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Abstract Recent work indicates that low-dimensional dynamics of neural and behavioral data are often preserved across days and subjects. However, extracting these preserved dynamics remains challenging: high-dimensional neural population activity and the recorded neuron populations vary across recording sessions. While existing modeling tools can improve alignment between neural and behavioral data, they often operate on a per-subject basis or discretize behavior into categories, disrupting its natural continuity and failing to capture the underlying dynamics. We introduceContrastiveAlignedNeuralDYnamics (CANDY), an end-to-end framework that aligns neural and behavioral data using rank-based contrastive learning, adapted for continuous behavioral variables, to project neural activity from different sessions onto a shared low-dimensional embedding space. CANDY fits a shared linear dynamical system to the aligned embeddings, enabling an interpretable model of the conserved temporal structure in the latent space. We validate CANDY on synthetic and real-world datasets spanning multiple species, behaviors, and recording modalities. Our results show that CANDY is able to learn aligned latent embeddings and preserved dynamics across neural recording sessions and subjects, and it achieves improved cross-session behavior decoding performance. We further show that the latent linear dynamical system generalizes to new sessions and subjects, achieving comparable or even superior behavior decoding performance to models trained from scratch. These advances enable robust cross-session behavioral decoding and offer a path towards identifying shared neural dynamics that underlie behavior across individuals and recording conditions. The code and two-photon imaging data of striatal neural activity that we acquired here are available athttps://github.com/schnitzer-lab/CANDY-public.git. 

Abstract The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently non-Euclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.