Vector Quantization in the Brain: Grid-like Codes in World Models

We sincerely thank the reviewer for your positive and insightful feedback. We are very encouraged by your recognition of our work's key strengths, including our unified GCQ model that integrates spatial and temporal quantization via an action-conditioned codebook, the interpretable latent structure yielding a cognitive map conducive to inverse modeling, the strong empirical performance, particularly on long-horizon prediction and planning, and our well-supported, data-driven hypothesis for the function of grid-cell codes. Your accurate summary and positive assessment are greatly appreciated.

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Response to Weaknesses

1. The reviewer correctly points out that the version of GCQ presented in this manuscript is designed for tasks with explicit, discrete action labels. We totally agree that extending GCQ to continuous or unknown action spaces is a valuable research direction. In fact, we have already begun theoretical and experimental explorations of GCQ variants for these scenarios.

   • For continuous action spaces, we plan to use a variant to replace the discrete bump shifts with rotation matching based on continuous rotation angles (similar to RoPE method) .
   • For unlabeled action spaces, we leverage the strong temporal correlation between adjacent frames in a sequence as a supervisory signal, enforcing that the codes for proximate frames should also be closely related.

   Our preliminary results indicate that these variants are not only feasible but also achieve promising performance. We consider this a significant avenue for future work and plan to report our findings in a subsequent publication.

2. The reviewer raises an important point about scaling to finer-grained or higher-dimensional action spaces. There are two primary mechanisms to scale the representative capacity of GCQ:

   • Increasing the number of CANNs ($m$).
   • Increasing the dimensionality of the attractors ($P$). As described in Line 123, our current work utilizes 2D attractors, where each has 5 potential transitions (four shifts and one stationary). With $m$ such CANNs, the model can represent $5^m$ distinct actions. If we use P-dimensional attractors, the number of states per CANN will become $(2P+1)$, yielding a total action space of $(2P+1)^m$. Therefore, GCQ can be scaled to higher-dimensional action spaces by adjusting both $m$ and $P$. For the experiments in this paper, which involve relatively low-dimensional actions, 2D attractors are sufficient.

3. We concur with the reviewer that evaluating GCQ on complex domains such as real-time video and continuous control is a great direction for extending our work. As mentioned in our response to point 1, the variant using continuous rotation matching is well-suited for continuous control tasks and it is designed with an efficient matching algorithm to maintain computational performance. We have already conducted preliminary experiments on datasets like TartanDrive, RECON, SCAND, and HuRoN with positive results, sorry we are unable to present figures here due to the rebuttal policy. The datasets suggested by the reviewer (e.g., BAIR robot pushing, Kinetics-600) are highly relevant, and we will consider them for future investigation.

4. We thank the reviewer for this excellent suggestion to test the model's robustness. We have performed supplementary experiments to analyze the impact of noisy action labels. The table below shows the FIDp score as a function of the noise level,

   Noise level	1%	2%	4%	8%	16%
   GCQ (112M)	44	44	46	52	58

   We believe these results effectively demonstrate the robustness of our method. We commit to including this analysis and discussion in the final version of the paper.

5. Minor (Line 276): We agree with the reviewer's observation and appreciate the suggestion for more precise phrasing. The statement "GCQ converged faster" is indeed not entirely accurate at that point of comparison. In the final version, we will revise this to state that "GCQ demonstrated superior performance in the early stages of training" and will consider adding a plot of the training loss over time to provide clearer evidence of its convergence behavior.

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Response to Questions

1. Yes, we can confirm that the number of CANNs ($m$) scales logarithmically with the number of representable actions. To provide a more detailed clarification: each 2D CANN is built upon two orthogonal bases, and each base supports transitions in two opposing directions. Together with the option to remain stationary, this yields a total of 5 discrete transitions per CANN (i.e., 2 bases × 2 directions + 1 stationary option). Since the $m$ CANNs operate independently, the total set of possible actions is the Cartesian product of the individual action sets. This results in a total of $5^m$ unique representable actions. Consequently, the number of CANNs required, $m$, is related to the total number of actions, $|A|$, by $m = \log_5(|A|)$.

