default (Paszke et al. 2019].
[i). Models were implemented
For the spatial objective (]). the input the feedforward network consisted of minibatches of continuous Cartesian coordinates, sampled randomly and uniformly within 2 x 2 square enclosure. For the conjunctive objective (2). the input to the network was a concatenation of randomly sampled Cartesian coordinates x, and uniformly sampled scalar context signals i.e. input cat(x, c). Context signals were sampled uniformly in the interval c E [-2,2] 2]. To increase the number of training samples, distances in either objective function were computed between minibatch elements.
The recurrent network consisted of a single vanilla recurrent layer equipped with the ReLU activation function, without added bias. Like the feedforward network, this network featured n. 256 recurrent units. In the purely spatial case, the recurrent state at a particular time was given by
p: ReLU(W, rea Pe- + Wv.),
8

---

where W E R is recurrent weight matrix, W E R",x2 an input weight matrix, and va Cartesian velocity inputs. In the case of conjunctive encoding, the imput at time was concatenation of the velocity and a (time-constant) scalar context signal. Note that this also adds an additional column to the input weight matrix. To mitigate vanishing and exploding gradients, the recurrent weight matrix was initialized to the identity, similar to Le et al. (2015) As with the feedforward network, losses were computed by comparing across trajectories (and contexts). The trajectory length was taken to be T = 10 timesteps. The initial recurrent state was computed by feeding the trajectory starting location (and optionally, the context) into a three-layer, densely connected network with 64, 64 and n. 256 units, respectively, each equipped with ReLU activation function. We also trained a recurrent network on long sequences, without explicit positional information coosition-indepenne state initialization, only velocity input) to demonstrate that our findings hold even when assumptions on model inputs are relaxed (see App. for details).
For the recurrent network, inputs consisted of velocity vectors along trajectories in the same square region as for the feedforward case. Trajectories were generated by creating Noundary-avoiding steps successively, A step was formed by sampling heading directions according to von Mises distribution, alongside step sizes drawn from a Rayleigh distribution. If the step landed the trajectory outside the enclosure, the velocity component normal to the wall was reversed, bouncing the trajectory off the boundary. Initial trajectory steps were sampled uniformly and randomly within the square arena. The von Mises scale was taken to be 4T, and the Rayleigh scale parameter 0.025. At training time, data was created on the fly due to the low computational cost. All networks were trained for a total of 60000 training steps, with a batch size of 64. For each model, used the Adam optimizer (Kingma and Ba 2017] with a learning rate of 10-4. Unless otherwise specified, all models were trained with 0.1, 3 0.5 and 0.25. To quantify place field numbers and sizes for trained representations, we applied the thresholding- and connected area-approach used by Harland et al.) [202
4.2 Spatial correlation & remapping
To evaluate representational changes in the face of changing context input, we ran the trained feedforward network on 32 linearly spaced contexts (in the range [-2, 21). Following Leutgeb [204]). we then computed the spatial correlation between unit ratemaps across contexts. Between- context spatial correlations were computed by correlating the ratemap of a unit in one context with the same unit's ratemap in another context in a binwise fashion. Ratemaps were only compared if a unit displayed non-zero activity in both contexts.
FollowinggLeutgeb et al.
To investigate whether the recurrent network encoded lower-dimensional structures across different contexts, we employed Uniform Manifold Approximation and Projection (UMAP) [McInnes et al., 2018). with default parameters. We first created unit ratemaps using 50000 10- t--timestep trajectories. and a resolution of 32 bins (in both x- and y direction) for 5 linearly spaced context signals (in the range [-2, 21). We then applied UMAP to the concatenated ratemaps and reduced the dimensionality of the ratemap population vector down to a two-dimensional space.
4.3 Orthogonal transformations
To explore how the spatial map of one context, Pay. can be transformed into the spatial map of another context, Poz. we applied orthogonal transformations. Our goal was to find an orthogonal transformation Q that minimizes the Frobenius norm ||= between the transformed spatial map from context C1 and the spatial map in context cz. Here, P E R",, x and the number of spatial bins Thins was chosen to be 128. This problem, known as the Procrustes orthogonal problem [Schönemann 1966], is defined by
minl - -Pezlle s.t. QTQ== I.
Q
Using the Singular Value Decompositon M = Pog PT. or can then be computed as Q = UVT.
ie. M UEVT, the orthogonal matrix Q
To explore whether global orthogonal transformations could be used to form new, distinct repre- sentations (similar to global remapping). we transformed learned representation using different global orthogonal transforms, and computed the spatial correlation between the original and trans- formed representations. Specifically, we considered random rotations, reflections and permutations as possible candidate transformations. To form n. dimeensiioi rotation matrices, we used the spe-

---

cot_ortho__aa group from the SciPy library VVrtanen et al. 2020). while reflection matrices were formed by interchanging two columns of such rotation matrix. Random permutation matrices were formed simply as permutations of the identity.
To determine whether orthogonal transformations could also he used generate new representations continuously, we conducted a simple numerical experiment, wherein existing representations were transformed iteratively into previously unexplored regions (akin an environment expansion). Specifically, we transformed population vectors at the end of an existing representation iteratively along the horizontal axis using the exponential map, i.e.
Pi+1 = e Pi
where 5 = (A - A ensures that e is orthogonal and Po is a learned population vector. For simplicity, we take A to be a randomly sampled permutation matrix. When extending the representation by distance of Ax| in one spatial dimension, we transform particular p. by setting
4 In 3)e |xx B)
d S'p:
which ensures that the similarity structure in the direction of transformation adheres approximately to (I) see App. [ooo details. We applied (4) iteratively population vectors of the feedforward network starting at the boundary of the environment, extending the representation to twice its horizontal length. The corresponding step length was equal to the bin width for original ratemaps, | = 2/n,, with
n2 = 32 being the number of bins in the horizontal direction.
4.4 Decoding
In model, each output unit is presumed to encode a spatial location near its peak activity. To determine the decoded position at a specific location x, we employed a weighted average of peak positions of the output units similar to Zhang et al. [9988. The weights correspond to the activity levels of the respective units at that location. The decoded position f(x) is thus given by:
En Pi)r; E" (x
f(x)
(5)
where f(x) denotes the decoded position at x, (>)) the activity of unit i at x, and r; indicates the peak location encoded by unit i. Notably, the decoding process does not incorporate all output units. Instead, units are prioritized based on their activity at position x, and only the top n most active units included in the decoding.
As an alternative scheme that exploits the similarity structure of the learned representations, we implemented simple population decoding procedure. This done by generating and saving "memory" of M population vectors {p." LM subsequent locations the location corresponding to the closest population vector in memory. For simplicity, we chose memory locations to be midpoints of unit ratemap bins, and memory population vectors the activity in that bin (for both feedforward and recurrent networks).
Ji=1 and corresponding locations {x²" LM Ji=1, and decoding
As a baseline, we also trained a linear decoder to predict Cartesian coordinates given network representations p. Decoder weights were initialized according to random uniform distribution, and trained using a batch size of 64, the Adam optimizer with a learning rate of 10-3 and 5000 (10000 for the RNN case) training steps. To compare decoding methods in the feedforward case, we chose a memory resolution of 32 bins (in both x- and y direction), and the same resolution to determine peak locations of units for Top n decoding. For the Top scheme we used = 44 based on the minimum decoding eтror across 10 models. The linear decoder was trained on population vectors of uniformly sampled locations. Finally, evaluated all methods on a grid of 129 129 points. For the RNN, we used a memory resolution of 32 bins, and 5000, 10-timestep trajectories were used to create population ratemaps. Both methods were evaluated on 256 trajectories with 20 timesteps each.
4.5 Figures& & code availability
Figures were created using BioRender.com, and code to reproduce findings and figures is available at http:p/p.....hc..oooo.oooooooaaa
10

