Modelling brain connectomes networks: Solv is a worthy competitor to hyperbolic geometry!

Modelling brain connectomes networks: Solv is a worthy competitor to hyperbolic geometry! Dorota Celi nska-Kopczy nska, Eryk Kopczy nski Institute of Informatics, University of Warsaw, Warsaw, Poland July 24, 2024 Abstract 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 con-nectome networks (8 species). To this end, we suggest an embedding algorithm based on Simulating Annealing that allows us to embed con-nectomes 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. 1 Introduction Connectomes are comprehensive maps of the neural connections in the brain. Understanding the interactions they shape is a key to understanding cognitive processes. Given their spatially embedded complexity, shaped by physical 1 arXiv:2407.16077v1 Therefore, a vast amount of recent research has been devoted to finding the appropriate embeddings for con-nectome networks. Recent studies (e.g., [WHKL22, AS20]) have advocated for the superiority of two-dimensional hyperbolic embeddings over Euclidean embeddings in modeling connectomes across species, especially human con-nectomes. However, those studies had limitations: they restricted the focus to Euclidean, hyperbolic, or spherical geometries, neglecting to explore other potential embedding spaces.