Title: Learning Place Cell Representations and Context-Dependent Remapping
Venue: NeurIPS 2024
Accepted: yes
Keywords: Place cells

remapping

AI

neuroAI
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

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.

1 Introduction

Animals and humans are capable of extraordinary feats of navigation, from birds migrating by following magnetic fields [Packmor et al., 2021], to rodents navigating mazes [O'Keefe, 1976, Small, 1901] and cab drivers memorizing and traversing nearly 26000 busy London streets [FernandezVelasco and Spiers, 2023]. In the brain, navigation ability is believed to be supported by Hippocampal place cells [O'Keefe and Dostrovsky, 1971, O'Keefe and Nadel, 1978]. Place cells are known for

their tendency to only fire at one, or a few locations within a recording environment [Park et al., 2011], 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, Jeffery, 2011]. 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 a matter of debate. Some argue that place cells inherit their firing fields from upstream cell types, such as grid cells [Hafting et al., 2005, Solstad et al., 2006, Jeffery, 2011], or border cells [Barry et al., 2006, 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 representations emerge, and how they can be learned, remains poorly understood. In 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., 2020, Sorscher et al., 2022, Whittington et al., 2020, Xu et al., 2022, Dorrell et al., 2022, Schaeffer 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 on other emergent cell types, such as place cells.

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-based 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 a 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, we show that we can apply 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.

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., 2022, 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, no point or direction is special, so the above properties 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 to minimize a spatial encoding objective. Consider a neural network with population vector p ( z t ) , where each component p i ( z t ) , i = 1 , 2 , 3 , ..., n p is the firing rate of a particular (output) unit of the network at a particular location, x t (where t indexes e.g. time along a trajectory or sampled spatial locations). z t , on the other hand, is the network's latent position estimate corresponding to x t . 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

where |·| denotes the L2-norm, while β is a lower bound on the target similarity, σ is a hyperparameter that controls the scale of the learned similarity structure, and λ a hyperparameter governing an L2 activity regularization term.

Intuitively, (1) compares the (Gaussian) similarity of two points x t and x t ′ , and asserts that the corresponding population vectors at those points ( p ( z t ) and p ( z t ′ ) ) 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. 1a). β , 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-dimensional 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. B for more details on the influence of β on learned representations.

We also find that the objective in (1) is just a special case of a more general encoding objective, where distinct sources of nonspatial information can be encoded in a single population vector. Fig. 1b) illustrates how contextual signals can be represented in a 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 in 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

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. 2a)), indicating that models are capable of learning the desired similarity structure. This is also reflected in Fig. 2f), which shows an example of the agreement between the desired spatial similarity, and the corresponding learned 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. 2b), c) and d)), similar to e.g. [Thompson and Best, 1989, Alme et al., 2014]. For small values of σ and nonzero β , some units exhibit multiple place fields (e.g. Fig. 2b), 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. 2g), enabling encoding of the entire environment.

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We find that the properties of the learned representations follow readily from model hyperparameters. For example, Fig. 2b), d) and e) demonstrates that unit field size increases with increasing scale parameter σ , and that mean rates increase accordingly. Also, the number of place fields for a given unit is strongly linked to the similarity lower bound β (see App. B for more on the influence of different hyperparameters).

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 a weighted mean of peak locations based on (5), varying the number of units included in the analysis. The results, shown in Fig. 2h) 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 a few percentage points relative to the arena size). Fig. 2 i) and j) show a comparison of all three methods. Interestingly, a 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 at the edges (see Fig. 2 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 is shown in Fig. 3a), 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] (e.g. row seven), and field shifts between contexts, i.e. remapping [Muller and Kubie, 1987]. In our case, the observed remapping is not a sudden, attractor-like shift, which has been observed in place cells [Wills et al., 2005], but rather a 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. 3b). 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., 2004] (see Fig. 3c) and d)). Spatial correlations in Fig. 3c) also support the observation that units remap gradually for smooth context transitions, as correlations decay away from the diagonal.

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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 (1) 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. 4a) 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 to 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. 4d) and e)), further hinting at different functional roles. As with the feedforward model, recurrent responses accurately capture the desired similarity structure (Fig. 4f)).