---

References
Charlotte B. Alme, Chenglin Miao, Karel Jezek, Alessandro Treves, Edvaard I. Moser, and May-Britt Moser, Place Cells in the Hippocampus: Eleven Maps for Eleven Rooms. Proceedings of the National Academy of Sciences, 111662):14428-184555 December 2014. ISSN 1027-8424, 1091-6490. doi: 10.1073/pnas. htt.aaa...211166... 1421056111. URL
https://pnas orggas.org/aa
1073/ 1421056111
Andrea Banino, Caswell Barry, Benigno Uria, Charles Blundell, Timothy Lillicrap, Piotr Mirowski, Alexander Pritzel, Martin J. Chadwick, Thomas Degris, Joseph Modayil, Greg Wayne, Hubert Soyer, Fabio Viola, Brian Zhang. Ross Goroshin, Neil Rabinowitz, Razvan Pascanu, Charlie Beattic, Stig Petersen, Amir Sadik, Stephen Gaffney, Helen King, Koray Kavukcuoglu, Demis Hassabis, Raia Hadsell, and Dharshan Kumaran. Vector Based Navigation Using Gikd-like Representations in Artificial Agents. Nature, 557 (7705):429433,, May 2018. ISSN 0028-0836, 1476-4687. doi: 10. 1588-118--01000 7/www. www.:nature... wtt.://eee.meaaaeeeeeeeeeeeeeeee http:/aarcceese1555555.e 001-44111800000002212200 018-0102-6
URIpttpp/
C. Barry, C. Lever, R. Hayman, T. Hartley, S. Burton, J. O'Keefe, K. Jeffery, and N. Burgess. The Boundary Vector Cell Model of Place Cell Firing and Spatial Memory. Reviews in the Neurosciences, 17(1-2), January 2006. ISSN 2191-0200, 0334-1763
doi: 10.11//REVNEURO.200....
17.1--..7. URL https
//www www.wddgruyter:...m com coom///oomeet//oii..... 1551:/0001115///EEEEEE 17.
Christopher J. Cueva and Xue-Xin Wei. Emergence of Grid-like Representations by Training Recurrent Neural Networks to Perform Spatial Localization. arXiv: 1803.07770, March 2018. http:/tt:///atriii org/ abs/1803 .07770
William Dorrell, Peter E. Latham, Timothy F. J. Behrens, and James R. Whittington. Actionable Neural Representations: Grid Cells from Minimal Constraints, September 2022. URL http:://axiii....../////// org/ abs/ 2209. 15563
2KKv:2209.11563 [q-bio].
Pablo Fernandez- Velasco and Hugo Spiers. Waylinding
and tundra: what traditional cultures
teach us about navigation. Trends in Cognitive Sciences, page $13364661333005511 October 2023. ISSN 13646613, doi: 10. ti.. 00.000....000,..... httltttts////iiiinnege http://ss.....aat.oo/.ooo/oooooa cleevviee..com/rrezeeeee/..L 1136466132500255
Torkel Hafting, Marianne Fyhn, Sturla Molden, Britt Moser, and Edvard Moser. Microstructure of spatial map in entorhinal cortex. Nature, 436(7052):801-8800 2005. ISSN 0028-0836, 1476-4687. doi: 10. 1038/nature03721. URL htt.t:ttt///wwtt nature. com/ httpa//ree..eeerrreeee/aaaarraaa
Bruce Harland, Marco Contreras, Madeline Souder, and Jcan-Mare Fellous, Dorsal CAI hippocampal place cells form multi-scal representation of megaspace, Current Bioloyy, 3100)) 2190.e6, May 2021. ISSN 09609822. doi: www.:///.....b2...... 50960982221004220
Kathryn J. Jeffery. Place Cells, Grid Cells, Attractors, and Remapping, Neural Plasticity, 2011: 11, 201 ISSN 2090-5904, 1687-5443. doi: 10515520011/1826661 URL http:://ww.wwt. www.ww.widdw....m. com/ joom/jouree journals/
hp/2011/182602/
Pentti Kanerva. Hyperdimensional Computing: An Introduction to Computing in Distributed Representa- tion with High-Dimensiont Random Vectors. Cognitive Computation, 1(2):139-159) June 2009. ISSN
1866-9956. 1866-9964. doi: 0000077125590009990000
http:/// htatt:::/iinkk.sringgrr springer com/10. 1007/
512559-009-9009-8
Diederik Kingma and Jimmy Ba. Adam: Method for Stochastic Optimization. arXiv: 1412.6980 [es]. January 2017. http: httttp://aaxii.rrgg abs/ 6980
Julija Krupic, Neil Burgess, and John O' Keefe. Neural Representations of Location Composed of Spatially
Periodic Bands. Science, 337(6096):853 857, August 2012. ISSN 0036-8075, 1095-9203. doi: 10.1126/ science. science.1222403..
http:://wwwwwccccccccorrrrrr//// wttp:///oo.oooo////oooooo/o///.. htt.://///.....///iiineeeee 1222403
Patrick Latuske, Olga Kornienko, Laura Kohler, and Kevin Allen. Hippocampal Remapping and Its
Entorhinal Origin. Frontiers in Behavioral Neuroscience. 11:253, January 2018 ISSN 1662-5153. doi: 10.3389/fbbch.2017.0025 2017.00253/full.
http::///joumall.fttttreessssss wwwg/ariilleeio...... att.aa/eeee./..33//tteeeee 3389/fnbeh
frontiersin
Quoe V. Le, Navdeep Jaitly, and Geoffrey E. Hinton. A Simple Way to Initialize Recurrent Networks of Rectified Linear Units. arXiv: 1504.00941 [cs] Issue: 150iv:1504.00941 Publisher: arXiv, April 2015. URL
http: /tap:///axxvv.oo// org/ hwwgg/abs/tie 1504. 00941

---

Jill K. Leutgeb. Stefan Leutgeb, Alessandro Treves, Retsina Meyer, Carol A. Barnes, Bruce L. McNaughton. May-Britt Moser, and Edvard L. Moser. Progressive transformation of hippocampal neuronal representations "morphed" environments. Neuron, 48(2):345-358, October 2005a. ISSN 0896-6273, doi: 10160.
neuron. 2005.09.000 URL http://////d doi. org/10.
neuron. 2005.08 0w..009.000 007
Stefan Leutgeb, Jill K. Leutgeb, Alessandro Treves, May-Britt Moser, and Edvard I. Moser. Distinct Ensemble Codes in Hippocampal Areas CA3 and CA1. Science, (06655688)11155-122577 August 2004. ISSN 0036-8075, 1095-9203. doi: 10. 10.112222111111 26/sciience. 100265.
Stefan Leutgeb, Jill K. Leutgeb, Carol A. Barnes, Edvard Moser, Bruce L. McNaughton, and May-Britt Moser: Independent Codes for Spatial and Episodic Memory in Hippocampal Neuronal Ensembles. Science, 309657334;619--22, July 2005b. ISSN 8075, 1095-9203. doi: 106/sc/nce 14037. URL
https://www.sche
www//oo///////oo /t1///.111111111 111///cceennceee 1114037
Leland McInnes, John Healy, Nathaniel Saul, and Lukas Großberger: Umap: Uniform manifold approximation
and projection. Journal of Open Source Software 3(29):661, 2018. doi: 10.21105/joss.0066.. URL
httpa:// lttrpa://doi.org/11
org/10 10. 21105/josa. 000661
Ru Muller and JI Kubie. The effects of changes in the environment the spatial firing of hippocampal complex- spike cells. The Journal of Neuroscience, 7(7):1951- July 1987. ISSN 0270-6474, 1529-2401. doi: 00522//IEERROSCL00000000000000 NNEUK0SCI.07-07001383
15233
J. O'Keefe and J. Dostrovsky. The Hippocampus as a Spatial Map. Preliminary Evidence from Unit Ac- tivity in the Freely-Moving Rat. Brain Research, 34(1):1111775. November 1971. ISSN 00068993. doi: 10.1006/00068990 1793388-1 URL 0006899371903581
elsevier wwlssvier.com/retriieeeeee cow./eetrieeeeppi//
John O'Keffe Place units in the hippocampus of the freely moving rat. Experimental Neurology, 51(1): 78-109, January 1976. ISSN 00144886. doi: 10.
URL htt.:tttt:////nkiigghha
l.ssvvier.com/rerrieeeeee http/////p.../014488/000055530
John O'Keefe and Neil Burgess. Geometric Determinants of the Place Fields of Hippocampal Neurons. doi: 10.1038/38142552 URL
Nature, ):11558)):42-485 May 1996. ISSN 0028-0836, 1476-4687.
sstp://www/ natura. com/ wwtaaaaeeeeeeeeeee.e.ee
John O' Keefe and Lynn Nadel. The Hippocampus a Cognitive Map. Oxford University Press, Oxford, 1978. ISBN 0- 0-19-857206-9.
Florian Packmor, Dmitry Kishkinev, Flora Bittermann, Barbara Kofler, Clara Machowetz, Thomas Zech- meister, Lucinda C. Zawadzki, Tim Guilford, and Richard A. Holland. A magnet attached to the forehead disrupts magnetic compass orientation in a migratory songbird. Journal of Experimen- tal Biology, 224(2)jjh243337, November 2021. ISSN 0022-0949, 1477-9145. doi: 10.12 jeh.243337. httpsttts:://oouaaaat biologists. www.///b//rttiee///444////////// /tt.aa/c/aa//2/2//22223333/22222 Jeb243337/273488 whee-fforreaae disrupts
EunHye Park, Dino Dvorak. and André A. Fenton. Ensemble Place Codes in Hippocampus: CAL, CA3, and Dentate Gyrus Place Cells Have Multiple Place Fields in Large Environments. PLDS ONE, 6(7):e2239, July 201 ISSN 1932-6203. doi: 10. wjooumall..onee002222.
URL https://dxx plos. org/10. 1371/
journal. pournal.pone..e www.lpooee..2224... 0022349
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Köpf, Edward Yang, Zach DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. PyTorch: An Imperative Style, High Performance Deep Learning Library. 2019. doi: 0144850//AXXI.1112.00000 URL :ttttpss///rxivv org/ hts///b////99 191tt.///.....0 01703 Publisher: arXiv Version Number: 1.
Markus Borud Pettersen, Vemund Sigmundson Schøyen, Anders Malthe-Sprensseen and Mikkel Elle Lepperød. Decoding the Cognitive map: Learning place cells and remapping, March 2024, Publisher: arXiv Version Number: 1.
Alexei Samsonovich and Bruce McNaughton. Path Integration and Cognitive Mapping Continuous Attractor Neural Network Model. The Journal of Neuroscience, 17(15):590-55 59200 August 1997. ISSN 0270-6474, 1529-2401, doi: 10. W..12///EEUROSCIIIIO 17- 15- -5-01100.9999977 URL httph:ttt//// /wwww jneurosci
hrg//ookkp//di//10 1123: 17 05900............ 1997
12