2. The reviewer's analysis of the computational cost as $O(mKn)$ is correct. However, we would like to clarify that this matching operation is highly parallelizable. When executed on a GPU, the computations across $m$, $K$, and $n$ can be performed in parallel, which significantly mitigates the computational burden and prevents it from becoming a bottleneck, even for larger-scale inputs. We include the relevant codes for the matching algorithm here:

def forward(self, latents: Tensor, label: Tensor) -> Tuple[Tensor, Tensor]:
    """
    Vector quantization for sequence data with parallel processing support for batch dimensions

    Args:
        latents: Tensor of shape [B x S x D x H x W]
        label: Tensor of shape [B x S x 4] representing the change from current frame to next frame

    Returns:
        quantized: Quantized tensor of shape [B x S x D x H x W]
        vq_loss: Vector quantization loss
    """
    B, S, D, H, W = latents.shape
    N = H * W // 4
    device = latents.device

    # Reshape to processable form
    latents = latents.view(B, S, D, 4, N).permute(0, 3, 4, 1, 2).contiguous()  # [B, 4, N, S, D]
    factor_latents = latents.view(B, 4, N, S * D)

    # Construct expanded embeddings without loops
    # label: [B, S, 4] -> rearrange to [B, 4, S]
    shift_amounts = (label.permute(0, 2, 1).long() * self.step)  # [B, 4, S]
    # Construct indices for rolling: for each shift_amount against self.embedding_weight
    # First generate indices [K], then calculate new indices via broadcasting: new_idx = (arange(K) - shift) % K
    k_idx = torch.arange(self.K, device=device).view(1, 1, 1, self.K)  # [1,1,1,K]
    # After expanding shift_amounts: [B,4,S,1]
    rolled_indices = (k_idx - shift_amounts.unsqueeze(-1)) % self.K  # [B, 4, S, K]
    # Using rolled_indices to extract new embeddings from embedding_weight, resulting shape [B,4,S,K,D]
    expanded_embed = self.embedding_weight[rolled_indices]  # [B,4,S,K,D]
    # Adjust dimensions: exchange [K] and [S] positions before merging S and D
    expanded_embed = expanded_embed.permute(0, 1, 3, 2, 4).contiguous()  # [B,4,K,S,D]
    # Finally reshape to [B, 4, K, S*D]
    expanded_embedding = expanded_embed.view(B, 4, self.K, S * D)

    # Calculate nearest neighbor indices
    # factor_latents: [B,4,N,S*D]; expanded_embedding: [B,4,K,S*D]
    A = factor_latents  # [B,4,N,S*D]
    B_expand = expanded_embedding  # [B,4,K,S*D]
    A_sq = (A ** 2).sum(dim=-1, keepdim=True)  # [B,4,N,1]
    B_sq = (B_expand ** 2).sum(dim=-1).unsqueeze(-2)  # [B,4,1,K]
    cross = 2 * torch.matmul(A, B_expand.transpose(-1, -2))  # [B,4,N,K]
    dist = A_sq + B_sq - cross  # [B,4,N,K]
    encoding_inds = dist.argmin(dim=-1)  # [B,4,N]

    # Sample vectors from expanded_embedding according to indices using torch.gather
    # expanded_embedding shape [B,4,K,S*D], sampling on dim=2
    encoding_inds_exp = encoding_inds.unsqueeze(-1).expand(-1, -1, -1, S * D)  # [B,4,N,S*D]
    embedding_results = torch.gather(expanded_embedding, 2, encoding_inds_exp)  # [B,4,N,S*D]

    commitment_loss = F.mse_loss(embedding_results.detach(), factor_latents)
    embedding_results = factor_latents + (embedding_results - factor_latents).detach()  # [B,4,N,S*D]
    embedding_results = embedding_results.view(B, 4, N, S, D)
    embedding_results = embedding_results.permute(0, 3, 4, 1, 2).contiguous().view(B, S, D, H, W)
    return embedding_results, commitment_loss

As illustrated in the code, we have replaced all iterative loops with matrix operations so the time complexity does not become a bottleneck in practice. However, we acknowledge that the space complexity does scale with the input size. Therefore, parameters such as $m$ and $K$ should be adjusted based on the available hardware resources during deployment to strike a balance between performance and memory footprint.

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Response to Limitations

• As detailed in our Response to Weaknesses 1, we are actively developing GCQ variants to address continuous action spaces and scenarios where action labels are unavailable. While these extensions are beyond the scope of the current paper, they are a primary focus of our future research.
• The strategy for scaling to very large discrete action spaces by adjusting the number of CANNs ($m$) and the attractor dimensionality ($P$) is outlined in our Response to Weaknesses 2.