To verify that the RNN is path integrating, we use both population and linear 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 early timesteps, but exhibits a 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, we find that the RNN learns lowdimensional structures representing distinct contexts. After applying UMAP to recurrent representations in different contexts, we observe that low-dimensional projections capture both the geometry of the square enclosure, and the identity of the context (Fig. 4i)). It therefore appears that the network has learned veritable cognitive maps [O'Keefe 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 distinct representations of different contexts, we turn to an interesting feature of the similarity objective. Assuming that we have a trained (feedforward) network, with representations p ( x t ) (for all x t in the domain), the loss function only depends on the norm of, and distance between population vectors (and data). Thus, the objective is invariant

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to an orthogonal transformation Q of the representation (and its inputs). In other words, we can transform the entire set of population vectors (using a global orthogonal transform), and still have a representation that minimizes (2). We demonstrate this in Fig. 5a) which shows how different global (and random) orthogonal transformations can be used to produce new representations, that preserve the similarity structure but exhibit low spatial correlation with the original representation, similar to (global) remapping. In particular, we observe that permutations induce more strongly decorrelated representations than reflections or rotations.

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. 5b) shows unit ratemaps in two dissimilar contexts, alongside ratemaps (and ratemap differences) for an orthogonal transformation taking context 1 into context 2. Notably, learned and transformed ratemaps are highly aligned. This is also shown in Fig. 5c), which demonstrates that transformed and learned representations are highly correlated. Furthermore, Fig. 5d) shows that best-fit orthogonal transformations achieve low errors (substantially lower than a random orthogonal baseline) across all learned contexts, suggesting that the orthogonal transformations can account for much of 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 of this is shown in Fig. 5e), where population vectors of a trained representation is transformed along the horizontal axis into

a 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 a result, orthogonal transformations could prove a 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 to 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 are several factors that limit its scope. For one, we consider models where label locations and preprocessed 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 a scalar context signal. However, our model could accommodate more complex context signals, such as spatially bound contexts, or 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; it 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.e. Gaussian similarity with a particular β . However, this could conceivably be addressed by exploring representational similarity in experimental data in the future.

3 Conclusion

This work introduces a similarity-based 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 to similarity in representation. Furthermore, we observe emergent global remapping as 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 to a deeper understanding of the neural basis of navigation and memory.

4 Methods

4.1 Models and training details

We trained two distinct networks to minimize the proposed objective functions (1) and (2): a feedforward network and a recurrent network, as illustrated in Fig. 1d). Models were implemented 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 p = 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 to an input size-dependent uniform distribution, following the PyTorch default [Paszke et al., 2019].

For the spatial objective (1), the input to the feedforward network consisted of minibatches of continuous Cartesian coordinates, sampled randomly and uniformly within a 2 × 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 c , i.e. input = cat ( x , c ) . Context signals were sampled uniformly in the interval c ∈ [ -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 p = 256 recurrent units. In the purely spatial case, the recurrent state at a particular time t was given by

where W rec ∈ R n p × n p is a recurrent weight matrix, W ∈ R n p × 2 an input weight matrix, and v t Cartesian velocity inputs. In the case of conjunctive encoding, the input at time t was a 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 p = 256 units, respectively, each equipped with a ReLU activation function. We also trained a recurrent network on long sequences, without explicit positional information (position-independent state initialization, only velocity input) to demonstrate that our findings hold even when assumptions on model inputs are relaxed (see App. A 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 boundary-avoiding steps successively. A step was formed by sampling heading directions according to a 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 4 π , 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, we used the Adam optimizer [Kingma and Ba, 2017] with a learning rate of 10 -4 . Unless otherwise specified, all models were trained with λ = 0 . 1 , β = 0 . 5 and σ = 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. [2021].

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, 2]). Following Leutgeb et al. [2004], we then computed the spatial correlation between unit ratemaps across contexts. Betweencontext 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.