---

Rylan Schaeffer, Mikail Khona, Tzuhsuan Ma, Cristóbal Eyzaguirre, Sanmi Koyejo, and lla Rani Fiete. Self- Supervised Learning of Representations for Space Generates Multi-Modular Grid Cells. 2023. doi: 10.48550/ REXIV.2311.02316 URL inttps://axxiv org/ a231111 2311 02316 Publisher: arXiv Version Number:
Peter H. Schönemann. A generalized solution of the orthogonal procrustes problem. Psychometrika. 31(():1-10. March 1966. ISSN 1860-0980. doi: 10.1007/BF0228451.
Vemund Schøyen, Markus Borud Pettersen, Konstantin Holzhausen, Mariannn Fyhn, Anders Malthe-Sorenssen, and Mikkel Elle Lepperød. Coherently remapping toroidal cells not Grid cells are responsible for path integration in virtual agents, iScience, 26(11): November 2023. ISSN 25890042. doi: 10. 1016/j.isci. 2023.108 108102. URL htttss://linkinghun elsevier www./eeeeeeeeeeeeeeeeeeeeeeeeeee http://wwwweepp////2222202222220
Willard S. Small. Experimental study of the mental processes of the rat. ii. The American Journal of Psychology. 12(2):206, January 1901. ISSN 0002-9556. doi: 10.2307/1412534, URL http: /ttp//xx.ooo.. org/ 10. 2307 141253
Trygve Solstad, Edvard 1. Moser, and Gaute T. Einevoll. From Grid Cells to Place Cells: A Mathematical Model. Hippocampus, 22:1)))62100000000 December 2006. ISSN 10509631, 10981063, doi: 10. 1002/hipo.20244 URL httpp:ttt: inel ibrrrrrrrarr www.arrwwwee..cco//oo///////////
ctm http:///1111111/ http://....02/p/pp.pp 20244
Ben Sorscher, Gabriel C. Mel. Samuel A. Ocko, Lisa M. Giocomo, and Surya Ganguli. A unified theory for the computational and mechanistic origins grid cells. Neuron, page S089662732200907, October 2022. ISSN 08966273. doi: wttp.//eeroo..0000 2022. 10.003. URI. Btl//ttttt:://kkiigghht wtw.:aaiiinnnub...eieeeeeee.eeee elaevier com/retrieee/
p31/50896627322000077
Lt Thompson and Pj Best. Place cells and silent cells in the hippocampus freely-behaving rats. The Journal of Neuroxcience, 9(7):2382-3900, July 1989. ISSN 0270-6474, 1529-2401, doi: 10.EURSJ.EU 09-07-02382.1989.
Benigno Uria, Borja Ibarz, Andrea Banino, Vinicius Zambaldi, Dharshan Kumaran, Demis Hassabis, Caswell Barry, and Charles Blundell. A model of egocentric allocentric understanding in mammalian brains. preprint, Neuroscience, November 2020. httttp:/bboorziv.orr/oooooo http/////.....11///2.0...1
11.378141
Pauli Virtanen, Ralf Gommers, Travis F. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser. Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, CJ Carey, Ilhan Polat, Yu Feng. Eric W. Moore, Jake VanderPlas, Denis Laxalde. Josef Perktold, Robert Cimrman, Ian Henriksen. E. A. Quintero, Charles Harris, Anne M. Archibald. Antônio H. Ribeiro, Fabian Pedregosa, Paul van Mulbregt, and SciPy 1.0 Contributors. SciPy 1.0: Fundamental Algorithms for Scientific Computing Python. Nature Methods, 17:261-272. 272, 2020, doi: 10. 1038//41592--00 -00---2---9-----2222
James C.R. Whittington. Timothy H. Muller. Shirley Mark, Guifen Chen, Caswell Barry, Neil Burgess, and Timothy F.J. Behrens. The Dolman-Fichenbann Machine: Unifying Space and Relational Memory through Generalization in the Hippocampal Formation. Cell, 1239-16122992233.2 00928674. doi: 10. 1016//.cell... cet...ll22220. cel1.2020. 10.0..
November 2020. ISSN
URL. htttp:://inkkigghhtt elsevier com/retrieve/
015/80092867420313888
Tom J. Wills, Colin Lever, Francesca Cacucci, Neil Burgess, and John Oeeeee, Attractor dynamics in the hippocampal representation of the local environment. Science, 308(572))87338776 2005. doi: 10.1126/ science.1 108905. URRI/htpps http:/wwww.ccincc... www.ecce.e.cccc..cc...o......e.. wttpeeer/////o////...ootte
1108905
Dehong Xu, Ruiqi Gao, Wen-Hao Zhang, Xue-Xin Wei, and Ying Nian Wu. Conformal Isometry of Lie Group Representation in Recurrent Network of Grid Cells, 2022. _eprrnt: 2210.02684.
Kechen Zhang, Iris Ginzburg, Bruce L. McNaughton, and Terrence Sejnowski. Interpreting neuronal population activity hy reconstruction: Unified framework with application to hippocampal place cells. Journal of Neurophysiology. 1912) 1998. doi: 10.1152/ja. 1998.79..... 79.1.7....0.77. https:: 1162/ jn. 1998, 79, 2, 1017 PMID: 9463459.
13

---

Appendix / supplemental material
A Learning spatial representations without explicit position information
For both feedforward and recurrent models in this work, we make use of explicit Cartesian coordinates train networks. In the real world, agents do have access exact coordinates labeling the environment, which could raise concerns about the transferability of our results to biological networks. We therefore train recurrent neural network without explicit positional information, show that our findings hold even when relaxing assumptions available inputs.
Concretely, consider, before, RNN with n 256 recurrent units. For this network, however, we initialize the network using trainable initial state that does not depend on data. We furthermore train this network long sequences (T 500 timesteps), and compute similarities time, i.c. within trajectory, rather than aggregating over multiple smaller trajectories. The input to the network just Cartesian velocities along such trajectories, and similarities used for training are computed only using relative distances.
trajectory
The results in Fig. Al show that the model also learns place and band-like representations and is able to learn the objective (see Fig. a)) b) and d)), similar to our other findings. As with other networks, can also decode positions from this network using either simple linear decoder, our population decoding scheme, demonstrating that the learned representations are decodable (Fig. )T))). Here, evaluate the model along the same 16 unseen long trajectories of 500 timesteps for both schemes. Training of the linear decoder was performed by repeatedly sampling population vectors using 500 timestees-trajecctoiies, for 5000 training steps using the Adam optimizer, learning rate of 10 10-3, and a batch size of 64. Additionally, we used 100 trajectories of 500 timesteps to form the memory vectors with resolution of 32x32 bins for the population decoding scheme (see Sec.
4.4). In both cases, we omitted the first 50 timesteps of the trajectories to avoid large initial errors caused by the network's 'ss lack of positional information.
14

---

A 0.5
0.05
C.D. () 05
61
unto
70-
Deaety
100
50
1003
Disamee?
2
0.05 5
0.D. (
h)
300
Peveersnn unto
70
100
4.6 6.6 Number Edlds
Nunber Bekds
Nunaber Bakks
Prppltatonn Victor Maguitude
0.5
Figure A1: Spatial representations without explicit position information. a) Ratemap examples of randomly selected units of the long-sequence recurrent network. b) Training loss of the long-sequence recurrent network (data created on the fly). c) Mean decoding error of a linear decoder and the population decoding scheme on 16 unseen long trajectories. The dashed line indicates naive case in which the decoded position is always at the center of the environment. d) Learned (left) and target (middle) similarity structure, alongside their difference (right) relative to center arena, for sengesequence recurrent network.
15

---

B Loss ablation and effects of B
We have previously noted that learned place field sizes are governed by the scale parameter o. To further explore the influence of hyperparameters on learned representations, we performed an ablation study, wherein we train feedforward networks with and 5 ablated. We also explore the effects of changing the similarity measure, from Gaussian (depending on the square of the distance between locationsfeerresenttiion) to exponential in the Euclidean distance, i.c. sim(a, b) x xe" e-la-bl
The results are shown in Fig. A2 With activity regularization (A 0), units exhibit multiple smaller, but more irregular place fields (Fig. h)).). Stronger population vector separation (8 = 0) leads highly unimodal representations, and larger place fields (Fig. A2 Notably, when both 3 and À are set to zero (Fig. A2e)), units no longer display place-like tuning, but rather appear to change linearly across the arena, possibly reflecting the positional input the feedforward network. Thus, place-like tuning in our model is dependent on either a non-zero similarity threshold, or activity regularization, but not both.
When similarities are computed using the Euclidean distance directly, (Fig. A2 we observe that network units still exhibit place-like spatial tuning. However, units now exhibit differing field sizes, with some large place-field units, and some units with smaller fields. Some units are also somewhat ttrpe-like. Thus, different similarity measures and distance functions can lead to different and possibly more expressive tuning profiles, which could be an interesting avenue for future investigation.
To better understand the role of 3 determining network representations, we first turn to an interesting fact of (normalized) vectors in higher dimensions; for large n, vectors on the n-sphere tend to reside at some intermediate distance from most other vectors; an example of this is shown in Fig. A3), where distances are computed for random vectors of increasing dimension on the n-sphere. We therefore consider that dissimilarity may be meaningfully defined by some intermediate level of vector separation (as intermediately separated vectors most common, and appear, in this sense effectively random).
Notably, this intuition can also be applied fairly directly to trained networks, as their population vectors tend to reside on the Spphee (Fig. /3p) likely related to the applied L2 activity regularization. Being high-dimensional by design, network population vectors will thus tend to follow a similar distance distribution to Fig. A3a), again suggesting that non-zero similarity threshold is sensible.
From a practical standpoint, a non-zero B allows for more expressive representations, as represen- tations may be reused across space and contexts, without incurring substantial loss. For example, if a unit fires at different, distant locations, this contributes to increased similarity. For 8 = 0 and sufficiently distant spatial locations, this situation would incur loss, whereas nonzero 8 allows for some degree of similarity. Notably, even though states are of intermediate similarity, they can still be accurately decoded (as we demonstrate in e.g. Fig. 2
We indeed find that nonzero 3 allows for more unit reuse across space. This can be seen in Fig. A3c), which demonstrates that higher values of / (and smaller σ)) lead the emergence of units with multiple place fields (see e.g. the case where a = 0.1, 3 = 0.5). We also observe a similar trend over different contexts, which is illustrated in Fig. A3.. In this case, units trained with 3 O are only active around a narrow set of context values, whereas for 3 > 0, units are active across a wide range of distinct contexts, and place fields shift between contexts, similar Hippocampal remapping.
16

