InputDSA : Demixing then comparing recurrent and externally driven dynamics

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 intrinsicallydriven 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 ¹.

1. Introduction

Identifying that two seemingly disparate complex systems have the same underlying structure is a 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 (e.g., brains, minds, computational models, or physical objects) is to compare the geometry of their states. Well-known methods to do so are Representational Similarity Analysis, Centered Kernel Alignment, Procrustes Analysis, Canonical Correlation Analysis, and Pearson Correlation (Kriegeskorte et al., 2008; Kornblith et al., 2019; Williams et al., 2022; Gallego et al., 2018; Raghu et al., 2017; Schrimpf et al., 2018). Neural networks can also be characterized by the topology of their activations, (Chaudhuri et al., 2019;Gardner et al., 2022; Lin & Kriegeskorte, 2024), a 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 al. 2023) offer an important complementary lens to structure characterization, by proposing a similarity metric on the level of dynamics. DSA provides a linear, efficient and theoretically grounded dynamical similarity metric that has been successfully applied to recurrent network dynamics, training dynamics, and biological neural data (Redman et al., 2024a; Huang et al., 2025; Codol et al., 2024a; Guilhot et al., 2024; Versteeg et al., 2025; Lazzari & Saxena, 2025). Briefly, DSA embeds nonlinear 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 et al., 2025; Chen et al., 2024; Nejatbakhsh et al., 2024) based on other computational techniques such as neural networks and shape metrics. Notably, none of these methods consider the effect of external input.

In neuroscientific settings such as central pattern generators or working memory circuits, dynamics may be treated as approximately autonomous (Marder & Bucher, 2001; Grillner, 2006; Kiehn, 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 is 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-autonomous, receiving sensory signals or communication from other subsystems (Eisen et al., 2025). 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 a 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), demonstrating that they can together provide complementary insights on both intrinsic as 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.

2. 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 $f:X\to X$ and $g:Y\to Y$ with mapping $\varphi :X\to Y$, (semi-) conjugacy is defined as:

$$g\circ \varphi =\varphi \circ f$$ (1)

The existence of such a mapping entails a 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 a function can be arbitrarily complex, which can make searching for the true conjugacy map challenging in 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 in some high-dimensional space, yielding dynamics models $x_{t+1}=A{x}_t$. Then similarity is defined on the linear operators using the following metrics:

$$\mathrm{DSA}\left(A_1,A_2\right):=\underset{C\in O(n)}{\min }{‖C{A}_1{C}^T-A_2‖}_F$$ (2)

$$\text{DSA}\left({\text{Λ}}_1,{\text{Λ}}_2\right):=\underset{P\in \text{Π}(n)}{\min }{‖P{\text{Λ}}_1{P}^T-{\text{Λ}}_2‖}_F$$ (3)

Where ${\text{Λ}}_i$ is the eigenvalue matrix of $A_i$, and $O(n)$, $\text{Π}(n)$ the groups of $n\times n$ dimensional orthogonal and permutation matrices. The latter metric was introduced by (Redman et al., 2024b) and is a special case of the former (Ostrow et al., 2023). These metrics are reminiscent of Procrustes Analysis, 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 Koopman Operator Theory based on the relationships between Koopman Operators of conjugate systems (Budišić et 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

$$\dot{x}=A_{1}x+B_{1}u(t)\dot{y}=A_{2}y+B_{2}u(t).$$ (4)

A key feature of non-autonomous systems is their controllability: the ability for an input sequence to drive the state to 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):

$$K_1(T)=\left(\begin{array}{lllll}{B}_1\hfill & A_1{B}_1\hfill & A_1^2{B}_1\hfill & \dots \hfill & A_1^{T-1}{B}_1\hfill \end{array}\right)$$ (5)

and its corresponding Gramian, which encodes the geometry of controllability.

$$W_c(T)=K_1{K}_1^T=\sum _{i=0}^T{A}_1^i{B}_1{\left(A_1^i{B}_1\right)}^T$$ (6)

Intuitively, directions with small eigenvalues are easier to control, because they are more responsive to the effect of input. Controllability, as measured by the eigenvalues of the Gramian, is only preserved under orthogonal transformations between state spaces:

$$y=Cx\Rightarrow A_1=C{A}_2{C}^T,B_1=C{B}_2{K}_1=C{K}_2$$ (7)

