Abstract: It is increasingly recognised that computations in the brain and artificial neural networks can be understood as outputs of a high-dimensional dynamical system conformed by the activity of large neural populations. Yet revealing the underpinning latent dynamical processes from data and interpreting their relevance in computational tasks remains challenging. A prominent line of research has observed that neural activity is often confined to low-dimensional smooth manifolds. However, there is a lack of theoretical frameworks for the unsupervised representation of neural dynamics that are interpretable based on behavioural variables, comparable across systems, and decodable to behaviour with high accuracy. In this talk, I will introduce Manifold Representation Basis Learning (MARBLE), an unsupervised representation-learning framework for non-linear dynamical systems. Our approach combines empirical dynamical modelling and geometric deep learning to decompose state space dynamics into a statistical distribution of local flow fields. I will show that this decomposition allows the development of unsupervised learning algorithms and obtaining latent representations that are preserved under different manifold embeddings. I will show that MARBLE representations give rise to a well-defined similarity metric between dynamical systems, e.g., different instances of recurrent neural networks and animals. Moreover, they are expressive enough to compare computations and detect continuous and discontinuous changes due to external variables. In neuroscience, I will show that these properties allow the discovery of interpretable representations of neural activity in motor, navigation and cognitive tasks and lead to significantly higher decoding performance than state-of-the-art. Our results suggest that using the manifold structure yields a new class of algorithms with higher performance and the ability to assimilate data across experiments