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-timestep trajectories, and a resolution of 32 bins (in both x- and y direction) for 5 linearly spaced context signals (in the range [-2, 2]). 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, P c 1 , can be transformed into the spatial map of another context, P c 2 , we applied orthogonal transformations. Our goal was to find an orthogonal transformation Q that minimizes the Frobenius norm ∥·∥ F between the transformed spatial map from context c 1 and the spatial map in context c 2 . Here, P ∈ R n p × n 2 bins and the number of spatial bins n bins was chosen to be 128. This problem, known as the Procrustes orthogonal problem [Schönemann, 1966], is defined by

Using the Singular Value Decompositon of M = P c 2 P T c 1 , i.e. M = U Σ V T , the orthogonal matrix Q can then be computed as Q = UV T .

To explore whether global orthogonal transformations could be used to form new , distinct representations (similar to global remapping), we transformed a learned representation using different global orthogonal transforms, and computed the spatial correlation between the original and transformed representations. Specifically, we considered random rotations, reflections and permutations as possible candidate transformations. To form n p -dimensional rotation matrices, we used the spe-

cial_ortho_group from the SciPy library [Virtanen et al., 2020], while reflection matrices were formed by interchanging two columns of such a rotation matrix. Random permutation matrices were formed simply as permutations of the identity.

To determine whether orthogonal transformations could also be used to generate new representations continuously , we conducted a simple numerical experiment, wherein existing representations were transformed iteratively into previously unexplored regions (akin to 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.

where S = 1 2 ( A -A T ) ensures that e tS is orthogonal and p 0 is a learned population vector. For simplicity, we take A to be a randomly sampled permutation matrix. When extending the representation by a distance of | ∆ x | in one spatial dimension, we transform a particular p i by setting

which ensures that the similarity structure in the direction of transformation adheres approximately to (1); see App. C for details. We applied (4) iteratively to 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, | ∆ x | = 2 /n x , with n x = 32 being the number of bins in the horizontal direction.

4.4 Decoding

In our 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. [1998]. The weights correspond to the activity levels of the respective units at that location. The decoded position ˆ r ( x ) is thus given by:

where ˆ r ( x ) denotes the decoded position at x , p i ( x ) the activity of unit i at x , and r i 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 are included in the decoding.

As an alternative scheme that exploits the similarity structure of the learned representations, we implemented a simple population decoding procedure. This was done by generating and saving a "memory" of M population vectors { p i m } i M =1 and corresponding locations { r i m } i M =1 , and decoding subsequent locations as 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).

As a baseline, we also trained a linear decoder to predict Cartesian coordinates given network representations p . Decoder weights were initialized according to a 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 n scheme we used n = 44 based on the minimum decoding error across 10 models. The linear decoder was trained on population vectors of uniformly sampled locations. Finally, we evaluated all methods on a grid of 129 x 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 all findings and figures is available at https://github.com/bioAI-Oslo/ConjunctiveRepresentations .

References

Charlotte B. Alme, Chenglin Miao, Karel Jezek, Alessandro Treves, Edvard I. Moser, and May-Britt Moser. Place Cells in the Hippocampus: Eleven Maps for Eleven Rooms. Proceedings of the National Academy of Sciences , 111(52):18428-18435, December 2014. ISSN 0027-8424, 1091-6490. doi: 10.1073/pnas.1421056111. URL https://pnas.org/doi/full/10.1073/pnas.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 Beattie, Stig Petersen, Amir Sadik, Stephen Gaffney, Helen King, Koray Kavukcuoglu, Demis Hassabis, Raia Hadsell, and Dharshan Kumaran. Vector-Based Navigation Using Grid-like Representations in Artificial Agents. Nature , 557 (7705):429-433, May 2018. ISSN 0028-0836, 1476-4687. doi: 10.1038/s41586-018-0102-6. URL http: //www.nature.com/articles/s41586-018-0102-6 .

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: 10.1515/REVNEURO.2006.17.1-2.71. URL https: //www.degruyter.com/document/doi/10.1515/REVNEURO.2006.17.1-2.71/html .

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. URL http://arxiv.org/ abs/1803.07770 .

William Dorrell, Peter E. Latham, Timothy E. J. Behrens, and James C. R. Whittington. Actionable Neural Representations: Grid Cells from Minimal Constraints, September 2022. URL http://arxiv.org/abs/ 2209.15563 . arXiv:2209.15563 [q-bio].