This motivates our proposed dissimilarity metric, which extends DSA:

$$\mathrm{InputDSA}\left(A_1,A_2,B_1,B_2,T\right)=\underset{C\in O(n)}{\min }\sum _{i=0}^T{‖C{A}_1^i{B}_1-A_2^i{B}_2‖}_F^2=\underset{C\in O(n)}{\min }{‖C{K}_1-K_2‖}_F^2$$ (8)

We also provide a theoretical extension of Eq. 3 in Appendix G.1, which we note is highly susceptible to numerical instability. Although Eq. 2 requires iterative optimization, Eq. 8 is solved via Procrustes alignment, which yields an exponential acceleration of prior work. We provide further theoretical discussion in Appendix G. After solving for $C^{*}$, we can additionally study the joint state and input DSA scores:

$${\text{InputDSA}}_{\text{state}}\left(A_1,A_2,C^{*}\right)={‖C^{*}{A}_1{C}^{{*}^T}-A_2‖}_F^2$$ (9)

$${\text{InputDSA}}_{\text{input}}\left(B_1,B_2,C^{*}\right)={‖C^{*}{B}_1-B_2‖}_F^2$$ (10)

If the inputs directly applied to the system are known, as in computational models, Eq. 8 is sufficient. However, when the true input is some modification of a 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. 8 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. (2023): identify the best linear approximation of an input-driven system, following which comparison can be done efficiently between 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.

2.3. Estimating Linear Operators

As in DSA, We fit linear operators via the Dynamic Mode Decomposition (DMD) family of methods. The DMD (Schmid, 2010; 2022) identifies the linear dynamics that best explain the data:

$$\varphi \left(x_{t+1}\right)=A\varphi \left(x_t\right).$$ (11)

Here $x_t$ represents the measured state of the system at time $t$, $\varphi$ is a 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 dynamical systems which encodes the linear dynamics of observables (functions that act on the state) under the system dynamics. Prior work has explored many different choices of $\varphi$. For example, $\varphi$ can be a kernel function, a delay embedding, or even a neural network (Williams et al., 2016; Brunton et al., 2017; Arbabi & Mezić, 2017; Lusch et al., 2018). Intuitively, the dimensionality 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 dynamics (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 & Solano-Castellanos, 2024; Haseli et al., 2025). Likewise, the DMD can be generalized to driven systems: When given control inputs $u_t$, we can instead apply DMD with control (DMDc, Proctor et al. 2016a;b):

$${\varphi }_1\left(x_{t+1}\right)=A{\varphi }_1\left(x_t\right)+B{\varphi }_2\left(u_t\right)$$ (12)

Here, ${\varphi }_1$ and ${\varphi }_2$ can be distinct nonlinearities. While DMDc was originally only applied with no nonlinearity $\left({\varphi }_1,{\varphi }_2=\text{Id}\right)$, it too can be freely generalized to high-dimensional nonlinear embeddings. For more algorithmic details, see Appendix B.

Issues of Partial Observation

While estimating $A$ and $B$ via DMDc is an intuitive extension to input-driven systems, it has a hidden failure mode in the analysis of partially-observed systems. This is particularly important in the analysis of neural data, in which a small subset of neurons in a 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 $t$ (instantaneously) and in future time steps through the unobserved state (Fig. 2A). This means that simply applying DMDc in this setting will result in the $B$ matrix becoming biased toward the intrinsic dynamics of the system. We develop a formal description of this problem in Appendix D. We solve this problem by introducing Subspace DMDc, a novel extension of Subspace DMD (Takeishi et al., 2017b) 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:

$$x_{t+1}=A{x}_t+B{u}_t{y}_t=C{x}_t$$ (13)

Figure 2: InputDSA SubspaceDMDc is robust to partial observation.

(A) Under partial observation, inputs can have effects on observed states (red nodes) in the future via the unobserved states (green nodes), thereby biasing estimates of input driven-dynamics. Purple arrows indicate this indirect propagation of input into the observed states. (B) Sample inputs and observed states from 4 dynamical systems, which have alternate pairings of the same intrinsic and input-driven dynamics denoted by arrows. (C) Sample state similarity matrices based on estimates from the DMD, DMDc and SubspaceDMDc on data generated as in (B). four iterations of each system are generated, each with unique inputs and initial conditions, resulting in 16 × 16 dimensional matrices. (D) Sample input distance matrices on the same data as in (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). As a baseline input-label silhouette score for DMD is computed on the state matrix with the ground-truth input similarity labels. Bars denote standard error across seeds. (F) Silhouette scores for each DMD and similarity type as the system is increased from 2-dimensional to 1000-dimensional. Each size was repeated across 20 random seeds. Shading denotes standard error.