---

135
Lime revoding daniita
8.20
180
8500
30
TKonko
Figure A2: Loss ablation and effect of similarity measure. a) Ratemaps of randomly selected feedforward network units, when ablating X. b) As in a), but for ablating B. c) As in a) and b), but for ablating both and A. d) Ratemaps trained feedforward units when the squared distance of the similarity measure is replaced by the Euclidean distance.
b)
= 0.5,
0.25
0, = 0.1, 0..5
3 0, À
0.25
elp-p'
Figure A3: Hyperdimensional computing and the effect of B. a) Histogram of squared Euclidean distances between 512 randomly sampled vectors of different number of dimensions (legend) on the corresponding ri-shher. b) Distribution of population vector norms for a trained feedforward network with 8 0.5, parameter configurations (inset). d) Ratemaps of randomly selected units of a trained feedforward network with À = 0.1, 0.25, across different contexts for different values of B. For each value of 3, one row represents one unit and each column one context value. Context values increase linearly from -2 (leftmost column) (rightmost column).
0.25, and À 0.1. Histograms of the number of place fields for different
17

---

C Extending existing representations by orthogonal transformations
In this section we demonstrate how we can extend existing representations learned by the neural network to novel representations, using continuous orthogonal transformations, To do so, we employ the exponential map. given by
Q(I) -
where t IR is parameter, and choose S IR"*** to be akew-symmetris matrix, which makes Q(t) orthogonal. To extend an existing (purely) spatial representation, we need to transform it in manner that respects ( For orthogonal transformations that preserve norms, the quantity of interest is therefore distances in the neural representation before and after transformation. For orthogonal transformations, we have that
Q(t)p 2pp 2pp
where is the population vector we wish to extend from. Using the Taylor expansion of the exponential map
Q Q 15 52 +...
2
and keeping terms up to second order, we have that
&² P Q(Ppp ~ 2p² 2(p² 1p"Sp Sp 2 5
and we can solve for the parameter approximately. Note that is skew-symmettic, so Sp and
p Q(t)pl
t
p² S2p
Notably, we can solve for the numerator term required to match the desired similarity structure in ie.. demand
Qp In((1 3)e 2④अ r-r'l + ().
and insert into the expression for I, yielding
In((1 - B)e 2 p-rl
+
pd S
We use this expression for the transformation parameter to extend a place-like representation hori- zontally into region that has not been previously visited by the network, and demonstrate that the similarity objective is still (approximately) satisfied. Notably, this approach generalizes outside the training domain of the network that generated the starting representation without additional training. However, since we perform otherwise unconstrained transformations, representations are no longer guaranteed to be non-negative. One should also note that the above derivation does ensure that the generated representations adhere to desired similarity structure the vertical direction. However, we observe empirically that this approximately the case (see similarity structure Fig. (B)), and hope to study the general case more closely in the future.
D Context-dependence of recurrent representations
Fig. shows ratemaps for 10 randomly selected recurrent units of a network trained to minimize during path integration evaluated across different contexts. As with the feedforward network, representations are place-like, and demonstrate multiple place fields, rate changes, and field shifts between contexts, similar to Hippocampal remapping.
18

---

Figure A4: Remapping Behavior in Recurrent Network. Ratemaps of recurrent network as a function of context, for a random selection of 10 units. Each row corresponds to one unit and each column to particular context value. Contexts increase linearly from -2 (leftmost column) to (rightmost column).
19

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: We propose similarity based objective and train neural networks minimize it. When this is done, networks learn place-like representations, conjunctive codes and even remapping.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No
NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
is fine to include aspirational goals as motivation as long as is clear that these goals not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification:
We have a separate subsection detailing limitations of our approach.
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create a separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (c.g.. independence assumptions, noiseless settings, model sell-speciiiaatinn asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-tex, system might not be used reliably to provide closed captions for online lectures because it fails handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and faimesss.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community, Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
20

---

Answer: [NA]
Justification: There are no purely theoretical results or theorems in our work; but we derive one application of orthogonal transformations representations. We provide full derivation complete with required assumptions in the appendix, and back up our result by computational experiments. We also back up claims relating to the proposed objective function by computational experiments.
Guidelines:
The answer NA means that the paper does not include theoretical results. All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced in the statement of any theorems.
The proofs can either appear the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proof's provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should properly referenced.
4. Experimental Result Reproducibility
Does the paper fully disclose all the information needed to reproduce the main perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided not)?
Question:
Answer: [Yes]
Justification: The methods and results fully describe the models and datasets, and is sufficient for reproducing our findings. We also link to code that can be used reproduce our findings.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution be necessary to either make it possible for others to replicate the model with the same dataset, provide access the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how replicate the results, access a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
specific model and empirical evaluation, may
While NeurlPS does not require releasing code, the conference does require submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make it clear how reproduce that algorithm. (b) If the contribution is primarily
model architecture, the paper should describe the architecture clearly and fully.
(c) the contribution is a new model (c.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome describe the partieular way they provide for reproducibiliity In the case of closed-source models, it may be that access to the model is limited in
21

---

some way (e.g., to registered users), but it should be possible for other researchers have some path reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
Answer: [Yes]
Justification:
The codebase is open source under MIT licence and is referenced in the methods section.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurlPS code and data submission guidelines (nttps://nnis.cc)
for more details.
While we encourage the release of code and data, we understand that this might not be possible, so "No" an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https
7 //nips. /tipss.c/ppb.i... httpcc/pccccccceeeeeeeeeeeeeeeee wwllcc/guiiss//CC htte:/gguuddde/oSSSubbbbiiii
for more details.
The authors should provide instructions on data access and preparation, including how access the raw data, preprocessed data, intermediate data, and generated data,
The authors should provide scripts to reproduce all experimental results the new proposed method and baselines. If only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible supplemental material (appended the paper) is recommended, but including URLs to data and code is permitted.
6. Experimental Setting/Details
Does the paper specify all the training and test details (c.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Question:
Answer: [Yes]
Justification: These are all described in the methods section.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, or as supplemental material.
7. Experiment Statistical Significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [NA]
Justification:
We do not carry out experiments do statistical comparisons between groups.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by emoor bars, confi- dence intervals, or statistical significance tests, least for the experiments that support the main claims of the paper.
22

---

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
It should be clear whether the error bar is the standard deviation the standard error of the mean.
It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not show in tables figures symmetric error bars that would yield results that are out range (e.g. negative error rates).
If error bars are reported in tables or plots, The authors should explain the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments Compute Resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Question:
Answer: [NA]
Justification:
The model is minimal, and runs on most modern laptops.
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount compute required for each of the individual experimental runs as well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make into the paper).
9. Code Of Ethics
Question: Does the research conducted the paper conform, in every respect, with the NeurlPS Code of Ethics https: tttt:ps://eeurrie hww.rps....../. wwt.rrss.ccppubic/eeeee hccpppblic/tthicccuieeeeell..eed
Answer: [Yes]
Justification:
We have reviewed, and hold that we conform to the NeurIPS Code of Ethics.
Guidelines:
The answer NA means that the authors have not reviewed the NeurlPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., there a special consid- cration due to laws or regulations their jurisdiction).
10. Broader Impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [NA]
Justification: As the research approaches fundamental questions neuroscience, potential socictal impacts are far removed and would be pure speculation.
Guidelines:
The answer NA means that there is no societal impact of the work performed.
23

---

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: There is little to no risk of misuse of our model, as it merely aims to explain particular spatial representations.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model. for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
recognize that providing effective safeguards challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification:
We have created the majority of the code, and referenced other libraries where appropriate.
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package or dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (c.g., CC-BY 4.0) should be included for each asset.

---

For scraped data from a particular source (e.g., website). the copyright and terms of service of that souree should be provided.
assets are released, the license, copyright information, and terms use in the datasets.paperssiii paperswi thcode. www.coo//aaaaa datasets package should be provided. For popular datasets,
has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
If this information is not available online, the authors are encouraged to reach out to the asset's s creators.
13. New Assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [Yes]
Justification:
The codebase is well documented.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/mmodll as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent obtained from people whose asset is used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsourcing and Research with Human Subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given participants and screenshots, if applicable, as well as details about compensation (if any)?
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then much detail as possible should be included in the main paper.
According to the NeurlPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
25