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Once again, we thank the reviewer for your positive assessment and constructive suggestions. We are particularly grateful for the suggestions regarding GCQ variants for continuous and unlabeled actions, which align perfectly with the focus of our future research. We hope our responses have addressed all your concerns. We welcome any further questions or suggestions.

Thank you for the valuable and constructive comments, the recognition of the novelty of our work on using GCQ for compressing observation–action sequences, and the acknowledgement of our method’s potential advantages for long-horizon prediction and goal-directed planning.

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Response to Weaknesses

Method

• We would like to clarify the set-up used for the VQ baseline. As stated in the Baselines section (line 201) of the manuscript, the model referred to as “VQ” is in fact a two-stage architecture, consisting of a VQ-VAE [1] followed by a UNet. We adopted this term as a shorthand for brevity, and we apologize if this naming led to any confusion.

   

  The rationale for this design is as follows: This two-stage architecture is essential for our world modeling task, which requires processing sequences of observations and actions. The VQ-VAE component handles the compression of static images into discrete latent codes ($o_t\to z_t$). However, to model the evolution of these codes over time, an explicit temporal model is necessary. The UNet's role, therefore, is to explicitly model the temporal dynamics across the sequence of these latent codes ($(z_t,a_t)\to z_{t+1})$. It is worth noting that unlike the VQ baseline model, our GCQ does not rely on a two-stage architecture, as it integrates spatiotemporal compression within a unified framework— which represents a key contribution of our work.

   

  It is important to note that this two-stage design is a standard and widely adopted practice in recent world modeling literature. Many state-of-the-art methods first use a VQ-VAE or VAE for spatial compression and then employ a separate, powerful sequence model (like a Transformer, RNN or UNet) to capture temporal dependencies [2][3]. Our choice of the baseline model is therefore consistent with established approaches in the field.

   

  To ensure this is clear to all readers, we will revise the manuscript to replace the "VQ" shorthand with the full term "VQ+UNet" throughout the text, tables, and figures. We hope this explanation clarifies our experimental setup, and we thank you for helping us improve the precision of our paper.

   

  In our main experiments, the specific UNet architecture we used is identical to the one proposed in the LAPO paper . To further strengthen our comparison and demonstrate the robustness of our approach, we also conducted additional experiments with a VQ+transformer baseline. For this, we replaced the UNet with a Transformer whose architecture is based on TransDreamer [2]. The results, summarized below, show that our proposed method outperforms both the UNet-based and the Transformer-based baselines. The table below shows the FIDp score as a function of the prediction step, similar to the analysis in Fig. 4D. Lower is better.

  Prediction Steps	1	2	4	8	16	32
  VQ+Unet (96M)	51	78	115	149	159	164
  VQ+Transformer (121M)	61	64	71	79	88	111
  GCQ (112M)	42	44	43	44	44	44

• Let us first clarify the notation. In the paper, each frame is compressed to $m$ codes and $e_j$ denotes the $j$-th code ($j=1,...,m$). $a_i$ refers to the action vector applied to all codes at time step $i$. When an action $a_i$ is applied to a latent state composed of $m$ codes, each code $e_j$ undergoes a transition determined by the $j$-th component of $a_i$, denoted $a_{i,j}$. To make this process clearer, we have revised the manuscript and replaced Equation (7) with the following:

  $$e_j \oplus a_{1:n-1} = \lbrace e_j, e_j + a_{1,j}, \dots, e_j + a_{1:n-1,j} \rbrace \tag{7*}$$

   

  As an example, for a latent state composed of $m = 3$ codes, consider a moving agent. We would first manually define the overall bump transitions corresponding to different movements. For instance, "move left" is defined as $a_{left} = [a_\theta^+, 0, 0]$ (the first bump moves slightly in the positive theta direction, while the other two bumps remain stationary). At time step $i$, the latent code is $s_i$, and the movement at the $i$-th time step is to the left, i.e., $a_i = a_{left}$. Then, $s_{i+1} = s_i + a_i$ (the first bump of $s_i$ moves slightly in the positive theta direction, while the other two bumps remain stationary).We hope this clarification and the updated equation address your concern.

This is a great suggestion—we will add a description of the typical VQ configuration prior to introducing GCQ, as we agree that such a comparison would benefit readers.

