Characterizing control between interacting subsystems with deep Jacobian estimation

2025-07-02

AI Summary

This study addresses the challenge of characterizing dynamic control relationships—specifically directionality, strength, and context dependence—among subsystems in nonlinear complex systems. To this end, we propose JacobianODE, the first data-driven framework capable of unbiased, direct estimation of the full Jacobian matrix of arbitrary nonlinear dynamical systems from time-series observations. Integrating deep neural networks with differential equation modeling, JacobianODE enables end-to-end learning of continuous, differentiable, high-dimensional Jacobian estimates, overcoming fundamental limitations of linear or local approximation methods. Empirically, it achieves state-of-the-art performance on high-dimensional chaotic systems and successfully uncovers time-varying control flows between perception and cognition modules within recurrent neural networks. These results enable causal intervention and precise regulation of network functionality, advancing both interpretability and controllability of complex nonlinear dynamics.

Technology Category

• Intelligent Robots: State Estimation
• Cognitive Modeling & Cognitive Systems: Adaptive Behavior
• Machine Learning: Causal Learning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Characterize nonlinear control between interacting biological subsystems
• Estimate Jacobians from time-series data for dynamical systems
• Understand and manipulate interactions in multi-area neural networks

Innovation

• Deep learning estimates Jacobian for nonlinear systems
• JacobianODE outperforms existing estimation methods
• Framework enables precise control of subsystem interactions