Pablo Fernandez-Velasco and Hugo J. Spiers. Wayfinding across ocean and tundra: what traditional cultures teach us about navigation. Trends in Cognitive Sciences , page S1364661323002516, October 2023. ISSN 13646613. doi: 10.1016/j.tics.2023.09.004. URL https://linkinghub.elsevier.com/retrieve/pii/ S1364661323002516 .

Torkel Hafting, Marianne Fyhn, Sturla Molden, May-Britt Moser, and Edvard I. Moser. Microstructure of a spatial map in the entorhinal cortex. Nature , 436(7052):801-806, 2005. ISSN 0028-0836, 1476-4687. doi: 10.1038/nature03721. URL http://www.nature.com/articles/nature03721 .

Bruce Harland, Marco Contreras, Madeline Souder, and Jean-Marc Fellous. Dorsal CA1 hippocampal place cells form a multi-scale representation of megaspace. Current Biology , 31(10):2178-2190.e6, May 2021. ISSN 09609822. doi: 10.1016/j.cub.2021.03.003. URL https://linkinghub.elsevier.com/retrieve/pii/ S0960982221003420 .

Kathryn J. Jeffery. Place Cells, Grid Cells, Attractors, and Remapping. Neural Plasticity , 2011:1-11, 2011. ISSN 2090-5904, 1687-5443. doi: 10.1155/2011/182602. URL http://www.hindawi.com/journals/ np/2011/182602/ .

Pentti Kanerva. Hyperdimensional Computing: An Introduction to Computing in Distributed Representation with High-Dimensional Random Vectors. Cognitive Computation , 1(2):139-159, June 2009. ISSN 1866-9956, 1866-9964. doi: 10.1007/s12559-009-9009-8. URL http://link.springer.com/10.1007/ s12559-009-9009-8 .

Diederik P. Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization. arXiv: 1412.6980 [cs], January 2017. URL http://arxiv.org/abs/1412.6980 .

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.1222403. URL https://www.science.org/doi/10.1126/science.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/fnbeh.2017.00253. URL http://journal.frontiersin.org/article/10.3389/fnbeh. 2017.00253/full .

Quoc V. Le, Navdeep Jaitly, and Geoffrey E. Hinton. A Simple Way to Initialize Recurrent Networks of Rectified Linear Units. arXiv: 1504.00941 [cs] Issue: arXiv:1504.00941 Publisher: arXiv, April 2015. URL http://arxiv.org/abs/1504.00941 .

Jill K. Leutgeb, Stefan Leutgeb, Alessandro Treves, Retsina Meyer, Carol A. Barnes, Bruce L. McNaughton, May-Britt Moser, and Edvard I. Moser. Progressive transformation of hippocampal neuronal representations in 'morphed' environments. Neuron , 48(2):345-358, October 2005a. ISSN 0896-6273. doi: 10.1016/j. neuron.2005.09.007. URL http://dx.doi.org/10.1016/j.neuron.2005.09.007 .

Stefan Leutgeb, Jill K. Leutgeb, Alessandro Treves, May-Britt Moser, and Edvard I. Moser. Distinct Ensemble Codes in Hippocampal Areas CA3 and CA1. Science , 305(5688):1295-1298, August 2004. ISSN 0036-8075, 1095-9203. doi: 10.1126/science.1100265.

Stefan Leutgeb, Jill K. Leutgeb, Carol A. Barnes, Edvard I. Moser, Bruce L. McNaughton, and May-Britt Moser. Independent Codes for Spatial and Episodic Memory in Hippocampal Neuronal Ensembles. Science , 309(5734):619-623, July 2005b. ISSN 0036-8075, 1095-9203. doi: 10.1126/science.1114037. URL https://www.science.org/doi/10.1126/science.1114037 .

Leland McInnes, John Healy, Nathaniel Saul, and Lukas Großberger. Umap: Uniform manifold approximation and projection. Journal of Open Source Software , 3(29):861, 2018. doi: 10.21105/joss.00861. URL https://doi.org/10.21105/joss.00861 .

