Original: Testing geometric representation hypotheses from simulated place cell recordings
Revised: Testing geometric representation hypotheses from simulated place cell recordings

Original: Editors: Sophia Sanborn, Christian Shewmake, Simone Azeglio, Arianna Di Bernardo, Nina Miolane
Revised: Editors: Sophia Sanborn, Christian Shewmake, Simone Azeglio, Arianna Di Bernardo, Nina Miolane

Original: Abstract
Revised: Abstract

Original: Hippocampal place cells can encode spatial locations of an animal in physical or taskrelevant spaces.
Revised: Hippocampal place cells can encode spatial locations of an animal in physical or taskrelevant spaces.

Original: We simulated place cell populations that encoded either Euclidean- or graph-based positions of a rat navigating to goal nodes in a maze with a graph topology, and used manifold learning methods such as UMAP and Autoencoders (AE) to analyze these neural population activities.
Revised: We simulated place cell populations that encoded either Euclidean- or graph-based positions of a rat navigating to goal nodes in a maze with a graph topology, and used manifold learning methods such as UMAP and Autoencoders (AE) to analyze these neural population activities.

Original: The structure of the latent spaces learned by the AE reflects their true geometric structure, while PCA fails to do so and UMAP is less robust to noise.
Revised: The structure of the latent spaces learned by the AE reflects their true geometric structure, while PCA fails to do so and UMAP is less robust to noise.

Original: Our results support future applications of AE architectures to decipher the geometry of spatial encoding in the brain.
Revised: Our results support future applications of AE architectures to decipher the geometry of spatial encoding in the brain

Original: 1. Introduction
Revised: 1. Introduction

Original: Animals including humans navigate by representing spatial locations in the combined neural activity of the hippocampal formation.
Revised: Animals including humans navigate by representing spatial locations in the combined neural activity of the hippocampal formation.

Original: Place cells represent spatial locations by firing when the animal is at a specific location in physical Euclidean space (at particular ( x, y ) coordinates) (O’Keefe and Dostrovsky, 1971; O’Keefe and Nadel, 1978).
Revised: Place cells represent spatial locations by firing when the animal is at a specific location in physical Euclidean space (at particular ( x, y ) coordinates) (O’Keefe and Dostrovsky, 1971; O’Keefe and Nadel, 1978).

Original: However, recent work has shown that when animals perform complex tasks, these neurons represent ‘places’ in spaces that are task-relevant but not always Euclidean (Behrens et al., 2018;
Revised: However, recent work has shown that when animals perform complex tasks, these neurons represent ‘places’ in spaces that are task-relevant but not always Euclidean (Behrens et al., 2018;

Original: Whittington et al., 2020).
Revised: Whittington et al., 2020).

Original: We are interested in developing a methodology that can resolve this apparent divide about the role of place cells.
Revised: We are interested in developing a methodology that can resolve this apparent divide about the role of place cells.

Original: To do so, we explore spatial navigation in the context of graphs, as simple topological objects defined only by the adjacency of a finite number of nodes connected by edges.
Revised: To do so, we explore spatial navigation in the context of graphs, as simple topological objects defined only by the adjacency of a finite number of nodes connected by edges.

Original: In a physical maze apparatus, we will present various routes that share a common underlying graph topology to a moving rat while recording from hippocampal place cells.
Revised: In a physical maze apparatus, we will present various routes that share a common underlying graph topology to a moving rat while recording from hippocampal place cells.

Original: Across trials, the topology of graphs remains the same whereas their physical Euclidean locations change.
Revised: Across trials, the topology of graphs remains the same whereas their physical Euclidean locations change.

Original: In this work, we simulate the activity of place cells whose firing rates conform to one of two competing hypotheses: they represent locations in
Revised: In this work, we simulate the activity of place cells whose firing rates conform to one of two competing hypotheses: they represent locations in

Original: Euclidean space as classically defined, or they represent locations in graph space, reflecting the structure of the task.
Revised: Euclidean space as classically defined, or they represent locations in graph space, reflecting the structure of the task.

Original: Upon using manifold learning methods such as UMAP and Autoencoder networks to encode the simulated neural activity in a latent space, we examined how such a latent representation can be identified with the two hypotheses.
Revised: Upon using manifold learning methods such as UMAP and Autoencoder networks to encode the simulated neural activity in a latent space, we examined how such a latent representation can be identified with the two hypotheses.

