REMI: Reconstructing Episodic Memory During Internally Driven Path Planning

Thank you for the insightful comments. We are glad you see this as an ambitious and unified theory for sensory perception, spatial representation, memory, and planning. We agree that a key novelty of this work is that planning involves explicit reconstruction of experiences and an embodied, imaginative process where the agent visualizes the sensory consequences of a potential path.

We appreciate your comments. Our responses are below.

On partial supervision of grid cells

Our core contributions do not depend on co-emergence of grid cells or idealized grid responses.

First, we are showing that if place cells (PCs) auto-associate grid cells and spatially modulated cells (SMCs), the agent can use sensory cues to recall GC patterns. Both recall and reconstruction of sensory experiences during planning work, as long as GC and SMC patterns are spatially stable.

We will revise Section 4 to include a brief mathematical argument showing that planning with grid cells does not require idealized responses. Within a single grid scale, the displacement vector can be decoded even if the field is sheared or compressed (in 1D, this changes scale $\ell$; in 2D, it reduces to two non-colinear 1D components). Larger displacements across multiple grid-scales in distorted grid-fields can be pieced together by the decoding process. Decoding is also robust with sub-optimal grid-scales. The equation on Line 276 can be generalized as $\hat{d} = \frac{1}{2\pi m} \cdot \left( \sum_{i=1}^{k} \ell_{i} \cdot Z_i + \sum_{j=k+1}^{m} \ell_{j} \cdot \Delta \phi_i \right)$ So long as there are sufficiently decodable scales, the predicted displacement reliably moves the agent toward the target.

Unlike place cells, grid cell patterns might not be learned but are believed to be self-organized by a pre-wired network. For instance, deleting an NMDA receptor subunit gene disrupts grid patterns without affecting other spatial cells (Gil et al., Nat Neurosci., 2018), suggesting a genetic basis for grid-like activity. Theoretical works also suggest that grid patterns emerge from dynamical attractor network (Fuhs & Touretzky, J. Neurosci., 2006; Burak & Fiete, PLoS Comput Biol, 2009; Gardner et al., Nature, 2022). Our supervised GC module could be aslo replaced by existing models (Cueva & Wei, ICLR, 2018; Kang & Balasubramanian, eLife, 2019; Sorscher et al., NeurIPS, 2019) without affecting our conclusions. We use grid cells simply as spatial pattern generator and ask: if such patterns exist, could the animal use them for recall and planning?

.... to the "ground truth" GC signals during both training and testing

To clarify, at training time, the GC sub-network receives the ground-truth GC signal as input at the first time-step, path-integrates the hidden state forward, and receives a loss that measures the deviation from the ground-truth GC signal at each time-step. At test time, it receives the ground-truth only at the first step. We discuss new experiments based on your suggestion in the next section.

Statistical validation for the experimental results perform extensive robustness analyses by systematically adding noise, geometric distortions (e.g., shear, compression) ...

We conducted additional experiments to systematically distort the simulated ground truth grid cells. We trained 18 HCMEC and corresponding planner models (10 to evaluate the impact of noise and 8 to assess geometric distortions). Each model was evaluated on 4,096 trials. A trial is successful if the final position is within 10 cm of the goal. On average, the goal is 57.54 ± 25.53 cm away from the start.

For the noise experiments, we added post-activation Gaussian noise (standard deviation 0.1 to 1.0) to all hidden units of the HCMEC at each time step during training. No noise was added at test time (planning phase).

Due to space constraints, we report results based on decoding from SMCs, though we verified that results are consistent when decoding from GCs. In all cases, the mean distance to the goal remains below 10 cm (in a 100 cm × 100 cm environment), and success rates remain high across noise levels. This suggests that the planner can effectively operate even with noisy GC representations.

Next, we added noise to the ground-truth GC signal that is used at the first time step during planning.

Additionally, we trained another 8 models without noise but with sheared grid fields.

We also trained 10 models with 60-degree grid cell lattices, without noise, using different random seeds. The mean distance to the goal was 3.14 ± 2.30 cm.

Experiments on real datasets: We highlight an additional contribution: as suggested by reviewer b2n6, we repeated our experiments in a visually realistic environment (Habitat-Sim).

