InputDSA: Demixing, then comparing recurrent and externally driven dynamics
Authors: Ann Huang
Mitchell Ostrow
Satpreet Harcharan Singh
Leo Kozachkov
Ila R Fiete
Kanaka Rajan
Venue: ICLR 2026 Poster (2026)
Keywords: dynamical systems
recurrent neural networks
neural dynamics
similarity metrics
computational neuroscience
Primary area: applications to neuroscience & cognitive science
TL;DR: We introduce InputDSA, a method for quantitatively comparing two input-driven dynamical systems, based on prior work in dynamical similarity, systems identification, and Koopman operator theory.

Abstract: In control problems and basic scientific modeling, it is important to compare observations with dynamical simulations. 
For example, comparing two neural systems can shed light on the nature of emergent computations in the brain and deep neural networks. Recently, Ostrow et al. (2023) introduced Dynamical Similarity Analysis (DSA), a method to measure the similarity of two systems based on their recurrent dynamics rather than geometry or topology. However, DSA does not consider how inputs affect the dynamics, meaning that two similar systems, if driven differently, may be classified as different. Because real-world dynamical systems are rarely autonomous, it is important to account for the effects of input drive. To this end, we introduce a novel metric for comparing both intrinsic (recurrent) and input-driven dynamics, called InputDSA (iDSA). InputDSA extends the DSA framework by estimating and comparing both input and intrinsic dynamic operators using a variant of Dynamic Mode Decomposition with control (DMDc) based on subspace identification. We demonstrate that InputDSA can successfully compare partially observed, input-driven systems from noisy data. We show that when the true inputs are unknown, surrogate inputs can be substituted without a major deterioration in similarity estimates. We apply InputDSA on Recurrent Neural Networks (RNNs) trained with Deep Reinforcement Learning, identifying that high-performing networks are dynamically similar to one another, while low-performing networks are more diverse. Lastly, we apply InputDSA to neural data recorded from rats performing a cognitive task, demonstrating that it identifies a transition from input-driven evidence accumulation to intrinsically-driven decision-making. Our work demonstrates that InputDSA is a robust and efficient method for comparing intrinsic dynamics and the effect of external input on dynamical systems.

Published as a conference paper at ICLR 2026
INPUTDSA : DEMIXING THEN COMPARING RECUR- RENT AND EXTERNALLY DRIVEN DYNAMICS
Ann Huang 12,3+, Mitchell Ostrow Satpreet H. Singh?.3, Leo Kozachkov", Ila Fiete", Kanaka Rajan
Harvard University Haarard Medical School Kempper Institute "Massachusetts Institute of Technology Brown University
(annhuang@g. harvard, ostrowwmit)..ed
Equual contribution
ABSTRACT
In control problems and basic scientific modeling. it is important to compare ob- servations with dynamical simulations. For example, comparing two neural sys- tems can shed light on the nature of emergent computations in the brain and deep neural networks. Recently, Ostrow et al. (2023) introduced Dynamical Similar- ity Analysis (DSA), a method to measure the similarity of two systems based on their recurrent dynamics rather than geometry or topology. However, DSA does not consider how inputs affect the dynamics, meaning that two similar systems, if driven differently, may be classified as different. Because real-world dynam- ical systems are rarely autonomous, it is important to account for the effects of input drive. To this end, we introduce a novel metric for comparing both intrinsic (recurrent) and input-driven dynamics, called InputDSA (iDSA). InputDSA ex- tends the DSA framework by estimating and comparing both input and intrinsic dynamic operators using variant of Dynamic Mode Decomposition with con- trol (DMDe) based on subspace identification. We demonstrate that InputDSA successfully compare partially observed, input-driven systems from noisy data. We show that when the true inputs are unknown, surrogate inputs can be substituted without a major deterioration in similarity estimates. We apply InputDSA on Recurrent Neural Networks (RNNs) trained with Deep Reinforce- ment Learning, identifying that perforrero networks are dynamically sim- ilar to one another, while low- www-perfoming networks are more diverse. Lastly, we apply InputDSA to neural data recorded from rats performing a cognitive task, demonstrating that it identifies a transition from input-driven evidence ac- cumulation to antrisicallly-drive I is robust and efficient method for comparing intrinsic dynamics and the effect external input on dynamical systems.
secision-making Our work demonstrates that
INTRODUCTION
Identifying that two seemingly disparate complex systems have the same underlying structure is widespread objective across many scientific fields, including deep learning (Huh et al., 2024). computational and systems neuroscience (Yamins et al., 2014 Aldarondo et al.) 2024] Prinz et al,,
2004]. and physics (Hohenberg & Halperin| 1977 Feigenbaum 1978). One common approach to characterizing the similarity of two systems (c.g., brains, minds, computational models, or physical objects) is to compare the geometry of their states. Well-known methods to do are Representa- tional Similarity Analysis, Centered Kernel Alignment, Procrustes Analysis, Canonical Correlation Analysis, and Pearson Correlation (Kriegeskorte et al. |0088] Kornblith et al.) 2019 Williams al., 2022 Gallego et al./ 2018;; Raghu et al. 2017: Schrimpf et al. 20188) Neural networks can also be characterized by the topology of their activations, (Chaudhuri
al..))00))) Gardner et al. 2022 Lin & Kriegeskorte] 2024), more invariant measure than geometry, which depends on the particular sampling of neurons from the network. However, common to all is that they do not capture similarity in temporal dynamics (Galgali et al. 2023 Maheswaranathan et al. 2019 Ostrow et al. 2023).
Metrics such as Dynamical Similarity Analysis (DSA, Ostrow et 2023) offer an important com-
plementary lens to structure characterization, by proposing similarity metric on the level of dy- namics. DSA provides an efficient and theoretically grounded dynamical similarity metric that has been successfully applied to recurrent network dynamics, training dynamics, and biological neural

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data [Redman al. 2024a; Huang et al. 2025 Codol al., 2024a| Guilhot et al. 2024 Versteeg iet al., 2025 Lazzari & Saxena 2025). Briefly, DSA nonlinearly embeds dynamics into a high-
dimensional space and estimates a linear state-transition operator from observed trajectories, which is then compared across systems. Recent work introduced other methods for dynamics comparison
(Redman et al. 2024b Vermani et al., 2024, Cotler et al. 2023] Gosztolai al. 2025] Chen al.)
2024 Nejatbakhsh et al., 2024) based on other computational techniques such as neural networks and shape metries. Notably, none of these methods consider the effect of external input.
In neuroscientific settings such as central pattern generators or working memory circuits, dynam- ics may be treated as approximately autonomous (Marder & Bucher 2001; Grillner [006 Kichn
2016 Fuster & Alexander 1971| Funahashi et al. 1989; Goldman Rakic 1995 Compte et al. 2000
Wang 1999). Prior methods work well for comparisons in these settings. However, when activity
the result of both intrinsic dynamics and input drive, comparisons can be confounded by inputs. Most systems of interest in neuroscience and machine learning are non non-autonomous, receiving sen- sory signals or communication from other subsystems (Eisen al., (025). They are driven by complex external inputs and can receive observations that are contingent on the systems' outputs (Madhav & Cowan 2020; Kao & Hennequin 2019, Rajan et al. 2010))
Despite the ubiquity of input, current dynamical similarity methods ignore input-driven dynamics and do not incorporate estimation of how inputs affect states. To bridge this gap, we introduce InputDSA (iDSA), a method that disentangles intrinsic dynamics from input-drive, thereby enabling joint or separate metric comparisons of input-driven and intrinsic dynamics. InputDSA extends the DSA framework by explicitly estimating both the intrinsic (state-transition) operator and the input- to-state mapping, which not only defines new notion of similarity that incorporates the effect of inputs, but also in turn improves estimation of the intrinsic operator.
Contributions We extend DSA to non- autonomous systems that are driven by external input, which we call InputDSA To do so, we develop a novel similarity metric and variant of the dynamic mode decomposition (DMD), demonstrat- ing that they can together provide complementary insights on both intrinsic well as input-driven dynamical similarity. We demonstrate InputDSA first on systems with known ground truth. We next show that similarity scores can be robust to surrogate or noisy inputs, provided that they have sufficient similarity to the real inputs. Finally, we apply InputDSA to two datasets: RNNs trained with Reinforcement Learning, and neural population data (spiking) from rats performing a sensory decision- making task. We show that InputDSA distinguishes high- from low-performing models and reveals how dynamics reorganize across different task periods.
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Figure 1: InputDSA schematic (1), state and input data are collected from two sys- tems. (2) data are embedded in high- dimensional space (3) linear state-space models are fit to the data (4) Controllability, state, and input similarity are computed on learned state-space models. Gray indicates extensions from DSA.
METHODS
2.1 DYNAMICAL SIMILARITY ANALYSIS (DSA)
In dynamical systems, a key notion of similarity is called topological conjugacy: the existence of a homeomorphism that maps trajectories of one system onto those of another. When two systems are conjugate, they have the same qualitative structure, including the same number and type of fixed points. Given two dynamical systems f X + X and g : Y Y with mapping & X + Y, (semi-) conjugacy is defined as:
(I) The existence of such mapping entails one-to-one alignment between topological features of each system such as invariant manifolds. Note that this is not geometric because distances and angles are not necessarily preserved under this mapping. In general, such function can be arbitrarily complex,
goó =

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which can make searching for the true conjugacy map challenging all but the simplest settings. DSA attempts to circumvent the optimization problem by approximating the Koopman Operator, which linearizes nonlinear dynamical systems via high-dimensional embeddings (Koopman 1931
Budišić et al. 2012). In the linear space, conjugacy maps are linear and therefore easier to identify. The methodology of DSA is therefore as follows: First, approximate your systems as linear some high-dimensional space, yielding dynamics models 1 Ax;- Then similarity is defined on the linear operators using the following metrics:
DSA(AI, A2) == min CEO(m)
|CA,CT - -22lFF
DSA(AI, A2) == min
[PA| PT - A2llF
PET)())
Where A, is the eigenvalue matrix of A,, and O(n), [I()) the groups of dimensional orthog-
onal and permutation matrices. The latter metric was introduced by (Redman et al. 2024b) and is special case of the former (Ostrow al. 2023). These metries
reminiscent of Procrustes Analy-
sis, which seeks an orthogonal transformation to align two data matrices, hence Ostrow et al. (2023) termed the first one Procrustes Analysis over Vector Fields. The latter metric is inspired from Koop- man Operator Theory based on the relationships between Koopman Operators of conjugate systems (Budišić al. 2012). Other notions of similarity on the Koopman Operator are defined in (Mezić
& Banaszuk 2004 Mezic 2016).
2.2 INPUTDSA
Inspired by DSA's approach for autonomous systems, consider two linear dynamical systems
(4) A key feature of input-driven systems is their controllability: the ability for an input sequence to drive the state arbitrary points in finite time. In linear systems, this is encoded in the T-step controllability matrix (with T typically taken as the dimension of the system):
i = A₁r + Biu(t) == A2y + B2u(t).
(T (B, A, ABB,
Ai-1B,)
and its corresponding Gramian, which encodes the geometry of controllability.
W - K = Σ A(
(6)
1=0
Intuitively, directions with small eigenvalues are casier to control, because they are more respon- sive to the effect of input. Controllability, as measured by the eigenvalues of the Gramian, is only preserved under orthogonal transformations between state spaces (Appendix G.2).
y = Cx = A = CA2CT, Bi = CB2 K1 = CK2
This motivates our proposed dissimilarity metric, which extends DSA:
InputDSA (A1, A2, B1, B2,
min CEO(m)
Σ [|CA - 13 B2 = min |CK1 K2 (8) CEO(A)
We also provide a thcoretical extension of Eq. 3 in Appendix G.I which we note is highly suscep-
tible numerical instability. Although Eq. requires iterative optimization, Eq. 8 is solved via Procrustes alignment, which yields an exponential acceleration of prior work. We provide theoreti- cal discussion Appendix G After solving for C*, we can study the joint state and input scores:
InputDSA state (A1, A2, C*) = |IC* A,C* AA21
InputDSA impur (B1, B2, C*) = IC* B, -B21
(10)
If the inputs directly applied to the system are known, in computational models, Eq. Kis sufficient. However, when the true input is some modification of surrogate input, it may be necessary to align the input as well. This is relevant in settings such as the comparison of two brain regions, when the surrogate input u is a behavioral or sensory variable that is transformed by upstream regions. We therefore can extend Eq. to consider joint alignment of the input, without significant differences in the optimization problem. For further technical details, see Appendix F
This metric motivates the following approach as in Ostrow et al. (002)): identify the best linear approximation of an input-driven system, following which comparison can be done efficiently be- tween the approximations. To do so. Ostrow et al.) (2023) applied the Dynamic Mode Decomposition
(Schmid] 2022), which we introduce and extend to fit our setting next.