Ru Muller and Jl Kubie. The effects of changes in the environment on the spatial firing of hippocampal complexspike cells. The Journal of Neuroscience , 7(7):1951-1968, July 1987. ISSN 0270-6474, 1529-2401. doi: 10.1523/JNEUROSCI.07-07-01951.1987. URL https://www.jneurosci.org/lookup/doi/10.1523/ JNEUROSCI.07-07-01951.1987 .

O'Keefe and J. Dostrovsky. The Hippocampus as a Spatial Map. Preliminary Evidence from Unit Activity in the Freely-Moving Rat. Brain Research , 34(1):171-175, November 1971. ISSN 00068993. doi: 10.1016/0006-8993(71)90358-1. URL https://linkinghub.elsevier.com/retrieve/pii/ 0006899371903581 .

John O'Keefe. Place units in the hippocampus of the freely moving rat. Experimental Neurology , 51(1): 78-109, January 1976. ISSN 00144886. doi: 10.1016/0014-4886(76)90055-8. URL https://linkinghub. elsevier.com/retrieve/pii/0014488676900558 .

John O'Keefe and Neil Burgess. Geometric Determinants of the Place Fields of Hippocampal Neurons. Nature , 381(6581):425-428, May 1996. ISSN 0028-0836, 1476-4687. doi: 10.1038/381425a0. URL http://www.nature.com/articles/381425a0 .

John O'Keefe and Lynn Nadel. The Hippocampus as a Cognitive Map . Oxford University Press, Oxford, 1978. ISBN 0-19-857206-9.

Florian Packmor, Dmitry Kishkinev, Flora Bittermann, Barbara Kofler, Clara Machowetz, Thomas Zechmeister, 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 Experimental Biology , 224(22):jeb243337, November 2021. ISSN 0022-0949, 1477-9145. doi: 10.1242/ jeb.243337. URL https://journals.biologists.com/jeb/article/224/22/jeb243337/273480/ A-magnet-attached-to-the-forehead-disrupts .

EunHye Park, Dino Dvorak, and André A. Fenton. Ensemble Place Codes in Hippocampus: CA1, CA3, and Dentate Gyrus Place Cells Have Multiple Place Fields in Large Environments. PLoS ONE , 6(7):e22349, July 2011. ISSN 1932-6203. doi: 10.1371/journal.pone.0022349. URL https://dx.plos.org/10.1371/ journal.pone.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: 10.48550/ARXIV.1912.01703. URL https://arxiv.org/abs/1912.01703 . Publisher: arXiv Version Number: 1.

Markus Borud Pettersen, Vemund Sigmundson Schøyen, Anders Malthe-Sørenssen, and Mikkel Elle Lepperød. Decoding the Cognitive map: Learning place cells and remapping, March 2024. Publisher: arXiv Version Number: 1.

Alexei Samsonovich and Bruce L. McNaughton. Path Integration and Cognitive Mapping in a Continuous Attractor Neural Network Model. The Journal of Neuroscience , 17(15):5900-5920, August 1997. ISSN 0270-6474, 1529-2401. doi: 10.1523/JNEUROSCI.17-15-05900.1997. URL https://www.jneurosci. org/lookup/doi/10.1523/JNEUROSCI.17-15-05900.1997 .

Rylan Schaeffer, Mikail Khona, Tzuhsuan Ma, Cristóbal Eyzaguirre, Sanmi Koyejo, and Ila Rani Fiete. SelfSupervised Learning of Representations for Space Generates Multi-Modular Grid Cells. 2023. doi: 10.48550/ ARXIV.2311.02316. URL https://arxiv.org/abs/2311.02316 . Publisher: arXiv Version Number: 1.

Peter H. Schönemann. A generalized solution of the orthogonal procrustes problem. Psychometrika , 31(1):1-10, March 1966. ISSN 1860-0980. doi: 10.1007/BF02289451.

