Code Repository for the Paper "Characterizing Neural Manifolds' Properties and Curvatures using Normalizing Flows" (PRX Life, 2026)

Code Repository for the Paper "Characterizing Neural Manifolds’ Properties and Curvatures using Normalizing Flows" (PRX Life, 2026)AbstractNeuronal activity is found to lie on low-dimensional manifolds embedded within the high-dimensional neuronspace. Variants of principal component analysis are frequently employed to assess these manifolds. Thesemethods are, however, limited by assuming a Gaussian data distribution and a flat manifold. In this study, weintroduce a method designed to satisfy three core objectives: (1) extract coordinated activity across neurons,described either statistically as correlations or geometrically as manifolds; (2) identify a small number of latentvariables capturing these structures; and (3) offer an analytical and interpretable framework characterizingstatistical properties by a characteristic function and describing manifold geometry through a collection of charts.To this end, we employ normalizing flows (NFs), which learn an underlying probability distribution of data byan invertible mapping between data and latent space. Their simplicity and ability to compute exact likelihoodsdistinguish them from other generative networks. We adjust the NF’s training objective to distinguish betweenrelevant (in manifold) and noise dimensions (out of manifold). Additionally, we find that different behavioralstates align with the components of the latent Gaussian mixture model, enabling their treatment as distinct curvedmanifolds. Subsequently, we approximate the network for each mixture component with a quadratic mapping,allowing us to characterize both neural manifold curvature and non-Gaussian correlations among recordingchannels. Applying the method to recordings in macaque visual cortex, we demonstrate that state-dependentmanifolds are curved and exhibit complex statistical dependencies. Our approach thus enables an expressivedescription of neural population activity, uncovering nonlinear interactions among groups of neurons.