InputDSA: Demixing, then comparing recurrent and externally driven dynamics¶

Conference: ICLR 2026
OpenReview: https://openreview.net/forum?id=2FrAvRz1E7
Code: To be confirmed
Area: Neural Dynamics / Representation and Dynamical Similarity
Keywords: Dynamical Similarity, DSA, Koopman, DMDc, Subspace Identification, Input-driven dynamics, RNN, Neural Data

TL;DR¶

The authors extend the Dynamical Similarity Analysis (DSA) method from autonomous to non-autonomous systems. By utilizing Subspace Dynamic Mode Decomposition with control (SubspaceDMDc) to simultaneously estimate the intrinsic operator $A$ and the input operator $B$, InputDSA enables the separate comparison of "intrinsic dynamics" and "input-driven dynamics" under partial observation, noise, or when only proxy inputs are available.

Background & Motivation¶

Background: Comparing whether two systems (brains, RNNs, physical systems) "perform the same function" is a common cross-disciplinary challenge. Dominant approaches compare state geometry (RSA / CKA / Procrustes / CCA) or topology, but none characterize temporal dynamics. Ostrow et al. (2023) introduced DSA, which uses Koopman theory to embed nonlinear dynamics into high-dimensional space and fits a linear transition operator $A$ for comparison. This has been applied to RNNs, training dynamics, and neural data.
Limitations of Prior Work: DSA and subsequent dynamical comparison methods assume the system is autonomous ($x_{t+1}=Ax_t$), ignoring external inputs. Real-world brains and machine learning systems are almost always non-autonomous, continuously receiving sensory signals, communicating across regions, or involving closed-loop feedback.
Key Challenge: Two systems with identical intrinsic dynamics may be judged "different" by DSA if driven by different inputs, and vice versa. Mixing input components pollutes the comparison of intrinsic dynamics and fails to evaluate whether systems respond to inputs in the same way.
Goal: To propose a similarity metric that can demix intrinsic and input-driven dynamics, allowing for joint or separate comparisons that remain robust under partial observation, noise, and unknown inputs.
Core Idea: [Demix then Compare] Explicitly estimate the state transition operator $A$ and the input-to-state mapping $B$, basing similarity on the controllability matrix. [Subspace Identification for Debiasing] Replace naive DMDc with subspace identification (N4SID / PO-MOESP) from classical control theory to solve the bias caused by partial observation. [Procrustes Acceleration] Simplify similarity optimization into a Procrustes alignment, achieving exponential speedup over the original DSA.

Method¶

Overall Architecture¶

InputDSA extends the DSA pipeline (embedding → fitting linear operators → comparing operators). For two systems, state and input data are collected, embedded into high-dimensional space, and a linear state-space model with control $x_{t+1}=Ax_t+Bu_t,\; y_t=Cx_t$ is fitted using SubspaceDMDc. Finally, controllability similarity, state similarity, and input similarity are computed based on the learned operators.

flowchart LR
    A[State + Input Data from Two Systems] --> B[High-dim Nonlinear Embedding φ]
    B --> C[SubspaceDMDc Fitting<br/>Subspace ID to estimate A, B]
    C --> D[Procrustes Alignment<br/>Solve for Optimal Orthogonal C*]
    D --> E1[State DSA: Intrinsic Similarity]
    D --> E2[Input DSA: Input Similarity]
    D --> E3[Controllability / Joint Similarity]

Key Designs¶

1. Controllability as a Similarity Metric: Including the Input Operator. DSA only defines similarity on $A$ as $\min_{C\in O(n)}\|CA_1C^T-A_2\|_F$. InputDSA recognizes that a key feature of input-driven systems is controllability—whether an input sequence can drive the state to any point in finite steps. This is characterized by the $T$-step controllability matrix $K(T)=[B,\,AB,\,A^2B,\dots,A^{T-1}B]$, which is preserved under orthogonal transformations of the state space ($A_1=CA_2C^T,\,B_1=CB_2 \Rightarrow K_1=CK_2$). The metric is thus extended to align the entire controllability matrix: $$\text{InputDSA}(A_1,A_2,B_1,B_2,T)=\min_{C\in O(n)}\sum_{i=0}^{T}\|CA_1^iB_1-A_2^iB_2\|_F^2=\min_{C\in O(n)}\|CK_1-K_2\|_F^2.$$ After finding the optimal $C^*$, two interpretable sub-scores are derived: the state score $\|C^*A_1C^{*T}-A_2\|_F^2$ and the input score $\|C^*B_1-B_2\|_F^2$.

2. Procrustes Acceleration: Converting Iterative Optimization to Closed-form Alignment. Standard DSA alignment for $A$ requires iterative optimization over the orthogonal group $O(n)$, which is computationally expensive. Since the new metric is formulated as $\|CK_1-K_2\|_F^2$, it is essentially an Orthogonal Procrustes problem, solvable via a single-step SVD, providing exponential speedup.