With only $y_t$ and $u_t$ observed. The situation of partial observability is a 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) on lifted states (thereby leveraging the power of nonlinear DMD algorithms such as Williams et al. 2016, although). For further technical details on the subspace identification algorithm, see Appendix Section E. We also highlight that subspace identification methods are designed to handle data with both observation and process noise, therefore providing InputDSA noise robustness (Verhaegen, 1994; Verhaegen & Verdult, 2007).

3. Experiments

3.1. InputDSA discriminates intrinsic dynamics from input-driven dynamics

To demonstrate that InputDSA can capture similarities in both intrinsic and input-driven dynamics, we simulated partially observed nonlinear discrete-time systems with the following equations:

$$x_{t+1}=A\left(x_t+gF\tanh \left(x_t\right)\right)+B\left(u_t+\tanh \left(u_t\right)\right)$$ (14)

$$y_t=\left(\begin{array}{ll}{\mathbf{I}}_d\hfill & {\mathbf{0}}_{n-d}\hfill \end{array}\right)x_t+{ϵ}_t$$ (15)

Where F and g are fixed across all simulations, and ${ϵ}_t$ is observation noise. We randomly sampled two matrices for $A\in {ℝ}^{n\times n}$, and two for $B\in {ℝ}^{n\times 1}$, from which we constructed 4 systems: Systems 1 and 2 (3 and 4) share the same intrinsic dynamics matrix $A_1\left(A_2\right)$, while Systems 1 and 3 (2 and 4) share the same input matrix $B_1\left(B_2\right)$. We randomly sampled low-pass filtered white noise as 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 $\left(x\in {ℝ}^{20}\right)$ systems and observed 2 dimensions $\left(y\in {ℝ}^2\right)$ for 5,000 time points. For simulation details, see Appendix J. 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 Arbabi & Mezić 2017; Brunton et al. 2017), (2) the state distance using a delay-embedded DMDc, (3) the state distance using the SubspaceDMDc, (4) the input distance using the DMDc, and (5) the input distance using the SubspaceDMDc. Note that DSA does not have the ability to compare inputs, so it is left out. For a discussion on hyperparameter tuning, see Appendix J). For the sake of space, we report the jointly optimized input distance (Input DSA, Eq. 10) and the individually optimized state distance (State DSA, Eq. 9) as these are the most interpretable, although the jointly optimized state distance was highly similar.

In Fig. 2B we visualize the observed input and one dimension of the observed output for a 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. 2C. While the DMD and the DMDc have notable structure pertaining to the true state similarity, the SubspaceDMDc 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-truth state labels, the DMD scores 0.6, DMDc scores 0.68, and the Subspace DMDc scores 0.94. In Fig. 2D, 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 a silhouette score of 0.19. The silhouette score of the SubspaceDMDc is 0.83, indicating robust separation. We also compute the total similarity matrices (Eq. 8, Appendix Fig. 6), for which the SubspaceDMDc reports correctly that each type of system is altogether unique. We swept over 100 seeds in Fig. 2E and found that the SubspaceDMDc-based InputDSA consistently yielded the best separability.