In our model, SMCs represent sensory observations. While the main text uses simulated sensory signals, here we replace them with features derived from visual observations in the Habitat Synthetic Scenes Dataset (HSSD). The agent captures images during exploration, which are processed by a vision encoder to generate SMC signals. We follow the same training procedure as in Section 4.4 and find the model successfully reconstructs visual features along the path.

The vision encoder produces features that can be decoded back into images, allowing visualization of scenes along the planned path. Intermediate SMC states generated during planning decode into images that closely match expected views at their corresponding planned locations. Key implementation details are as follows:

• We discretize the environment into an $X_w \times X_h$ grid and capture $W \times H$ allocentric panorama images at each location.

• We fine-tuned a visual encoder (a modified Masked AutoEncoder; He et al., 2021) on images from this simulation. It encodes images into compact features that can be decoded back: $D(E(I)) \approx I$, where $E: \mathbb{R}^{W \times H} \to \mathbb{R}^{512}$ and $D: \mathbb{R}^{512} \to \mathbb{R}^{W \times H}$.

• Each image is encoded into a 512-dim vector used as the SMC response, enabling training of HCMEC and the planner as before.

• During planning, the HCMEC updates the SMC region to intermediate states ${z_0, \dots, z_T}$, where $z_i \in \mathbb{R}^{512}$. Decoding these yields a sequence of images that closely resemble the expected views at planned locations.

Successor Representation

We will remove the sentence “SR-based models require visiting all locations…” on Line 57.

Introduction “Successor representation (SR) frameworks (e.g., Stachenfeld et al., Nature Neurosci., 2017; Levenstein et al., bioRxiv, 2024) propose that hippocampal place cells (HPCs) encode expected future state occupancy under a given policy, enabling flexible behavior across related tasks. These models provide compelling accounts of generalization in structured environments and task transfer (Barreto et al., NeurIPS, 2017; Abdolsheh et al., ICML, 2021). However, most SR models often emphasize discrete state spaces represented by HPCs, which exhibit strong contextual tuning. In contrast, grid cells in the medial entorhinal cortex (MEC) display continuous and periodic spatial structure that supports vector-based navigation. It is less context-specific and potentially allows greater generalization across environments. If grid cells encode spatial transition likelihoods, as suggested by the stability of their phase relationships, they may therefore offer a complementary substrate for planning.”

Discussion “Our framework provides a complementary perspective to existing SR models. In REMI, HPCs must predict both GC and SMC responses at the next timestep in order to pattern-complete. Since HPCs are direction-agnostic, accurately reconstructing GC and SMC activity requires encoding possible state transitions. Because adjacent locations tend to have higher transition probability (Stachenfeld et al., 2017), our model implicitly reflects the SR framework of HPCs, where transition structure is embedded in the pattern-completion dynamics. This may be validated experimentally: selectively disrupting HPC-to-MEC projections may degrade planning under REMI, but not SR frameworks.”

Conceptual leap from the discrete-state Markovian planning model

Our discussion of the Markov chain serves primarily as a conceptual framework to explain how the RNN might generalize from short to long trajectories, and why such generalization depends on the periodic grid structure. It is challenging to directly verify whether the RNN instantiates the specific Markov chain we describe. We will revise the paragraph on Line 238 to clarify this intent.

We thank the reviewer b4bC for their insightful comments which we address below.

The novelty of the contribution

The main contribution of this paper is to answer the questions: "Why do animals maintain both place cells and grid cells?" and "What can animals do with these two representations to support internally motivated navigation tasks?" The answers to these questions form the core contributions of this work:

• First, we propose that place cells serve to auto-associate the spatially modulated cells (SMCs) and grid cells (GCs), such that the two types of cells are tightly coupled within an environment and dynamically rewired across environments. Here, we do not intend to answer why place cells emerge, but rather to refine the functional role of place cells proposed in previous theories.

• While this is an extension of previous place cell emergence theories, it makes predictions that cannot be captured by prior models, e.g., auto-associating SMCs and GCs allows retrieval of GC patterns at goal locations.

• We subsequently show that using grid cells for planning is more efficient than direct planning with place cells or SMCs. Then, we explain why direct planning with place cells and SMCs, while feasible, may not be necessary. This auto-association also allows reconstruction of sensory experience from GC patterns, and we would like to highlight that this is not a previously known prediction and is a core contribution we aim to emphasize in this paper.