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2.3 ESTIMATING LINEAR OPERATORS
As in DSA, We fit linear operators via the Dynamic Mode Decomposition (DMD) family of meth-
ods. The DMD (Schmid 2010 2022) identifies the linear dynamics that best explain the data:
((x++)) (x)()))
(11)
Here Et represents the measured state of the system at time f, Φ is nonlinear embedding of the data that typically expands the dimensionality of the state space, and A is a matrix that is identified using some variant of least-squares regression. The goal of the Dynamic Mode Decomposition is to approximate the Koopman Operator (Koopman 1931]. a theoretical object that exists for all dy- namical systems which encodes the linear dynamics of observables (functions that act the true system state) under the system dynamics. Prior work has explored many different choices of Φ. For example, @ can be kernel function, delay embedding, or even neural network (Williams al., 2016 Brunton et al.] 2017: Arbabi & Mezić 2017] [ Lusch et al. 2018). Intuitively, the dimension- ality expansion acts similarly to the kernel trick (Smola & Schölkopf 1998). where embedding into higher dimensions 'unfolds' the nonlinearity. The DMD can be applied in non-autonomous systems, although this risks mixing driving and intrinsic dynamies (Proctor et al., 2016a).
Incorporating Control into DMD and Koopman While the original Koopman theory assumed
autonomous dynamics, prior work has sought to incorporate control into the theory (Korda & Mezić,
2018 Proctor et al., 2016b; Strässer et al. 2025 Asada &
2024 Haseli et al,
Likewise, the DMD can be generalized to driven systems: When given control inputs uy we can instead apply DMD with control (DMDc. Proctor al. [2016a
2025
Φ (I++1) = Adh + Bib((u))
(12)
Here, Φ1 and 62 can be distinct nonlinearities. While DMDc was originally only applied with no nonlinearity (Ø1,02 Ø2 Id), it too can be generalized to high-dimensional nonlinear embeddings. As in DMDc, we assume that inputs are known or a useful surrogate can be constructed (Section 3.2)
Further algorithmic details are in Appendix B
Issues of Partial Observation While estimating A and B via DMDc is an intuitive extension to input-driven systems, has a hidden failure mode in the analysis of partially-observed systems. This is particularly important in the analysis of neural data, in which small subset of neurons in vast population are recorded. Generically, an input-driven system that is partially observed receives input to both the observed and unobserved components. The input at time t therefore affects the observed state at time / (instantaneously) and in future time steps through the unobserved state (Fig. [TA). This means that simply applying DMDc in this setting will bias the B matrix towards the intrinsic dynamics of the system. We develop a formal description of this problem for linear systems in Appendix D We solve this problem by introducing Subspace DMDc, an extension of Subspace DMD (aaeeishi et al.) (017b) that incorporates input. In brief, Subspace DMDc utilizes subspace identification algorithms from classical control theory (Verhaegen & Verdult 2007). which seek to identify linear dynamical systems of the form:
It+1 = Ax + Bug
yt = Ca.
(13)
Here, only ye and in are observed. The situation of partial observability is special case of this
problem. In practice, we use the well-known N4SID or PO-MOESP algorithms to estimate A and B (Van Overschee & De Moor 1994, Verhaegen 1994) in Eq. 13 on lifted states (thereby leveraging the power of nonlinear DMD algorithms such as Williams et al. 2016). The algorithmic idea behind subspace identification is similar to that of instrumental variable regression: the lifted state data to-be-predicted (future data 7)+1) is projected onto the basis of the past input and lifted state data (ye, u4) before estimating a A and B via linear regression. This has the added benefit of projecting out observation and process noise, thereby providing noise robustness (Verhaegen) 1994 Verhaegen & Verdult 2007). For technical details on the subspace identification algorithm, see Appendix
Tuning Sub for InputDSA SubspaceDMMD has three key hyperparameters: the rank of the linear operator, the number of delays used in subspace identification, and the type of nonlinear basis used for linearizing the dynamics. In practice, the most general method we use to tune Sub is to pick the smallest rank model that best predicts future states of the

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system. Because SubspaceDMDc must infer the latent state from prior data, we utilize Kalman Filtering (Kalman) 1963) for efficient next-step prediction. This is effectively done with the Akaike Information Criterion (AIC) along with other metries we discuss in Appendix! !
EXPERIMENTS
3.1 INPUTDSA DISCRIMINATES INTRINSIC DYNAMICS FROM INPUT-DRIVEN DYNAMICS
To demonstrate that InpuIDSA can capture similarities in both intrinsic and input-driven dynamics, simulated partially observed random RNNs with the following equations:
I++1 A(r + gF tanh(x:)) + B( + tanh(u,))
(14)
y/ (Ia On-d) 2t t
(15)
Where F and g are fixed across all simulations, and e, is observation noise. We randomly sampled two matrices for A E R and two for B E gnxl, from which we constructed 4 systems: Systems and 2 (3 and 4) share the same intrinsic dynamics matrix A1 (Ag), while Systems and (2 and 4) share the same input matrix B1 (Bz). We randomly sampled low-pass filtered white noise the input drive (four times for each system), each with random initial conditions, yielding 16 systems in each distance matrix. In our experiments, we simulated 20-dimensional (x: IR22)) systems and observed 2 dimensions (y E R²) for 5,000 time points. For simulation details, see Appendix [ We computed 5 distance matrices for each dataset, across 100 random seeds: (1), the DSA score using a delay-embedded DMD (Hankel DMD, or Hankel Alternative View of Koopman AArbaii & Mezić 2017 Brunton et al. 2017. . (2) the state distance using delay-embedde state distance using the Subspace DMDe, (4) the input distance using the DMDc, and (5) the input distance using the SubspaceDMDe. Note that DSA does not have the ability to compare inputs, so it is left out. For a discussion on hyperparameter tuning, see Appendix D. For the sake of space, we report the jointly optimized input distance (Input DSA, Eq. (T0) and the individually optimized state distance (State DSA, Eq. (9) these are the most interpretable, although the jointly optimized state distance was highly similar.
DMDc, (3) the
In Fig. 2 we visualize the observed input and one dimension of the observed output for sample set of systems, noting that it is not obvious at all a priori, let alone from the geometry. of any similarity relationships. We present sample state distance matrices from one random seed in Fig. 22.. While the DMD and the DMDc have notable structure pertaining to the true state similarity, the SubspaceDMDe similarity scores are noticeably sharper. Quantifying these matrices with the silhouette score (a measure of cluster separability and dispersal, 1.0 is best) utilizing ground-trute state labels, the DMD scores 0.6, DMDc scores 0.68, and the Subspace DMDc scores 0.94. In Fig. D. we present the respective input scores for each method. As predicted by our previous discussion on the effects of partial observation on input matrix estimation, the input DSA score computed with DMDc does not align with ground truth, reporting silhouette score of 0.19. The silhouette score of the SubspaceDMDe is 0.83, indicating robust separation. We also compute the total similarity matrices (Eq. 8. Appendix Fig. 6 for which the SubspaceDMDe reports correctly that each type of system is altogether unique. We swept over 100 seeds in Fig. 2 and found that the SubspaceDMDe--base
InputDSA consistently yielded the best separability.
To assess the effect of partial observation, we ran the above analysis for different-sized systems (ranging from 1000 dimensions) with only observed dimensions, for which we present the average silhouette scores for InputDSA in Fig. 2 The state similarity scores for each method gracefully degrade with the total state size, and SubspaceDMDe has noticeable improvement over the other methods. The DMDc input score appears to never be robust. However, the SubspaceDMDe input similarity is robust across all system sizes. We also assess the effect of hyperparameters, input dimensionality, and process noise on SubspaceDMDe (Appendix Figs. 8 9 10 respectively). Together, this suggests that Sub can be used to measure the dynamical similarity of partially observed, input-driven, noisy dynamical systems.
3.2 ROBUSTNESS TO INPUT NOISE AND TRANSFORMATION
In real-world settings such as neural populations, the true inputs driving the system are rarely acces- sible. Instead, what we observe are often noisy partial measurements, limited by sensor resolution,
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A
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Figure 2: InputDSA SubspaceDMDc is robust to partial observation (A) Inputs can affect ob- served states (red nodes) the future via the unobserved states (green nodes), biasing estimates of input driven-dynamics. Purple arrows indicate indirect propagation of input. (B) Sample inputs and observed states from 4 dynamical systems, which have alternate pairings the same intrinsic and input-driven dynamics denoted by arrows. (C) Sample state distance matrices based the DMD, DMDc and SubspaceDMDe on data generated as in (B). Four iterations of each system are gener- ated, each with unique inputs and initial conditions. (D) Sample input distance matrices the same data as (C). The DMD does not learn an input operator.
(E) Aggregate silhouette scores of each similarity matrix across 100 random seeds, each generated as in (C,D). Baseline input-label silhou- ette score for DMD is computed on the state matrix with the ground-truth input labels. Bars denote standard error. (F) Silhouette scores for each DMD and similarity type as the system is increased from 2- to 1000-dimensional. Each size was repeated across 20 seeds. Shading denotes standard error, asterisks indicate that SubspaceDMDe scores are significantly greater than both DMD and DMDc (one-side Mann-Whitney U-Test).
sampling rates, or inherent partial observability. As a consequence, researchers often rely on behav- ioral variables, task instructions, or environmental features as proxies when modeling neural circuits (Vinograd al. 2024 Sani et al. 2024 Burak & Fiete 2009b Schaeffer et al. 2020, Mante al., the true inputs are unknown, can
2013b). This raises key question for applying InputDSA surrogate inputs that are correlated with the ground truth still yield accurate distance estimates?
We begin by examining how well the true InputDSA distance matrix when the provided input is noise-corrupted. We repeated the simulation and comparison in Fig. 22,, this time applying different types of noise perturbations to the input used in Subspace DMDD (example in 3/A). For complete details the noise perturbations, by different real-world situations, such as partial observation, temporal smoothing, or multiplicative Gaussian noise, and repeated each perturbation across range of parameters (e.g. standard deviation in the noise settings or filter width in the smoothing setting).
Appendix M We applied 10 types of perturbations inspired
To measure the deviation of the signal consistently across perturbation types, we compute the signal- to-error ratio (SER) for each perturbation: given a time series X E p and its perturbed version Var(X)
X = f(),, SER is defined as SER(X, X) = Var(X-X) SER generalizes signal-to-noise, ratio for non-additive perturbations. Despite the prevalence of noisy inputs, we found that InputDSA distances remain robust, decaying slowly below the SER < threshold (Fig. 3): High SERs lead to high correlations with ground truth distances, and correlations tend remain above r > 0.75 even as SER approaches 1. This robustness arises due to the delay embedding and reduced-rank regression in Subspace DMDe: delay embedding incorporates the history of inputs, while reduced- rank regression removes noisy modes with spurious correlations.
To generalize this analysis to more complicated transformations. we repeated the analysis using inputs transformed by random polynomials (Fig. 3C.D). Specifically, we sampled 500 random 4-th order polynomials with coefficients drawn uniformly from 1-0.1, 1,0.1.1.... 0.1]. which we applied dimension- wise to the inputs as new perturbation. To generate inputs with higher SERs, we also generated 200 polynomials where the linear coefficient was fixed at 0.9, while all other coefficients were sampled

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from the same range. As in the previous analysis, we find a similar pattern across SER: the state DSA correlations are the most robust, followed by the combined and the imput DSA scores. This suggests that up to reasonable SER, the InputDSA scores are robust input perturbations.
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Figure 3: InputDSA provides robust distance estimates under input noise and surrogate in- puts. (A) Example of multiplicative Gaussian noise added to input data. (B) Effect of different noise perturbations on the InputDSA similarity matrices in Fig. 2 (see Appendix Section M for further technical details on the noise). The y-axis indicates the correlation between the InputDSA matrices given the true input and the perturbed input. The x-axis indicates the signal to error ratio /ar(()/Vaa((X
//r(X)) /aa( X - x). From left to right: joint controllability DSA (Eq. 8), jointly optimized state
DSA (Eq. | jointly optimized input DSA (Eq. TÖp. (C) Example of a polynomial function applied to the same input as in (A). (D) Similar analysis as in (B), with various random polynomial functions applicd to the input. (E) Random target task schematic. (F) We compare RNNs dynamics across multiple time points in training with InputDSA. We study changes in the distance matrix when applying surrogate inputs. (G) Example first Principal Component for different surrogate inputs and their correlation with the true input (Obs). (H) Correlation between InputDSA distances estimated using the ground truth input and surrogate inputs. Error bars indicate standard error across 10 train- ing runs. Jointly optimized state and input DSA are presented.