---

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
26


Reviews:
[{"review_id": "CvicS3ooo6", "reviewer": "Reviewer_wYa3", "summary": "How do place cell responses to location and context emerge? The authors propose that place cell response emerge from a similarity-based object function that encourages places that are closer in space to be encoded as more similar representations. Specifically, the authors find that their model learns place-like representations (operationally defined here as the peak activation locations spanning the training arena and being able to decode position from the model's output). They also find that, when the model and objective function are extended to include context, the model's rate maps change as a function of context, and these changes in the model's representations (due to context changes) are orthogonal transformations.", "questions": "Q1. Why are there different inputs given to the feedforward vs. recurrent models?\nQ2. Relatedly, does the objective function get modified for the recurrent model (since its inputs are velocity rather than the position coordinates)?\nQ3. If the feedforward model (without context) only receives Cartesian coordinates as inputs (input dimensionality of 2), why does the model need multiple layers with 64 and 128 hidden units, respectively? In the world of machine learning, the model is quite small, but for only encoding 2 numbers, it might be over-parameterized.\nQ4: What is the difference between Fig 4g and 4h?", "limitations": "See weaknesses.", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "Overall, I applaud the effort to use machine learning techniques as task-performing models of neural activity.\n\nThe paper offers an interesting perspective on remapping representations based on context information. The idea that remapping can be accomplished by orthogonal transformations could also be applied in future work to other types of neural networks and/or modeling different brain regions.\n\nThe writing is clear and well-organized. The figures are excellent, well-made, and easy to understand.", "weaknesses": "- My major concern is that the input to the feedforward network is the ground truth location (the animal's position as Cartesian coordinates). It's not particularly impressive that a model that directly receives position information is able to encode position information, nor that we can decode the position from its representation. If their main research question is \"how place cell responses emerge\" it doesn't make sense to feed them place information directly.\n\n- Beyond the potential redundancy issue of having place information be both the input to and output from the models, the bigger issue is that their model does not mimic the task that the brain (or place cells more specifically) need to learn. The brain needs to figure out location through its sensory data without being given veridical Cartesian coordinates. The recurrent model (which only has access to velocity information) is much more interesting, but it only used in one out of 4 experiments.\n\n- The authors describe their objective function as \"a minimal, similarity-based objective,\" but it doesn't seem particularly minimal. Their objective function include five different constraints (page 2, lines 65-69), not including the version for incorporating context information. It would be helpful for the authors to spend more space in the paper describing their objective function in a way that highlights its simplicity.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer very much for their well-structured and insightful feedback. In the following, we want to address some of the concerns and answer the questions.\n\nWe absolutely understand the concern about the access to euclidean position in the feedforward model. Here, the RNN model offers a variant where explicit positions in the input are not needed. We show in Fig. 4 that we get very similar representations with this model suggesting that the model can learn to path integrate and solve the objective using only relative displacements. Such signals have already shown to exist in MEC, for example in the form of speed cells (see [Kropff et al. 2015](https://www.nature.com/articles/nature14622)) and head direction cells (see [Muller et al. 1996](https://www.sciencedirect.com/science/article/pii/S0959438896800730?via%3Dihub)). Even for the labels in the objective function one could simply use the relative distance ($x_t-x'_t$) instead of the explicit cartesian coordinates. The choice to use those was mainly to keep things simple. Again, the above mentioned internal signals could provide this information. In addition to the results presented in the manuscript, we have included results for a variant of our RNN model in the attached PDF (see Fig. 6). This model does not even need access to the initial position and therefore can be trained and used without any explicit coordinates, relying only on relative distances. We believe that this makes our model much more biologically plausible as this is, as you say, closer to the problem the brain has to solve. \n\nYour comment on the simplicity of the objective function is a very good point. We are planning to spend more time on making the purpose of the different terms more clear in the camera-ready version. To us, the simplicity lies in the fact that basically we are just saying similar locations (or contexts) in the outside world should also be similar in neural space, as captured in equation (2). \n\nIt is worth mentioning that the position and the context signal are incorporated in the loss in the very same manner. To prove our main point, the exponential terms alone are enough. The regularization term helps ensure rather small activations, while the $\\beta$ parameter allows our neural representation to keep a certain similarity even for very distant physical locations. This leads to the nice property that neurons can be reused (as reusing the same neuron in a different context always leads to a certain similarity).\n\nAnswer to the questions.\n1. We are feeding in different inputs in order to show that the model does not necessarily need explicit euclidean coordinates but can also rely on relative distances (velocity) and path integrate those to solve the problem of representing close physical point similarly and far points dissimilarly, hence solving the same objective function. By doing so we are trying to provide a more biologically plausible model that does not need the ground truth location. In the new version we attempt to make the motivation behind this more clear.\n2. The objective function is the same for both models. So even though the inputs are velocities in the RNN case, the model still has to solve the problem of representing similar euclidean coordinates similarly and dissimilar ones dissimilarly. The only labels required are distances and the initial position. However, as we show in Fig. 6 (in the additional figure panel provided along with this rebuttal), when using longer trajectories our model can even solve the problem without access to the initial position, and therefore does not need any explicit position information. \n3. That's a very good question. Based on the theory on deep neural networks acting as universal function approximators, we wanted to make sure our network is deep and expressive enough. You are absolutely right in that our network is over parameterized. Here, our goal was not to come up with a low-dimensional solution but instead have the neural network learn a representation similar to those of Hippocampal cells.\n4. Thank you for pointing out the lack of clarity in this figure. In Fig 4g we are showing multiple single trajectories in the UMAP space while in Fig. 4h we sampled from a lot of trajectories and averaged the binned activity over positions. We will make sure this is more clear in the camera-ready version.", "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": "vdvheCp5VX", "reviewer": "Reviewer_mSmA", "summary": "The paper explores neural network models that mimic hippocampal place cells by learning spatial and contextual representations through a similarity-based objective function. It demonstrates how both feedforward and recurrent neural networks can encode and remap these representations in response to context changes, akin to biological place cells. The study also examines the application of orthogonal transformations to generate new spatial maps from learned representations without retraining.", "questions": "Please refer to weaknesses 2,3", "limitations": "The authors have adequately discussed limitations of their work.", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The objective function proposed in this paper, aimed at maintaining similarity between representational and physical spaces, is both intuitive and persuasive. It successfully achieves localized representation while enabling the network to have remapping capabilities.\n2. The idea of using orthogonal transformations for remapping is intriguing and insightful.\n3. The authors have a clear understanding of the model's limitations and have discussed them in detail.", "weaknesses": "1. The paper's claim of 'conjunctive representation' seems somewhat overstated. In my view, conjunctive representation should involve a joint representation of location and localized sensory stimuli, whereas this paper merely represents context as a new dimension, enabling remapping under different contexts. I believe that remapping is not equivalent to conjunctive coding.\n2. In the experimental setup of this paper, the shape of the environment remains unchanged (2x2), with only the context cues changing. For real animal experiments, would such a setup typically lead to rate remapping instead of global remapping?\n3. The core of the proposed objective function is to maintain similarity before and after representation, similar to UMAP, which aims to preserve geometric features of data points before and after dimension reduction. Using UMAP to validate the model is not very convincing. Have the authors validated their model using other dimensionality reduction methods?\n4. If, as the authors speculate, place cells remapped through orthogonal transformations, one would expect to observe no change in the pairwise correlations between neurons before and after remapping, which seems not to have been observed experimentally.\n5. As the authors themselves discuss, the approach proposed in the paper still lacks biological plausibility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer very much for their well-structured and fair feedback. In the following, we want to address some of the concerns and answer the questions that arose.\n\nRegarding the listed weaknesses we have a few comments that we hope might clarify a few things and answer some of the questions asked by the reviewer.\n1. We absolutely agree that remapping and conjunctive coding are not the same thing. In the current version of the article, we only use a global context signal and show how in this simple case the conjunctive encoding of space and global context can lead to remapping. One can, however, also imagine a case in which the context signal itself depends on the location ($c(x)$) and then becomes a localized context. From that definition, one can derive many interesting cases and experiments which we plan to further investigate in the future. We understand, however, how this part about the conjunctive coding might be unclear and will make sure to make this more clear in the camera-ready version.\n2. It's true that global remapping is usually associated with a change in the geometry and rate remapping with contextual changes. However, for rate remapping, the contextual changes are typically rather small. For bigger contextual changes (such as smell and room color), partial remapping has been observed which includes the relocation of place fields of cells (see [Latuske et al. 2018](https://www.frontiersin.org/journals/behavioral-neuroscience/articles/10.3389/fnbeh.2017.00253/full) for a good overview on those different types of remapping). An example can be found in [Leutgeb S. et al. (2005)](https://www.frontiersin.org/journals/behavioral-neuroscience/articles/10.3389/fnbeh.2017.00253/full#B40 \"https://www.frontiersin.org/journals/behavioral-neuroscience/articles/10.3389/fnbeh.2017.00253/full#b40\"). Here, they observe partial and even global remapping when switching from one room to another but keeping the enclosure constant (hence, no change in geometry). Overall, there are different results from different labs making it hard to say what exactly the requirement is for at least partial remapping (remapping that includes change in the position) to happen. We plan to to further discuss that in our article. Thank you very much for pointing that out.\n3. The purpose of using UMAP was to visualize the learned similarity structure. We aimed to validate the models learned representations by showing the following:\n\t1. The loss saturates at a low value (showing that the objective was learned). It is also worth mentioning that the data is generated on the fly (we are not using a pre-created dataset) so it is basically a test loss. \n\t2. The learned representations are a good representation of space as the location can be decoded. \n\t3. The representations are place-like.  \n  We have included a further validation using PCA in the attached PDF (see Fig. 3). Here, we show that while the dimensionality for each spatial map (one for each context) is roughly the same (90% explained variance with around 8 components), while the dimensionality for all spatial maps combined is much higher (<50% of variance explained with 12 components). Therefore, the spatial maps for different contexts have different low-dimensional features and are indeed different.\n4. You are right, and we agree that one would not expect to see any pairwise correlations between neurons. In Fig. 3b we show that for the learned representations of different contexts the place fields of neurons are indeed uncorrelated between contexts, as expected. When we talk about rotations (or orthogonal transformations) later what we mean is the rotation of the entire population vectors. In that case, the pairwise correlations (before and after) of single neurons are not preserved. In fact, one could use a full permutation of the population vector as special case of an orthogonal transformation. In that case, since the place fields of different place cells are in different locations, the neurons' place fields would be uncorrelated. We hope this answers the question, and we are of course happy to provide more details on how we went about the rotations if anything is still unclear.\n5. We agree that especially the use of explicit cartesian coordinates is biologically implausible. We just want to point out that in the RNN case we are able to use only relative distances in the input (velocity signal). Furthermore, for the ground truth labels, we also just need relative distances as opposed to actual euclidean coordinates (see the $x_t-x'_t$ term in the objective function). Internal signals of relative distance traveled have been shown to exist in the brain, for example in the form of speed cells (see [Kropff et al. 2015](https://www.nature.com/articles/nature14622)) and head direction cells (see [Muller et al. 1996](https://www.sciencedirect.com/science/article/pii/S0959438896800730?via%3Dihub)). Lastly, we want to mention that we can and have also trained an RNN model that does not require the initial location as an initialization but instead is initialized with a zero-state. We included a figure showing the resulting representations in the attached PDF (see Fig. 6). In conclusion, while in the naive feed forward case we do rely on implausible explicit positions, there are ways to make our model work with only relative distances which make it a lot more biologically plausible and leads to similar representations. These models solve the same objective function given in equation 2.", "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": "uhFDZK7JNg", "reviewer": "Reviewer_QBGM", "summary": "This paper studies the emergence of “conjunctive representations” in the context of place cell representations. The authors investigate how their proposed similarity-based objective function produces context-dependent place cell representations that exhibit remapping behaviors across contexts. The proposed objective function further produces representations that are invariant to orthogonal transformations across contexts.", "questions": "1. Minor point: One of the assumptions the authors include were that unit activations are bounded, yet the activation function they used was a ReLU (unbounded). Does this assumption actually matter in practice? Also, I think a reference for the assumptions of their loss function (lines 65-69) would be helpful.\n\n2. Minor question: The authors mention that the spatial grid was 2x2 square enclosure. Initially, it read to me that this space was actually a 2x2 matrix, with 4 total points to sample from. But I assume that this space is sampled continuously within a grid, rather than just having four points in total. \n\n3. Details on exactly how the loss was computed for a trajectory was unclear. For a given trajectory, which has 10 time points, was a distance metric computed between every pair of time points? Or was it just between the initial point from t_0 and t_k, for k=1,…,10, and then averaged across all pairwise losses?\n\n4. Related to weakness #2 – do the authors think their results would hold across environments of different shapes?