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

Zenodo (CERN European Organization for Nuclear Research)

Abstract

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.

Authors 8

1. Peter Bouss Aachen

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute for Advanced Simulation (IAS-6), Jülich Research Centre, Jülich, Germany

   RWTH Aachen University, Aachen, Germany

2. Sandra Nestler Aachen

   Forschungszentrum Jülich · Technion – Israel Institute of Technology · RWTH Aachen University

   Affiliation as printed

   Institute for Advanced Simulation (IAS-6), Jülich Research Centre, Jülich, Germany

   RWTH Aachen University, Aachen, Germany

   Rappaport Faculty of Medicine and Network Biology Research Laboratory, Technion - Israel Institute of Technology, Haifa, Israel

3. Kirsten Fischer Aachen

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute for Advanced Simulation (IAS-6), Jülich Research Centre, Jülich, Germany

   RWTH Aachen University, Aachen, Germany

4. Claudia Merger Aachen Department of Physics Faculty 1

   Scuola Internazionale Superiore di Studi Avanzati · Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute for Advanced Simulation (IAS-6), Jülich Research Centre, Jülich, Germany

   Department of Physics, Faculty 1, RWTH Aachen University, Aachen, Germany

   International School of Advanced Studies (SISSA), Trieste, Italy

5. Lars Schutzeichel Aachen

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute for Advanced Simulation (IAS-6), Jülich Research Centre, Jülich, Germany

   RWTH Aachen University, Aachen, Germany

   Institute of Biological Information Processing (IBI-3), Jülich Research Centre, Jülich, Germany

6. Simon Musall Aachen Institute of Biology 2

   University of Bonn · Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute of Biological Information Processing (IBI-3), Jülich Research Centre, Jülich, Germany

   Institute of Biology 2, RWTH Aachen University, Aachen, Germany

   Faculty of Medicine, Institute of Experimental Epileptology and Cognition Research, University of Bonn, Bonn, Germany

7. Alexandre René Aachen Faculty of Computer Science

   University of Ottawa · RWTH Aachen University

   Affiliation as printed

   https://orcid.org/0000-0003-3795-5073

   Department of Physics, University of Ottawa, Ottawa, Canada

   Faculty of Computer Science, RWTH Aachen University, Aachen, Germany

8. Moritz Helias Aachen Department of Physics Faculty 1

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute for Advanced Simulation (IAS-6), Jülich Research Centre, Jülich, Germany

   Department of Physics, Faculty 1, RWTH Aachen University, Aachen, Germany