Title: RNNs perform task computations by dynamically warping neural representations
Venue: NeurIPS 2025
Accepted: yes
Keywords: RNN

Computational Neuroscience

Dynamical Systems

Neural Representations
Abstract: Analysing how neural networks represent data features in their activations can help interpret how they perform tasks. Hence, a long line of work has focused on mathematically characterising the geometry of such "neural representations." In parallel, machine learning has seen a surge of interest in understanding how dynamical systems perform computations on time-varying input data. Yet, the link between computation-through-dynamics and representational geometry remains poorly understood. Here, we hypothesise that recurrent neural networks (RNNs) perform computations by dynamically warping their representations of task variables. To test this hypothesis, we develop a Riemannian geometric framework that enables the derivation of the manifold topology and geometry of a dynamical system from the manifold of its inputs. By characterising the time-varying geometry of RNNs, we show that dynamic warping is a fundamental feature of their computations.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces a framework for understanding computation in recurrent neural networks (RNNs) based on Riemannian geometry. The central hypothesis is that RNNs perform computations by dynamically warping the geometry of their internal neural representations. The authors develop a mathematical formalism to test this hypothesis. First, they show that if the time-varying inputs to a dynamical system lie on an $m$-dimensional manifold, the trajectories of the state are constrained to a manifold of at most $m+1$ dimensions. Then, they define a \"representational metric\" on this neural activity manifold, derived as a pullback from the state space. This metric quantifies the local \"warping\" of the representation space and can be computed by solving an adjoint differential equation alongside the main RNN dynamics.\n\nThe framework is applied to two toy tasks: In the first task, the model has to output the sign of one of two real inputs, selected based on a (one-hot) contextual clue. The analysis reveals that the network learns to warp its representational manifold to selectively compress the dimension corresponding to the irrelevant input, while expanding the dimension for the relevant input near the decision boundary (close to zero). In the second task, the model has to reproduce two inputs after a time delay. The authors show that the network activity lies on a toroidal manifold, which dynamically changes its intrinsic geometry to selectively encode and recall items from memory.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n1. The paper is well-written and organized. Many helpful illustrations allow the reader to develop an intuitive understanding of the relevant concepts and ideas.\n2. The mathematical framework is rigorously developed and intuitively makes sense. It goes beyond an analysis of the dynamis near fixed points.\n\n**Weaknesses**\n1. The authors only apply their framework to very small models and simple toy tasks. While this is helpful to illustrate central concepts, an application to larger models trained to solve real-world tasks is lacking. Without this, it remains unclear whether the framework can be used to gain an understanding of the inner workings of more complex models.\n2. While the framework provides an account of how a model's internal representations are transformed, the link between geometry and computation seems correlational: While the geometry warps *as* the model performs the task, it is not evident to me that this geometric transformation *is* the computation itself, versus a by-product of another underlying process (i.e. the \"actual computation\"). The presented framework is currently post-hoc, and it is not yet clear if it can be used prescriptively to design or train models more effectively.\n\nWhile I find the general premise of the paper very interesting, my main concern (and why I did not rate it higher) is that it feels \"incomplete\" to me. The paper does not apply the developed framework to any \"real problem\" and does not demonstrate what kind of improvements or advancements it enables. I think the paper would be much more impactful if it demonstrated a few applications beyond toy examples.\n\nQUESTIONS:\n1. Have you considered applying your framework to RNNs trained on real-world tasks, for example, for natural language processing? I think it could be very insightful to explore how different architectures (e.g., LSTMs vs. SSMs) solve tasks. Maybe it would even be possible to extend the framework, so that it can be applied to (autoregressive) transformers? They can be considered as a special kind of RNN, where the size of the internal state (the KV-cache) grows over time (this paper https://arxiv.org/abs/2006.16236 is helpful to make this analogy clearer).