\n\n5. Minor question: What is the rationale for only including the top n active units for decoding? I’m surprised that adding additional units would reduce the overall decodability of place. In principle, the decoder would only need to pay attention (ie have high coefs) for the units that are relevant, while ignoring other units.\n\n6. Related to weakness # 3 – what is Fig. 5 trying to convey? I found the figure a bit obscure.\n\n7. Another result that I think would have been nice to see (while bolstering the claims of the paper), is to see a continual learning experiment in the context of demonstrating that once place cells are found for one context, due to remapping, would producing place cells for a new context develop faster (i.e., with greater sample efficiency), since all that would be required to learn is the orthogonal transform?", "limitations": "1. As mentioned in other sections, the primary theoretical limitation is that it’s unclear how the objective function that discovers place cells (and context-dependent remappings) is biologically/ecologically valid, since it is highly supervised. An ideal objective function would be to discover one based on form of self-supervised loss. \n\n2. Another primary limitation is the fact that all environments (across contexts) are the same shape, thus it is difficult to intuit whether the findings are generalizable across differently shaped environments.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "In general, I enjoyed this paper. It was well-written, thought-provoking, and for the most part, the results are clear. The results were not overstated, and the figures (for the most part) accurately conveyed the findings of the paper. The similarity-based objective function was also intuitive, and has sparked some considerations (personally) on how to implement such similarity-based objective functions in a self-supervised manner. I particularly found the UMAP representations visualized in Fig. 4g and h to be particularly helpful in conveying the primary claims of the paper.\n\nThe intuition provided of their loss function (lines 81-84) were particularly helpful.", "weaknesses": "1. The authors allude to this primary weakness in their limitations section, but it is worth mentioning again: The supervised nature of the task makes it difficult to understand how biologically/ecologically valid their proposed objective function is. In some ways, the formulation of the task is such that the agent already needs to have an ‘oracle’ understanding of the environment it will inhabit/traverse. Equation 2 requires quite a lot of supervised/labelled information.\n\n2. It appears that the shapes of all environments (across contexts) are the same, i.e., grids of the same size. This makes it hard to infer how general the findings are across contexts with differently shaped grids. For example, would there still be 1-1 remapping (i.e., via some orthogonal transform) across contexts with differently shaped environments (e.g., square -> circle, or square of [2,2] size to [1,1] size)? It would have been interesting to see these differently shaped environments in the UMAP in Fig. 4g and h, and whether the environment-specific shapes would be preserved in that figure.\n\n3. I found Figure 5 to be confusing. For example, what do panels 5c and d add? It is known that the product of QQ^T produces the identity, since that is the definition of an orthogonal transformation. If the goal of the figure is to convey that remapping occurs via orthogonal transformation, then it might be more helpful to just illustrate the norms of the two maps (across contexts) before and after the orthogonal transform. Perhaps as a control, it would be helpful to see what the norms would look like against some baseline, such as a random orthogonal matrix (or random matrix in general). I didn’t find the visualization of the orthogonal matrices themselves to be particularly helpful or intuitive.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer very much for their well-structured and insightful feedback. In the following, we want to address some of the concerns and answer the questions.\n\nMentioned weaknesses\n1. We agree that especially the use of explicit cartesian coordinates is biologically implausible. We just want to point out that in the RNN case we are able to use only relative distances in the input (velocity signal), and solve the same objective. Furthermore, for the ground truth labels, we also just need relative distances as opposed to actual euclidean coordinates (see the $x_t-x'_t$ term in the objective function). Internal signals of relative distance traveled have been shown to exist in the brain, for example in the form of speed cells (see [Kropff et al. 2015](https://www.nature.com/articles/nature14622)) and head direction cells (see [Muller et al. 1996](https://www.sciencedirect.com/science/article/pii/S0959438896800730?via%3Dihub)). Lastly, we want to mention that we can and have also trained an RNN model that does not require the initial location as an initialization but instead is initialized with a zero-state. We included a figure showing the resulting representations in the attached PDF (see Fig. 6). In conclusion, while in the naive feed forward case we do rely on implausible explicit positions, there are ways to make our model work with only relative distances which make it a lot more biologically plausible and leads to very similar representations. Note that this model can also easily be extended to accommedate sensory inputs.\n2. That's a good point. In general, we can imagine two different cases here. First, when we use our model (for example the feed-forward one) as is and change the geometry. In that case, we would just sample from a different subset of cartesian coordinates and get similar representations to what we presented. We have included an example of a circular environment in the attached PDF (see Fig. 4). The second case would be to actually study how place cells morph / change in response to geometric changes which we believe is what the reviewer is referring to. In that case, the RNN model is more interesting and we could think about training it on one environment and then performing inference on a different geometry. Even though this was beyond the scope of this work, we agree with the reviewer in that it would be very interesting to see the changes in representations. \n3. Thanks for pointing that out. Upon reviewing the figure we agree with the reviewer in that it would be useful to actually show the norm before and after applying the transformation. We will include that in the camera-ready version. Our idea behind showing the product $QQ^T$ was that the transformation is actually learned using the objective function shown in equation 5. Therefore, we wanted to validate that it is indeed orthogonal.\n\nQuestions\n1. That's a good point. We should have made more clear how our requirements are linked to the actual objective function. We will change that in the camera-ready version. The purpose of the ReLU is to make sure we only have non-negative activations (assumption 5). By bounding we here mean that the activity stays within a reasonable range, which is fullfilled by adding the regularization term in our loss function, making sure the model learns representations with low activity levels. We included this assumption since biological systems are generally energy efficient and cannot have unbounded activity. We did not use sigmoid because it tends to produce difficulties during training like vanishing gradients. \n2. Yes, absolutely. We sampled uniformly and continuously from the grid (-1 to 1 in x and y direction), and did not just use 4 points.\n3. Thanks for pointing that out. For one batch of trajectories $(batch size, timesteps, n)$ we flattened the batch size and timesteps dimension and then computed the pairwise distance between all points, so across time and trajectories, and averaged over all of those. We will make it more clear in the camera-ready version.\n4. See point 2.\n5. In this work, we used a non-trainable decoder, similar to [Zhang et al., 1998](https://journals.physiology.org/doi/full/10.1152/jn.1998.79.2.1017?rfr_dat=cr_pub++0pubmed&url_ver=Z39.88-2003&rfr_id=ori%3Arid%3Acrossref.org). However, you are right that for a trainable one, more units should not make the decoding worse. We tried this for the RNN, and the results are shown in the attached PDF (see Fig. 2). We will make this more clear in the camera-ready version.\n\n6. The figure is trying to convey two main points:\n\t1. We can learn an orthogonal transformation using the described method (see learning objective in equation 2), apply it to our representations and get a different representation that still minimizes our objective in the same way (because distances are preserved). 2. We show that we can compute transformations that take you from one learned representation (in one context) to another. Thus, the behavior learned by the network can be summarized by an orthogonal transformation.\n7. We absolutely agree and we are planning to look into continuous learning settings in the future. It is now already possible to just apply an orthogonal transformation to get new spatial representations (a new spatial map) right away where the similarity structure is (by definition) preserved. That's what we show in Fig 5c-e. In those cases, the spatial representations do not have to be learned but can just be computed using the transformation. After showing that, one can imagine that we could learn one initial representation and then just produce new ones using orthogonal transformations instead of re-learning them. Relatedly, we identified two interesting questions which we also plan to study further in future work: 1. Is there structure in the orthogonal transformations? 2. What upstream representations are needed to induce the orthogonal transformations in this manner?", "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": "1QXoNlpgtx", "reviewer": "Reviewer_qPN7", "summary": "This paper proposes a new model of hippocampus neurons by devising a new objective function that enforces particular distances between latent codes associated with different position inputs. A Gaussian distance weighting function is proposed. Experiments show that the resulting model produces hippocampus-like position selectivity in the context of learning from single observations or trajectories and demonstrates remapping-like change in selectivity of model units with drifting context values.", "questions": "- why is there a $\\beta$ term in equations 1 and 2? Looks like it's not needed \n\n- line 265: \"restricted orthogonal transformations could prove a viable way of modeling Hippocampal remapping.\" how would this approach be useful? Please explain.", "limitations": "Limitations were discussed", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The topic of modeling hippocampal representations is important. \n\n- Paper is well written and easy to follow.\n\n- Comprehensive experiments and analyses are conducted. Figures are also nicely made.", "weaknesses": "- My biggest issue with the proposed method is that it explicitly requires position information in its training and inference which is completely non-biologically plausible. The core of the problem of learning hippocampus like representation is doing so without access to position information. There are already some models that can learn such representations through predictive coding from observations and velocities alone (Whittington et al 2020 and 2021) or with relaxed assumptions (relative positions Schaeffer et al. 2023) which substantially diminishes the significance of the current work. \n\nWhittington, J. C., Warren, J., & Behrens, T. E. (2021). Relating transformers to models and neural representations of the hippocampal formation. arXiv preprint arXiv:2112.04035.\n\n- line 27: the preceding sentences speak of selectivity to spatial and non spatial features separately. Include refs for conjunctive coding in hippocampus. \n\ne.g. in primates \nGulli, R.A., Duong, L.R., Corrigan, B.W. et al. Context-dependent representations of objects and space in the primate hippocampus during virtual navigation. Nat Neurosci 23, 103–112 (2020). https://doi.org/10.1038/s41593-019-0548-3\n\nin rodents: \nAnderson, M. I., & Jeffery, K. J. (2003). Heterogeneous modulation of place cell firing by changes in context. Journal of Neuroscience, 23(26), 8827-8835.\n\n\n- A scalar context signal is used in all experiments. Can all contexts be represented as a scalar? what are the limitations associated with this assumption? \n\n- decoding appraoch: I’m not familiar with this decoding approached and I don’t understand its implementation or logic from the description either. Why not using a simple linear readout to decode the position. \n\n- section 3.4: more explanation is needed for motivating this analysis. maybe a copule of sentences at the beginning of section 3.4 since it is not clear why that is a useful property until all the results are presented\n\n- section 3.4: what is the computational value of having the rotation invariance property? is learning an orthogonal transformation easier than learning the FF projection? Does it help with learning a task? Would an agent be able to learn tasks faster with that property? Putting this into the context of behavior would be helpful", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer very much for their insightful feedback. Below, we address some of the concerns and questions asked.\n\nWe understand your concern about the biological plausibility of using Cartesian coordinates. It is true that in the FF case we use explicit position information. However, we used the FF model because of its simplicity and it is worth mentioning that our approach also works using only relative distances in the RNN case. Here, the input is just a velocity signal inspired by biological speed cells (see [Kropff et al. 2015](https://www.nature.com/articles/nature14622)) and head direction cells (see [Muller et al. 1996](https://www.sciencedirect.com/science/article/pii/S0959438896800730?via%3Dihub)). For the labels in the objective function we only need relative distances as well ($x_t-x'_t$). Lastly, while we initialize the RNN with the initial position in the current version, we show that a model trained on longer trajectories can also learn the objective without positional initialization. This version of our model does not use sensory information. We have added a figure to the appended PDF (see Fig. 6), showcasing how the RNN without any positional information does in fact learn similar representations as the networks used in the article. In conclusion, we can say that while in the simplified case we do use explicit position information, those are not necessary to learn the objective and arrive at the presented representations. \n\nWe also understand the reviewer’s concern about the novelty of the learned spatial representations, and agree that interesting models with diverse cell types have been explored in other works. However, we would argue that our work is novel in several ways: For one, we focus specifically on the encoding properties that hippocampal representations should have, whereas most other works tend to focus more on grid cells and complex, joint models of the entorhinal-hippocampal circuit. Relatedly, our model is explainable, in the sense that learned representations can be understood directly from the objective function. Lastly, but perhaps our biggest novelty, is incorporating (possibly) non-spatial context information in the same model, and showing in an interpretable manner how Hippocampal remapping occurs as a consequence of this form of encoding. We would also highlight our finding that orthogonal transformations leave the objective invariant, as a novelty of this work. \n\nWe thank the reviewer for pointing out that we should include references for conjunctive coding in Hippocampus. That's a very good point, and we will make sure to have those in place in the camera-ready version.\n\nThe point about the context signal is very interesting. In our current version, we have always assumed a scalar signal for simplicity which might not be biologically plausible. It would mean that the entire context is encoded as a one-dimensional signal. However, our approach easily extends to higher-dimensional context signals. As we just use distances between two context signals (in the same way as we use distances in position, see equation 2 for reference) one can imagine the context to be encoded by more neurons in high-dimensional space. Likewise, one could also consider different forms of contexts signals, such as discrete ones. \n\nWe regret that our description the decoding was unclear. Essentially, we use a weighted average of the peak locations of nearby place cells to decode position. The weight of each place cell is given by its activity in the particular location to be decoded, similar to e.g. [Zhang et al., 1998](https://journals.physiology.org/doi/full/10.1152/jn.1998.79.2.1017?rfr_dat=cr_pub++0pubmed&url_ver=Z39.88-2003&rfr_id=ori%3Arid%3Acrossref.org). For reference, we also trained a linear decoder and included the result in the attached PDF (Fig. 2).\n\nRegarding your comments on section 3.4: we will make sure to give a stronger motivation at the start of this section. We see an advantage in using orthogonal transformations as they preserve distances and leave the result of the objective function unchanged. So once we have found a representation that is a solution to the objective, we can form new representations using orthogonal transformation which are still guaranteed to be a good solution. In the context of behavior what this means is that we might just have to come up with a good representation of space once and then using orthogonal transformations we could form new spatial maps without having to relearn them from scratch. These maps could be used to represent different contexts. However, whether this manifests in terms of faster learning is something we want to explore further in the future.