Next, we evaluated whether talk-relevant surrogate inputs could be used in place of ground truth, instead of perturbed versions of the true input. We analyzed trained RNNs from the Random Target Reach task (Fig. 3 Codol et al. 2024b), a widely used paradigm for studying neural control of movement from which rich neural and behavioral dynamics emerge (Hatsopoulos et al., 2007: ; Flint jet al.) 2012: Churchland et al. 2012). Across 20 epochs equally spaced in training, we recorded the RNN's 's hidden states, observations (the true input), actions (bchavioral output), and other task variables FF only 5 epochs shown for visualization purpose). For a detailed description of the task and training. see Appendix [ Passing the hidden states of the RNN and the ground truth inputs through InputDSA. we obtained two distance matrices that characterize how the network's 's intrinsic and input-driven dynamics change over learning. We repeated this process for various task-related surrogate inputs: RNN output (actions), position, velocity, task instruction, and various combina- tions. We also included random inputs sampled from the uniform distribution on [0, 1] as baseline. Among surrogates, the actions maintain the highest trial-averaged correlation with the ground truth input (Fig. BG). We find that InputDSA intrinsic (state) distances estimated with surrogate inputs have strong correlation with the ground truth distance, even with random inputs (Fig. 3H. For input- driven comparisons, more highla-correlated surrogates tend to yield more accurate similarities, with the RNN's combined action and positions providing strong correlations with the ground truth dis- tance (Fig. BH). Overall, our analysis suggests that state similarities are robust to perturbations of many different types, while the combined and input similarities are still robust, albeit less so.
4 APPLICATIONS
4.1 INPUTDSA TRACKS THE EVOLUTION OF INDIVIDUAL DIFFERENCE OVER LEARNING
In closed-loop Reinforcement Learning (RL) environments, stochastic action selection and small differences in policies can shift the distribution of sensory inputs encountered across training. To understand divergence between agents, it is crucial to study how inputs interact with dynamics and
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shape agent performance. The Plume Tracking task (Fig. (A provides an ideal testbed because the agents must balance between memory-based intrinsic dynamics with stimulus-driven responses.
In this task, artificial flies (RNNs) trained by deep RL navigate the source of simu- lated turbulent odor plume in a windy 2D arena. At each timestep, the agent senses only local cues (intermittent odor concentration and wind direction) and takes actions to move its position. Due to the stochastic nature of sensory observations and exploration. agents di- verge across training. producing a wide variation of success rates (Fig. 3B). This raises a key question: do performance differences reflect variations in intrinsic dynamics (the abil- ity to form and maintain task-relevant representations) or input-driven responses to stimuli?
Plume tracking
B
Top agints
We trained 15 independent agents on the Plume Tracking task. We selected the five best-performing ("Top") agents with 65% to 20% success rate locating the odor source across 200 evaluation episodes, and five worst-performing ("Bottom") agents who never succeeded on any episode (Fig. 4B). Applying InputDSA revealed that the input-driven dynamics of the Top agents were significantly more similar to each other and clearly sepa- rated from those of the Bottom agents, whereas intrinsic dy- namics were not significantly different between groups (Fig. 4D). This suggests that suc- cessful plume tracking heavily depends rapid input-driven responses to wind direction and odor concentration. To probe how the input-driven dynamics differ between Top and Bottom agents, we examined the singu- lar values of the input-mapping B in Fig. 4.. Singular values of the operator quantify how strongly input directions are injected into RNN state space. We found that the singular values of B for Top agents decay more slowly than for Bottom agents. This implies that inputs excite more dimensions of the RNN in Top agents (Fig. 4B), allowing them to directly control each dimension with the input, rather than indirectly through other dimensions.
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Figure 4: InputDSA identifies how successful and unsuccess- ful agents differ over training. (A) The Plume Tracking envi- ronment schematic adapted from/Singh et al. (2023). (B) Average performance (success rate) of 15 independently trained agents. The 5 most performant ("Top") and 5 failed ("Bottom")) agents are studied further.
(C) Neural dynamics of trained agents are or- ganized in a cow-dimensional space and reflective of behaviorally relevant variable (i.e. the odor concentration).
(D) Average dis- tance computed within the 5 Top Agents, within the 5 Bottom agents, and across groups (Top-Bottom). (E) The singular value spectrum of the mapping operator B from Top and Bottom agents. (F) The evolution of similarity within and across groups over learning. Shaded area indicates standard error.
We next ask how individual variability in neural dynamics evolves during training. To this end, we computed pairwise dynamical similarity among Top and Bottom agents every 20 gradient updates (Fig. ( While within- group input similarity decreases over training for both Top and Bottom, the Top agents ultimately converge to more consistent set of input-driven dynamics, whereas the Bot- tom agents diverge toward heterogeneous, idiosyncratic dynamics. This is reminiscent of the "Anna Karenina principle". in which effective solutions are similar to each other, while worse ones are highly varied. We repeated this comparison using DMDc, finding that identified similar structure in the input-driven dynamics but not the intrinsic dynamics (Appendix Fig. 17..
4.2 INPUTDSA CAPTURES DIFFERENCES IN NEURAL POPULATION DYNAMICS ACROSS TIME
Lastly, we apply InputDSA to a recently published dataset in which neural population activities were recorded from six frontal and striatal regions with Neuropixels probes during an auditory evidence

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accumulation task (Luo et al. 2025 Fig. 5A). During this task, rats were trained listen auditory pulses from speakers on the either side of the animal, and to turn to the side with more auditory pulses. This dataset contains 12 rats across 115 daily sessions with a median of 327 neurons recorded and 455 trials completed per session. We chose 4 rats with more than 15 recorded sessions for our analysis ensure accurate estimation of neural dynamics. In the original study, the authors define the neural time of commitment (πTc) the internal moment during perceptual decision- making when an animal has effectively committed to a choice (Fig. 3). To examine how neural population dynamies reorganize across this point, we applied InputDSA to neural activities during the stimulus presentation period before and after the nTc. Spiking activity was binned in 50 ms windows, smoothed with causal Gaussian kernel (a 250 ms), and dimensionality reduced with PCA to preserve 99% of variance. The activity was then embedded into three dimensions using Isomap, and InputDSA inputs as two-dimensional time series encoding the number of auditory pulses from the left and right within each bin. An analysis of the input revealed that this input structure was sufficient to robustly identify a B operator (i.e it is persistently exciting, Appendix Fig. 21
applied with hyperparameters detailed in Appendix Q We construct the
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Figure 5: InputDSA quantifies differences in neural population dynamics across task epoch. (A) Auditory evidence accumulation task schematic (adapted from Luo et al. 2025). (B) Trial- averaged neural trajectories visualized in the top two Principal Components. Stars indicate "neu- ral time of commitment" (nTc): the time point when the curvature of trial-averaged trajectories is maximum (marked by stars). (C) Similarity of neural dynamics before and after the nTc for rat T223. Bars denote standard error across 21 sessions.
(D) Distribution of top real eigenvalues of state-transition matrix and fit power law for pre- vs. post-commitment activity. Left, sample distribution. Right, distribution of power law exponents across sessions. Dots denote individual sessions, lines indicate paired periods within session, likewise in E and F. (E) Effects of intrinsic and input-driven dynamies in pre post periods, measured by L2-norm of the SubspaceDMDe op- erator. (F) Distribution of Frobenius norms of Controllability Gramians for SubspaceDMDo models for all sessions of one rat. p 0.001, Mann Whitney U-Test.
Comparing neural dynamics before and after the nTc ("Pre" vs. "Post"), we found significant shifts in both intrinsic and input-driven dynamics, consistent with the changes at nTc reported in (Luo et al. 2025) (Fig. 55). To probe how the intrinsic dynamics change, we analyzed the eigenspectrum of the state-transition matrix A estimated by SubspaceDMDe before and after the nTc. Each spec-
trum was fit with a power law A; & ( where X are the i-th eigenvalue sorted in descending order. We found that the post-commitment periods consistently showed smaller 8, indicating slower decay and thus longer-lasting intrinsic dynamics (Fig. 5D). This is directly related to the control- lability of the dynamical system, which describes how easy it for an input sequence to drive the system to arbitrary points in state space [Luenberger] 1979). Smaller DMD eigenvalues implics greater input controllability, which would be expected for a more input-driven system as [uo et al.) (2025) identified is the case in the pre-n''cc regime. Likewise, the average magnitude of the intrinsic dynamics strengthen while the input-driven dynamics weaken in the Post-n'Te period, reflecting a transition into more autonomous, less input-sensitive regime after the nTc (Fig. 5D).

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Lastly, we computed the Frobenius norms of the controllability Gramians for each SubspaceDMDo model, which measures how easy it is to control the system into arbitrary directions. In Fig. [ we report the distribution of norms for one rat pre- and post-nTT. The distribution significantly decays (p < 0.001, Mann-Whitney U-test), indicating that the neural dynamics become less input- controllable over time. We found similar results for all other animals in the dataset (Appendix Fig- 22). The median percent change in controllability energy was -47.63% (standard error 15.15%).TR To- gether, these results suggest that population activity undergoes a regime shift at the nTc: transition- ceedencc-accumulltiinn phase into an intrinsically dominated, decision- commitment phase, as suggested by (Luo et al. 2025). Applying the DMDe for comparison did not reveal similar results, almost certainly due to partial observation (Fig. 19
ing from input-driven,
DISCUSSION
We introduced a paarrtically-motivattee method (InpuiDSA) to quantitatively compare the intrinsic dynamics and effects of input between two dynamical systems, from data alone. We extended the DSA framework (Ostrow et al., 2023) to account for input-driven systems, which required a novel variant of the Dynamic Mode Decomposition with Control (Proctor et al. 201 called Subspace DMDc. We also developed a novel optimization algorithm for our similarity metric that is multiple orders of magnitude faster than prior work.
We demonstrated that InputDSSA can effectively estimate similarity from partially-observed systems (Fig. 2). which is necessary when dealing with most physical and biological systems. In many settings, the true input is not known (for example the signal from one brain region to another), but we demonstrated that even approximate or noisy inputs can provide reasonable input and intrinsic similarity estimates (Fig. 3). Since many models in computational neuroscience tend to utilize proxy inputs (Nair et al., [023 Sohn et al., 2019: Burak & Fiete| 2009a Mante et al.) [0033 Sussillo ee al. 2015), our work provides principled methodological support to this practice. Inputs could
also be estimated via another computational method (e-g. [errihh et al. 2222 [uo et al. 2025) before applying InputDSA. As Fig. suggests, even utilizing weakly correlated proxy inputs can increase the robustness of the intrinsic comparison with InpuIDSA.
InputDSA could be used for further validation of computational models with perturbation as in O'Shea al. (2022). Known optogenetic or electrical impulse perturbations could be applied to both a model and biological neural circuit, following which both their internal dynamics and im- pulse responses could be compared. This can provide more stringent tests than comparing intrinsic dynamics alone. Other subspace identification methods could be used in place of SubspaceIMMDe, such as Eigensystem Realization ARR,, Juang & Pappa| 1985). In a similar vein, InputDSS could potentially be used to identify the information content in cross-brain-regin tiple models could be constructed with different surrogate inputs, and the most similar input should have the lowest input distance to the data (Fig. 3
communication mul-
Although we only applied InputDSA to biological neural data and recurrent neural networks, can be applied to any time series data. Indeed, the constraints the method are based the capabilities of systems identification and Koopman Operator approximation. For example, if the input is not persistently exciting, state modes will be under-approximated. If a viable basis is not identified, the linear model may not be able to capture enough structure for effective comparison. However, there exists wide range of work in both fields designed to tackle these problems (Wul et al. 2021; Colbrook et al., 2023 Takeishi et al., 2017a; Ichinaga et al., (024)). It is also worth noting that near hear-perfect estimation is not necessary for informative comparison.
