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arxiv: 2507.01946 · v2 · submitted 2025-07-02 · 🧬 q-bio.QM · cs.LG· math.DS· q-bio.NC

Characterizing control between interacting subsystems with deep Jacobian estimation

Adam J. Eisen , Mitchell Ostrow , Sarthak Chandra , Leo Kozachkov , Earl K. Miller , Ila R. Fiete This is my paper

Pith reviewed 2026-05-19 06:06 UTC · model grok-4.3

classification 🧬 q-bio.QM cs.LGmath.DSq-bio.NC
keywords Jacobian estimationnonlinear controldynamical systemsrecurrent neural networkssubsystem interactionsworking memorydeep learningneural dynamics
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The pith

JacobianODE estimates the state-dependent Jacobian from time-series data to quantify how one subsystem controls another in nonlinear dynamics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a data-driven method to characterize control between interacting subsystems by recovering the Jacobian of their joint nonlinear dynamics. JacobianODE uses deep learning to estimate this Jacobian directly from observed trajectories without requiring the underlying equations or linear approximations. In a multi-area recurrent neural network trained on a working memory selection task, the estimates show that the sensory area progressively gains greater influence over the cognitive area. The same Jacobian estimates enable construction of control signals that precisely steer the network's internal states and outputs.

Core claim

We devise JacobianODE, a neural architecture that learns the full Jacobian matrix of arbitrary dynamical systems from time-series data alone by embedding consistency with the system's differential structure into the training objective. Application to a trained multi-area RNN demonstrates that control from the sensory area to the cognitive area strengthens across learning epochs on the working-memory task. The recovered Jacobians further permit direct application of control inputs that manipulate the RNN's behavior with high precision.

What carries the argument

The Jacobian matrix of the joint vector field, which at each point in state space encodes the instantaneous linear effect of perturbations in one subsystem on the rate of change of the other.

If this is right

  • Direction, strength, and context dependence of control between subsystems become measurable in fully nonlinear regimes.
  • Control relationships in recurrent networks can shift measurably as the network learns a task.
  • Estimated Jacobians supply the linear maps needed to design targeted interventions that alter system trajectories.
  • The approach extends control analysis to high-dimensional chaotic systems where linear methods fail.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same estimation pipeline could identify which genes exert dominant control over expression dynamics in regulatory networks.
  • Applied to simultaneous recordings from multiple brain regions, it could map effective connectivity without assuming linear or stationary interactions.
  • Real-time deployment might support closed-loop interventions that steer biological or artificial systems using only streaming observations.

Load-bearing premise

Time-series observations alone contain sufficient information to recover an accurate Jacobian of the underlying nonlinear dynamics without additional structural assumptions or access to the true vector field.

What would settle it

Generate trajectories from a known system such as the Lorenz attractor, apply JacobianODE to the data only, and compare the estimated Jacobian against the analytically known Jacobian at the same points; large pointwise errors would falsify accurate recovery.

Figures

Figures reproduced from arXiv: 2507.01946 by Adam J. Eisen, Earl K. Miller, Ila R. Fiete, Leo Kozachkov, Mitchell Ostrow, Sarthak Chandra.