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Experiments

First, we would like to point out that Table 1 reports PSNR—a widely used metric for reconstruction error that serves as a perceptual measure of image compression performance. Also the FID scores reported in Table 1, Figure 4(A)(B)(C)(D), and Figure 7(A)(B) are also standard and widely adopted metrics for assessing the quality of generated images.

Additionally, we have conducted supplementary experiments measuring the planning success rate, and the results are shown in the table below(2DMaze dataset):

We will incorporate these results into the final version of the paper. We hope these additional experiments help substantiate the effectiveness of GCQ.

1. In the case of GCQ, the model size refers to the encoder–decoder backbone, as the codebook in GCQ is non-trainable and does not contain learnable parameters. In contrast, the VQ baseline requires both a VQ-VAE and an additional network (e.g., UNet) to complete the experimental tasks—VQ-VAE alone is insufficient. Therefore, we reported the total parameter count of this two-stage pipeline. Importantly, GCQ and VQ are matched in terms of compression ratio and codebook size, ensuring a fair comparison. The reported model sizes reflect all learnable parameters in the entire systems.

2. In fact, GCQ alone is fully capable of learning action embeddings and accomplishing all tasks in our experiments, including long-horizon prediction, goal-directed planning, and inverse modeling. This is a key advantage of GCQ over VQ. While VQ requires additional modules—such as the UNet from LAPO—to construct a two-stage world model, GCQ integrates spatiotemporal compression into a single, unified framework without relying on any auxiliary architecture.

   Importantly, LAPO components are used only for the VQ-based baselines and play no role whatsoever in the implementation, training, or evaluation of GCQ. We hope this clarification resolves any potential misunderstanding.

3. For the GSV dataset, we used images seen during training as the goals for goal-directed planning. In contrast, for the 2DMaze dataset, we used goals that were unseen during training. This approach was chosen because the GSV dataset has a complex data distribution, and the limited amount of data used for our training makes generalization challenging. In our future work, we plan to leverage a larger dataset to enable planning to unseen locations within the GSV dataset.

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Finally, we once again thank you for recognizing the innovative nature and potential impact of our work. We hope our responses address your concerns and misunderstandings regarding both the methodology and experimental design. We also appreciate your constructive suggestions, and we commit to incorporating the corrected equation and additional experiments into the final version of the paper.

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[1] Neural discrete representation learning.

[2] Transdreamer: Reinforcement learning with transformer world models.

[3] Efficient World Models with Context-Aware Tokenization.

We would like to express our sincere gratitude for your thorough review of our rebuttal and additional experiments. We are glad to hear that our response has addressed your concerns, and we deeply appreciate you raising the overall score for our paper.

Regarding your valuable suggestion, we confirm that we will add the pseudocode for both the baseline VQ-based method and our GCQ in the revised version of the paper. We agree this will significantly enhance the clarity of our work.

We are also very encouraged by your interest in our future work on extending GCQ to continuous or unknown action spaces. Your feedback gives us more confidence in pursuing this direction.

Once again, thank you for your support and constructive guidance throughout this process. We look forward to preparing the revised version.

Thank you for your positive and insightful review. It is rewarding to hear that you found our proposal 'creative and technically appealing' and the mathematical derivations of 'solid technical quality'. We are glad that you recognized our work has the potential to inspire new latent-space designs, especially for challenging long-horizon prediction tasks. We believe this brain-inspired perspective allows us to introduce a novel and effective architecture to the machine learning community. Your appreciation—acknowledging that our 'conceptual novelty is genuine', the theoretical derivations show 'solid technical quality', and the paper is 'readable'—are greatly encouraging to us.

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Responses to Questions

• We thank the reviewer for this insightful question. It allows us to clarify the conceptual framing of our work and its relationship with computational neuroscience.

   

  To begin, we'd like to outline the distinct functional roles of the two cell types in question. Grid cells are fundamentally associated with path integration [1]. Their activity is conditioned by self-motion (i.e., they are bound to action) and they provide a universal spatial metric that does not globally remap across different environments. In contrast, place cells are crucial for localization by recognizing specific places. Their firing is driven by environmental sensory cues, causing them to remap when the environment changes [2], and they are not directly bound to specific actions.

   

  Given this functional distinction, the design of our model and its core objective naturally align with the principles of grid cells. The main reason for not considering place-like cells is that our model aims to build an action-conditioned world model, a task we see as a generalization of the principle underlying grid cells. Specifically, the world model learns to predict a future latent state based on the current state and a given action ($s_{t+1} = s_t+a_t$) as expressed by Eq.(6-7). This process of modeling the environment's transition dynamics is a generalized form of path integration. Consequently, the latent representation our model utilizes is naturally a grid-cell-like code that is bound to the applied action.