InputDSA has other limitations. The method assumes additive input, which may not be able to ap- proximate the effects of multiplicative input (Logiaco et al.) [021]] Shine al.) 2021). Disentangling the contribution of state and input can also be challenging or intractable when they are synchro- nized (Rajan et al. 2010) or the input is linear function of the state (Verhaegen Verdult 2007] although methods exist for subspace identification in closed loop Van Der Veen et al. 2013). From a computational complexity standpoint, the bottleneck is fitting the SubspaceDMDe, as comparison is extremely fast. Regardless, we found that even for reasonably sized systems (e.g. 50 dimensions, 10,000 timepoints) and hyperparameters (100 delays), the method requires a O(1 minute) on M1 Pro Mac, and is even faster GPU.
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ACKNOWLEDGEMENTS
We thank members of the Rajan and Ficte labs for helpful discussions. Funded by NIH (RF1DA056403 to K.R.), James S. McDonnell Foundation (220020466 to K.R.), Simons Foun- dation (Pilot Extension-000033322000
K.R.). McKnight Endowment Fund (K.R.), CIFAR Azrieli Global Scholar Program (K.R.), NSF (2046583 to K.R.), Harvard Medical School Neurobiology Lefler Small Grant Award (K.R.). Harvard Medical School Dean's Innovation Award (K.R. and S.H.S.), and Alice and Joseph Brooks Fund Postdoctoral Fellowship (S.H.S.). A.H is supported by the Kempner Graduate Fellowship. M.O. is funded by the NSF GRFP.
REPRODUCIBILITY STATEMENT
The description of all algorithms utilized in our paper is sufficiently detailed in order to reproduce InputDSA (pseudocode, mathematical details, hyperparameter tuning) and our results. We built upon publically-availabee code from Ostrow et al. (202)) to create InpuIDSA For our RNN studies, we used open source code, and have detailed the hyperparameters we used in the simulations as well as in our analyses. For our neural data analysis, we used public data and processed it generically, thereby making reproduction easy, if desirable. We are thankful for prior open source work and we will release our method open source upon acceptance. We hope InputDSA is not only used and but also improved upon.
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Appendix
OUTLINE
LLM Usage Statement
B.
Dynamic Mode Decomposition With Control (DMDc)
C.
Relationship Between DMD (Regular) and DMDe
Partial Observation Induces Biases in Input Operator B
E.
De-Biasing under Partial Observation With Subspace Identification
F.
Misaligned Input Spaces
G.
Solving for Optimal Orthogonal C' Efficiently
H. InputDSA Pseudocode
Hyperparameter Tuning for InputDSA
Partially Observed System Comparison: Further Detail
K.
Process Noise Effect on InputDSA]
L.
Establishing a Noise Floor for InputDSA Score
M. Input Noise Generation
N.
Ordinary Least Squares Biases in Estimates of A in the Presence of Input Noise
Random Target Reach Task
P. Plume Tracking Task
Neural Dataset
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A LLM USAGE STATEMENT
We used LLMs in preliminary phases of conducting this research, in particular for brainstorming research ideas and literature review, as well as writing simple boilerplate code (e.g. plotting). All code, math, and writing was checked by at least one author before including it in the paper.
B DYNAMIC MODE DECOMPOSITION WITH CONTROL (DMDc)
Dynamic Mode Decomposition with control (DMDc) (Proctor et al. 2016a) extends standard DMD to dynamical systems with external inputs. It provides a data-driven approximation of both the intrinsic dynamics A and input couplings B, enabling system identification and forecasting for non- autonomous dynamical systems. Here, we briefly review the formulation of DMDc. For full details, please refer to Proctor et al. (2)1 In practice, we can apply DMDc whenever the state is fully observed. When this is not the case, refer to Sections and
We consider the input- driven linear model
Fk+1 = Ar + Buk,
A E ER*" B BERUX,
(16)
where Ta E R are state snapshots and up Rp input signals. For an input-driven dynamical system, we collect pairs of the system states and input signals into
X [
T2
Im-1],
(17)
X' [22
I3
mm]
(18)
U [ui
U2
Um-1],
(19)
where X,X E [=x(m=1) and U (m
1),
We can rewrite equation into into
X
X GD [A
B
(20)
U
where Ω E [(=+p)x(m-1) and G E (=x(atp) The optimal operator then obtained by solving
G arg min = X'
(21)
&
where (.)+ denotes the Moore-Penrose pseudoinverse.
Let the truncated SVD of Ω be
2 ÜVLV
at ~ト~
(22)
Partition Ü into state and input blocks:
Ü. E Rax U, RRxP.
(23)
Ú.
The system matrices are then estimated
A X'VE--
B = B-X'VÊ-'U
(24)
We can further project A and B into the system's state space using
X U,E,V
U, Rerr
(25)
Ā -= AUU
B B=U
(26)
is useful to perform SVD independently on X and U. assuming there is minimal correlation among the variables. This is especially useful when using nonlinear embeddings such as delay embeddings in the regression. This changes the algorithm of DMDc but not significantly. In particular, we can
now write:
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Uz
0 Un
Ex
VT V
Ω
(27)
En
This enables us to pick ranks separately for X and U components. In practice, apply the techniques used HAVOK (Brunton et al. 2017) to estimate the DMD. We do regression in the eigen-time-delay (pca-whitened) spaces of X and U (Hankelized), which allows to select ranks separately for the X and the U space.
B. ON NONLINEAR EMBEDDINGS IN DMDC
In the standard DMDe formulation (above), an SVD is taken across Ω, which concatenates the state data X with the input data U. Although this has the benefit of whitening across all regressors, bias the estimation of A and B depending on the relative scalings and dimensionalities of X and U. This has a critical effect when applying high-dimensional nonlinear embeddings to only X (U) individually, as the SVD will be increasingly dominated by signal from X (U) if the data sufficiently rich. Therefore, whenever we apply delay embeddings or other nonlinear embeddings X, we do so commensurately to
C RELATIONSHIP BETWEEN DMD (REGULAR) AND DMDC
x 1 (5Ix U.U) 1:m
xTU)=
X (UT)
Xat1
As A= (X-x)- X
S = U'
A
(X7 (
1
Xnt1
1 = -x-xx-x)---'X A,S A Xn41 A
A As - S. A, X - A
D PARTIAL OBSERVATION INDUCES BIASES IN INPUT OPERATOR
Consider partially observed linear system:
-AAr0
Aoo Anc
Aou
x x
[Bo Bu
Aua
ut-1
(28)
t-1
We observe states xº This system can also be formulated as a Vector-Auutrregressive model with exogenous inputs (VAR-X). To see this formulation, we recursively substitute the definition of z with its dynamical equation, hence arriving at formulation of x as function of past observed states and inputs:
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5 = Aoti-1 1 + Aouti + Bolls
(29)
Anti-1 + Aon AonlAuof?2 + Amill + B
(30)
(31)
= Aoo
+
A (Auoti + Buut-i) Bou
(32)
We take an infinite sum here for completeness, but in practice i can be capped up to marginal error based on the decay rates (eigenvalues) of A,, We can write this equation as a function of the delay- embedded observed state and inputs:
27
T [Ao AouAuo AouAnAuo
AouA
|uo)
F7-1
(33)
+ +[Bo Ao AouAnBu... B.r.B....AAA..
A Bu]
14-1
(34)
Lur-d
These equations show that when performing regression in DMDc partially-observed, delay- emhedded data, the estimates of B become biased by the intrinsic dynamics in the unobserved states. Biases in B emerges when utilizing delay embeddings from the above formulation. Although we display the formal connection with linear systems above, it is simple observe that the same problem occurs with nonlinear dynamics as well.
dimensionality expansions,
E DE-BIASING B UNDER PARTIAL OBSERVATION WITH SUBSPACE IDENTIFICATION
In this section, introduce SubspaceDMMDe natural extension of two DMD models in the liter- ature: Subspace DMD (Takeishi et al.] 2017b) and DMDe (Proctor et al. 2016a). SubspaceDMDc has notable difference from SubspaceDMD. as Takeishi al. (2017b) utilize the subspace identi- fication approach to handle observation noise, whereas we utilize subspace identification to handle input affecting future timesteps (although we gain noise robustness through similar means). In the control theory literature, there a number of subspace identification algorithms, two of the most famous are Multivariable Output-Error State sPace (MOESP) modeling and Numerical Algorithms for Subspace State Space System Identification (N4SID) (Verhaegen & Verdult 2007 Verhaegen 1994 Van Overschee & De Moor 1994). In order to be brief, we will discuss only N4SID, which is the method chose to implement. In general, the algorithms have similar behavior, except on ill- conditioned data. Practically speaking, either method could be used in DSA; it up to the user and their respective performances the dataset. The extension of these methods to SubspaceDMMDo is the introduction of a lifting feature space: polynomials, kernels, random feature maps, neural networks, nonlinear features can be used in order find best-predicting nonlinear basis upon which the features evolve linearly.
E. SUBSPACE DMD
Subspace DMD Takeishi al. 2017b) designed handle the estimation of the Koopman op- erator given data that is contaminated with observational and process noise. Assuming that the dynamics and the noise are independent, one can project out the contribution of the noise in the data and leave only the component that is explainable with past data (via delay embedding, step 2 of the algorithm below). We assume real data, although the method works for complex data as well. Al Algorithm in the paper reads:
1. Construct data matrices Yp [Y] YT7 Y, [g(xz) ... g(I+-m41)
Y
YT
where Y,
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2. Compute the orthogonal projection of the future data onto the past data: O YyPy where the projector Pyς = Yp (Y,,))
3. Compute the compact SVD (e.g., the SVD with no zero rows or columns): O = UUS,V V and define Uę2. U22 by taking the first and last rows of Ug. This is done in order to split the projection matrix into the observability matrix and the state matrix: O = rX, up right/ left multiplication by invertible matrix. The observability matrix looks like CAa)). Because this matrix encodes the time-shifted structure of the dynamics, we split into the top and last rows to get Uqi and Ug2 upon which we do
r -= (C CA
reduced-rank regression in the next step.
4. Compute the compact SVD of UT USVT and define the operator Ā UVV-L
5. If desirable, dynamic modes are defined as w A A-' for eigenvalues X, eigen- vectors w of A.
E.2 N4SID
Numerical Algorithms for Subspace State Space System Identification (N4SID) (Van Overschee & De Moor 1994) utilizes a similar approach as the above jointly estimate A, B, C, operators in a state space model from data Y and U. Here we briefly describe the algorithm that we apply to estimate A and B that are used for comparison of partially observed systems, first defined by for our implementation of n4sid. For the Subspace DMDc, we lift a nonlinear space before state estimation.
Overschee & Moor (1994). We used code from https://github.ccmm wttp./o//g//ooooosiiioo///o/////
For state estimation to succeed, standard conditions on the data state and input apply. In particular (1) the state vector is sufficiently excited (it explores all relevant dimensions of the state space). or the system reachable, (2) the input sequence persistently exciting, i.e., the Hankel matrix of the inputs is full rank, and (3) there is no linear state feedback, i.e. the state and the input are not collinear. Note that nonlinear feedback is permissible provided they are not collinear. Prediction in the SuDSpaceDMDe is done with Kalman filtering, because state estimation must first take place.
Briefly, we explain the key computations behind N4SID. There are two slightly different approaches. The first algorithm is similar in spirit to Subspace DMD which we detail here:
E.2.1 PROJECTION- BASSED N4SID
As above, we create a Hankel data matrix of the observations, but also the input too, splitting this into past and future. First, project out the data explained by U, in the future observations Yj, but also the past observations and inputs Zp UUp YpJ. thereby removing its influence. Then to remove measurement and process noise biases, we project the future states onto the space explainable by the states and inputs in the past, Zp. This yields our matrix O = ГX, which we split using SVD as before to get r. the extended observability matrix, and the states X up to similarity. Noting again that our extended observability matrix has time-shifted structure, we can perform regression on the shifted components of X given the instantaneous U, to arrive A, The observability matrix also encodes C' in its top rows, which we can directly read out. However, we found this algorithm in practice to be less stable than the next one.
In pseudocode form, we have the following:
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Algorithm Subspace DMD with Control (N4SID on lifted states)
Require: Output data Y E RP XN Input data U E RRxx,, past window p. future window f, system order n, regularization
Ensure: Estimated system matrices Á, A,
1: procedure BUILDHANKELMATRICESS U,,,
2:
T + N p - ++1
3: Construct Hankel matrices Yp, Up. y, U
T Y
4: Zp
f
5: return (Y y, 2,,2p?7 T)
6: end procedure
7: procedure OBLIIQUEPROJECTIONYY U X, L U7 XI) U 8: II 17
9:
Y Y
10:
pp Uy
11:
O y y
b Oblique projection via pseudoinverse
12: return O
13:
end procedure
procedure ESTTAANTSSTTEEFRRRPRREECCICC
15:
U, Vo t SV)(O)
Truncate to rank n: Un,S,, nn
F + UnVS,
> Estimated observability matrix
18:
X- vS,
Estimated state sequence
19:
return (,)
end procedure
21: Y, U, Zp. < BULDDHANKELLATRICCSSYYY. U.P.