• Finally, we extend the previous theoretical framework of grid-cell-based navigation (Bush et al., Using Grid Cells for Navigation, Neuron, 2015). Our extended framework provides a feasible implementation for both brain and ANNs without requiring an exponential number of neurons to encode pairwise displacements.

Experiments on practical and real datasets: We would also like to highlight our additional contribution that, as suggested by reviewer b2n6, we conducted experiments on realistic datasets using the Habitat Synthetic Scenes Dataset (HSSD) in Habitat-Sim, a visually realistic simulation environment.

In our model, SMCs represent sensory observations. While the main text uses simulated sensory signals, here we replace them with features derived from visual observations from the Habitat Synthetic Scenes Dataset (HSSD): the agent captures images during exploration, which are passed through a vision encoder to generate SMC signals. We follow the same training procedure for HCMEC and planner models as in Section 4.4 and find that the model successfully reconstructs visual features from this realistic dataset.

Importantly, our vision encoder produces features decodable back into images, allowing visualization of scenes along the planned path. All intermediate SMC states generated during planning can be decoded into images that closely match the expected scenes at their respective planned locations. We provide key implementation details below:

• We discretize the environment into an $X_w \times X_h$ spatial grid and capture $W \times H$ allocentric panorama images at each grid point.

• We fine-tuned a generalized visual encoder (a modified Masked AutoEncoder (MAE; He et al., 2021)) on images from this specific simulation gym. This MAE learns to encode images into compact feature representations, which can later be decoded back to the original images. Compactly, it can be expressed as $D(E(I)) \approx I$, where $I$ is the image, $E: \mathbb{R}^{W \times H} \to \mathbb{R}^{512}$ is the encoder, and $D: \mathbb{R}^{512} \to \mathbb{R}^{W \times H}$ is the decoder.

• At each location, the image is encoded into a 512-dim vector used as the SMC response, enabling training of HCMEC and the planner as in the main text.

• During planning, the HCMEC updates the SMC region to intermediate states ${z_0, \dots, z_T}$, where $z_i \in \mathbb{R}^{512}$. Decoding them with $D$ yields images ${I_0, \dots, I_T}$ that closely resemble the expected views at the corresponding planned locations.

Integrating goal retrieval into the path-planning process

In the main text, we separated retrieval from planning to maintain a clearer separation of concerns across sections. Following the reviewer’s suggestion, we conducted an additional experiment that integrates the recall process directly into planning. We will include this additional result in the supplemental materials.

Detailed training procedures

Supplemental Materials contain a detailed description of the simulation and training procedure. This is referred to on Line 127, 176, and 183. We will make this more prominent. We will also open-source our training code and model weights to ensure reproducibility.

Figure clarity and defining all notations

Thanks. We will revise Fig 2. We would be grateful if you can point us to notations that were not defined, and fix the issue.

On partial supervision of grid cells

Our core contributions and claims do not depend on co-emergence of grid cells, or having idealized grid cell responses.

First, we are showing that if place cells (PCs) auto-associate grid cells and spatially modulated cells (SMCs), the agent can use sensory cues to recall GC patterns. Both recall and reconstruction of sensory experiences during planning work, as long as GC and SMC patterns are spatially stable.

We will modify Section 4 to include a short mathematical argument that shows that planning with grid cells does not require idealized GC representations. It will consist of the following claims. Within a single grid-scale, the displacement vector can be correctly decoded even if the grid field undergoes shearing and compression (in 1D these two are directly related to grid scale, while the 2D case can be decomposed into 1D cases on non-colinear axes). Larger displacements across multiple grid-scales in sheared and compressed grid-fields can be pieced together by the decoding process. Decoding is also robust with sub-optimal grid-scales. The equation on Line 276 can be written in a more general form $\hat{d} = \frac{1}{2\pi m} \cdot \left( \sum_{i=1}^{k} \ell_{i} \cdot Z_i + \sum_{j=k+1}^{m} \ell_{j} \cdot \Delta \phi_i \right)$Under optimal scaling, $\ell_i = \ell_{0} s^i$ but so long as there are sufficiently many scales, the predicted displacement is guaranteed to move the animal closer to the target.