   

  The emergence of place cells, on the other hand, is tightly linked to binding specific sensory inputs (e.g., visual landmarks) to locations. Our model is not optimized for this sensory-associative function. By focusing on the generalized transition dynamics, our architecture is designed to factor out environment-specific sensory details in favor of a universal state representation. Therefore, the absence of place cells is confirmatory evidence that our architecture has effectively isolated and replicated the specific neural computational mechanism we target for: a universal, action-based state prediction.

   

  We agree that integrating a place-cell-like mechanism is a valuable direction for future work. As the reviewer suggests, this would sharpen the model's biological grounding and expand its capabilities. For instance, in future work on SLAM tasks where recognizing and relocalizing within previously seen environments is crucial, a place-cell-like module would be essential. This module could help bind the grid-like representations from our current modules to a specific environmental context, providing a mechanism for global remapping. However, this is beyond the scope of our current contribution.

• Regarding your second point, we would like to provide a clear justification for our choice of hard-coded actions. Our approach is grounded in experimental evidence showing that grid-like bumps shift in correlation with an animal’s movements [3]. In our model, this principle translates to the mapping between an action and its corresponding displacement in the latent CANN space. While the scaling factor of this mapping could be learnable, we opted to fix it in our design.

   

  To validate this choice, we conducted an ablation study where the scaling factor was learnable. The results showed a decline in performance compared to using hard-coded actions. We hypothesize that this is because a non-stationary (i.e., constantly changing) latent transition destabilizes the training process. A constantly shifting transition makes it significantly more difficult for the model to learn the mapping between latent states and observations, thus hindering convergence. The outcomes of this ablation are shown in the table below (For FIDp, lower is better. For PSNRp, higher is better).

  	FIDp	PSNRp
  GCQ (fixed)	43.41	27.77
  GCQ (learnable)	47.76	24.48

We hope this explanation and ablation study address your two main questions. We will incorporate these clarifications into the final version of the paper to address the ambiguity. Please don’t hesitate to discuss them with us if further questions arise. Now, we turn to your remaining questions.

• We agree with your recommendation to move the continuous attractor derivations from Section 3 to the Methods section in the final version for improved clarity and narrative flow.

• Regarding Figure 2, we will revise it to add an inset zoom on the “multi-CANN codebook to decoder” path, to more clearly indicate that the blank rows correspond to separate CANNs. (Apologies that we cannot include the revised figure here due to the rebuttal policy.)

• Thank you for pointing this out. We will add reference [4] to support the statement on “instability issues commonly seen in current world models.” Additionally, we would like to point out that the experiment shown in Figure 4(D) provides empirical evidence, showing that the performance of the VQ method degrades as the prediction horizon increases.

• We will also revise the phrasing in the last paragraph to adopt a more academic tone.

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Responses to Weaknesses

• As you suggested, we have conducted a comparative experiment with a stronger benchmark by replacing the UNet with a Transformer whose architecture is based on TransDreamer [6] (Transformer-SSM is a variant of RSSM). The table below shows the FIDp score as a function of the prediction step, similar to the analysis in Fig. 4D. Lower is better.

  Prediction Steps	1	2	4	8	16	32
  VQ+Unet (96M)	51	78	115	149	159	164
  VQ+Transformer (121M)	61	64	71	79	88	111
  GCQ (112M)	42	44	43	44	44	44

  Additionally, we report the planning success rate as a quantitative metric for evaluation(2DMaze dataset).

  	542M	307M	140M
  seen goal	97/100	93/100	68/100
  unseen goal	91/100	88/100	65/100

  We believe these additional benchmarks strengthen our model's claims. While we were unable to complete the full statistical uncertainty analysis due to time constraints, we commit to including these results and the complete analysis in the final version of the paper.

  We also commit to including more detailed hyper-parameters, compute budget, and codebook ablations in the final appendix. We have submitted our experimental code in the Supplementary Material to ensure reproducibility.

• The remaining points you raised—including relocating technically dense description to the Methods section, providing further discussion on place cells and hard-coded actions, citing instability issues in world models along with empirical support, and adopting more academic language in the discussion—have all been addressed in detail in the Responses to Questions section above. We also commit to carefully proofreading the final manuscript to eliminate 'minor spelling and grammatical errors'. We sincerely thank you for your attentive review.