Original: 2. Methods 2.1. Behavioural and neural data generation
Revised: 2. Methods 2.1. Behavioural and neural data generation

Original: Our neurophysiological experiments will utilize a novel dynamic maze apparatus, which consists of a 7 × 7 grid of octagonal tiles (Figure 1a).
Revised: Our neurophysiological experiments will utilize a novel dynamic maze apparatus, which consists of a 7 × 7 grid of octagonal tiles (Figure 1a).

Original: Any desired maze configuration can be formed by opening and closing specific gates (Figure S1).
Revised: Any desired maze configuration can be formed by opening and closing specific gates (Figure S1).

Original: In the maze, rats will navigate to goal positions to receive reward.
Revised: In the maze, rats will navigate to goal positions to receive reward.

Original: We simulated these goal-seeking behavioural trajectories and corresponding place cell firing data.
Revised: We simulated these goal-seeking behavioural trajectories and corresponding place cell firing data.

Original: For a given configuration, we modeled the motion of a rat by setting up a goal position g ∈ R 2 and simulating a discrete random walk that is biased towards g (detailed in Appendix A).
Revised: For a given configuration, we modeled the motion of a rat by setting up a goal position g ∈ R 2 and simulating a discrete random walk that is biased towards g (detailed in Appendix A).

Original: We simulated the time series of the neural state associated with such a trajectory as the firing rate of N neurons.
Revised: We simulated the time series of the neural state associated with such a trajectory as the firing rate of N neurons.

Original: To do so, we first assigned a specific field center position c to each neuron.
Revised: To do so, we first assigned a specific field center position c to each neuron.

Original: For a given position x of the rat, the firing rate f
Revised: For a given position x of the rat, the firing rate f

Original: ( x, c ) is
Revised: ( x, c ) is

Original: [Equation]
Revised: [Equation]

Original: where d is a distance function over the maze, F max > 0 is the maximum firing rate of the neuron, σ > 0, ε ≥ 0 and N (0 , 1) > 0 is Gaussian white noise.
Revised: where d is a distance function over the maze, F max > 0 is the maximum firing rate of the neuron, σ > 0, ε ≥ 0 and N (0 , 1) > 0 is Gaussian white noise.

Original: In practice, we used
Revised: In practice, we used

Original: F max = 40 Hz , σ = 0 .
Revised: F max = 40 Hz , σ = 0 .

Original: 3 and ε = 0 or ε = 0 .
Revised: 3 and ε = 0 or ε = 0 .

Original: 1 (for additive noise).
Revised: 1 (for additive noise).

Original: σ parameterizes the place field size.
Revised: σ parameterizes the place field size.

Original: Parameters were chosen so that the fields have biologically realistic firing rates and areas (Huxter et al., 2003; Neher et al., 2017).
Revised: Parameters were chosen so that the fields have biologically realistic firing rates and areas (Huxter et al., 2003; Neher et al., 2017).

Original: The distance function d ( . ) depends on the geometry that the place fields represent, which is derived from one of the two hypotheses:
Revised: The distance function d ( . ) depends on the geometry that the place fields represent, which is derived from one of the two hypotheses:

Original: Euclidean hypothesis : Neurons fire according to 2D Euclidean distance from the field center, such that d = ∥ · ∥ 2 , or d ( x, y ) =
Revised: Euclidean hypothesis : Neurons fire according to 2D Euclidean distance from the field center, such that d = ∥ · ∥ 2 , or d ( x, y ) =

Original: (cid:112) ||
Revised: (cid:112) ||

Original: x − y
Revised: x − y

Original: ||
Revised: ||

Original: Graph hypothesis : Neurons fire locally according to the Euclidean distances as in 1, but within a range defined by the number of tiles connecting x and y .
Revised: Graph hypothesis : Neurons fire locally according to the Euclidean distances as in 1, but within a range defined by the number of tiles connecting x and y .

