Network geometry of the Drosophila brain

TL;DR

This work employs a hyperbolic embedding approach that maps the neural network onto a point cloud in the two-dimensional hyperbolic space and applies the well-known Euclidean network embedding approach Node2vec, where the dimension of the embedding space, $d$ can be set arbitrarily.

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.

Figures (6)

• Figure 1: Embeddings of the Drosophila brain network. In panels a) and d) (left column) we show a 2d Cartesian projection of the original neuron coordinates in the 3d Euclidean space. Panels b) and e) (middle column) display the hyperbolic embedding of the network in the 2d native disk representation according to CLOVE, where node size is an indicator of the degree. In panels c) and f) (right column) we display the 64d Euclidean embedding with Node2vec, projected to 2d using UMAP. In panels a), b), c) (top row) nodes highlighted in red correspond to neurons belonging to the "central" super class, whereas in panels d), e), f) (bottom row) the red node colour indicates that the given neuron belongs to the "optic" super class.
• Figure 2: Embedding quality scores compared across embedding dimensions. We show the scores obtained for Node2vec as a function of the embedding dimension with line plots, compared with the result for the 2d hyperbolic embedding (solid horizontal line) and for the original 3d neuron coordinates (dashed horizontal line). a) The Mapping Accuracy. b) The scores related to greedy navigation: the Greedy Routing Succes Rate (light blue) the Greedy Routing Score (burgundy) and the Greedy Routing Efficiency (light green). c) The Area Under the Receiver Operating Characterstic Curve for edge prediction, EPAUC (light blue) and the Edge Prediction Precision (burgundy). d) The Edge Prediction Recall results for EPR20 (dark green) and EPR5 (light green).
• Figure S1: Embeddings of the Drosophila brain network highlighting the "ascending" super class. In panel a) we show a 2d Cartesian projection of the original neuron coordinates in the 3d Euclidean space. Panel b) displays the Hyperbolic embedding of the network in the 2d native disk representation according to CLOVE. In panel c) we display the 64d Euclidean embedding with node2vec, projected to 2d using UMAP.
• Figure S2: Embeddings of the Drosophila brain network highlighting the "sensory" super class. In panel a) we show a 2d Cartesian projection of the original neuron coordinates in the 3d Euclidean space. Panel b) displays the Hyperbolic embedding of the network in the 2d native disk representation according to CLOVE. In panel c) we display the 64d Euclidean embedding with node2vec, projected to 2d using UMAP.
• Figure S3: Embeddings of the Drosophila brain network highlighting "visual projection" super class. In panel a) we show a 2d Cartesian projection of the original neuron coordinates in the 3d Euclidean space. Panel b) displays the Hyperbolic embedding of the network in the 2d native disk representation according to CLOVE. In panel c) we display the 64d Euclidean embedding with node2vec, projected to 2d using UMAP.