Figure 1
Figure 1. Figure 1: Schematic overview of control-theoretic framework applied to neural interactions. (A) Control theory generalizes across diverse systems. (B) Illustration of interareal control, highlighting how neural activity in one area directly influences dynamics in another. One approach to extend control-theoretic analyses to nonlinear systems is by linearizing the nonlinear dynamics through Taylor expansion, which in… view at source ↗
Figure 2
Figure 2. Figure 2: Analytical framework for pairwise interacting subsystem control. Trajectory dynamics (left) are locally linearized via Jacobians (middle), explicitly separating within-area (diagonal blocks) and interareal (off-diagonal blocks) dynamics (right). These separated dynamics can be used to construct interaction-specific control systems. 2.1 Characterizing control between subsystems with Jacobian linearization C… view at source ↗
Figure 3
Figure 3. Figure 3: Jacobian estimation with JacobianODEs. (A) Path integration of the Jacobian predicts future states. (B) Generalized teacher forcing stabilizes trajectory predictions during training. (C) Loop-closure constraints enforce consistency of Jacobian estimates. (D) Training pipeline, combining neural Jacobian estimation, path integration, teacher forcing, and self-supervised loop-closure loss. 3.1 Parameterizing … view at source ↗
Figure 4
Figure 4. Figure 4: JacobianODE surpasses benchmark models on chaotic dynamical systems. Error bars indicate standard deviation, with statistics computed over five different model initializations. (A,B) State-space trajectories for (A) Lorenz and (B) 64-dimensional Lorenz96 systems, with 10 time-step predictions. (C,D) Accuracy of 10 time-step trajectory predictions at varying noise levels. (E,F) Comparison of Jacobian estima… view at source ↗
Figure 5
Figure 5. Figure 5: JacobianODE accurately infers trained RNN Jacobians. Error bars indicate standard deviation, with statistics computed over five different model initializations. (A) Task schematic, involving stimuli presentation, delays, cueing, and response. (B) RNN architecture with distinct visual and cognitive areas interacting. (C–E) Structure of trained weight matrices: (C) recurrent weights within RNN, (D) input con… view at source ↗
Figure 6
Figure 6. Figure 6: JacobianODE reveals differential interareal reachability. Error bars indicate standard deviation for (A) and (C), and standard error for (B),(E), and (F), with statistics computed over trajectories. (A) Comparison of interareal reachability between ground truth and JacobianODE estimates. (B) Temporal evolution of reachability (Gramian trace) throughout the delay period. (C) Comparison of reachability in ea… view at source ↗
read the original abstract

Biological function arises through the dynamical interactions of multiple subsystems, including those between brain areas, within gene regulatory networks, and more. A common approach to understanding these systems is to model the dynamics of each subsystem and characterize communication between them. An alternative approach is through the lens of control theory: how the subsystems control one another. This approach involves inferring the directionality, strength, and contextual modulation of control between subsystems. However, methods for understanding subsystem control are typically linear and cannot adequately describe the rich contextual effects enabled by nonlinear complex systems. To bridge this gap, we devise a data-driven nonlinear control-theoretic framework to characterize subsystem interactions via the Jacobian of the dynamics. We address the challenge of learning Jacobians from time-series data by proposing the JacobianODE, a deep learning method that leverages properties of the Jacobian to directly estimate it for arbitrary dynamical systems from data alone. We show that JacobianODEs outperform existing Jacobian estimation methods on challenging systems, including high-dimensional chaos. Applying our approach to a multi-area recurrent neural network (RNN) trained on a working memory selection task, we show that the "sensory" area gains greater control over the "cognitive" area over learning. Furthermore, we leverage the JacobianODE to directly control the trained RNN, enabling precise manipulation of its behavior. Our work lays the foundation for a theoretically grounded and data-driven understanding of interactions among biological subsystems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes JacobianODE, a deep learning architecture that estimates the Jacobian of an unknown nonlinear dynamical system directly from time-series observations by exploiting the chain-rule structure of the variational equation. The authors report that JacobianODE outperforms prior Jacobian estimators on high-dimensional chaotic benchmarks and then apply the method to a multi-area RNN trained on a working-memory selection task, claiming that the sensory area acquires increasing control over the cognitive area across training epochs. They further demonstrate that the learned Jacobian can be used to synthesize control inputs that manipulate the RNN's behavior.

Significance. If the data-driven Jacobian estimates prove accurate, the framework offers a principled route to quantify directed, context-dependent control between subsystems in high-dimensional biological networks without requiring an explicit model of each subsystem. The combination of variational ODE integration with deep networks and the downstream control demonstration are technically distinctive and could be useful for analyzing multi-area neural recordings or gene-regulatory networks once the accuracy concerns are addressed.

