Comparing Dynamical Models Through Diffeomorphic Vector Field Alignment

Dynamical systems models such as recurrent neural networks (RNNs) are increasingly popular in theoretical neuroscience as a vehicle 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: (1) comparison of learned dynamics across models is difficult because a priori there is no enforced equivalence of their coordinate systems, and (2) identification of mechanistically important low-dimensional motifs (e.g., limit sets) is analytically 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 a set of models exhibits topological equivalence, that is, their dynamics are mechanistically similar despite differences in their coordinate systems. A by-product of this methodology 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 magnetic resonance imaging recordings, we illustrated that DFORM can identify limit cycles from high-dimensional data-driven models, which agreed well with prior numerical analysis.