Non-Euclidean brains

Finding suitable embeddings for connectomes (spatially embedded complex networks that map neural connections in the brain) is crucial for analyzing and understanding cognitive processes. Recent studies have found two-dimensional hyperbolic embeddings superior to Euclidean embeddings in modeling connectomes across species, especially human connectomes. However, those studies had limitations: geometries other than Euclidean, hyperbolic, or spherical were not considered. Following William Thurston’s suggestion that the networks of neurons in the brain could be successfully represented in Solv geometry, we study the goodness-of-fit of the embeddings for 21 connectome networks (8 species). To this end, we suggest an embedding algorithm based on Simulating Annealing that allows us to embed connectomes to Euclidean, Spherical, Hyperbolic, Solv, Nil, and product geometries. Our algorithm tends to find better embeddings than the state-of-the-art, even in the hyperbolic case. Our findings suggest that while three-dimensional hyperbolic embeddings yield the best results in many cases, Solv embeddings perform reasonably well. This is a visualization accompanying our ECAI 2024 paper “Modelling brain connectomes networks: Solv is a worthy competitor to hyperbolic geometry!“ (arXiv: ) Geometries are visualized as follows: Euclidean 3D -- obvious hyperbolic 3D -- Poincaré ball (except first-person perspective for H3 manifold) Nil, Solv -- the screen XYZ coordinates correspond to the Lie logarithm of the point (in case of Nil, this is the same model as in “Nil geometry explained“ -- the geodesic ball is longer along the ’Z’ axis, in the visualization we rotate around the Y axis) H2xR -- azimuthal equidistant (the distance and direction from the center are mapped faithfully) Twist (twisted product of H2xR) -- each layer uses azimuthal equidistant projection Spherical 3D -- azimuthal equidistant projection hyperbolic 2D -- Poincaré disk Edges are drawn as geodesics (except Solv). All nodes are drawn as balls of the same size (so their size and distortion can be to understand the scaling of the projection). Our embedder is based on the maximum likelihood method, assuming that the probability that two edges in distance d is connected is (independently) 1/(1 \exp((d-R)/T)). (I.e., the parameters R, T, and positions of nodes are placed in such a way that the probability of obtaining connections and non-connections like in the actual dataset is maximized.) :NLL (Normalized Log-likelihood), MAP, IMR (inverse MeanRank), SC (greedy success rate), and IST (inverse greedy stretch) are various quality measures from the literature, normalized to [0,1]. For every connectome, we show the geometries which are in top 3 according to some measure (according to the Copeland voting rule). Music: Somatic Cosmos by Timo Petmanson (petmanson) the Sphere by Jakub Steiner (jimmac) Lost Mountain by Lincoln Domina (HyperRogue soundtrack) YouTube compression is not great with such a visualization. Try selecting a higher quality in YouTube, or go here: and download.