Unlike place cells the grid cell patterns may not be learned but are believed to be dynamically self-organized via a network that was pre-wired by evolution. For instance, genetically deleting the gene encoding a critical NMDA receptor subunit specifically disrupts grid patterns without affecting other spatial cell types (Gil et al., Nat Neurosci., 2018), which indicates that the underlying molecular components and the capacity to form grid-like patterns are at least partly encoded in the genome. Existing theoretical literature also provides evidence suggesting the grid cells may not be learned but are made by a pre-wired dynamical attractor network (Fuhs & Touretzky, J. Neurosci. 2006; Burak & Fiete, PLoS Comp. Bio., 2009; Gardner et al., Nature. 2022). The supervised grid cell region can be directly replaced with one of the existing computational and theoretical models (Cueva & Wei, ICLR, 2018; Kang & Balasubramian, eLife, 2019; Sorscher et al. NeurIPS, 2019) of grid cells without affecting our claims of recall and reconstruction during planning. Here we only use the grid cells as pattern generators that generate grid-like patterns (idealized or not), and ask, “if we have these patterns, could the animal use them for recall and planning”.

Path integration mechanism and projection from speed and direction cells ...

We describe the path integration mechanism in Section 2.1.1. Specifically, speed and direction signals are directly input into designated speed and direction cells, which in turn project to grid cells through learned recurrent connections embedded within the single recurrent weight matrix (Eq. 2; Figure 1b, d). Grid cells are trained to infer the ground-truth GC responses from their initial state and the sequence of speed and direction signals relayed via these input cell types. While we did not explicitly analyze the emergence of conjunctive cells in our model, we observed band-like tuning patterns in the direction cells. Due to NeurIPS rebuttal constraints, we are unable to include these visualizations here but will provide them in the supplementary materials.

Why is the SMC trained separately on masked sensory inputs?

The SMC is not trained separately on masked sensory inputs. We extend classic auto-association theories of place cells by proposing that they auto-associate not only SMCs but also grid cell patterns. This leads to two key predictions: sensory cues can recall GC patterns, and intermediate GC states during planning can be used to reconstruct sensory experiences.

To trigger auto-association, the original signals must be distorted. Prior work (Wang et al., Time Makes Space, NeurIPS 2024) used masking for this purpose. In our model, SMCs simulate sensory input and GCs simulate path integration. Since these signals are not causally linked, we apply independently sampled masking ratios. The training process remains fully integrated, not separated into stages.

Has the planner RNN been analyzed or dissected to confirm that it implements the proposed planning mechanism described in Sections 4.1–4.3?

The discussion in Section 4.1 and 4.3 is validated by the fact that we can decode back the location and trajectory of the animal correctly using SMCs and GCs. Section 4.2 provides a conceptual framework to think of how the RNN could generalize to planning for long trajectories after learning to plan for short trajectories. It is difficult to check the mechanism of the RNN, e.g., whether or not it instantiates a Markov chain that we have alluded to. We will preface the paragraph on Line 238 to emphasize this point.

We thank reviewer b2n6 for their thoughtful review and respond below to the comments

Definition of auto-association

We adopt the term from prior literature on the hippocampus's role in memory (Treves & Rolls, Hippocampus, 1994; ET. Rolls. Front. Syst. Neurosci. 2013.). Formally, we define an auto-association system as a function $f: \mathbb{R}^n \to \mathbb{R}^n$ such that $f(\mathbf{x}') = \mathbf{x}$ if $\mathbf{x}’ \approx \mathbf{x}$. In the scope of this paper, consider two vectors representing the Spatially Modulated Cells (SMCs) and Grid Cells (GCs), denoted as $\mathbf{x_{\text{smc}}}$ and $\mathbf{x_{\text{gc}}}$, respectively. These vectors, at any point during navigation, may be masked into $\mathbf{x_{\text{smc}}}'$ and $\mathbf{x_{\text{gc}}}'$ due to imperfect observation or path integration. We suggest that place cells serve as the function $f(\cdot)$ such that $f([\mathbf{x_{\text{smc}}}' ; \mathbf{x_{\text{gc}}}' ]) = [\mathbf{x_{\text{smc}}} ; \mathbf{x_{\text{gc}}} ]$.

Experiments on practical and real datasets

We conducted additional experiments using the Habitat Synthetic Scenes Dataset (HSSD) in Habitat-Sim, a visually realistic simulation environment. In our model, SMCs represent sensory observations. While the main text uses simulated sensory signals, here we replace them with features derived from visual observations from the Habitat Synthetic Scenes Dataset (HSSD): the agent captures images during exploration, which are passed through a vision encoder to generate SMC signals. We follow the same training procedure for HCMEC and planner models as in Section 4.4 and find that the model successfully reconstructs visual features from this realistic dataset.