22: O + OBLIQUEPROOECCTOONNY, Zp.
f X ESTTMATTSSATEEFROPPPRRRCTTOOOOO
24:
25:
Align data for regression
26: Xeurrent f X|:
27:
X -- [T
28: Umid U Ul:pp:
29:
Solve for system matrices
X Xuument
31:
[
B
Xnext
Unid
32:
Č +- first Pour of f
33:
return Ć.C
F MISALIGNED INPUT SPACES
For any orthogonal matrix C, the following equivalence holds:
y == Cr = CA,C" x CB, u(t)
(35)
Now consider the where inputs equivalent in each system, but that they also related by coordinate transform:
Uny (t)
Then Eq.
38
resolves to:
CA x CBBC, (1)
(36)
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This motivates the dissimilarity metric that seeks to jointly optimize C' and C., with the second term in Eq. generalizzin to
min Cueo(n)
|ICB -B2ll
when = 0, equation 8 is the so-called two-sided Procrustes problem, which when solved jointly resolves to comparing the singular values B1, B2: ||Σ - which can be computed efficiently. When * 0, the two minimizations need to be jointly optimized. The method of optimization from Ostrow et al. (2023) can be effectively generalized to do so, with note that this is a larger optimization problem and requires longer optimization time (but see next section).
the inputs that are directly applied to the system are known, as in RNN or RL models (& far u)), this joint optimization procedure can be discarded. Likewise, when the inputs aligned in time, Procrustes or other spatial alignment methods can be directly applied to the inputs first. Note that this input comparison does directly compare the dynamics of the inpur, but rather how the input is read into the system. If one is interested in comparing the dynamics of the input as well, then DSA can be run on the input directly.
G SOLVING FOR OPTIMAL ORTHOGONAL C' EFFICIENTLY
The InputISSA formulation allows for efficient solving of the optimal C E O()). Recall that
Ay,
min ||A CA,CT CEO(m)
(37)
a non-convex optimization problem, and hence has to be solved iteratively Ostrow et al. (2023) However, the addition of the control constraint, I|B - C'Byl yl le means that we can solve this prob- lem using convex optimization for a = 0.5. Observe that under similarity,
-= CAC
CB = AB CAB
This suggests that we can identify C via Procrustes alignment on the controllability matrix K [B AB A²B ...AB)) B)B), where A E Rux,
min CEO(m)
K1 - CK2l 1
(38)
The minimizer C has closed-form solution via orthogonal Procrustes. Likewise, jointly aligning
the input dimension via C₂ (Appendix Sec. F) can be done in closed form as well via the two-sided Procrustes solution. This results in acceleration of multiple orders, with the computation of C taking O(1 millisecond), as opposed to O(1 second).
However, this formulation can result in C* that are biased towards the controllable directions. ie. B can have an inordinate effect or can dominate. In practice, we found that using this approach with ground truth C resulted the state similarity score becoming biased near dimension 30 (that is. A E 1230300, While this is still quite large, and the biases are small (average deviation (0.01)) per element), we can do better. We can add further constraints to by noticing that A also holds in the previous implication under similarity:
-= CAC" B CB
- ĀB - CA
Thus, we can concatenate these powers as well to K, giving:
K (B AB
ATB
ABB AT"
Where the metric is once again Eq.
38 This improves the optimization stability on until at least dimension 150 for O(0.001) error per element error, which is more than enough in practice for InputDSA. We have the following lemma which states that this metric captures equivalency between two linear systems.

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Lemma G.1. Given two linear systems It+1 Ag + uy and Mt+1 Aule By Eq. 38 equal zero if and only if Cx for some C
Proof. Let first consider the forward direction. Assuming y = C* then we have the equivalence relationships A C² AyC" and B₃ =C' By Applying this relationship Kas have
K₂ (c (CBB, By
C Ag C' O B,
=0" Ky For which min -Kyl Now consider the reverse direction. We can expand Eq. 38 as:
evidently C
|K, - CKyl 1/p 1, CBB,l l + IIA,B, CA,BBIIA | - C, By P +...
For min ||,, CK2 I = 0. each subterm must be zero for minimizer Č. This immedi- ately gives Br B.. Inspecting the next term, we substitute this relationship, giving
||,,B CA,,, ||A,BB, - B,l = =I(AA0- - CA,) A,C ČA,
(39)
(40)
(41)
(42)
With the last step following from By * 0. This in turn gives A₂ ČA,CTT. We similarly apply this reasoning to the next expression, which gives the same result. Reversing the previous logic, we have 1++1 = CA, ÖT x, + CB,uu, = y= CT x.
For given A, B, the standard right Procrustes problem is written as:
C* argmin ||CA - BB
(43)
argmax C.A, B >== Tr((CA)*
(44)
Writing the form of this problem with K1, K2, can separate individual elements in the Frobe- nius inner product, giving
K2,, >F
B, + + Bg, ATJ By >F
(45)
<C,,A,,,(ABBB))) >F + <CC, By(A Ba)
(46)
1-0
=< C.
B.) + AT BA B.Y >F
(47)
1=
With the last steps due linearity the inner product and the second step using the trace permuta- tion identity. This gives the maximum over C' E ((n) to be
CU=UVT where Σ A. + B.) = USV i=0
In practice, taking large matrices A to many powers results in numerical instability issues, especially when Amaz (A) > 1. Algorithmically, we check the condition number of A before choosing to include the term in the controllability matrix. If it is too small or too large, we stop.
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Lemma G.2. In linear systems, controllabiliit is only preserved under orthogonal changes of basis, = Cz, where C' E O(n) (if and only if).
Proof. We define the controllability Gramian for the system 21+1
Axx Bug as:
T
W₂ =
Σ A'B(A'))=
Σ (A))
(48)
==
Consider the change of basis y - Cr. The dynamics become:
It+1 = C(Ax, + Bu))
= CCACC D1) + CBu
-(CAC- + (CB))
We compute the Gramian for y, denoted W', using the transformed matrices (CAC-1) and (CB).
Note the identity (CAC-1)² = C-T C--AA"C
Substituting these into the sum:
W CICAC"YCCBICBTTCCCOYY)
Σ CIB i=0
T
A'BB" (1P (1 CT
= c
This simplifies to the congruence relation:
Wy - CW
(49)
Direction (If): If CC (i.e., C E (())), then CT = C- Equation 49
becomes:
y CW,C-1
This implies W, z W, (similarity), so the spectrum is preserved.
Direction 2 (Only If): If we require preservation of the spectrum, we must have W, _ W.. By definition of similarity, this means there exists a relationship:
W, = CW2CO-1
(50)
Equating our derived result (49 with the requirement (5)):
CW,C² CW
Assuming W. is full rank (invertible), we can cancel CW2, from the left:
CH =
Multiplying by C on the right:
c²C=1
Thus, preservation requires to orthogonal (C E O(n)).
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G.1 GENERALIZING THE WASSERSTEIN DISTANCE FOR INPUTDSA
Recall the Wasserstein distance over DMD eigenvalues,
DSA(A1, A2) :-
min || - A2lle Pell())
(51)
This metric respects the notion of equivalency under general similarity transforms, A for invertible C's, given that only eigenvalues are preserved under these transformations. We would like identify a similar metric for input driven systems. To motivate our metric, consider applying a diagonalizing transform to the dynamics of our input-driven system:
C'AC-1
T++1 = Ar, + Bu,
(52)
= VAV + Bue
(53)
V 3e+1 = AV-1x, - V- Bu,
(54)
We observe that the corresponding feature of the input to each eigenvalue À is the vectors V-1 B, which we henceforth term the eigenmode-input interaction matrix. We can easily show that these features are invariant any invertible transform. Given transform A + C'AC-1, CB,
VA CVAV- : -AA-1
(55)
Hence V CV
(56)
Thus, natural extension to Eq. 51 is the joint Wasserstein distance over [A, (V-) Denoting A the set of eigenvalues for system one, and denoting π a permutation map.
- A2 +1 -1-1aaa a))(a (V1 B InpuIDSA(A ,A,,,,,,,,,,,, V.VVV'',, B1, V..V2''B, B2, = min -
(57)
This metric is intuitive: V- B describes how input direction interacts with the independent eigen- modes, which is related to the controllability of that mode. However, this metric has numerical sta- bility issues. First, eigenvalues can only be identified up an arbitrary phase-hence, to study instead the norms of each eigenmode-input interaction, I(V B1):B ):]2. This loses informa- tion but works reasonably for small systems. Identifying the eigenvectors of an arbitrary matrix is challenging for poorly-comee corlyycoonitiioed matrices. Hence, we suggest evaluating the conditioning of the DMD matrix before applying this metric.
forced
H INPUTDSA PSEUDOCODE
Algorithm 2 InputDSA
Require:
X1, X2 E Raxtxd, U1, U2 E Ruxtxt number of delays q, nonlinear lifting functions Φ1, 42 rank for state-space
1:
Ensure: Similarity transform distance d between the two dynamical systems
2: A1, Bj + SUBSPACEDMIC (
()
3: Ag, Bg SUBSPACEDMID (X22.
4: d = min CEO(n) a C - -A2
CCBC B2llF
5: return C.EO(M)
I HYPERPARAMETER TUNING FOR INPUTDSA
Delay In InputDSA. the delay parameter controls the size of the delay embedding used to estimate the dynamics operator. If too few delays are chosen, the embedding may distort the data and amplify noise. Conversely, too many delays fold the data into unnecessarily high dimensions, making it more difficult to model the dynamics with DMD (Ostrow et al. 2024).
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Algorithm Dynamic Mode Decomposition with Control (DMDc)
Require: Delay-embedded states H inputs Hy, truncation ranks Fally Sstate ridge regularization À
Ensure: Dynamics operator and B
1: procedure BUILDSNAPSHOTS Hu) 11 X+ < [x
2: X_ - Hx 1
3: U- Hu 2 1
X_
4: Q+ U-
5: return (X+, X-,,,
6: end procedure
7: procedure SVDs(X+, +,
8: (Up, Ep. V/)) t SVD(22
9: Partition Up = Up1
into state/input blocks
Up2
(U,, E,, V,) SVD(
11:
return (UpI, Un2, Ep, V U,)
12:
end procedure
13: procedure REDUCERANK(U,I, U22, Ep. Vo,, U,)
14:
Truncate rant: Upl +- Un1 (i, 1:ran). U22 U22/5, 1:ra), V Voli 1 ::ran] Ex Ep[1:
ranl
Truncate to Tstate: U, 55,+UU, Uy 1:::::1110 ssstte
16:
return (Up1, Up2. Ep, Vp, Uy)
end procedure
18:
procedure COMPUTEOPERATOR((A, Vp, Eps Upl U22 Up, 2)
19:
(A) diag
+
20:
A + X (A
p p2
21:
B x V,, UA)
22:
Project to the state space À + AU,, B+ U. B return (Á,
23:
end procedure
24: Xt.X_ BUILDSNAPSHOTS(H,, Hx, HU)
25: UpL Up, Ex,, U, + SVDS(X., Q)
26: Up,, Ep, V, REDUCHRANK()
27: A, COMPUTEEPPERATORSS (X+, Vp, Lp, UpI Up2, N),
28: return A, B
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Rank SubspaceDMDe involves one rank parameter r, corresponding to the dimensionality of the latent state space. In practice, selecting an slightly higher than the true state dimension often yields a better estimation of the matrix.
Hyperparameter tuning pipeline We suggest jointly optimize the delay and the rank r according the following criteria:
Prediction accuracy: The delay embedding should enable accurate modeling of the dy- namics. To evaluate this, split the dataset into training and test sets, fit InputDSA (via SubspaceDMDe) on the training set, and assess performance on the test set using the mean absolute standardized error (MASE), standard metric for time-series forecasting. MASE compares the forecast error of the model against that of a naive persistence baseline pre- dictor and is defined
T 21=1
MASE T
Σizl V--1
A value MASE < 1 indicates that DMDe predicts next-step activity (using the estimated operators A and B) more accurately than simply copying the current time step.
