# Beyond Geometry: Comparing the Temporal Structure of Computation in Neural Circuits with Dynamical Similarity Analysis

arXiv: https://arxiv.org/abs/2306.10168 · HTML: https://mlsanity.com/paper/2306.10168/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 69 references that are in the map.
- First posted on arXiv 16 Jun 2023.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [The Forward-Forward Algorithm: Some Preliminary Investigations](https://mlsanity.com/paper/2212.13345/) (2022)
- [Generalized Shape Metrics on Neural Representations](https://mlsanity.com/paper/2110.14739/) (2021)
- [Universality and individuality in neural dynamics across large populations of recurrent networks](https://mlsanity.com/paper/1907.08549/) (2019)
- [Attractor and integrator networks in the brain](https://mlsanity.com/paper/2112.03978/) (2021)
- [Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition](https://mlsanity.com/paper/1409.5496/) (2014)
- [Recurrent switching linear dynamical systems](https://mlsanity.com/paper/1610.08466/) (2016)
- [Neural tuning and representational geometry](https://mlsanity.com/paper/2104.09743/) (2021)
- [Two Routes to Scalable Credit Assignment without Weight Symmetry](https://mlsanity.com/paper/2003.01513/) (2020)
- [The Influence of Learning Rule on Representation Dynamics in Wide Neural Networks](https://mlsanity.com/paper/2210.02157/) (2022)
- [Representational dissimilarity metric spaces for stochastic neural networks](https://mlsanity.com/paper/2211.11665/) (2022)
- [How to Train Your Wide Neural Network Without Backprop: An Input-Weight Alignment Perspective](https://mlsanity.com/paper/2106.08453/) (2021)
- [Identifying Learning Rules From Neural Network Observables](https://mlsanity.com/paper/2010.11765/) (2020)
- [Koopman Operator Theory for Nonlinear Dynamic Modeling using Dynamic Mode Decomposition](https://mlsanity.com/paper/2110.08442/) (2021)

## Built on by (most cited first, 23 of 23)

- [Block-Recurrent Dynamics in Vision Transformers](https://mlsanity.com/paper/2512.19941/) (2025)
- [Self-supervised contrastive learning performs non-linear system identification](https://mlsanity.com/paper/2410.14673/) (2024)
- [Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups](https://mlsanity.com/paper/2410.14477/) (2024)
- [InputDSA: Demixing then Comparing Recurrent and Externally Driven Dynamics](https://mlsanity.com/paper/2510.25943/) (2025)
- [Delay Embedding Theory of Neural Sequence Models](https://mlsanity.com/paper/2406.11993/) (2024)
- [Deep RL Needs Deep Behavior Analysis: Exploring Implicit Planning by Model-Free Agents in Open-Ended Environments](https://mlsanity.com/paper/2506.06981/) (2025)
- [DFORM: Diffeomorphic vector field alignment for assessing dynamics across learned models](https://mlsanity.com/paper/2402.09735/) (2024)
- [Neural timescales from a computational perspective](https://mlsanity.com/paper/2409.02684/) (2024)
- [Geometry-aware similarity metrics for neural representations on Riemannian and statistical manifolds](https://mlsanity.com/paper/2603.28764/) (2026)
- [Fading memory as inductive bias in residual recurrent networks](https://mlsanity.com/paper/2307.14823/) (2023)
- [Phase codes emerge in recurrent neural networks optimized for modular arithmetic](https://mlsanity.com/paper/2310.07908/) (2023)
- [A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems](https://mlsanity.com/paper/2509.24920/) (2025)
- [What Makes Two Language Models Think Alike?](https://mlsanity.com/paper/2406.12620/) (2024)
- [Modeling Neural Activity with Conditionally Linear Dynamical Systems](https://mlsanity.com/paper/2502.18347/) (2025)
- [Finding separatrices of dynamical flows with Deep Koopman Eigenfunctions](https://mlsanity.com/paper/2505.15231/) (2025)
- [Can Biologically Plausible Temporal Credit Assignment Rules Match BPTT for Neural Similarity? E-prop as an Example](https://mlsanity.com/paper/2506.06904/) (2025)
- [Koopman Learning with Episodic Memory](https://mlsanity.com/paper/2311.12615/) (2023)
- [Measuring and Controlling Solution Degeneracy across Task-Trained Recurrent Neural Networks](https://mlsanity.com/paper/2410.03972/) (2024)
- [Characterizing Nonlinear Dynamics via Smooth Prototype Equivalences](https://mlsanity.com/paper/2503.10336/) (2025)
- [Fast dynamical similarity analysis](https://mlsanity.com/paper/2511.22828/) (2025)
- [Interpreting Reinforcement Learning Model Behavior via Koopman with Control](https://mlsanity.com/paper/2603.19968/) (2026)
- [Learning Options for Compositional Motor Control with Adapter Banks](https://mlsanity.com/paper/2609.17042/) (2026)
- [Analyzing the Dynamics of WSBM-based Echo State Networks using InputDSA](https://mlsanity.com/paper/2610.15380/) (2026)

## Abstract (arXiv)

How can we tell whether two neural networks utilize the same internal processes for a particular computation? This question is pertinent for multiple subfields of neuroscience and machine learning, including neuroAI, mechanistic interpretability, and brain-machine interfaces. Standard approaches for comparing neural networks focus on the spatial geometry of latent states. Yet in recurrent networks, computations are implemented at the level of dynamics, and two networks performing the same computation with equivalent dynamics need not exhibit the same geometry. To bridge this gap, we introduce a novel similarity metric that compares two systems at the level of their dynamics, called Dynamical Similarity Analysis (DSA). Our method incorporates two components: Using recent advances in data-driven dynamical systems theory, we learn a high-dimensional linear system that accurately captures core features of the original nonlinear dynamics. Next, we compare different systems passed through this embedding using a novel extension of Procrustes Analysis that accounts for how vector fields change under orthogonal transformation. In four case studies, we demonstrate that our method disentangles conjugate and non-conjugate recurrent neural networks (RNNs), while geometric methods fall short. We additionally show that our method can distinguish learning rules in an unsupervised manner. Our method opens the door to comparative analyses of the essential temporal structure of computation in neural circuits.