Importantly, our vision encoder produces features decodable back into images, allowing visualization of scenes along the planned path. All intermediate SMC states generated during planning can be decoded into images that closely match the expected scenes at their respective planned locations. We provide key implementation details below:

• We discretize the environment into an $X_w \times X_h$ spatial grid and capture $W \times H$ allocentric panorama images at each grid point.

• We fine-tuned a generalized visual encoder (a modified Masked AutoEncoder (MAE; He et al., 2021)) on images from this specific simulation gym. This MAE learns to encode images into compact feature representations, which can later be decoded back to the original images. Compactly, it can be expressed as $D(E(I)) \approx I$, where $I$ is the image, $E: \mathbb{R}^{W \times H} \to \mathbb{R}\_+^{512}$ is the encoder, and $D: \mathbb{R}\_+^{512} \to \mathbb{R}^{W \times H}$ is the decoder.

• At each location, the image is encoded into a 512-dim vector used as the SMC response, enabling training of HCMEC and the planner as in the main text.

• During planning, the HCMEC updates the SMC region to intermediate states ${z_0, \dots, z_T}$, where $z_i \in \mathbb{R}^{512}$. Decoding them with $D$ yields images ${I_0, \dots, I_T}$ that closely resemble the expected views at the corresponding planned locations.

Comparison to existing RL-based navigation methods

We intend this work primarily to suggest answers to the question of why animals maintain two systems to encode space. We propose that grid cell representations serve as efficient, generalizable encodings for planning across diverse environments, but they lack sensory detail and therefore require hippocampal place cells (HPCs) to pattern-complete the corresponding sensory information along the planned path. The most relevant metric for this study is the successful path-planning ratio; we presented an extensive test of the success planning ratio of the HCMEC model in the last section of this rebuttal.

In the paper, to emphasize biological realism, we adopt a simplistic continuous-time RNN for modeling. However, the proposed framework is directly applicable to more advanced architectures such as GRUs or LSTMs, which may yield improved performance. We now additionally trained a variant with a simple layer normalization step before the hidden state update, which significantly increases the success rate to 0.99, making the approach potentially relevant for robotics applications.

Empirical evidence supporting HPC reconstructs sensory experiences during planning

To address this, we first test whether hippocampal place cells (HPCs) specifically contribute to trajectory reconstruction. This can be evaluated by checking whether intermediate HPC states during planning decode into a valid trajectory. For each planning trial, we record all HPC states and decode each time step into a location using nearest neighbor search over ratemaps aggregated during testing.

While the main text focuses on decoding using spatially modulated cells (SMCs) and grid cells, here we compare planning performance across 4096 trials using HPC-decoded versus SMC-decoded trajectories. We find that HPC-decoded trajectories also reliably reach the goal:

A complementary perspective is to test whether the reconstructed experience can be decoded beyond nearest neighbor search. The experiment in Experiments on practical and real datasets provides strong support: we train a compact image encoder-decoder, replace SMC signals with encoded image features, and find that the reconstructed sensory signals along the planned path can be decoded back into meaningful images.

Performance varies with different levels of partial supervision or noise

We conducted additional experiments to systematically distort the simulated ground truth grid cells. We trained 18 HCMEC and corresponding planner models (10 to evaluate the impact of noise and 8 to assess geometric distortions). Each model was evaluated on 4,096 planning trials. A trial is considered successful if the agent's final position is within 10 cm of the goal. Success rate is computed over all trials. On average, the goal is 57.54 ± 25.53 cm away from the start.

For the noise experiments, we added post-activation Gaussian noise (standard deviation 0.1 to 1.0) to all hidden units of the HCMEC at each time step during training. No noise was added at test time (planning phase).

Due to space constraints, we report results based on decoding from SMCs, though we verified that results are consistent when decoding from GCs. In all cases, the mean distance to the goal remains below 10 cm (in a 100 cm × 100 cm environment), and success rates remain high across noise levels. This suggests that the planner can effectively operate even with noisy GC representations.

Next, we added noise to the ground-truth GC signal that is used at the first time step during planning.