3. SubspaceDMDc: Solving Pollution from Partial Observation. Using naive DMDc ($\phi(x_{t+1})=A\phi(x_t)+B\phi(u_t)$) in partial observation scenarios (e.g., recording only a subset of neurons) is problematic. Inputs act directly on the observed dimensions and indirectly through unobserved dimensions. Naive estimation incorrectly attributes these indirect effects to intrinsic dynamics, biasing $B$ toward $A$. SubspaceDMDc incorporates control signals into subspace DMD, using N4SID / PO-MOESP algorithms to identify systems where only $y_t, u_t$ are observed. This method projects future (lifted) states onto the basis of past inputs and states, effectively filtering out observation and process noise.

4. Proxy Inputs and Hyperparameter Selection. In experimental settings, "true inputs" are often unavailable. InputDSA extends the metric to include joint alignment for inputs, allowing the use of proxy inputs (e.g., behavioral variables or task instructions). Hyperparameters (operator rank, delay steps, nonlinear basis) are determined by minimizing rank while maximizing future state prediction, utilizing Kalman filtering and AIC criteria.

Key Experimental Results¶

Main Results: Distinguishing Intrinsic vs. Input Dynamics under Partial Observation¶

On a 20-dimensional random RNN (only 2 dimensions observed, 5000 time points), four types of systems were constructed (sharing $A$ or $B$). Evaluation utilized silhouette scores (clustering separability, 1.0 is best):

Method	State Similarity Silhouette	Input Similarity Silhouette
DMD (Hankel, DSA)	0.60	— (Cannot compare inputs)
DMDc	0.68	0.19 (Fails under partial observation)
SubspaceDMDc (Ours)	0.94	0.83

As system dimensions scaled from 2 up to 1000 (with only 2 observed), SubspaceDMDc significantly outperformed other methods. Input scores for DMDc were consistently non-robust, while SubspaceDMDc remained robust across all scales.

Robustness: Noise and Proxy Inputs¶

With SNR (SER) near 1, correlation with ground truth distance remained $r>0.75$, thanks to delay embeddings and low-rank regression.
In a Random Target Reach task, intrinsic (state) distances remained strongly correlated with ground truth even when using random proxy inputs, whereas input distances were most accurate when using task-relevant proxies (action + position).

Application I: Individual Differences in RL Agents (Plume Tracking)¶

InputDSA revealed that the input-driven dynamics of successful (Top) agents were significantly more similar and clearly separated from unsuccessful (Bottom) agents, while intrinsic dynamics showed no significant group difference.
Successful tracking depends primarily on rapid input response to wind/odor. Top agents converged toward consistent input dynamics, whereas Bottom agents diverged into idiosyncratic dynamics.

Application II: Rat Neural Population Dynamics Across Task Phases¶

Neuropixels recordings from 6 brain regions were analyzed during an auditory evidence accumulation task.
Comparing phases around the "neural Time of Commitment" (nTc): Both state and input dynamics changed significantly. Post-nTc, the intrinsic dynamics became more persistent (smaller power-law index $\beta$ for eigenvalues of $A$), and the Frobenius norm of the controllability Gramian decreased significantly ($p < 0.001$, median energy drop ~47.6%).
Key Finding: Population activity undergoes a mechanistic switch at nTc from an input-driven accumulation phase to an intrinsically dominated commitment phase. Naive DMDc fails to recover this structure due to partial observation.

Highlights & Insights¶

Input as a First-Class Citizen: First framework to explicitly model and separately compare input-driven dynamics, answering whether two systems respond to inputs in the same way.
Controllability as a Unified Language: Controllability matrices and Gramians bridge abstract similarity with neuroscientific narratives (e.g., evidence accumulation vs. commitment).
Validation of Proxy Inputs: Theoretical and experimental evidence supports the use of behavioral/sensory proxies for robust intrinsic similarity estimation in computational neuroscience.
Simultaneous Debiasing and Speedup: Subspace identification corrects partial observation bias and filters noise, while Procrustes alignment provides massive computational acceleration.

Limitations & Future Work¶

Additive Inputs Only: Assumes $Ax + Bu$, which may not capture multiplicative mechanisms like gain modulation.
Entanglement in Closed Loops: If inputs are linear functions of states (closed-loop), separating contributions is difficult.
Identifiability Conditions: Requires "persistent excitation" from inputs; otherwise, state modes might be underestimated.
Computational Bottleneck: While comparison is fast, fitting SubspaceDMDc remains the primary overhead.

Geometry/Topology Similarity: Complements RSA, CKA, and Procrustes by adding temporal dynamics.
DSA Lineage: Directly extends the work of Ostrow et al. (2023) by addressing the omission of external inputs.
Control Theory Integration: Builds upon DMDc (Proctor 2016) and Subspace DMD (Takeishi 2017), utilizing N4SID/PO-MOESP.

Rating¶

Novelty: ⭐⭐⭐⭐⭐ First to demix and compare input-driven dynamics with debiased subspace ID and Procrustes speedup.
Experimental Thoroughness: ⭐⭐⭐⭐⭐ Comprehensive validation across synthetic systems, RL agents, and real neural data.
Writing Quality: ⭐⭐⭐⭐ Clear motivation and strong link between theory and experiments.
Value: ⭐⭐⭐⭐⭐ Provides a robust, efficient tool for comparing brain regions, model validation, and training dynamics.