Model complexity: The estimated operators A and B should not be overly complex or dominated by spurious features (e.g., many small eigenvalues clustered near zero). To assess this, compute the Akaike Information Criterion (AIC) for next-step prediction the test set. AIC balances predictive accuracy against model complexity and, in our setting, is given by
1 N
2(r² + 1) N
AIC In
u3)²
Overall, we aim to select rank that is small enough avoid inflating the AIC, while still yielding good predictive accuracy (i.e., low MASE). In many low-dimensional dynamical systems, both AIC
and MASE exhibit a characteristic elbow-shaped curve (for example, see Figure P.T and Figure Q We recommend selecting the rank at this elbow point, and then choosing delay that yields a low MASE at that rank.
J PARTIALLY OBSERVED SYSTEM COMPARISON FURTHER DETAIL
We discretely simulated the following equations (repeated from 14
T++1 A(xe + gF gFggFaaanii (anh(xx)) + B(u, + tanh(u.))
(58)
yt (Ia On-d) Ze ct
(59)
We generated two matrices AI, A2. sampling each element i.i.d. from standard normal distribution. To enforce stability of these matrices, we globally rescaled the matrices by a term p/ Amans where 1. We arbitrarily picked P1 0.92
Amaz is the max eigenvalue of the sampled matrix and and P2 = 0.82 to ensure a significant difference in the intrinsic dynamics, but not so large as to make the data obviously different. We set 0.1 for each system, and fixed F to be the matrix defined as Fij = 6.5ōe5a where d the number of observed states in the observation matrix C = (La On-d)- We sampled B1, B2 from normal distributions well, with BL, N(..91). B2,, ((,992) setting 91 0.5.g2 = 2.0. We sampled E; ~ ((0,0.01) for each observed index for each time- point.
Across Figs. Pb.c.d.e we simulated 20-dimensional systems with only 2 dimensions observed, for 5000 timepoints. For every type of DMD, we applied delay embeddings of size 150, and fit state space dynamics matrices with rank 20. We chose these parameters by inspecting the spectral distribution of the estimated observability matrix (line of Algorithm across multiple delays. We added delays under the largest modes before the spectral drop-off point stopped changing (similar to the idea of a false neighbors analysis, Kennel et al. 1992). then picked the elbow of that curve. We selected the maximum of those values for each of the four systems. We observe these curves in Fig.
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DMDc
SuDspaceDMDD
Joint DSA
Joint DSA
3.8
0.62
1
System
2
System
2
3
3
4
2
System
System
Figure 6: Joint InputDSA Comparison using DMDc and SubspaceDMDD The sum jointly- optimized state and input distances is presented here, with 0.5. Comparisons generated on the same dataset as in Fig. 2 and d. DMDe Silhouette score on state, input: 0.235, 0.088. SubspaceDMDe Silhouette score state, input: 0.368, 0.55.
However, we note that InputDSA is robust to number of different ranks (Fig. 8 both larger and smaller than the true system size.
We computed silhouette score using Scikit-Learn on the precomputed InputDSS distance. Based on some given label (here, state input ground-truth similarity), the dataset is divided into subsets
C₁, C2..... C, with each data point Ti, 32 ...IN belonging to one subset. Define the labels (cluster index) of each point cı2 C2 CN. Next, the mean intra-cluster and the minimum mean inter-cluster distance is computed for each data point:
a(1) [Ceill
Σ
d(z,,z))
Cees Coter
b(1) = min
Σ
d(z,,
f Cj
c(k)=}
Where (Catt) denotes the cardinality of the set, and d(:,:) denotes the distance function to be used. In our setting, we use the InputDSA input or state distances for d. Lastly, the silhouette score is computing as:
b(i) - a(i) S = N max(a(i),b(i))
N
The silhouette score approaches when all points in each class are strongly separated and there is minimal distance between the points within each class, while it is 0 if the inter- and intra-cluster distances are equivalent. It is notable that a silhouette score of 0.7 can correspond to perfect lin- classification of all classes, as deviations from 1.0 can be caused by within-class variance that remains non-overlapping with other classes.
J.1 ROBUSTNESS TO HYPERPARAMETERS
Here, using partially observed nonlinear systems with known ground-truth similarity structure, we test InputDSA's robustness to changes in hyperparameters. For 20-dimensional systems with only observed dimensions, we vary both the delay and the rank used in fitting InputDSA. As shown in Fig. g rank and delay affect state similarity more than input similarity, while the Silhouette
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System
System 2
Delays
Eigenvalue
10
10
25
50
10
75
100
125
150
175
200
System 3
System 4
Eigenvalue
10
10-1
100
200
100
200
Mode Number
Mode Number
Figure 7: Spectral content of the Extended Observability Matrix from Subspace DMDe for each system in Fig. across multiple delays. Dotted line indicates rank 20.
score for input similarity remains nearly invariant across hyperparameters. When the chosen rank too small relative to the true system dimensionality (N 20), the learned dynamics operator A cannot adequately capture the state dynamics, yielding low Silhouette scores the fitted distance matrix. Once the rank is large enough to capture the high-variance directions of the system, even if still lower than the true dimensionality, the Silhouette score approaches 1, indicating nearly perfect clustering accuracy. In other words, the rank used for InputDSA does not need to match the ground- truth dimensionality; it only needs be high enough capture the dominant dynamical modes. Beyond this threshold, the estimated similarity structure becomes robust to small increases in rank. However, in real-world noisy systems, choosing a rank that too high may incorporate spurious, noise-driven dimensions into the dynamic operator. We therefore recommend selecting rank via hyperparameter sweep as described in Appendix
For the delay, following (Ostrow et al., 2024), choosing too few delays distort the embedding and amplify noise, whereas too many delays fold the data into unnecessarily high dimensions, making harder for DMD to model the dynamics effectively. Here, we demonstrate that InputDSA's estimated similarity structure robust to delay choice within a reasonable range- as long as the delay is sufficiently large for the delay-embedded state span rich enough basis fit linear dynamic operator.
J.2 EFFECT OF INPUT DIMENSIONALITY ON SYSTEM IDENTIFCATION
In Fig. 2 we analyzed partially observable systems driven by low-pass filtered one-dimensional white noise. Here, we vary the dimensionality of the input and evaluate how the Silhouette score (computed against ground-truth labels) changes partial observability increases. using InputDSA distance estimates based on Subspace DMDc. As shown in Fig. ( increasing the dimensionality of the input provides only modest improvement in recovering the latent state similarity structure, while simultaneously reducing the identifiability of the input-driven dynamics.
K PROCESS NOISE EFFECT ON INPUTDSA
In real-world systems such as neural circuits, process noise is pervasive: intrinsic stochasticity in the biophysics, fluctuations synaptic and network activity, and unobserved perturbations all contribute to variability in the dynamics. We tested InputDSA's robust to process noise added to the state in
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State
Input
1.0
1.0
Silhouette Score
0.5
0.5
0.0
0.0
10
20
10
20
Rank
Rank
1.0 -
1.0
Silhouette Score
0.5
0.5
SubspaceDMDc DMDc DMD
0.0
0.0
50
150 200 Delay
100
50 100
150 200 Delay
Figure 8: Effect of rank and delay in each DMD algorithm on clustering scores, utilizing 20 di- mensional systems with 2 dimensions observed, and 1000 datapoints per dataset. When varying the rank, we fix the number of delays be 150 and when varying the delay, we fix the rank be the true rank of the system, i.e. N 20.
State
Silhouette Score
0
Input dim=1
Input dim=3
Input
Input dim=5
1-
0
10
102
103
Number of States
Figure 9: Effect of input dimensionality on InputDSA performance under partial observabil- ity. Silhouette scores for InputDSS based on Subspace DMDc the system increased from 2-dimensional to 1000-dimensional. while only 2 dimensions observed. Each size was repeated across 20 random seeds. Shading denotes standard error.
Section 3.1 We again consider the partially observed nonlinear dynamical system parameterized by this set of equations:
Et+1 = A(x, + gf tanh(x;)) + B(u, + tanh(u)) +
(60)
= (Ia 0,-d) Nz +
(61)
Here, Vi (0, 2) is the process noise added the latent state at time We generate the data
exactly as in Section 3.1 The state-space similarity structure estimated by InputDSA remains highly robust process noise, exhibiting an almost unchanged Silhouette score as increases (Fig. T03). On the other hand, the Silhouette score of the input-driven similarity structure decreases smoothly
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with increasing o, reflecting graceful degradation in the identifiability of input-driven dynamics under higher process noise.
A
B
System
System
Silhouette score
1.0
State
Input
0.5
o=0
0=5.0
0.0
1
4
2
4
C
= 0.00
o= 1.00
a - 2.00
3.00
4.00
5.00
1
System
2-
State
System
Input
4 3 2
3 System
3 System
System
System
System
System
Figure 10: InputDSA remains robust to moderate levels of process noise. (A) Example latent state activity on over time sampled from two different dynamical systems, overlaid with the state trajectory under process noise sampled from .oO,, o)) where = 5.0. (B) Silhouette score against the groundtruth label when the systems are clustered based on the InputDSA state input similarity sturcture. (C) Example distance matrices underlying varying levels of o.
L ESTABLISHING A NOISE FL.OR FOR INPUTDSA SCORE
It is often desirable to establish statistically insignificant noise floor for the InputDSA score, below which two systems can considered effectively identical. estimate this noise floor, we can split the state and input time series of a single system into two halves (or multiple segments, if suffi- cient data are available) and compute the InputDSS score between these segments. The resulting within-system scores provide empirical noise floor against which cross-system comparisons can be evaluated.
Here, using the partially observed nonlinear system examples in Fig. we computed and com- pared three quantities for the state distance and input distance (1) Split-half: the split-half noise floor, established by computing the distance between segments of state and input trajectory of the same system driven by identical input, (2) Same: the distance between systems sharing identical groundtruth state / input dynamic operator, but driven by inputs with different random initial ditions, (3) Different: the distance between systems with different state / input dynamic operator. In other words. both Split-half and Same are within-system distances, while Different measures cross-system distances estimated by differnt methods.
As shown in Fig- the Split-half and Same distances are typically similar, and both are markedly smaller than the Different distances. This provides a basic sanity check that the comparison methods tested here can reliably distinguish between different dynamical systems. One exception is the input distance estimated by DMDe, where the Split-half and Different scores are comparable, indicating that DMDe fails to recover input-driven dynamics correctly partially observed nonlinear systems.
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DMD
DMDc
SubspaceDMDc
State
State
State
2
Distance
Distance
Distance
Split-half
Same
Different
Split-half
Same
Different
Split-half
Same
Different
Not Available
Input
Input
Distance
Distance
0.25
0.00
Split-half
Same
Different
Split-half
Same
Different
Figure 11: Noise floor analysis for state and input distances in partially observed nonlinear systems. We compare three quantities: (1) Split-half, the within-system noise floor estimated by comparing segments of the same trajectory; (2) Same, the distance between systems sharing identical state/input dynamics but driven by different input initializations; and (3) Different, the distance between systems with different underlying dynamics. For both state and input distances, Split- half and Same are similarly small and substantially below Different, demonstrating that the tested methods reliably distinguish different dynamical systems. DMDe is an exception for input distance, where Split-half and Different are comparable, indicating poor recovery of input-driven dynamics under partial observability.
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M INPUT NOISE GENERATION
To assess the robustness of InputDSA to noisy or corrupted inputs, we systematically added different types of noise or transformations to the input time series of the nonlinear dynamical systems we created. Below we describe how each type of noise was generated. We visualize examples of the noise-corrupted input in Fig. 12 and Fig. 13
Gaussian (white) noise.
White Gaussian noise was added independently to each imput channel:
i(t) = u(t) + N((,o
where a is set by the noise level.
Pink noise. Pink (1/f) noise was generated in the frequency domain with power spectrum propor- tional to 1/f" (with = 1 by default), then inverse Fourier transformed and scaled to the desired amplitude.
Rotation.
For two-dimensional input signals, applied random planar rotation:
coo 0 sin
sin 0 cos 0
a(t) R()) u(t),
R(0)
with rotation angle f proportional to the noise level.
Low-pass filtering. Inputs were smoothed using a digital Butterworth low-pass filter with cutoff frequeney set by the noise level. Larger values corresponded to stronger filtering.
Multiplicative noise.
Each input channel was scaled by random Gaussian factor:
ū(t) = u(t) n, ~~,,0))"
where a is set by the noise level.
Uniform noise. Additive noise sampled uniformly from [-a, a) was added to each channel, where a is the noise level.
Impulse noise. At each time point, with probability p, impulse of magnitude toa (set by the noise level) was added to the input.