To assess the effect of partial observation, we ran the above analysis for different-sized systems (ranging from 2 to 1000 dimensions) with only 2 observed dimensions, for which we present the average silhouette scores for InputDSA in Fig. 2F. The state similarity scores for each method gracefully degrade with the total state size, and SubspaceDMDc has a noticeable improvement over the other methods. The DMDc input score appears to never be robust. However, the SubspaceDMDc input similarity is robust across all system sizes. This suggests that SubspaceDMDc can be used to measure the dynamical similarity of partially observed, input-driven 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 accessible. Instead, what we observe are often noisy or partial measurements, limited by sensor resolution, sampling rates, or inherent partial observability. As a consequence, researchers often rely on behavioral variables, task instructions, or environmental features as proxies when modeling neural circuits (Vinograd et al., 2024; Sani et al., 2024; Burak & Fiete, 2009b; Schaeffer et al., 2020; Mante et al., 2013b). This raises a key question for applying InputDSA : if the true inputs are unknown, can 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. 2C, this time applying different types of noise perturbations to the input used in SubspaceDMDc (example in 3A). For complete details on the noise perturbations, see Appendix K. We applied 10 types of perturbations inspired by different real-world situations, such as partial observation, temporal smoothing, or multiplicative Gaussian noise, and repeated each perturbation across a range of parameters (e.g. standard deviation in the noise settings or filter width in the smoothing setting). 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 $X\in {ℝ}^{t\times d}$ and its perturbed version $\tilde{X}=f(X)$, SER is defined as

$$SER(X,\tilde{X})=\frac{\mathrm{Var}(X)}{\mathrm{Var}(\tilde{X}-X)}$$

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 $SER<1$ threshold (Fig. 3B): 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 DMDc: 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 [−0.1,0.1], which we applied dimension-wise to the inputs as a 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 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 input DSA scores. Together, this suggests that up to reasonable expectations (SER greater than or close to 1), the InputDSA scores are robust to generic perturbations on the input data.

Figure 3: InputDSA provides robust distance estimates under input noise and surrogate inputs.

(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 K 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 $\mathrm{Var}(\text{X})/\mathrm{Var}(\tilde{X}-X)$. From left to right: joint controllability DSA (Eq. 8), jointly optimized state DSA (Eq. 9; jointly optimized input DSA (Eq. 10). (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 applied 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 training runs. Jointly optimized state and input DSA are presented.

Next, we evaluated whether task-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. 3E, 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 et al., 2012; Churchland et al., 2012). Across 20 epochs equally spaced in training, we recorded the RNN’s hidden states, observations (the true input), actions (behavioral output), and other task variables (Fig. 3F, only 5 epochs shown for visualization purpose). For a detailed description of the task and training, see Appendix M. 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 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 combinations. We also included random inputs sampled from the uniform distribution on [0,1] as a baseline. Among surrogates, the actions maintain the highest trial-averaged correlation with the ground truth input (Fig. 3G). 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 highly-correlated surrogates tend to yield more accurate similarities, with the RNN’s combined action and positions providing strong correlations with the ground truth distance (Fig. 3H). Overall, our robustness analysis suggests that state similarities are highly 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 how inputs interact with network dynamics and shape agent performance. The Plume Tracking task (Fig. 4A) provides an ideal testbed because the agents must balance between memory-based intrinsic dynamics with stimulus-driven responses.

Figure 4: InputDSA identifies how successful and unsuccessful agents differ over training.

(A) The Plume Tracking environment 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 organized in a low-dimensional space and reflective of behaviorally relevant variable (i.e. the odor concentration). (D) Average distance computed within the 5 Top Agents, within the 5 Bottom agents, and across groups (Top–Bottom). (E) The singular value spectrum of the input-mapping operator $B$ from Top and Bottom agents. (F) The evolution of similarity within and across groups over learning. Shaded area indicates standard error.

In this task, artificial flies (RNNs) trained by deep RL navigate to the source of a simulated 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 diverge 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 ability to form and maintain task-relevant representations) or input-driven responses to stimuli?

We trained 15 independent agents on the Plume Tracking task. We selected the five best-performing (“Top”) agents with 65% to 20% success rate at 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 separated from those of the Bottom agents, whereas intrinsic dynamics were not significantly different between groups (Fig. 4D). This suggests that successful plume tracking heavily depends rapid responses to wind direction and odor concentration. To probe how the input-driven dynamics differ between Top and Bottom agents, we examined the singular values of the input–mapping $B$ in Fig. 4E. 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. 4E), thereby allowing them to more effectively incorporate recent information to inform action selection in the future.

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. 4F). While within–group input similarity decreases over training for both Top and Bottom, the Top agents ultimately converge to a more consistent set of input-driven dynamics, whereas the Bottom 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.

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 accumulation task (Luo et al. 2025, Fig. 5A).

