Comparing Dynamical Models Through Diffeomorphic Vector Field Alignment

Area of Science:

• Theoretical Neuroscience
• Computational Neuroscience
• Dynamical Systems Theory

Background:

• Recurrent neural networks (RNNs) are valuable tools in theoretical neuroscience for hypothesis generation and data analysis.
• Evaluating the dynamics of these models is crucial for understanding their learned mechanisms.
• Challenges include comparing dynamics across models with different coordinate systems and identifying low-dimensional motifs in high-dimensional systems.

Purpose of the Study:

• To introduce a comprehensive framework, DFORM (diffeomorphic vector field alignment for learned models), to address challenges in evaluating dynamical systems models.
• To enable comparison of learned dynamics across models by aligning their state spaces.
• To facilitate the identification of mechanistically important low-dimensional dynamical motifs.

Main Methods:

• DFORM learns nonlinear coordinate transformations to align state spaces of dynamical systems.
• It achieves maximally one-to-one trajectory alignment between systems.
• The framework is applied to assess topological equivalence and locate dynamical motifs.

Main Results:

• DFORM successfully identified linear and nonlinear coordinate transformations in various systems, including RNNs.
• It quantified similarity between topologically distinct systems.
• DFORM located important dynamical motifs like invariant manifolds and saddle limit sets in high-dimensional models.
• Limit cycles were identified in RNNs trained on fMRI data, aligning with prior analyses.

Conclusions:

• DFORM provides a robust method for comparing dynamical systems and understanding their underlying mechanisms.
• The framework facilitates the discovery of key dynamical structures within complex, high-dimensional models.
• DFORM has practical applications in analyzing data-driven models, such as those derived from neuroimaging data.