\n2. Do you see a path toward using these geometric insights prescriptively? For instance, could the representational metric be incorporated into the training process (perhaps as a sort of regularizer), to encourage a model to learn specific geometric structures and potentially improve learning speed and/or generalization?\n3. Under Limitations, you mention the degeneracy of the map from network parameters to geometry. Have you investigated whether different parameter initializations converge to solutions with similar geometric properties? Perhaps more complicated tasks are necessary to observe \"distinct solutions\" (for the investigated toy tasks, it is difficult to think of different solutions).\n\nTypos (these do not influence the rating):\n\n- line 231: \"context-depenent\" should be \"context-dependent\"\n- caption of Figure 6b: \"is lies\" should be \"lies\"\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors present a formal framework for understanding computation through dynamics present in recurrent neural network architectures with time-varying inputs. They study the dimensionality and topology of network dynamics with respect to the structure of the input data for both contextual decision making and working memory tasks. On simple models of a contextual classification and memory task, they demonstrate how the geometry of the dynamics is warped to allow the network to discard (contextually defined) irrelevant information through dynamics.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n- The formal study of time-varying inputs in RNN dynamical systems is indeed something which is lacking in the literature, and a valuable contribution, especially to the computational neuroscience community (which the authors appear well aware of). \n- The framework seems broadly applicable to the study of nonlinear dynamics, and therefore has the potential to yield novel insights not possible with current methods.\n- The paper is well written and organized for readability and understanding. For example, the review of core concepts for static neural representations in Section 2 is a welcome addition.\n- The proposed framework and viewpoint for understanding computation in recurrent neural networks is novel to the best of my knowledge and appears to open many doors for further understanding how RNNs perform computation -- the authors also clearly demonstrate this potential through their chosen examples.\n\n\n**Weaknesses**\n- The paper seems to have a relatively significant connection to the recent work on Low-Rank RNNs from Valente, Pillow & Ostojic, (Neurips, 2022). The authors cite this paper, yet have no further discussion on the relationship to that theory.\n\nQUESTIONS:\n- In section 2, does the choice of Tanh non-linearity have a significant effect on the warping of the space near the decision boundary compared to a ReLU? More generally, can the authors comment on how the choice of piece-wise linear non-linearity will impact their results compared with 'sigmoid-like' non-linearities? \n- It appears a proof of Theorem 3.4 is lacking from the supplementary material. Is this somehow given by 3.3? Should it then be written as a corollary?\n- Can the authors comment on the relationship of their theory to that of Low-Rank RNNS? Adding a discussion of this to the paper would be beneficial for the community.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper authors use the framework of Riemannian geometry to analyse how RNNs compute through their dynamics. Through their analyses they identify that the computations of RNNs can be captured through the warping of their representational spaces. Authors trace this warping across a couple of established examples of RNN computations.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths \n\nAttempt to unify established examples of computations of RNNs through a coherent perspective of manifold warping \n\nStrong theoretical analyses of the expected dimensionality of RNN signals as a function of input signals \n\nSystematic application of differential geometric tools to RNN analysis \n\nBeautiful presentation of your results.\n\nWeaknesses \n\nWhy is this a new framework?: Authors claim to introduce a new framework but fail to explain in detail how their usage of Riemannian geometry goes beyond existing approaches for analysing dynamic representations.  \n\nInconsistent metric definition: The specific metric they develop based on Riemannian geometry claims to be based on pullback construction, but they fail to define or consistently apply a single coherent metric across analyses. Different sections seem to use different mathematical objects all labeled as 'representational metrics' without clear relationships between formulations. This plays into a larger issue that each of the individual sections seem to have substance by themselves but I am struggling to identify the links between them. This is not helped by the fact that authors do not provide a joint summary of findings across sections at the start of the discussion. \n\nWhat new findings are possible with the new framework: While perhaps the authors outline a new way of thinking about dynamic representations, they seem to not be able to use this analysis framework to derive