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Response to Limitations

As addressed in Response to Questions(point two), we provide a detailed justification for the hard-coding of action sets, along with ablation results comparing to a learnable mapping variant.

As clarified in Response to Weaknesses, our work focuses on world modeling, and we included an additional benchmark using a stronger modern baseline (TransDreamer). Our experiments on goal-directed planning and inverse modeling are common downstream tasks for world models [5]. In future work, we plan to incorporate other modules such as place cells and boundary cells to support a broader range of downstream RL tasks and SLAM.

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In summary, we sincerely thank you again for your positive assessment and constructive suggestions on our work. We have responded thoroughly to your concerns, especially the two main clarification requests mentioned in your Questions. We hope our responses help to increase the perceived biological plausibility and methodological rigor of the work. We commit to implementing all proposed revisions in the final version of the paper. We hope our responses have answered all your questions and please feel free to reach out if any further clarification is needed.

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[1] Microstructure of a spatial map in the entorhinal cortex.

[2] Hippocampal remapping and grid realignment in entorhinal cortex.

[3] Path integration and the neural basis of the'cognitive map'.

[4] Facing off world model backbones: Rnns, transformers, and s4.

[5] Learning to act without actions.

[6] TRANSDREAMER: REINFORCEMENT LEARNING WITH TRANSFORMER WORLD MODELS.

Thanks for your positive and constructive reviews. We are grateful for your recognition of our work's originality and primary contributions—the quantization method and the concept of attractor-as-code—and for your encouraging comments on that our model has potential to be used as a world model and that our experiments are well designed.

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Response to Weakness

1.  Thank you for the constructive suggestion. We agree that using more standard terminology improves the paper's clarity. Accordingly, we will revise the manuscript by replacing 'cognitive map' with 'attractor landscape' throughout the text. We believe this change makes the concept more accessible to broader audience, and we appreciate your suggestion on this.

2.  We agree to highlight the focus of our work is on dynamic/world model. Indeed, a key contribution of our work is replacing the conventional codebook in VQ with attractors (GCQ), thereby directly compressing spatiotemporal information in sequential data. As a result, GCQ can handle all tasks that standard VQ can, such as image, video, speech, action, and multimodal data. In the design of our experiments we chose to demonstrate GCQ’s strengths in the context of world modeling, as it is the most natural application. We appreciate this insightful suggestion and agree that emphasizing this aspect more strongly would benefit readers. We will revise both the title and introduction accordingly in the final version to underscore GCQ’s role as a world model.

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Response to Questions

Thanks for raising this insightful question. Your intuition is absolutely correct, and you’ve highlighted a fascinating aspect of attractor networks. While the term "attractor" implies stability, the structure of that stability is not limited to a single, fixed point. This leads to different types of representations:

• Discrete Attractors: In many classic networks (like Hopfield networks), the system stabilizes into one of several isolated, discrete points in the state space. These are ideal for representing discrete tokens.

• Continuous Attractors: It is also possible for attractors to form continuous structures (e.g., a line or a ring of stable states). These are perfectly suited for encoding continuous-valued tokens.

This leads directly to your main point: yes, our method can be used for a mixture of continuous and discrete tokens, specifically, continuous attractors for continuous-valued tokens and discrete attractors for discrete tokens.

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Response to Limitations

We agree that quantization is needed in many contexts. Thus, our method has broad applicability and potential contributions across a range of domains, including those mentioned in the response to Weakness 2. In this paper, we chose to focus our experiments on dynamic/world model settings where GCQ’s advantages are most direct. We also demonstrate an additional application to image compression in Appendix A. We will add discussions about these potential applications in the revised manuscript, and in future work, we will explore them through extensive experiments.

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Once again, we sincerely thank you for your recognition of our contributions, the novelty of our idea, and the rigor of our experimental studies. Your positive evaluation and feedback are invaluable to us. We also appreciate your constructive suggestions on terminology, presentation, and broader applicability. We are committed to incorporating the corresponding revisions in the final version of the paper.

I appreciate the detailed reply from the author and answered all my questions

I looked the authors' replies to my questions , as well as to other reviewers' questions, especially reviewer imb7 and wNwm. The quality of the work is high and the quality of the reply are decent. Therefore, I like to raise my score and wish this work can help improve discrete tokenization methods used widely in generative model community