major comments (2)
  1. [RNN application / results on multi-area network] RNN application section: the headline claim that the sensory area gains greater control over the cognitive area rests on block norms or singular values of the estimated inter-area Jacobian. Because the RNN is fully specified, its exact Jacobian is available via automatic differentiation at every training epoch. A quantitative comparison (e.g., Frobenius or operator-norm error, or direct overlay of the true vs. estimated sensory-to-cognitive block) between JacobianODE and the ground-truth Jacobian is required to establish that the reported monotonic trend is not an artifact of estimation bias or metric sensitivity.
  2. [Methods] Methods / JacobianODE training: the loss is defined on observed trajectories and their variational derivatives, yet no ablation is shown that isolates the contribution of the Jacobian-specific regularization terms versus a standard neural ODE. Without this, it remains unclear whether the reported gains on chaotic systems and the control trend in the RNN are driven by the architectural innovations or by generic sequence modeling capacity.
minor comments (2)
  1. [Abstract / Results] Abstract and results: quantitative error bars, number of random seeds, and statistical tests for the outperformance claims on chaotic systems and for the control trend are not reported; these should be added.
  2. [Results] Notation: the symbol and precise definition of the control metric (e.g., block norm, principal singular value) used to quantify “greater control” should be stated explicitly in the main text rather than only in supplementary material.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and insightful comments. We address each major point below and will revise the manuscript to incorporate the suggested analyses.

read point-by-point responses

  1. Referee: [RNN application / results on multi-area network] RNN application section: the headline claim that the sensory area gains greater control over the cognitive area rests on block norms or singular values of the estimated inter-area Jacobian. Because the RNN is fully specified, its exact Jacobian is available via automatic differentiation at every training epoch. A quantitative comparison (e.g., Frobenius or operator-norm error, or direct overlay of the true vs. estimated sensory-to-cognitive block) between JacobianODE and the ground-truth Jacobian is required to establish that the reported monotonic trend is not an artifact of estimation bias or metric sensitivity.

    Authors: We agree that a direct comparison to the ground-truth Jacobian is valuable for validating the RNN results. Because the network is fully specified, the exact Jacobian is computable via automatic differentiation at each epoch. In the revised manuscript we will add quantitative error metrics (Frobenius norm and operator-norm differences) between the estimated sensory-to-cognitive block and the true block, together with a direct overlay plot at representative epochs. This will confirm that estimation error remains low and does not artifactually produce the reported monotonic increase in control. revision: yes

  2. Referee: [Methods] Methods / JacobianODE training: the loss is defined on observed trajectories and their variational derivatives, yet no ablation is shown that isolates the contribution of the Jacobian-specific regularization terms versus a standard neural ODE. Without this, it remains unclear whether the reported gains on chaotic systems and the control trend in the RNN are driven by the architectural innovations or by generic sequence modeling capacity.

    Authors: We acknowledge the utility of such an ablation. In the revised manuscript we will include a controlled comparison on the high-dimensional chaotic benchmarks in which a standard Neural ODE is trained using only the trajectory-matching term of the loss, without the variational-equation or Jacobian-regularization components. Performance on Jacobian estimation accuracy will be reported side-by-side with the full JacobianODE, thereby isolating the contribution of the proposed architectural and loss innovations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained against external benchmarks.

full rationale

The paper defines JacobianODE as a data-driven deep learning estimator trained directly on observed trajectories to recover Jacobians of arbitrary nonlinear systems, with performance validated on held-out high-dimensional chaotic benchmarks independent of the target RNN application. The central empirical claim—that sensory-to-cognitive control increases over learning—is obtained by applying the trained estimator to simulated multi-area RNN trajectories and computing control metrics from the resulting Jacobian blocks; this does not reduce by construction to the training loss, to any fitted parameter renamed as a prediction, or to a self-citation chain whose cited result itself depends on the present work. No self-definitional loops, ansatz smuggling, or renaming of known results appear in the provided derivation steps. The approach therefore retains independent content from its inputs and external validation sets.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework assumes that the Jacobian of an unknown nonlinear vector field can be recovered to useful accuracy from finite noisy time series via a deep network whose architecture and training procedure are chosen by the authors. No explicit free parameters beyond standard deep-learning hyperparameters are named in the abstract.

axioms (1)
  • domain assumption Time-series data generated by the true (unknown) dynamics contain enough information to identify the instantaneous Jacobian at observed states.
    Invoked when stating that JacobianODE learns the Jacobian 'from data alone' without the governing equations.
invented entities (1)
  • JacobianODE no independent evidence
    purpose: Deep network that directly outputs the Jacobian matrix of an arbitrary dynamical system from time-series observations.
    New architecture introduced to solve the Jacobian estimation problem; no independent evidence of correctness outside the reported experiments is provided in the abstract.

pith-pipeline@v0.9.0 · 5807 in / 1424 out tokens · 52080 ms · 2026-05-19T06:06:40.309642+00:00 · methodology

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