\n\nQuestions:\n1. You are right in that the $\\beta$ parameter is not necessary for the basic idea of our approach. However, we found that using $\\beta > 0$ leads to the nice property that place cells and fields can be reused in different contexts. That is because reusing a place cell from context A in context B will make the population vectors of these two contexts somewhat similar. Essentially, $\\beta>0$ allows population vectors of different contexts to still have some non-zero similarity. Note that these (somewhat similar) population vectors can still be distinguished; they do not need be completely orthogonal, and can thus still be used for discriminating locations/contexts. We will motivate $\\beta$ more clearly in the revised manuscript. \n2. Mechanisms behind remapping are still poorly understood. We have shown that the distance preserving property is useful for the model we studied. From there, we believe there are some interesting questions that follow that could help in studying remapping : 1. Is there structure in the orthogonal transformations? 2. What (upstream) representations are needed to induce the orthogonal transformations in this manner? 3. Can this property also be shown experimentally?", "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": "j2apf1o83B", "reviewer": "Reviewer_83Ga", "summary": "This paper shows that tuning profiles similar to those of hippocampal place cells can emerge in neural networks trained to minimize a relatively minimalistic spatial encoding objective. In both a feedforward and recurrent neural network, the authors show that the trained network's units are tuned to one or a few spatial locations, similarly to place cells, and that they exhibit remapping-like phenomena when the context is changed. Interestingly, this remapping appears to be a continuous shift rather than a sudden switch. In the case of a recurrent neural network, the units' response profile resembles \"band-like\" activations that have also been observed in the hippocampal formation. They also show that the networks' learned spatial maps are rotation-equivariant, as it is possible to learn orthogonal matrices that transform the map in one context to that in another context.", "questions": "- On page 4, line 120, the authors write: \"distances in either objective function were computed between minibatch elements\" does this mean that all possible pairs of datapoints were considered? If not, were they randomly sampled?\n- It is not clear how the training procedure differs between the feedforward and recurrent models: if I understand correctly, the feedforward model received datapoints that were essentially random samples (batches of isolated x, y, c), while the recurrent model received temporally structured sequences. What was the purpose, then, of still sampling relatively smooth trajectories in training the feedforward model? Was the idea to have batches that are not IID, but somewhat \"contextualized\" (in each given batch, samples are correlated to each other) as in some continual learning experiments?\n- Partially related to the previous question: what was the underlying \"world model\" behind using a randomly sampled context scalar at each datapoint? I would understand if input datapoints were just IID samples, in that case the idea would be that the same coordinates \"mean\" different things in different contexts, leading the network to learn separate spatial maps for the different contexts. But if there is some sequential structure in the data, does this correspond to a world in which an agent moves smoothly in space, but with a constantly varying context? And thereby, even visiting the same location repeatedly leads to a completely different experience? This seems counterintuitive given that contexts tend to be stable in the real world. I'm interested in what the reasoning was behind this choice, or perhaps I might have misunderstood the training procedure.", "limitations": "Limitations are properly acknowledged. The ones that have not been discussed are listed in the \"weaknesses\" section.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper elegantly focuses on a simple loss function, applied to standard neural network architectures. This makes the underlying assumptions clear, and helps to interpret the findings.\n- The paper is clearly written, making the methods and analyses easy to understand.\n- The analyses on the trained networks are thorough and creative.", "weaknesses": "- The title is vague and not very informative about the content of the paper. If possible, I would recommend changing it, to at least include a mention of hippocampal place cells, which are the main topic. The term \"conjunctive\" is also not particularly informative, as it suggests that the focus is on the binding between contextual information and spatial locations. Rather, context information as used here seems more like a tool to probe the networks' ability to maintain orthogonal spatial maps in different contexts. So perhaps a mention of \"context-specific\" spatial maps in the title would be more faithful to the paper's content.\n- The paper misses a \"related work\" section. Of course, some related work is discussed in the introduction, but a more explicit and complete roadmap at the beginning would help, especially for non-expert readers, as the literature on the response profiles of hippocampal cells is extremely large and diverse. Throughout the paper, relevant empirical findings are cited in relation to the present results, e.g. band cells related to the RNN findings, or gradual remapping between contexts related to the context switching analysis (figure 3). Without a more thorough overview of relevant findings, however, it is hard to assess whether these response patterns would be expected to emerge in any model of the hippocampal formation, whether they are idiosyncratic findings that only appeared in a few studies, and similar questions. Adding this overview would help the reader to assess this work's significance and novelty.\n- More generally, given the huge variety of response profiles that have been reported in the hippocampus, it would be helpful if the model's limitations with respect to empirical predictions were discussed more clearly. For example, how should the finding of gradual transition between contexts be interpreted? As the authors write, it is consistent with some empirical findings and inconsistent with others. Similarly, network units display peaks at multiple locations in space when the value of $\\sigma$ is small. Is this dependence on the receptive field's size to be expected, given previous findings?\n- The role of individual components of the loss function is not investigated (through ablation experiments), or at least discussed. In particular: (1) what is the role of the exponential (Gaussian) drop-off of the loss with spatial distance? Would simply plugging into the loss function e.g. the mean square error between the estimated and true position lead to similar results? One could argue that using the exponential drop-off in the loss function is a way to hard-code the place-cell-like response profile, and thus that this response profile is not a true emergent phenomenon. I am not saying that this is the case, but it would be good to add a discussion of the exponential's role, possibly with an ablation analysis showing its importance. (2) the use of a second-order loss is one of the main intuitions of the paper, as it displays several desirable properties which the authors discuss, such as rotation and translation invariance. However, the paper is missing an explicit comparison with a first-order objective whereby the network is trained to make absolute locations close together that are close to each other in space. What would happen with such a loss? How would the resulting response profile look? Explicitly testing this alternative loss could be a nice addition, but if this is not reasonable, an explicit discussion of the role of second-order similarity would be good. (3) the role of the lower bound $\\beta$ is briefly mentioned, in relation to the hyperdimensional computing notion of nearly orthogonal vectors, but this analogy is not explored any further. Does near-orthogonality play a functional role here as well? Is it related to the separation between distinct contexts?\n- In figures 2f, 4f and 5b, the difference between the learned spatial similarity structure and the objective is plotted, but this is not commented. What is the take-home message conveyed by these plots? Since the authors chose to show this information, a few words about what it might mean would be good.\n- In figure 2g, the position decoding error (Euclidean distance) is plotted. However, it is hard to make sense of this measure without an upper bound. For example, what would be the expected error if before computing the weighted sum in (6) the units were shuffled? This is just an example of a possible error upper bound, and might not be the most pertinent, but comparing the empirical error to an upper bound would help evaluate the quality of the decoding.\n- The recurrent neural network (RNN), besides having a different architecture, also receives input encoded in a different format, as velocity vectors rather than coordinates. I imagine that the purpose of this was to help it learn to path-integrate along the trajectories. However, this difference in input format makes it hard to determine whether the different response profiles (place vs. band) are due to the architecture or the input. Velocity vectors might already provide some minimal cues for the network to learn a more spatially distributed response. Would the response profiles of the RNN look the same if it received coordinates as input, like the feedforward network? And vice versa, would the feedforward network show more band-like responses if it received velocities?\n- The band response profiles of the RNN seem to display less periodicity than those reported in Krupic et al. 2012. In that paper, the authors used a 2D Fourier analysis to determine the amount of periodicity in the band cells. Reproducing a similar analysis here would give some important insight into the significance of this finding and its relation to experimental data. Moreover, it would also be informative to report the proportion of units that displayed this kind of response profile. Was it close to that reported in Krupic et al.' s paper (44%)?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their positive and insightful comments.\n\nThank you for your perspective on the title. As for the term conjunctive, we did initially focus on spatially bounded contexts, in addition to global ones. However, we ended up limiting the scope of this paper slightly, and we will revise the title to reflect this.  \n\nYour point on relevant literature is a good one. We will, within space constraints, add an overview of relevant findings and models. \n\nOn experimental findings:\nAs you point out, it is difficult to manage all the (sometimes inconclusive) experimental findings, but we will definitely attempt to make this discussion more lucid. As for the gradual/sharp context transitions, our (possible) interpretation is that real-world contexts could be represented continuously, but that this only manifests under very controlled conditions, where the animal has access (and is attentive) to the environment. In our case, we feed in exact context signals, but in the future, it could be very interesting to let e.g. context inference be a part of the task, or adding uncertainties to the context signals.\n\nOn the role of $\\sigma$:\nIn our model, $\\sigma$ primarily governs the size of the place field (or, more specifically, the spatial scale at which population vectors should be similar). On the other hand, $\\beta$ controls the similarity for distant points (both in space or context). We find that when *$\\beta$* is somewhat large, the network is more likely to display multiple firing fields. However, as you point out, we also find that $\\sigma$ influences the number of place subfields. A smaller $\\sigma$ could alternatively be used to model place field behavior in larger environments. In these cases, multiple firing fields have been observed experimentally. See e.g. “Dorsal CA1 Hippocampal Place Cells Forms a Multi-Scale Representation of Megaspace” by Harland et al. (2021). However, in that work they observe multiple spatial scales in large spaces. How that relates to our work, could present an interesting avenue for future work. \n\nOn loss ablation/evaluation:\nThese are all excellent suggestions, which could strengthen our results. We will strive to do ablation studies, and compare with a first-order loss in the camera-ready version. Regarding your question; were you referring to the squared difference between distances (in representation and space)? If so, we would very much like to look into this simplified model in the future. We believe that decoding the position and computing the error between true and predicted locations might not lead to similar results, however, as it would likely require more constraints to avoid having the network learn a \"trivial\" Cartesian solutions. See e.g. Cueva and Wei (2018) (https://arxiv.org/abs/1803.07770) where they study this problem!\n\nOn beta/hyperdimensional computing:\nWe found that using $\\beta > 0$ leads to the nice property that place cells and fields can be reused in different contexts. That is because reusing a place cell from context A in context B will make the population vectors of these two contexts somewhat similar. Essentially, \nthis allows population vectors of different contexts to still have some non-zero similarity. Note that these (somewhat similar) population vectors can still be distinguished; they do not need be completely orthogonal, and can thus still be used for discriminating locations/contexts. We will motivate $\\beta$ more clearly in the revised manuscript. See also Fig. 1 in the attached PDF.\n\nOn figures 2f, 4f, 5b:\nWe used those plots along with the loss in order to validate our model. We show that the error between the similarity structure imposed by the objective function and the actual learned similarity structure is low over space and that the model has indeed learned the objective. We refer to these figures briefly and will make this more clear in the revised manuscript.\n\nOn decoding error:\nThanks for pointing that out. We agree that it might be hard to interpret the errors without an upper bound. We therefore included a plot that shows exactly this in Fig. 5 (of the attached PDF) where we also show the decoding error for shuffled peak locations. \n\nOn the use of different input models/inputs:\nYou are right in that we did want the model to learn to path integrate and solve the same objective. We believe that this makes our model more biologically plausible as we do not rely on explicit position information, only on velocity inputs similar to speed (see [Kropff et al. 2015](https://www.nature.com/articles/nature14622)) and head direction cells (see [Muller et al. 1996](https://www.sciencedirect.com/science/article/pii/S0959438896800730?via%3Dihub)). Just to make sure we understand the question correctly: When you say training the feedforward on velocity signals do you have in mind providing the current position along with the velocity signal at each step, and predict the next state?\n\nOn band representations:\nThis is a good point; we can determine the periodicity of the learned bands and compare with experiments in time for the camera-ready version.\n\nQuestions:\nOn distances:\nYes exactly, we computed all pairwise distances across minibatch elements (and timepoints in the RNN case).\n\nOn training:\nIn the feedforward case, all samples are IID both for the context and the spatial dimension. Here, we did not sample trajectories but sampled uniformly and continuously within the spatial and contextual range and then computed all pairwise distances. In the RNN case the context was constant along each trajectory but randomly sampled (uniformly and continuously) for each trajectory. Thank you for pointing that out. We will make this more clear in the camera-ready version.\n\nOn context signals:\nSee answer above. You are right, learning different spatial maps was exactly what we hoped for and therefore, we used IID samples. Along trajectories in the RNN case the context signal was constant. We will make sure to clarify this.", "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:
We would like to sincerely thank for their positive and knowledgable feedback. We attached a PDF including supplementary figures to address some of your major concerns. Those figures are referenced in our individual rebuttals. Notably, we have included ratemaps of an RNN model that has been trained on only relative distance, without initializing it with the initial euclidean position. In this case, we do not need explicit coordinates, making it more biologically plausible. Note that this model can also easily be extended to accommedate sensory inputs. We believe that including your suggestions in our revised camera-ready version will strengthen manuscript even further. We hope we were able to address all questions and concerns in a satisfactory manner. Of course, we are happy to discuss any additional points or answer questions that you might still have.