Baseline drift. A slow oscillatory drift was added to each channel:
d(t) A sin 1 sin(4n
where A is the drift amplitude (noise level) and f is low drift rate.
Partial observability. random fraction of input time series was masked with zeros, with mask- ing probability given by the noise level.
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Table 1: Noise levels used in experiments for each noise type.
Noise type
Levels used
White (Gaussian)
0.1, 0.5, I, 1.2, 2,
Pink
0.01, 0.1, 0.2, 0.3, 0.5,
Rotation
0.2, 0.5, 0.8,
Low-pass
0.2, 0.5
Multiplicative
0.1, 0.2, 0.5, 1, 3,,,
Uniform
0.01, 0.5, 1.2, 1.5,
Impulse
0.5, 1,
Baseline drift
0.5, I, 1.2,
0.3, 0.5, 0.7, 0.9, 0.98 Partial observability
Temporal smoothing
5, 10, 20, 40,
Gaussian noise
0.1
0.5
1.2
1.5
SER - 9.81
2.5
SER 2.05
SER 100
SER 0.82
SER 0.86
SER 0.51
2.5-
Original
0.-
takky
120
50
100
100
50
100
10
16n
Scale 0.001
Scale 0.01
Scale 0.1
2.5
Original
2.55
10c
Rotation
2.5
0.0-
Original
100
-2.5
16c
Time
Time
Time
Time
Lowpass filtering
Cutoff
SER 1.05
SER 1.00
302
360
Time
Time
Multiplicative noise
0.1
0.2
0.5
12
1.5 SEN - 0.66
SER - 9.27
2.5
Selt 5.02
2.5
SER 2.00
SER
1.01
SER 0.54
Input
Origral Hakey
25
30 Time
100
200
50
100
100
50
20 Time
150
Figure 12: Effect of different noise types and levels on the input time series of the nonlinear dy- namical system as Fig. The plots show activity along four example observation dimensions received by the networks during an example trial. Here, we show gaussian white noise, pink noise, rotation, low-pass filter, and multiplicate noise applied the input.
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Uniform noise
0.5 SPR 3:42
SPR 170.12
2.5
SPR 37.21
2.5
1.75
SPR
1.40
2.5-
SER 3.16
-2.
50
100
Lio
50 Time
100
50 Time
300
50 Time
100
30 Time
15o
Time
Time
Impulse noise
Ampiltudie
Amplitude
Ampittate -
Amplitude -
Amplitute
SER 8.43
3.40
SER 2.14
SER 1.61
SER 1.07
My Aynt
Original Moliyy
2.5
30c
100
300
300
100
Arrdiiade 0.5
Ampiltude 1.25
Ampliude
Ampil -
Ampliudde -
Arpilitude
On oi-grral Nelliy
0.0
150
Time
Time
Partial Observation
Pimastongt 0.3
Pimasstegi
Pimesking) 0.5
Pimankingl =
Pimaskingi
Pimassting
98
Original
150
ddo
bb
15o
Timm
Time
Tima
Time
Time
Time
Temporal Smoothing
SPI
SEB
SPR
SPB - 1.03
SPR
SPR
1.00
50
300
150
30
100
50
300
1000
100
Time
Thrmw
Tim
Tirm
Timm
Figure 13: Effect of different noise types and levels the input time series of the nonlinear dynami- cal system as in Fig. 2 The plots show activity along four example observation dimensions received by the networks during example trial. Here, show noise sampled from uniform range, impulse noise, random drift, partial observability, and temporal smoothing applied to the input.
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M.1
EFFECT TIME-SHIFTED INPUT ON INPUTDSA DISTANCE ESTIMATION
Distance correlaiio
Combined
State
Input
1.0
0.5
0.0
10
10
20
0
10
20
Time steps shifted
Time steps shifted
Time steps shifted
Figure 14:
Correlation between the InputDSA distance matrix estimated using the ground truth input, and using time-shifted version of the true input.
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N ORDINARY LEAST SQUARES BIASES ESTIMATES OF A IN THE PRESENCE OF INPUT NOISE
Using two datasets: the nonlinear dynamical systems as in Fig. and the RNNS trained on Random Target Task as in Fig. we show that increasing the noise variance systematically contracts the singular spectrum of toward zero. Because the true input effect underfit, the SubspaceDMDe regression inflates the real part of A's eigenvalues to absorb the variance in the inputs that correlates with the state.
Parameterized nonlinear system
trained on Random Target task
Gaussian noise
Pink noise
Eigenvalues of A
Singular values of B
Eigenvalues of A
Singular values of
Singular Value
0.2
Singular Value
10
0.5
Im(A)
Im(A)
10"2_
0.1
O-0
ma0
p-0.5
10-1
pu0.11
-0.5
p-0.5
-1.0
10
0 Re(A)
10
Singular Value Index
Re(A)
Singular Value
Figure 15: The eigenspectrum of the A operator and singular spectrum of the B operator when the input time-series is corrupted by Gaussian or pink noise different variance.
Here, we also present short theoretical discussion of this effect.
We consider the system
Y = AX + BU,
where
Y R 7 are the next states,
X E BaxT are the current states,
U E m=x are the inputs.
R"X* and B E g=xm
We observe noisy inputs
Ú-U+ E,
where E is input noise. The regression becomes
Y R Áx Bü.
We solve the regression problem with OLS by first stacking the regressors:
X
The OLS estimator is
[A B = YZ (ZZ 1ZZ
Expanding Z,
X
XUT + XET UXT + TXT UUT +UE + EUT + EET
XXT
22 T
X
Assuming E is zero-mean and independent,
E|ZZ
Dre Eur
Dru
Enu + Exe
Enp + Exn B
BY
Ens + Enn BT
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To gain intuition, we consider the scalar case where Ozz Var(z),
Oun = Var(u),
dru = (C,
Jee = Var(E).
The least-squares estimates
à a +
Ozz aTaa + Jee) ) 0
azz
Fzz (u + Tee) 0?
can see that for b, large dee in the denominator attenuates b toward zero. For å, the direction of the signs of b, Jan and relative weights of daz and Oau- In particular, when
the bias is dependent b > O and Oee is large, à can be inflated. Intuitively, when the state and imput are strongly positively correlated and the imput drives the state in the same direction, à can be overestimated to absorb the shared variance the input.
O RANDOM TARGET REACH TASK
We trained recurrent neural network (RNN) policies perform random target reaching task in the MotorNet simulation environment (Codol et al. 2024b). We used code from httggghhww...ooooooooottett////e
The effector point-mass skeleton attached to 4 muscles and controlled by muscle activations. The environment provided sequence of random goals and fingertip states. The objective of the policy was to min- imize the distance between fingertip position and target over the course of each episode. At each time step, the model receives a 12-dimensional observation consisting of the proprioceptive input, visual input, and the last action taken by the model. The action space is 4-dimensional consisting each muscle. Each network consisted of a single recurrent layer (64 hidden units) followed by linear readout and sigmoid nonlinearity to produce bounded muscle activations. Training was carried out using the Adam optimizer with learning rate 0.001. The loss function was the mean LI distance between fingertip trajectories and target trajectories across timesteps. We visualize the groundtruth input alongside different types of surrogate inputs during an example trial in Fig. 16
Re l luoo ntMass24 model, 21)
of the activation
Observations
Actions
Positions
20
0.5
Magnitude
0.5
0.0
-0.5
-20
0.0
20
40
0
20
40
20
Goals
Velocities
Random
1.04
Magnitude
0.0
0.0
0.5
-0.5
-0.5
0.0
20
40
0
20
40
20
40
Time Step
Time Step
Time Step
Figure 16: The true input (Observations) and different types of surrogate inputs during an example trial of the Random Target task.
P PLUME TRACKING TASK
We used the plume tracking task implemented in Singh al. (2023) and the training code in In short, the plume tracking environment is
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2D arena where an odor source emits puffss carried downwind by steady flow. The wind can be constant, switch once, or switch multiple times during a trial. Each puff diffuses and drifts, pro- ducing intermittent odor encounters like in real plumes. The agent uses an actor-critic architecture with vanilla RNN backbone, followed by separate two-layer MLPs for the actor and critic. At each timestep it receives three inputs: egocentric wind direction along the x-axis, wind direction along the y-axis, and local odor concentration. Based on its internal state, the actor outputs two-dimensional action specifying turn rate and forward speed.
A
InputDSA (Subspace DMDc)
InputDSA (DMDc)
DSA (DMD)
Intrinsic
Input
Intrinsic
Input
Intrinsic
Distance
0.50
1.0-
0.1-
0.2
0.25
0.5-
0.0
0.00
Top-ipp
Top-Buttom
Bottem-Buttom
Top Top
Tap Buttom
Battom Battom-Buttom Bottom
Top-lop
Tap Hottom
Bottom Hortom
Top Top
Top- Top-Battom
Bottom-Borte Bartt--Brttmm
Top Top-bp
Top-Bond Bop-Bortom
Bottam Bottam-Buttom Bottom
B
InputDSA (Subspace DMDc)
InputDSA (DMDc)
DSA (DMD)
Intrinsic
Input
Intrinsic
Input
Intrinsic
0.275
0.8-
Distance
0.250
0.12
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14-
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o P Updates
Updates
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Figure 17: Full comparison of applying Subspace DMDc, DMDc, and DMD to the plume tracking dataset. (A) Average distance computed within the 5 Top Agents, within the 5 Bottom agents, and across groups (Top-Bottom). (B) The evolution similarity within and across groups over learning. Shaded area indicates standard error.
P.I HYPERPARAMETER SWEEPING
use for InputDSSA, we conducted a sweep of ranks and delays, described by Sec. [ By picking the minimum / elbow of the prediction error curves (AIC, MASE), we choose a delay of 40 and rank of 50 for all InpuIDSA computations on this dataset. We also computed non-normality score (the commutator score, ||AA" A Al). which mea-
To assess the optimal hyperparameters
10a
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Non-non nom--ommlllll score
n delays
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delays
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Figure 18: Hyperparameter sweep over number of delays and model rank on the plume tracking dataset.
sures (as described) the degree which matrix is non-normal (with normality being defined as AT AA')). This measures the relevancy of non-normality in the prediction of the SubspaceD- MDe model, which motivates the use of aligning the dynamics over orthogonal matrices. rather than invertible matrices although any invertible matrix is a coordinate transform, the dynamical system's transient before can change when the transform is non-orthogonal. Hence, capturing the full dynamical similarity of two systems can entail comparing up to orthogonal transform in these settings. Here, we find that rank of 50 with a delay of has a non-normality score close 2.5, indicating that transient dynamics can be significantly effected.
41

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Published as a conference paper at ICLR 2026
Q
NEURAL DATASET
The dataset
published
with
Luo
et
(2025)
can
be
found
here:
http://aatatyaaloooydtatttooo 11.1110..5066..0e Sttxxyadddd33xxmm
InputDSA (Subspace DMDc)
InputDSA (DMDc)
DSA (DMD)
Intrinsic
Input
Intrinsic
Input
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Distance
1.0
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Figure 19: Full comparison of applying Subspace DMDc, DMDc, and DMD to the Luo et al. (2025) dataset. Similarity of neural dynamics before and after the nTe for T223. Bars denote standard error across 21 sessions.
Q.1 HYPERPARAMETER SWEEPING
We chose delay of for the delay embedding and a rank of 6 for the reduced-rank regression on this dataset.
n. delays 3
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Figure 20: Hyperparameter sweep over number of delays and model rank on the processed Luo et
al. (2025) dataset.
Q.2 SURROGATE INPUTS USED IN NEURAL DATASET ARE PERSISTENTLY EXCITING
To whether these inputs were rich enough robustly model the input operator, we computed the standard measure of persistent excitation used in control theory to answer this question. An input persistently exciting if the matrix
M(t) - J( a(t)a(t)
positive definite, i.e., the covariance matrix of the embedded input is full rank. In this analysis, we used models with 5 delays and rank 6 (neural states were 3 dimensional and inputs were dimensional), so this covariance matrix needs to be at least rank 6. We computed this using the delay-embedded input as ū for each session of two rats, 39 in total with 100 trials on average per session (Fig. 21). In each session, the 6th singular value this covariance matrix much greater than 0 for each session, with the 10% quantile having value of 0.22 and the 1% quantile being 0.01. This suggests that even the input is simple, persistently exciting enough to robustly identify useful B operator, given that there are large number of trials.