Original: That is, d ( x, y ) =
Revised: That is, d ( x, y ) =

Original: (cid:112) ||
Revised: (cid:112) ||

Original: x − y
Revised: x − y

Original: || 

Revised: ||

Original: if d g ( x, y ) ≤ 1 and ∞ elsewhere, where d g ( x, y ) is the number of tiles in the shortest path along the graph between x and y (Figure 1b, second panel).
Revised: if d g ( x, y ) ≤ 1 and ∞ elsewhere, where d g ( x, y ) is the number of tiles in the shortest path along the graph between x and y (Figure 1b, second panel).

Original: Under the Euclidean hypothesis, the place field centers c
Revised: Under the Euclidean hypothesis, the place field centers c

Original: are distributed randomly across
Revised: are distributed randomly across

Original: the
Revised: the

Original: whole
Revised: whole

Original: platform (
Revised: platform (

Original: Figure
Revised: Figure

Original: 1b)
Revised: 1b)

Original: in
Revised: in

Original: a
Revised: a

Original: particular
Revised: particular

Original: maze configuration,
Revised: maze configuration,

Original: and remain in the same
Revised: and remain in the same

Original: physical position
Revised: physical position

Original: across
Revised: across

Original: configurations.
Revised: configurations.

Original: Under
Revised: Under

Original: the graph hypothesis, the physical positions of the place fields change across configurations, to preserve the same relative position in the graph (see Appendix B).
Revised: the graph hypothesis, the physical positions of the place fields change across configurations, to preserve the same relative position in the graph (see Appendix B).

Original: 2.2. Autoencoder design
Revised: 2.2. Autoencoder design

Original: We applied Autoencoders (AEs) (Tschannen et al., 2018) to our datasets.
Revised: We applied Autoencoders (AEs) (Tschannen et al., 2018) to our datasets.

Original: The AEs take a neural state S ∈ R N as input (rescaled firing rate of N neurons ∈ [0 , 1] N ) and learn an embedding in a 3-dimensional latent space (sufficient to capture the 2D Euclidean space and any graph topology in our apparatus).
Revised: The AEs take a neural state S ∈ R N as input (rescaled firing rate of N neurons ∈ [0 , 1] N ) and learn an embedding in a 3-dimensional latent space (sufficient to capture the 2D Euclidean space and any graph topology in our apparatus).

Original: The encoder and decoder are fully connected networks, with 4 hidden layers of sizes 64, 32, 16 and 8, and leaky ReLU activation functions.
Revised: The encoder and decoder are fully connected networks, with 4 hidden layers of sizes 64, 32, 16 and 8, and leaky ReLU activation functions.

Original: The encoder and decoder output layers are respectively linear and sigmoidal.
Revised: The encoder and decoder output layers are respectively linear and sigmoidal.

Original: The data is split into training (70%), validation (20%) and testing (10%) sets, with training performed in batches of 32 data points, using the ADAM optimizer.
Revised: The data is split into training (70%), validation (20%) and testing (10%) sets, with training performed in batches of 32 data points, using the ADAM optimizer.

Original: 2.3. Implementation and Code Availability
Revised: 2.3. Implementation and Code Availability

Original: The code for simulations was developed using Python 3.8.0.
Revised: The code for simulations was developed using Python 3.8.0.

Original: The Autoencoder design and training was performed using PyTorch 1.11.0 and PyTorch-Lightning 1.6.5.
Revised: The Autoencoder design and training was performed using PyTorch 1.11.0 and PyTorch-Lightning 1.6.5.

Original: The code and datasets are available in our Github repository.
Revised: The code and datasets are available in our Github repository.

Original: 3. Results 3.1. Simulating place field cells firing according to different representations
Revised: 3. Results 3.1. Simulating place field cells firing according to different representations

Original: We considered 25 maze configurations (3 examples shown in Figure 1a).
Revised: We considered 25 maze configurations (3 examples shown in Figure 1a).

Original: While different in shape, these mazes describe the same graph topology, which consists of a home tile connected to a central choice tile that connects to 3 other goal tiles.
Revised: While different in shape, these mazes describe the same graph topology, which consists of a home tile connected to a central choice tile that connects to 3 other goal tiles.

Original: The goal tiles sometimes overlap across configurations in Euclidean space (e.g. goals 2,3,2 in configurations 1,2,3, resp.), which allows for dissociating if the neural representation conforms to the Euclidean or graph hypothesis.
Revised: The goal tiles sometimes overlap across configurations in Euclidean space (e.g. goals 2,3,2 in configurations 1,2,3, resp.), which allows for dissociating if the neural representation conforms to the Euclidean or graph hypothesis.

