Network geometry of the Drosophila brain

Abstract

The recent reconstruction of the Drosophila brain provides a neural network of unprecedented size and level of details. In this work, we study the geometrical properties of this system by applying network embedding techniques to the graph of synaptic connections. Since previous analysis have revealed an inhomogeneous degree distribution, we first employ a hyperbolic embedding approach that maps the neural network onto a point cloud in the two-dimensional hyperbolic space. In general, hyperbolic embedding methods exploit the exponentially growing volume of hyperbolic space with increasing distance from the origin, allowing for an approximately uniform spatial distribution of nodes even in scale-free, small-world networks. By evaluating multiple embedding quality metrics, we find that the network structure is well captured by the resulting two-dimensional hyperbolic embedding, and in fact is more congruent with this representation than with the original neuron coordinates in three-dimensional Euclidean space. In order to examine the network geometry in a broader context, we also apply the well-known Euclidean network embedding approach Node2vec, where the dimension of the embedding space, $d$ can be set arbitrarily. In 3 dimensions, the Euclidean embedding of the network yields lower quality scores compared to the original neuron coordinates. However, as a function of the embedding dimension the scores show an improving tendency, surpassing the level of the 2d hyperbolic embedding roughly at $d=16$, and reaching a maximum around $d=64$. Since network embeddings can serve as valuable inputs for a variety of downstream machine learning tasks, our results offer new perspectives on the structure and representation of this recently revealed and biologically significant neural network.