42

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Published as a conference paper at ICLR 2026
Number of pulses heard
Mean: 2.712, std:3.13
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1.04
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2-
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CDF
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input covariance matrix
Figure 21: Persistent Excitation analysis of the Luo et al. (2025) surrogate input. The input data represents the total number of pulses coming from each speaker within each time bin of the trial. Left: sample input for one trial. Middle: distribution of the ranks of the embedded input covariance matrix for 80 individual sessions across rats. SubspaceDMDe models are chosen be rank 6, so all sessions are persistently exciting. Right: Cumulative distribution of the 6th singular value of the embedded input covariance matrix, indicating that these values are significantly greater than 0 for all sessions.
T176
T219
T223
T240
1e-4
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Figure 22: Distribution of Frobenius norms of Controllability Gramians for Subspace models for all sessions of all rats. "rerresenss p < 0.05 and *** represents p 0.001 using Mann Whimey
U-Test.
Q.3
CONTROLLABILITY ENERGY FOR ALL RATS
Q.4
SUBSPACE ANGLE
Given two dynamics matrices Ag, Ay: an orthonormal basis for each (Ag, Ay) is first computed (for example, via SVD or QR decomposition). Then, the subspace angles are computed as:
A A, UEVT
(62)
0: arccos(a:)
(63)
Where σ; is the i-th singular value defined by E. 0; is defined as the i -th principal an- gle. We report the maximum principal angle between dynamics operators, using the scipylliaag..uuspace.nngg function.
43

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Published as a conference paper at ICLR 2026
X -0.476
T176
Frequency
15
T219
T223
T240
Median
5
0
Normalized [opoeKooll - KroeaKrolll
Figure 23:
Normalized change in controllability energy across the nTe poot post IE pre / Ht/////Kreee Kreel aggregated across four rats.
63.1
100
46.7%
Subspace Angle
80
40
20
0.0
1.2º
Apre-A Aore-Apre
Apre
Apre-Apost Apost Apre B Bpre- B pre B Bpre- Bpre-Bpost post
Figure 24: Subspace angles of the input and state operators within and between time periods for rat T223. Apre Apre (likewise Bpre) denotes the noise floor via split-halves comparison.
44


Reviews:
[{"review_id": "UqVlOFWcWS", "reviewer": "Reviewer_3e6A", "summary": "This paper introduces InputDSA, which aims to extend Dynamical Similarity Analysis (DSA) to systems with external inputs. The method fits, for each system, a linear controlled latent state-space model h_{t+1}=Ah_t+Bu_t via a subspace ID-style procedure, and then defines a similarity metric that compares the recurrent term A (“intrinsic dynamics”) and the input term B (“input-driven dynamics”) across systems after alignment. The problem is important, and the core idea, explicitly separating internal dynamics from input drive and comparing both, is novel and interesting.", "questions": "See Weaknesses above.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The synthetic experiments are well designed and give evidence that under good excitation conditions and with partial observability, the proposed estimator can recover similar recurrent structure vs similar input structure. The applications to trained agents and to neural population data are very appealing and show how the method could validate and generate hypotheses about “what part of the dynamics actually matters.”", "weaknesses": "There are some serious concerns with the clarity and interpretability of the current manuscript:\n\n1.\tThe main text never clearly defines what state is being modeled. In practice, the method learns a latent linear model (A,B) on top of lifted observations via Subspace DMDc. But the paper (especially schematic Fig. 1 and Sec. 2) blurs together true system state, observed activity, and this identified latent state, and then talks about A and B as if they were the system’s real intrinsic vs input-driven dynamics. As written, it’s not clear what is actually being compared from the main text alone.\n\n2.\tMany essential details of the identification procedure (how A,B are estimated, model order selection, assumptions needed for demixing) are only in the appendix. From the main body, it is difficult to understand the method and pipeline.\n\n3.\tMore important: the application to neural data is not well justified, considering the fact that inputs in neural tasks are usually unknown/partial. In real neural recordings, the “inputs” we can measure are usually low-dimensional task variables, not persistently exciting signals. The paper acknowledges this and presents a robustness test with surrogate/noisy inputs in Sec. 3.2, and that test is done in an RNN control setting with richer continuous inputs. However, moving to neural data in Sec. 4.2 and the claimed interpretation “the circuit becomes intrinsically driven after commitment, whether the A/B split is still meaningful becomes unclear since the inputs here are pretty abstract and simple.\"\n\nOverall, I think the idea of this paper is very interesting, and the paper brings some valuable contributions. But the paper in its current form oversells its interpretability on neural data, and the core method is under-specified in the main text. I would like to see a revision that: (i) clearly defines the latent state and is explicit that InputDSA compares identified models; (ii) makes the system ID procedure understandable without digging too much through the appendix; and (iii) tones down the neural interpretation in light of partial/weak inputs.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1J6Wn2rJsl", "reviewer": "Reviewer_R5Xx", "summary": "The paper proposed a new method, inputDSA, to compare between observed trajectories of two dynamical systems. It is an extension to DSA that addresses the important point of external input. The intuition behind both methods is that lifting the observations to a highD space can make the dynamics linear (which is correct for infinite dimensions via Koopman theory). Comparison in the linear domain is relatively easy. Because neuroscience applications typically have external input, and this can completely bias the comparison process, the current work shows how to incorporate inputs into the framework. This has the added benefit of making the algorithm more efficient. The authors demonstrate the method on several synthetic examples, and on experimental neural data.", "questions": "The scope of applicability is not clearly defined. When should this fail? What are the conditions (reachability, full rank) – and how can you verify this. For instance, how do you know it works for your examples?\n\nEq 1. This is not a symmetric measure. For instance, the dimensions can be different. Might be useful to discuss what happens if internal/input dimensions differ between systems.\n\nEq 10 – what is the motivation for this form of dynamics?\n\nWhat is the meaning of the numerical value of scores? The original DSA paper used embedding in a lowD space, whereas you mostly use silhouette. What is the motivation for this choice?\n\nFig 2F – Partial observability degrades state estimation (top panel). From Figure 2A, one can think that having access to the true input is a way to rescue this estimation, because we have indirect information of the unobserved units. There is a small increase in the state silhouette score, but I wonder how much can be rescued. For instance, if the input is more than 1-dimensional – does this help? \nNo reference to Fig 3AB from main text\n\nFig 3D – perhaps use bars, violin, or anything else that does not connect the dots, when the X axis is categorical.\n\nFig 3D – why is the intrinsic estimation unaffected by wrong (random) inputs? Are there inputs (e.g. anti-correlated) that will hurt the intrinsic correlation?\n\nLine 338 typo? “crucial to how inputs”\n\nPlume tracking: “This implies that inputs excite more dimensions of the RNN in Top agents (Fig. 4E), thereby allowing them to more effectively incorporate recent information to inform action selection in the future.” – I’m not sure I follow the logic here. Why should high-D input be more effective in incorporating information? Shouldn’t this be linked to the relation between these directions and those of the intrinsic dynamics? If these higher dimensions are orthogonal to internal dynamics, for instance, then it should just be noise.\n\nNeural data: Aligning to the ntc point. If I understand correctly, this uses the result from a different pipeline (identifying this point), and then does a PSTH using this shift. Perhaps worth emphasizing.\n\nNeural activity: why the need to use PCA and isomap for preprocessing? Does using Isomap change any assumptions about the dynamical system? For instance, it could be that the discarded dimensions are important for the closure of the dynamics.\n\nLine 436: power law fit. Dynamics is usually dominated by the slowest mode. Why use this method of fitting? While several slow modes can interact, the very fast modes are not expected to be relevant here – but they are used in the fit. Also – integration (pre) is expected to have line attractor like dynamics, or very slow intrinsics dynamics. Isn’t the result opposite to this intuition? If so, do you know why?\n\nLine 442: reflected – reflecting\n\nAppendix K tables are in the next page, which is confusing.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "This is an important problem. With the availability of highD neural recordings, inference of dynamical systems from data is increasingly done using many methods. DSA is more of a comparison then direct inference, but it contributes to this field. External input is a crucial aspect of many of these methods.\n\nThe paper is clearly written, explains the problem, the approach and the results. \n\nThe focus on partial observation is highly relevant to possible applications, and done systematically.", "weaknesses": "The interaction between A and B are almost not discussed. Appendix J touches upon it, but the results in the main text mostly show input similarity and state similarity. For instance, in the neural data example, Figure 5F shows angles within A or B. But an interesting question is whether the input arrives to directions that “matter” to the intrinsic dynamics. Can that be measured using eigenvectors of A and their relation to B? \n\nThe limited effect of wrong inputs on state estimation. Results such as Figure 3D show that getting the input completely wrong has little effect on the quality of intrinsic (A) estimation. This seems somewhat contradictory to the original motivation of the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HWrnZnxbPq", "reviewer": "Reviewer_kqZq", "summary": "The paper proposes InputDSA (iDSA), extending Dynamical Similarity Analysis (DSA) to non-autonomous systems by explicitly modeling both (i) intrinsic/recurrent dynamics and (ii) input-to-state mappings. Estimation uses a new Subspace DMDc (a subspace-identification variant of DMD with control) to mitigate the partial-observation bias that makes vanilla DMDc fail. Applications include synthetic systems, RL-trained RNN agents, and neural data show shifts from input-driven to intrinsic dynamics.", "questions": "Please see weakness.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. This paper solves a clear problem gap - ‘Most dynamics-similarity methods ignore inputs.’\n2. The proposed method has partial-observation robustness. Subspace DMDc directly addresses the bias where inputs instantaneously affect observed states and, via unobserved states, which is an excellent point for neural data.", "weaknesses": "1. Since InputDSA mixes intrinsic and input dynamics (via alpha in equation 6), it would be useful to show how sensitive results are to alpha. In other words, how to choose alpha when applying this method? \n2. The paper should clarify how the delay parameter d is determined or selected in practice.\n3. The results lack statistical uncertainty estimates on results, such as statistical uncertainty test on silhouette scores.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qmVM32QA6L", "reviewer": "Reviewer_GeRu", "summary": "This work proposes InputDSA, an extension of DSA which considers inputs when defining a distance measure between data produced by two dynamical systems. To accomplish this, subspace DMDc is proposed which leverages results from prior works (N4SID) for subspace identification. A variety of synthetic and real data experiments are considered to show the utility of InputDSA.", "questions": "- In Figure 2d, the scale between input DSA for DMDc and SubspaceDMDc is very different. How do we determine what magnitude of effect is required to conclude that there is a difference between two systems?\n- How does the distance metric change for each experiment for different values of $\\alpha$? Is this an important hyperparameter to tune or can we always use $\\alpha=0.5$? When would we set $\\alpha \\neq 0.5$?\n- How does the window size affect the InputDSA distance metric? Does window size need to be carefully selected to get reliable results?\n- How do you select the rank of the latent space? Is the InputDSA still reliable when there is a mismatch between true and estimated rank?\n- Is the InputDSA metric differentiable? What is does the computational cost of computing InputDSA scale with larger latent spaces and longer time series?\n- In experiment 3.2, is the inputDSA metric also robust to phase changes of the input?\n- is it always the case that we want to consider rotated state dynamics as similar?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "- The text is overall easy to follow and well-written.\n- Does a good job at pointing out a technical gap, that DSA and its extensions have not explicitly considered differences in the effect of external inputs.\n- The experimentation is very diverse and does a good job at demonstrating practical use cases for computational neuroscience.\n- Good description of hyperparameters in the appendix.", "weaknesses": "- The baselines seem insufficient in experiments 3.2, 4.1, and 4.2 since they primarily compare against only DSA which does not learn an input operator. As a result, it is unclear whether the practical performance improvements for each scenario is due to the proposed SubspaceDMDc or due to explicitly considering input in the metric formulation or both. \n- Synthetic experiments do not consider process noise (noise over time added to the latent state) which is an important characteristic of neural signals due to their intrinsic stochasticity. Could the authors provide experiments that clarify whether InputDSA is also robust to increasing levels of process noise or not? A simple example would be experiment 3.1 where we also consider $x_{t+1} = A(x_t + gF {\\rm tanh} (x_t)) + B(u_t + {\\rm tanh}(u_t)) + \\nu_t$ where $\\nu_t \\sim \\mathcal{N}(0,\\sigma^2)$. \n\n\nMinor\n- Naming the approach “InputDSA” (used in Fig 2 caption) and the distance between input matrices $B$ as “input DSA” (second term in eq. 6 used in Fig 2d,e, and f title) may cause some confusion. \n- $\\alpha$ is used twice which may cause some confusion. In line 437 as the exponent of the power law, and in equation 6 as a balancing coefficient.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)