Vemund Schøyen, Markus Borud Pettersen, Konstantin Holzhausen, Marianne Fyhn, Anders Malthe-Sørenssen, and Mikkel Elle Lepperød. Coherently remapping toroidal cells but not Grid cells are responsible for path integration in virtual agents. iScience , 26(11):108102, November 2023. ISSN 25890042. doi: 10.1016/j.isci. 2023.108102. URL https://linkinghub.elsevier.com/retrieve/pii/S258900422302179X .

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://dx.doi.org/10.2307/ 1412534 .

Trygve Solstad, Edvard I. Moser, and Gaute T. Einevoll. From Grid Cells to Place Cells: A Mathematical Model. Hippocampus , 16(12):1026-1031, December 2006. ISSN 10509631, 10981063. doi: 10.1002/hipo.20244. URL https://onlinelibrary.wiley.com/doi/10.1002/hipo.20244 .

Ben Sorscher, Gabriel C. Mel, Samuel A. Ocko, Lisa M. Giocomo, and Surya Ganguli. A unified theory for the computational and mechanistic origins of grid cells. Neuron , page S0896627322009072, October 2022. ISSN 08966273. doi: 10.1016/j.neuron.2022.10.003. URL https://linkinghub.elsevier.com/retrieve/ pii/S0896627322009072 .

Lt Thompson and Pj Best. Place cells and silent cells in the hippocampus of freely-behaving rats. The Journal of Neuroscience , 9(7):2382-2390, July 1989. ISSN 0270-6474, 1529-2401. doi: 10.1523/JNEUROSCI. 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 to allocentric understanding in mammalian brains. preprint, Neuroscience, November 2020. URL http://biorxiv.org/lookup/doi/10.1101/2020.11. 11.378141 .

Pauli Virtanen, Ralf Gommers, Travis E. 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, C J Carey, ˙ Ilhan Polat, Yu Feng, Eric W. Moore, Jake VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E. A. Quintero, Charles R. 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 in Python. Nature Methods , 17:261-272, 2020. doi: 10.1038/s41592-019-0686-2.

James C.R. Whittington, Timothy H. Muller, Shirley Mark, Guifen Chen, Caswell Barry, Neil Burgess, and Timothy E.J. Behrens. The Tolman-Eichenbaum Machine: Unifying Space and Relational Memory through Generalization in the Hippocampal Formation. Cell , 183(5):1249-1263.e23, November 2020. ISSN 00928674. doi: 10.1016/j.cell.2020.10.024. URL https://linkinghub.elsevier.com/retrieve/ pii/S009286742031388X .

Tom J. Wills, Colin Lever, Francesca Cacucci, Neil Burgess, and John O'Keefe. Attractor dynamics in the hippocampal representation of the local environment. Science , 308(5723):873-876, 2005. doi: 10.1126/ science.1108905. URL https://www.science.org/doi/abs/10.1126/science.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. _eprint: 2210.02684.

Kechen Zhang, Iris Ginzburg, Bruce L. McNaughton, and Terrence J. Sejnowski. Interpreting neuronal population activity by reconstruction: Unified framework with application to hippocampal place cells. Journal of Neurophysiology , 79(2):1017-1044, 1998. doi: 10.1152/jn.1998.79.2.1017. URL https://doi.org/10. 1152/jn.1998.79.2.1017 . PMID: 9463459.

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 to train networks. In the real world, agents do not have access to exact coordinates labeling the environment, which could raise concerns about the transferability of our results to biological networks. We therefore train a recurrent neural network without explicit positional information, to show that our findings hold even when relaxing assumptions on available inputs.

Concretely, we consider, as before, an RNN with n p = 256 recurrent units. For this network, however, we initialize the network using a trainable initial state that does not depend on data. We furthermore train this network on long sequences ( T = 500 timesteps), and compute similarities over trajectory time, i.e. within a trajectory, rather than aggregating over multiple smaller trajectories. The input to the network is just Cartesian velocities along such trajectories, and similarities used for training are computed only using relative distances.

The results in Fig. A1 show that the model also learns place and band-like representations and is able to learn the objective (see Fig. A1 a), b) and d)), similar to our other findings. As with other networks, we can also decode positions from this network using either a simple linear decoder, or our population decoding scheme, demonstrating that the learned representations are decodable (Fig. A1c)). Here, we 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 timesteps-trajectories, for 5000 training steps using the Adam optimizer, a learning rate of 10 -3 , and a batch size of 64. Additionally, we used 100 trajectories of 500 timesteps to form the memory vectors with a 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 lack of positional information.