Meta-review: This paper introduces a novel objective function for modeling spatial representation akin to that of the hippocampus by forcing the spatially close positions to have similar latent representations. Place cell-like representations emerged in the model and context information was encoded as an orthogonal transformation within the same representational space.

The reviewers unanimously acknowledged the intriguing ideas presented in the paper, particularly the encoding of context information as an orthogonal transformation. However, several major concerns were raised by most of the reviewers:
1) Position information as input: A key issue highlighted by some of the reviewers is that the position information is provided as input to the model, making it unsurprising that the network develops a representation of this information. This is a significant criticism. The authors addressed this by including results that use relative distance instead of absolute position information. However, this solution divided the reviewers, with some agreeing and others disagreeing that it adequately addresses the criticism.

2) Remapping vs. conjunctive coding: Two reviewers pointed out that the conjunctive coding discussed in the paper is not the same as what is commonly understood in the field. They suggested changing the title to better reflect the content of the work. The authors agreed to use a more informative title in the camera-ready version.

3) Limited environment geometry: The use of only one environment geometry raises concerns about the generalizability of the results across different contexts and grid shapes.

While the paper offers several novel contributions, the reviewers noted that the claims require further experiments and controls to be fully substantiated. That being said, given that the majority of reviewers leaned toward acceptance, I recommend this paper be accepted for a poster presentation. I strongly encourage readers to carefully review the discussion between the reviewers and the authors.
Decision: Accept (poster)