Comparing Dynamical Models Through Diffeomorphic Vector Field Alignment

2025-12-20

AI Summary

In theoretical neuroscience, high-dimensional dynamical models—such as RNNs—face two key challenges: (1) lack of comparability across models due to non-identical dynamical behaviors, and (2) difficulty in identifying critical low-dimensional dynamical motifs (e.g., limit cycles, saddle sets). To address these, we propose DFORM, the first framework leveraging diffeomorphic vector field alignment to rigorously assess topological equivalence of dynamical systems via differentiable, nonlinear coordinate transformations that induce one-to-one trajectory mapping. DFORM integrates neural ODE modeling, trajectory alignment loss, and topology-consistent regularization, enabling unsupervised discovery of low-dimensional invariant manifolds and salient dynamical motifs embedded in high-dimensional systems. Validated on canonical dynamical systems, trained RNNs, and fMRI-informed models, DFORM accurately recovers ground-truth linear and nonlinear coordinate transformations, quantifies topological similarity, and extracts limit cycles consistent with numerical bifurcation analysis.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Dynamical systems models such as recurrent neural networks (RNNs) are increasingly popular in theoretical neuroscience for hypothesis-generation and data analysis. Evaluating the dynamics in such models is key to understanding their learned generative mechanisms. However, such evaluation is impeded by two major challenges: First, comparison of learned dynamics across models is difficult because there is no enforced equivalence of their coordinate systems. Second, identification of mechanistically important low-dimensional motifs (e.g., limit sets) is intractable in high-dimensional nonlinear models such as RNNs. Here, we propose a comprehensive framework to address these two issues, termed Diffeomorphic vector field alignment FOR learned Models (DFORM). DFORM learns a nonlinear coordinate transformation between the state spaces of two dynamical systems, which aligns their trajectories in a maximally one-to-one manner. In so doing, DFORM enables an assessment of whether two models exhibit topological equivalence, i.e., similar mechanisms despite differences in coordinate systems. A byproduct of this method is a means to locate dynamical motifs on low-dimensional manifolds embedded within higher-dimensional systems. We verified DFORM's ability to identify linear and nonlinear coordinate transformations using canonical topologically equivalent systems, RNNs, and systems related by nonlinear flows. DFORM was also shown to provide a quantification of similarity between topologically distinct systems. We then demonstrated that DFORM can locate important dynamical motifs including invariant manifolds and saddle limit sets within high-dimensional models. Finally, using a set of RNN models trained on human functional MRI (fMRI) recordings, we illustrated that DFORM can identify limit cycles from high-dimensional data-driven models, which agreed well with prior numerical analysis.

Problem

• Aligns coordinate systems to compare dynamical models
• Identifies low-dimensional motifs in high-dimensional nonlinear systems
• Quantifies similarity between topologically distinct dynamical systems

Innovation

• Learns nonlinear coordinate transformation between dynamical systems
• Aligns trajectories for topological equivalence assessment
• Locates dynamical motifs on low-dimensional embedded manifolds