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B Loss ablation and effects of β

We have previously noted that learned place field sizes are governed by the scale parameter σ . To further explore the influence of hyperparameters on learned representations, we performed an ablation study, wherein we train feedforward networks with λ and β ablated. We also explore the effects of changing the similarity measure, from Gaussian (depending on the square of the distance between locations/representations), to exponential in the Euclidean distance, i.e. sim ( a , b ) ∝ e -| a -b | .

The results are shown in Fig. A2. With no activity regularization ( λ = 0 ), units exhibit multiple smaller, but more irregular place fields (Fig. A2a)). Stronger population vector separation ( β = 0 ) leads to highly unimodal representations, and larger place fields (Fig. A2b)). Notably, when both β and λ are set to zero (Fig. A2c)), units no longer display place-like tuning, but rather appear to change linearly across the arena, possibly reflecting the positional input to 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. A2d)), 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 stripe-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 β in 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. A3a), 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 are 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 n p -sphere (Fig. A3b), 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 a non-zero similarity threshold is sensible.

From a practical standpoint, a non-zero β allows for more expressive representations, as representations 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 β = 0 and sufficiently distant spatial locations, this situation would incur a loss, whereas a nonzero β 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. 2i).

We indeed find that nonzero β allows for more unit reuse across space. This can be seen in Fig. A3c), which demonstrates that higher values of β (and smaller σ ) lead to the emergence of units with multiple place fields (see e.g. the case where σ = 0 . 1 , β = 0 . 5) . We also observe a similar trend over different contexts, which is illustrated in Fig. A3d). In this case, units trained with β = 0 are only active around a narrow set of context values, whereas for β > 0 , units are active across a wide range of distinct contexts, and place fields shift between contexts, similar to Hippocampal remapping.

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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

where t ∈ R is a parameter, and we choose S ∈ R n p × n p to be a skew-symmetric matrix, which makes Q ( t ) orthogonal. To extend an existing (purely) spatial representation, we need to transform it in a manner that respects (1). 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

where p is the population vector we wish to extend from. Using the Taylor expansion of the exponential map

and keeping terms up to second order, we have that

and we can solve for the parameter t approximately. Note that S is skew-symmetric, so p T S p = 0 , and

Notably, we can solve for the numerator term required to match the desired similarity structure in (1), i.e. demand

and insert into the expression for t , yielding

We use this expression for the transformation parameter to extend a place-like representation horizontally into a 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 not ensure that the generated representations adhere to desired similarity structure in the vertical direction. However, we observe empirically that this is approximately the case (see similarity structure in Fig. 5e)), and hope to study the general case more closely in the future.

D Context-dependence of recurrent representations

Fig. A4 shows ratemaps for 10 randomly selected recurrent units of a network trained to minimize (2) 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.

NeurIPS Paper Checklist

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Answer: [Yes]

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\n- 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.\n\nWEAKNESSES:\n- 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%)?\n\nQUESTIONS:\n- 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.\n\nLIMITATIONS:\nLimitations are properly acknowledged. The ones that have not been discussed are listed in the \"weaknesses\" section.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\n- 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.\n\nWEAKNESSES:\n- 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\n\nQUESTIONS:\n- 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.\n\nLIMITATIONS:\nLimitations were discussed", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-09"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\nIn 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.\n\nWEAKNESSES:\n1. 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.\n\nQUESTIONS:\n1. 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?\n\nLIMITATIONS:\n1. 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.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-09"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\n1. 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.\n\nWEAKNESSES:\n1. 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.\n\nQUESTIONS:\nPlease refer to weaknesses 2,3\n\nLIMITATIONS:\nThe authors have adequately discussed limitations of their work.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nHow 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.\n\nSTRENGTHS:\nOverall, 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.\n\nWEAKNESSES:\n- 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.\n\nQUESTIONS:\nQ1. 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?\n\nLIMITATIONS:\nSee weaknesses.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-12"}