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) Trialaveraged neural trajectories visualized in the top two Principal Components. Stars indicate a ”neural 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 $A$ 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) Normalized effects of intrinsic and input-driven dynamics in pre vs. post periods. (F) Subspace angles of the input and state operators within and between time periods. $A_{pre}-A_{pre}$ (likewise $B_{pre}$) denotes the noise floor via split-halves comparison.

During this task, rats were trained to listen to 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 20 recorded sessions for our analysis to ensure accurate estimation of neural dynamics. In the original study, the authors define the neural time of commitment (nTc) as the internal moment during perceptual decision-making when an animal has effectively committed to a choice (Fig. 5B). To examine how neural population dynamics reorganize across this point, we applied InputDSA to neural activities aligned to the $nTc$. Spiking activity was binned in 50 ms windows, smoothed with a causal Gaussian kernel ($\sigma =250$ ms), and dimensionality reduced with PCA to preserve 99% of variance. The activity was then embedded into three dimensions using Isomap, and InputDSA was applied with hyperparameters detailed in Appendix O. We construct the inputs as two-dimensional time series encoding the number of auditory pulses from the left and right within each bin.

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. 5C). To probe how the intrinsic dynamics change, we analyzed the eigenspectrum of the state-transition matrix $A$ estimated by SubspaceDMDc before and after the $nTc$. Each spectrum was fit with a power law ${\lambda }_i\propto {\left(\frac{i}{N}\right)}^{\alpha }$, where ${\lambda }_i$ are the $i$-th eigenvalue sorted in descending order. We found that the post-commitment periods consistently showed smaller $\alpha$, indicating slower decay and thus longer-lasting intrinsic dynamics (Fig. 5D). This is directly related to the controllability of the dynamical system, which describes how easy it is for an input sequence to drive the system to arbitrary points in state space (Luenberger, 1979). Smaller DMD eigenvalues implies greater input controllability, which would be expected for a more input-driven system as Luo et al. (2025) identified is the case in the pre-$nTc$ regime.

Likewise, the average magnitude of the intrinsic dynamics strengthen while the input-driven dynamics weaken in the Post-nTC period, reflecting a transition into more autonomous, less input-sensitive regime after the $nTc$ (Fig. 5E). By measuring the subspace angle between the $A$ and $B$ operators across the $nTc$, we find that the neural dynamics occupy different subspaces across the $nTc$, reflected a reorganization of dynamics at the decision-commitment time (Fig. 5F). Together, these results suggest that population activity undergoes a regime shift at the $nTc$: transitioning from an input-driven, evidence-accumulation phase into an intrinsically dominated, decision-commitment phase, as suggested by (Luo et al., 2025).

5. Discussion

We introduced a theoretically-motivated method (InputDSA ) 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., 2016a) 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 InputDSA 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., 2023; Sohn et al., 2019; Burak & Fiete, 2009a; Mante et al., 2013a; Sussillo et al., 2015), our work provides principled methodological support to this practice. Inputs could also be estimated via another computational method (e.g.Perich et al. 2020; Luo et al. 2025) before applying InputDSA. As Fig. 3 suggests, even utilizing weakly correlated proxy inputs can increase the robustness of the intrinsic comparison with InputDSA.

InputDSA could be used for further validation of computational models with perturbation as in O’Shea et 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 impulse 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 SubspaceDMDc, such as Eigensystem Realization (ERA, Juang & Pappa 1985). In a similar vein, InputDSA could potentially be used to identify the information content in cross-brain-region communication – multiple models could be constructed with different surrogate inputs, and the most similar input should have the lowest input distance to the data (Fig. 3).

Although we only applied InputDSA to biological neural data and recurrent neural networks, it can be applied to any time series data. Indeed, the constraints on the method are based on 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 a wide range of work in both fields designed to tackle these problems (Wu et al., 2021; Colbrook et al., 2023; Takeishi et al., 2017a; Ichinaga et al., 2024). It is also worth noting that near-perfect estimation is not necessary for informative comparison.

InputDSA has other limitations. The method assumes additive input, which may not be able to approximate the effects of multiplicative input (Logiaco et al., 2021; Shine et al., 2021). Disentangling the contribution of state and input can also be challenging or intractable when they are synchronized (Rajan et al., 2010) or the input is a 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 SubspaceDMDc, 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 on a GPU.

Figure 1: InputDSA schematic.

(1), state and input data are collected from two systems. (2) data are embedded in a 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.