Additionally, we trained another 8 models without noise but with sheared grid fields.

We thank reviewer u9fY for their thoughtful review and the endorsement of our work. We are excited to see that you think this is a compelling theoretical framework that unifies the functional roles of grid cells and place cells. As you have mentioned, we indeed attempt to propose this framework to construct a system-level theory to explain the empirical works on each neural sub-region, as collectively these patterns must be preserved by evolution due to their functional importance.

Experiments on practical and real datasets: We would first like to highlight an additional contribution: as suggested by reviewer b2n6, we conducted experiments in a visually realistic simulation environment (Habitat-Sim) using realistic datasets.

In our model, SMCs represent sensory observations. While the main text uses simulated sensory signals, here we replace them with features derived from visual observations from the Habitat Synthetic Scenes Dataset (HSSD): the agent captures images during exploration, which are passed through a vision encoder to generate SMC signals. We follow the same training procedure for HCMEC and planner models as in Section 4.4 and find that the model successfully reconstructs visual features from this realistic dataset.

Importantly, our vision encoder produces features decodable back into images, allowing visualization of scenes along the planned path. All intermediate SMC states generated during planning can be decoded into images that closely match the expected scenes at their respective planned locations. We provide key implementation details below:

• We discretize the environment into an $X_w \times X_h$ spatial grid and capture $W \times H$ allocentric panorama images at each grid point.

• We fine-tuned a generalized visual encoder (a modified Masked AutoEncoder (MAE; He et al., 2021)) on images from this specific simulation gym. This MAE learns to encode images into compact feature representations, which can later be decoded back to the original images. Compactly, it can be expressed as $D(E(I)) \approx I$, where $I$ is the image, $E: \mathbb{R}^{W \times H} \to \mathbb{R}^{512}$ is the encoder, and $D: \mathbb{R}^{512} \to \mathbb{R}^{W \times H}$ is the decoder.

• At each location, the image is encoded into a 512-dim vector used as the SMC response, enabling training of HCMEC and the planner as in the main text.

• During planning, the HCMEC updates the SMC region to intermediate states ${z_0, \dots, z_T}$, where $z_i \in \mathbb{R}^{512}$. Decoding them with $D$ yields images ${I_0, \dots, I_T}$ closely resemble the expected views at the corresponding planned locations.

Here we address the questions in the review:

It was unclear if the planning network is thought of as part of the same single-layer RNN:

We train the planning and HCMEC networks in a two-stage setting primarily because they may correspond to two separate behavioral processes. HCMEC training models the phase when the animal explores the environment and builds representations of it, whereas the planner models how, once a spatial representation is formed, the animal might virtually navigate on it to plan paths. At the implementation level, however, this is equivalent to initializing a single network with both HCMEC and planner units, then alternating between training to encode the space and to plan for navigation.

Additionally, we used the term "single layer" to emphasize that in the HCMEC model, both HC and MEC can project to each other, unlike prior models that typically assume unidirectional information flow. As the reviewer noted, we are leveraging the fact that any arbitrarily complex RNN can be represented as a single-layer RNN. This allows us to model bidirectional interactions between multiple neural subpopulations using a single recurrent weight matrix. These interactions are entirely determined by the optimization objective, without the need to define or handcraft structures.

On partial supervision of grid cells

Our core contributions and claims do not depend on co-emergence of grid cells, or having idealized grid cell responses.

First, we are showing that if place cells (PCs) auto-associate grid cells and spatially modulated cells (SMCs), the agent can use sensory cues to recall GC patterns. Both recall and reconstruction of sensory experiences during planning work, as long as GC and SMC patterns are spatially stable.

We will modify Section 4 to include a short mathematical argument that shows that planning with grid cells does not require idealized GC representations. It will consist of the following claims. Within a single grid-scale, the displacement vector can be correctly decoded even if the grid field undergoes shearing and compression (in 1D these two are directly related to grid scale, while the 2D case can be decomposed into 1D cases on non-colinear axes). Larger displacements across multiple grid-scales in sheared and compressed grid-fields can be pieced together by the decoding process. Decoding is also robust with sub-optimal grid-scales. The equation on Line 276 can be written in a more general form $\hat{d} = \frac{1}{2\pi m} \cdot \left( \sum_{i=1}^{k} \ell_{i} \cdot Z_i + \sum_{j=k+1}^{m} \ell_{j} \cdot \Delta \phi_i \right)$Under optimal scaling, $\ell_i = \ell_{0} s^i$ but so long as there are sufficiently many scales, the predicted displacement is guaranteed to move the animal closer to the target.

