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

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.

• The paper presents a novel simulated annealing algorithm that outperforms hyperbolic embeddings on metrics like mAP, MeanRank, routing success, and stretch.
• It systematically evaluates 21 connectome networks across multiple Thurston geometries, highlighting Solv's robust and competitive performance.
• The research offers actionable insights for improved visualization and analysis of complex brain networks using alternative geometric models.

Modelling Brain Connectome Networks: Solv as a Competitor to Hyperbolic Geometry

The paper presents an exploration into non-Euclidean geometric embeddings for brain connectomes, extending the analysis beyond the traditionally studied Euclidean, hyperbolic, and spherical geometries. It evaluates several Thurston geometries, including Solv and product geometries, based on their goodness-of-fit for embedding connectomes.

Key Contributions

1. Novel Embedding Algorithm: The paper introduces an embedding algorithm based on Simulated Annealing (SA). This algorithm has been demonstrated to outperform existing methods for hyperbolic embeddings according to standard measures such as mean average precision (mAP), mean rank (MeanRank), greedy routing success rate (SC), and stretch (IST). Specifically, the algorithm's flexible parameter optimization significantly enhances the embedding quality.
2. Comprehensive Geometric Evaluation: The study systematically evaluates embedding performance across 21 connectome networks from 8 species using a range of Thurston geometries, including Euclidean, spherical, hyperbolic, Solv, Nil, and product geometries.
3. Theoretical and Practical Insights: The findings indicate that three-dimensional hyperbolic geometries often yield the best embeddings. However, Solv geometries also demonstrate substantial effectiveness, suggesting that it is a viable competitor to hyperbolic embeddings.

Methodological Insights

Algorithm Details

The proposed algorithm maps connectome nodes to various points in a given geometric space, embedded within a grid, and then optimizes the embeddings using a likelihood maximization approach. Specifically:

• Grid Construction and Distance Computation: The grid sizes and distances between points are computed based on natural grids in the selected geometry. For instance, in Solv, the algorithm approximates longer distances using Dijkstra's algorithm due to the complexity of distance computations.
• Simulated Annealing Procedure: The algorithm applies an SA approach to refine the embeddings iteratively, starting with high-temperature values to explore the solution space extensively before gradually cooling down to refine the embeddings near local optima. The values for hyperparameters $R$ and $T$ influencing the probability of edge formation are periodically recalibrated for improved fitting.

Geometric Context and Visualization

The embeddings were visualized in two and three dimensions, providing a clear depiction of how different geometries manage the complex hierarchical and spatial constraints of connectome networks. For instance, star-like structures in rat nervous systems were shown to fit well within hyperbolic geometries.

Experimental Findings

Performance Comparison

The performance of the proposed SA-based embeddings was benchmarked using several criteria:

• Mean Average Precision (mAP) and Normalized Log-Likelihood (NLL): The paper shows that proposed embeddings outperform state-of-the-art methods on these measures for most connectomes.
• Greedy Routing Efficiency: Hyperbolic and especially three-dimensional hyperbolic embeddings score highly in routing measures, but Solv and product geometries also perform admirably well.

Robustness and Variability

The study conducts extensive robustness checks, demonstrating that the SA implementation and grid density significantly influence the results. The precision of embedding performance is notably higher with finer grids and longer annealing periods.

Implications and Future Directions

The practical implications of this research are profound, particularly for the analysis and visualization of brain networks:

• Visualization: The high-quality embeddings can support better visual analytics of connectomes, aiding neuroscientific research and potentially informing clinical diagnostics.
• Network Science Applications: The findings suggest that alternative non-Euclidean geometries, especially Solv, warrant attention beyond traditional hyperbolic and Euclidean models for various network-related applications.

Conclusion

Overall, the study advances knowledge in the field of connectome analysis by demonstrating the viability of various Thurston geometries, especially Solv, as formidable alternatives to hyperbolic embeddings. Future research could explore the anatomical and functional implications of these findings in more detail and extend the embedding strategies to other types of complex networks.