Original: To compare the firing rates produced under these two hypotheses, we considered two sets of N = 100 place cells sampled across the grid.
Revised: To compare the firing rates produced under these two hypotheses, we considered two sets of N = 100 place cells sampled across the grid.

Original: Our final dataset thus consisted of the firing rates of 100 neurons for 2 trajectories × 3 goal locations × 25 maze configurations × 2 hypotheses × 2 noise conditions ( ε = 0 and ε = 0 .
Revised: Our final dataset thus consisted of the firing rates of 100 neurons for 2 trajectories × 3 goal locations × 25 maze configurations × 2 hypotheses × 2 noise conditions ( ε = 0 and ε = 0 .

Original: 1) (e.g. Figure 1c).
Revised: 1) (e.g. Figure 1c).

Original: Using an Auto-Encoder to represent neural firing rate geometry
Revised: Using an Auto-Encoder to represent neural firing rate geometry

Original: We separately trained two AEs, described in Section 2.2, on the graph and Euclidean datasets without noise ( ε = 0), to learn a representation of the neural states within a latent space (shown in Figure S2).
Revised: We separately trained two AEs, described in Section 2.2, on the graph and Euclidean datasets without noise ( ε = 0), to learn a representation of the neural states within a latent space (shown in Figure S2).

Original: As we set the dimension of the latent space to 3, we found that after running standard PCA, 96 . 3% and 95 .
Revised: As we set the dimension of the latent space to 3, we found that after running standard PCA, 96 . 3% and 95 .

Original: 8% of the variance of the data in the graph and
Revised: 8% of the variance of the data in the graph and

Original: Euclidean AE latent space were explained by the first two principal components (PC’s), suggesting that the rat’s motion on the 2D platform is well captured.
Revised: Euclidean AE latent space were explained by the first two principal components (PC’s), suggesting that the rat’s motion on the 2D platform is well captured.

Original: We further confirmed this result by visualizing the latent states associated with the trajectories on the 2 PC’s in Figure 2a-b.
Revised: We further confirmed this result by visualizing the latent states associated with the trajectories on the 2 PC’s in Figure 2a-b.

Original: For the Euclidean data, the goal location shared across all configurations appears as a single location in the latent space (see Figure 2a).
Revised: For the Euclidean data, the goal location shared across all configurations appears as a single location in the latent space (see Figure 2a).

Original: Similarly, for the graph data, the same goal, although being located at different positions across configurations, appears at relatively close positions in the latent space (see Figure 2b).
Revised: Similarly, for the graph data, the same goal, although being located at different positions across configurations, appears at relatively close positions in the latent space (see Figure 2b).

Original: Finally, we compared in Figure 2c the pairwise Euclidean distances in the latent space (for 100 sets of 10 4 sampled pairs of data points) to the corresponding pairwise Euclidean and normalized graph distances in the maze (for a definition of the normalized graph distance, see Appendix C).
Revised: Finally, we compared in Figure 2c the pairwise Euclidean distances in the latent space (for 100 sets of 10 4 sampled pairs of data points) to the corresponding pairwise Euclidean and normalized graph distances in the maze (for a definition of the normalized graph distance, see Appendix C).

Original: With a better correlation achieved when using the Euclidean (normalized graph) distance for the
Revised: With a better correlation achieved when using the Euclidean (normalized graph) distance for the

Original: Euclidean (graph) dataset, this comparison shows that the standard metric in the learned latent space reflects the choice of the neural representation.
Revised: Euclidean (graph) dataset, this comparison shows that the standard metric in the learned latent space reflects the choice of the neural representation.

Original: In contrast, we failed to observe such a separation using PCA instead of AEs (see Figure S5).
Revised: In contrast, we failed to observe such a separation using PCA instead of AEs (see Figure S5).

Original: 3.3. Using UMAP to represent neural firing rate geometry
Revised: 3.3. Using UMAP to represent neural firing rate geometry

Original: We used UMAP (McInnes et al., 2018) to embed neural data into a 2-dimensional latent space.
Revised: We used UMAP (McInnes et al., 2018) to embed neural data into a 2-dimensional latent space.