Unlike place cells the grid cell patterns may not be learned but are believed to be dynamically self-organized via a network that was pre-wired by evolution. For instance, genetically deleting the gene encoding a critical NMDA receptor subunit specifically disrupts grid patterns without affecting other spatial cell types (Gil et al., Nat Neurosci., 2018), which indicates that the underlying molecular components and the capacity to form grid-like patterns are at least partly encoded in the genome. Existing theoretical literature also provides evidence suggesting the grid cells may not be learned but are made by a pre-wired dynamical attractor network (Fuhs & Touretzky, J. Neurosci. 2006; Burak & Fiete, PLoS Comp. Bio., 2009; Gardner et al., Nature. 2022). The supervised grid cell region can be directly replaced with one of the existing computational and theoretical models (Cueva & Wei, ICLR, 2018; Kang & Balasubramian, eLife, 2019; Sorscher et al. NeurIPS, 2019) of grid cells without affecting our claims of recall and reconstruction during planning. Here we only use the grid cells as pattern generators that generate grid-like patterns (idealized or not), and ask, “if we have these patterns, could the animal use them for recall and planning”.

Connection to Successor Features

Here, we analyze why, at the implementation level, our network must encode future state. In REMI, HPCs must predict both GC and SMC responses at the next timestep in order to pattern-complete. Since HPCs are direction-agnostic, accurately reconstructing GC and SMC activity requires encoding possible state transitions. Because adjacent locations tend to have higher transition probability (Stachenfeld et al., 2017), our model implicitly reflects the successor representation framework of HPCs, where transition structure is embedded in the pattern-completion dynamics.

Regarding our discussion of grid cell transition probabilities, we primarily use it as a conceptual framework to understand how the RNN might generalize from planning short trajectories to longer ones, and why such generalization is only feasible with the periodic grid structure. It is difficult to check the mechanism of the RNN, e.g., whether or not it instantiates a Markov chain that we have alluded to. We will preface the paragraph on Line 238 to emphasize this point.

Path Integration Accuracy

We tested the path integration accuracy of models with (Figure 1d) and without (Figure 1b) Spatially Modulated Cells (SMCs). Both models contain the same number of Grid, Direction, and Speed Cells, with the full HCMEC model additionally including SMCs and Place Cells. The HCMEC model was trained with a masking ratio for each trial sampled uniformly from $\mathcal{U}[0, 1]$.

To test path integration, the GC subregion was initialized with the ground truth grid state. During testing, the model received unmasked speed and direction inputs. In the full HCMEC model, the SMC subregion additionally received SMC signals. Grid cell subregions were expected to update their internal grid states accordingly.

We evaluated each model over 100 seconds across 512 independent trials. GC responses were decoded into spatial coordinates using nearest neighbor search, and errors were computed as Euclidean distances (cm) from the ground truth at [1, 5, 10, 20, 50, 100] seconds in a 100 cm × 100 cm environment (1cm/spatial pixel):

• Full HCMEC model: [1.96 ± 1.58, 2.63 ± 3.26, 3.14 ± 6.32, 3.76 ± 8.23, 7.67 ± 17.06, 11.43 ± 22.47] (in cm)
• Grid cell only model: [1.89 ± 2.92, 2.76 ± 5.34, 3.37 ± 7.11, 4.86 ± 10.77, 5.17 ± 12.39, 5.28 ± 12.74] (in cm)

Both models show high path-integration accuracy over time. The error may result from the discretization of space and time in our simulation, which introduces small inaccuracies that accumulate during path integration. In contrast, biological systems may use additional signals such as boundary cells for error correction (Eli et al. Dynamic self-organized error-correction of grid cells. 2018). Future work may incorporate such signals to improve accuracy. The full HCMEC model shows slightly lower path-integration accuracy than the GC-only model, likely due to training with masked velocity, speed, and direction inputs.

All experiments were conducted with agents moving at speeds sampled from a right-skewed log-normal distribution with mean and standard deviation of 10 cm/s. Since speed affects the temporal resolution of input signals, our results are robust across different timescales.