Fast dynamical similarity analysis

2025-11-27

AI Summary

Traditional neural dynamical similarity analysis methods neglect intrinsic dynamical structure and suffer from computational inefficiency. To address this, we propose an efficient framework for comparing neural dynamical representations. First, nonlinear neural activity is mapped into a globally linear space via Hankel delay embedding. Second, the optimal embedding order is selected automatically, and lightweight affine constraints replace strict orthogonality constraints, substantially reducing optimization complexity. Third, a conjugacy-invariant dynamical similarity metric is defined by jointly minimizing the distance between dynamical matrices and applying data-driven singular value thresholding. Our method preserves the original dynamical invariance and sensitivity while accelerating computation by over an order of magnitude. It enables scalable, cross-modal comparison of dynamical structures across large-scale neural recordings and computational models.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

To understand how neural systems process information, it is often essential to compare one circuit with another, one brain with another, or data with a model. Traditional similarity measures ignore the dynamical processes underlying neural representations. Dynamical similarity methods offer a framework to compare the temporal structure of dynamical systems by embedding their (possibly) nonlinear dynamics into a globally linear space and there computing conjugacy metrics. However, identifying the best embedding and computing these metrics can be computationally slow. Here we introduce fast Dynamical Similarity Analysis (fastDSA), which is computationally far more efficient than previous methods while maintaining their accuracy and robustness. FastDSA introduces two key components that boost efficiency: (1) automatic selection of the effective model order of the Hankel (delay) embedding from the data via a data-driven singular-value threshold that identifies the informative subspace and discards noise to lower computational cost without sacrificing signal, and (2) a novel optimization procedure and objective, which replaces the slow exact orthogonality constraint in finding a minimal distance between dynamics matrices with a lightweight process to keep the search close to the space of orthogonal transformations. We demonstrate that fastDSA is at least an order of magnitude faster than the previous methods. Furthermore, we demonstrate that fastDSA has the properties of its ancestor, including its invariances and sensitivities to system dynamics. FastDSA, therefore, provides a computationally efficient and accurate method for dynamical similarity analysis.

Problem

• Compares dynamical systems' temporal structures efficiently
• Accelerates similarity analysis while preserving accuracy and robustness
• Reduces computational cost via data-driven embedding and optimization

Innovation

• Automatic data-driven embedding selection reduces computational cost
• Lightweight optimization replaces slow exact orthogonality constraints
• Maintains accuracy and robustness while being significantly faster