New CSA method fixes flaw in comparing neural dynamical systems

A new paper from Godara, Tay, and Mattar tackles a fundamental problem in neuroscience and machine learning: determining whether two dynamical systems implement the same computation, despite differences in coordinates or measurements. The current state-of-the-art, Dynamical Similarity Analysis (DSA), aligns finite-dimensional Koopman approximations using orthogonal similarity transformations. However, the authors prove that orthogonal alignment is neither necessary nor sufficient for topological conjugacy—conjugate systems may require a non-orthogonal basis-transfer matrix that DSA cannot capture, while non-conjugate systems may have orthogonally equivalent Koopman operators that DSA fails to distinguish.

To address this, the team introduces Conjugacy-based Similarity Analysis (CSA). CSA restricts alignments to those induced by candidate state-space bijections rather than arbitrary orthogonal matrices. They prove that CSA's fitted alignment is the finite-data projection of the composition operator associated with the candidate bijection. Using controlled examples, they demonstrate why this distinction matters when observable dictionaries are chosen explicitly or implicitly from data. This work clarifies what Koopman-based similarity measures must ensure to support claims of identifying conjugacies between computational systems, with direct implications for comparing neural population dynamics across brains, tasks, and species.