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

TL;DR

This work expands the landscape of connectome embeddings beyond Euclidean and 2D hyperbolic spaces by evaluating 21 connectomes across 8 species within 15 Thurston geometries, including the anisotropic Solv geometry. It introduces a Simulated Annealing–based embedding algorithm that places nodes on a grid in each geometry, with edge formation modeled by a probabilistic rule $p(i,j)=rac{1}{1+ ext{exp}((d-R)/T)}$ and log-likelihood maximization to guide placement. Across extensive experiments, the authors show that 3D hyperbolic embeddings often perform best, but Solv and Twist geometries remain competitive and even superior in certain measures or networks, challenging the notion of a universal winner. They provide comprehensive robustness analyses (grid size, iterations, distance definitions) and demonstrate that geometry choice may depend on connectome function, with implications for visualization and analysis of brain networks. Overall, the study broadens geometric modeling of brain connectomes and introduces a versatile SA-based framework compatible with a wide range of Thurston geometries.

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 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.

Figures (16)

• Figure 1: Tessellations of the hyperbolic plane. From left to right: (a) bitruncated order-3 heptagonal tiling ($\{7,3\}$), (b) infinite-order triangular tiling ($\{3,\infty\}$), (c) binary tiling.
• Figure 2: A 2D projection of our embeddings. See brainvideo for 3D visualizations.
• Figure 3: Our best embeddings -- log-likelihood. Top = best embedding obtained, bottom = worst embedding obtained, * = fine grid.
• Figure 4: Our best embeddings -- mAP. Top = best embedding obtained, bottom = worst embedding obtained, * = fine grid.
• Figure 5: Our best embeddings -- MeanRank. Top = best embedding obtained, bottom = worst embedding obtained, * = fine grid.