Original: Trajectories and pairwise distances show results comparable to the AEs.
Revised: Trajectories and pairwise distances show results comparable to the AEs.

Original: Similarities between trajectories towards the same goal are less apparent (Figure 3a-b).
Revised: Similarities between trajectories towards the same goal are less apparent (Figure 3a-b).

Original: Correlations obtained with the pairwise distance metrics are better (Figure 3c).
Revised: Correlations obtained with the pairwise distance metrics are better (Figure 3c).

Original: UMAP fails to capture the structure in noisy Euclidean data while it does not degrade severely in noisy graph data (see Figures S6 and S7).
Revised: UMAP fails to capture the structure in noisy Euclidean data while it does not degrade severely in noisy graph data (see Figures S6 and S7).

Original: The Euclidean data is sparser since place field centers are distributed across the whole platform rather than a subset of tiles.
Revised: The Euclidean data is sparser since place field centers are distributed across the whole platform rather than a subset of tiles.

Original: This suggests that AEs are better suited for representing the structure of sparser codes in noisy real-world data.
Revised: This suggests that AEs are better suited for representing the structure of sparser codes in noisy real-world data.

Original: 4. Conclusion
Revised: 4. Conclusion

Original: We presented two models of place cells in a maze, associated with Euclidean or graph representations.
Revised: We presented two models of place cells in a maze, associated with Euclidean or graph representations.

Original: Applying UMAP and AE to a set of 25 maze configurations, we found that the latent representation reflected the geometry inherent to the way population neural activity was simulated.
Revised: Applying UMAP and AE to a set of 25 maze configurations, we found that the latent representation reflected the geometry inherent to the way population neural activity was simulated.

Original: This suggests that UMAP and AE architectures can discriminate between geometric representations and outperform linear methods such as PCA.
Revised: This suggests that UMAP and AE architectures can discriminate between geometric representations and outperform linear methods such as PCA.

Original: AEs may outperform UMAP in the presence of noise.
Revised: AEs may outperform UMAP in the presence of noise.

Original: We will more thoroughly explore the performance of different architectures (e.g. Miolane and Holmes (2019)) on more complex tasks; this would require a more refined and parametric quantification of the geometry and topology of the latent space, beyond the correlations presented in this work.
Revised: We will more thoroughly explore the performance of different architectures (e.g. Miolane and Holmes (2019)) on more complex tasks; this would require a more refined and parametric quantification of the geometry and topology of the latent space, beyond the correlations presented in this work.

Original: In addition, a more realistic model of neural activity should be considered, by taking into account remapping of the place cells, path integration, and temporal dynamics.
Revised: In addition, a more realistic model of neural activity should be considered, by taking into account remapping of the place cells, path integration, and temporal dynamics.

Original: Appendix A. Simulating rat trajectories using a biased random walk
Revised: Appendix A. Simulating rat trajectories using a biased random walk

Original: For a given maze configuration and goal location g ∈ R 2 , we generate a rat moving trajectory as a discrete-time random walk over the maze.
Revised: For a given maze configuration and goal location g ∈ R 2 , we generate a rat moving trajectory as a discrete-time random walk over the maze.

Original: More precisely, given the rat at position x ( n ) ( n ∈ N ), we simulate its next position as x ( n + 1) =
Revised: More precisely, given the rat at position x ( n ) ( n ∈ N ), we simulate its next position as x ( n + 1) =

Original: x ( n ) + l v , where l ∼ U ( l min , l max ) is the displacement length, uniformly sampled between 0 < l min < l max , and v is a random unit vector, such that
Revised: x ( n ) + l v , where l ∼ U ( l min , l max ) is the displacement length, uniformly sampled between 0 < l min < l max , and v is a random unit vector, such that

Original: [Equation]
Revised: [Equation]

Original: where p drift ∈ (0 , 1) is the probability that the rat moves towards the goal, with v g ( x ( n )) being the unit vector towards the centre of the next tile to be taken on the shortest path from x ( n ) to g , and v r being a random unit vector whose direction is uniformly sampled over (0 , 2 π ).
Revised: where p drift ∈ (0 , 1) is the probability that the rat moves towards the goal, with v g ( x ( n )) being the unit vector towards the centre of the next tile to be taken on the shortest path from x ( n ) to g , and v r being a random unit vector whose direction is uniformly sampled over (0 , 2 π ).