findings which have not been reported in the literature yet. The claimed 'dynamic warping' phenomena mimics established findings about RNN learning dynamics (e.g., emergence of structured attractors, dimensionality reduction during training) using different mathematical vocabulary, without providing new computational insights. For discussion of toroidal representation there are existing works (https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1011852, https://www.nature.com/articles/s41583-023-00693-x). For discussion of collapse of information over non-relevant dimensions, this is already shown in Sussillo's work and also in https://www.biorxiv.org/content/10.1101/2023.04.24.538054v2.abstract. In addition, Riemannian geometry has been used already for analysing (static) representations in https://proceedings.neurips.cc/paper_files/paper/2022/hash/4b3cc0d1c897ebcf71aca92a4a26ac83-Abstract-Conference.html.\n\nQUESTIONS:\nFramework definition: In the abstract authors write \"To test this hypothesis, we develop a Riemannian geometric framework\" -- please comment on what you mean with developing a framework, given that Riemannian geometry has been used already. What specific organizing principles, predictive capabilities, or systematic methodologies does your framework provide beyond applying existing mathematical tools? This links to the criteria of clarity and originality. \n\nMetric definition and consistency: Is your view of your work that you specifically introduce a metric for warping that can be applied across examples? As stated in \"We introduce the representational metric defined as the pullback of the metric on the neural space to the input manifold\". I am struggling to see where you define a metric which is then coherently used across examples. Please clarify the definition of the metric and explain how this metric highlights mechanisms of computations in later examples. This links to the criteria of clarity and quality. \n\nMechanistic insights: My reading is that your investigations did not directly reveal any new mechanism about computations in dynamics but more focuses on the introduction of a metric? Please correct me if you have a specific theoretical result that I am overlooking. This links to significance. \n\nOverall contributions: Please provide an overall conclusion statement of your learning across investigations at the start of the 'Conclusions' section. Currently you only mention how Riemannian geometry can be applied across use cases, but it is unclear to me how your work contributes to this application of Riemannian geometry beyond demonstrating that existing mathematical tools can be applied to RNNs. This links to clarity and significance.  \n\nAdditionally, note typo in line 165: “neurosciencce”.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work studies the geometry of the neural manifold and how it is shaped by a low-dimensional input manifold in an RNN dynamically. Specifically, the authors define a metric on the manifold that quantifies its shape and the evolution of its geometry across time. There are two main conclusions: contextual inputs warp (stretch) this manifold over time to disregard irrelevant inputs. Second, in the working memory task, the torus manifold is warped across time in the encoding phase, but it remains the same during delay.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**: \n\nThe paper is very well written and is backed by mathematical proofs for the arguments made. The problem considered is also an important one: studying how neural manifolds evolve over time. The visuals in the figures make the point go across better!\n\n\n**Weaknesses**: \n\nThe main theorem about the outputs of RNN being low-dimensional if the input is low-dimensional seems trivial. Can the authors state if there are any special dynamical systems where this will not be satisfied, and/or explain why this is non-trivial?\n\nThe tasks considered in this work are low-dimensional, which might not be true for RNNs designed for ML. I understand the neuroscience motivation behind considering these tasks, but it would be interesting to see if this warping occurs in some task where the manifold is higher dimensional. It might not be feasible here, but a discussion about whether the conclusions drawn here would hold for a higher-dimensional RNN doing something more complicated, or state what the limitations of this approach are.\n\nAlthough there’s a calculation about the curvature of the torus but it seems no insights were drawn from it. So I’m not entirely sure of what the utility of that calculation is.\n\nQUESTIONS:\nIt would be nice to have some more intuition about what the metric represents. \n\nIn Fig.4f, representing the warping of the manifold, is it an illustration or an actual calculation of the geodesic gridlines?\n\nWorking memory is not just about maintenance but also about the manipulation of neural activity. How would you think the manifold would evolve, and how would your conclusions change if you considered a task with manipulation?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}