Original: In practice, we set p drift = 0 .
Revised: In practice, we set p drift = 0 .

Original: 1, l min = 0 .
Revised: 1, l min = 0 .

Original: 009, l max = 0 .
Revised: 009, l max = 0 .

Original: 011 and the maze octagonal tiles are inscribed in 1 × 1 squares (for some examples of trajectories, see Figures 1a and 2b).
Revised: 011 and the maze octagonal tiles are inscribed in 1 × 1 squares (for some examples of trajectories, see Figures 1a and 2b).

Original: Appendix B. Sampling place field locations under the graph hypothesis
Revised: Appendix B. Sampling place field locations under the graph hypothesis

Original: Under the Euclidean hypothesis, we sample the place cell centers over the whole grid once, so they remain invariant with respect to the maze configurations.
Revised: Under the Euclidean hypothesis, we sample the place cell centers over the whole grid once, so they remain invariant with respect to the maze configurations.

Original: However, under the graph hypothesis, each center that gets sampled has to remain invariant according to the graph topology, and thus translates, rotates and scales accordingly across the maze configurations.
Revised: However, under the graph hypothesis, each center that gets sampled has to remain invariant according to the graph topology, and thus translates, rotates and scales accordingly across the maze configurations.

Original: This section describes our procedure to sample the centers under the graph hypothesis.
Revised: This section describes our procedure to sample the centers under the graph hypothesis.

Original: With all the mazes in our study mapping to the same graph topology (see Section 3.1 and Figure 1a), we first label the coordinates of the tile centers as (see Figure S3):
Revised: With all the mazes in our study mapping to the same graph topology (see Section 3.1 and Figure 1a), we first label the coordinates of the tile centers as (see Figure S3):

Original: • v 0 the center of the home tile.
Revised: • v 0 the center of the home tile.

Original: • v 1 , v 2 and v 3 the centers of the goal tiles.
Revised: • v 1 , v 2 and v 3 the centers of the goal tiles.

Original: • v 4 the center of the central tile.
Revised: • v 4 the center of the central tile.

Original: We then introduce the 4 “edge” vectors ( e i ) =
Revised: We then introduce the 4 “edge” vectors ( e i ) =

Original: −−→ v 4 v i , with the corresponding direct orthogonal unit vectors ( e ⊥ i ).
Revised: −−→ v 4 v i , with the corresponding direct orthogonal unit vectors ( e ⊥ i ).

Original: For any maze configuration, we sample a place field center as c = v 4 + δ e k + ρ e ⊥ k , where
Revised: For any maze configuration, we sample a place field center as c = v 4 + δ e k + ρ e ⊥ k , where

Original: • k ∼ U ( { 0 , 1 , 2 , 3 } ) is the index of the sampled edge e k .
Revised: • k ∼ U ( { 0 , 1 , 2 , 3 } ) is the index of the sampled edge e k .

Original: • δ ∼ U (0 , 1) is the relative distance of the cell along the sampled edge.
Revised: • δ ∼ U (0 , 1) is the relative distance of the cell along the sampled edge.

Original: • ρ ∼ U ( − 1 , 1) is the lateral shift associated with the sampled edge.
Revised: • ρ ∼ U ( − 1 , 1) is the lateral shift associated with the sampled edge.

Original: Appendix C. Normalized graph distance
Revised: Appendix C. Normalized graph distance

Original: We introduce a normalized graph distance to measure the graph distance between two points located in different maze configurations from our study.
Revised: We introduce a normalized graph distance to measure the graph distance between two points located in different maze configurations from our study.

Original: For two points x and y in different maze configurations, with corresponding edge indices ( k x and k y ), and relative positions along edges ( δ x and δ y ) (see Appendix B), the normalized graph distance between x and y is given by
Revised: For two points x and y in different maze configurations, with corresponding edge indices ( k x and k y ), and relative positions along edges ( δ x and δ y ) (see Appendix B), the normalized graph distance between x and y is given by

Original: [Equation]
Revised: [Equation]

Original: Supplementary Figures
Revised: Supplementary Figures