Navigable maps of structural brain networks across species

Abstract

Connectomes are spatially embedded networks whose architecture has been shaped by physical constraints and communication needs throughout evolution. Using a decentralized navigation protocol, we investigate the relationship between the structure of the connectomes of different species and their spatial layout. As a navigation strategy, we use greedy routing where nearest neighbors, in terms of geometric distance, are visited. We measure the fraction of successful greedy paths and their length as compared to shortest paths in the topology of connectomes. In Euclidean space, we find a striking difference between the navigability properties of mammalian and non-mammalian species, which implies the inability of Euclidean distances to fully explain the structural organization of their connectomes. In contrast, we find that hyperbolic space, the effective geometry of complex networks, provides almost perfectly navigable maps of connectomes for all species, meaning that hyperbolic distances are exceptionally congruent with the structure of connectomes. Hyperbolic maps therefore offer a quantitative meaningful representation of connectomes that suggests a new cartography of the brain based on the combination of its connectivity with its effective geometry rather than on its anatomy only. Hyperbolic maps also provide a universal framework to study decentralized communication processes in connectomes of different species and at different scales on an equal footing.

Fig 1. Illustration of a successful greedy path on a network embedded in the 2D-plane (indicated by the grid; distances correspond to the length of the straight line between two nodes).

The corresponding topological and geometrical stretches are respectively 1.33 and approximatively 0.91, thus illustrating how geometrical stretch can be lower than 1. Notice that the shortest and the greedy paths would coincide if the role of the source and of the target were exchanged (i.e., seeking a greedy path from the red node to the blue one instead).

Fig 2

(a) Success rate (SR) of the greedy routing protocol obtained for connectomes for which Euclidean distance between each pair of nodes in the anatomical embedding is available, as well as for randomized versions of these connectomes generated using the three null models presented in the Materials and Methods section. The x-axis is broken to highlight the difference between connectomes whose resolution is at the neuron levels (left) from the ones where nodes correspond to coarse-grained regions of the brain (right). Within these two resolution categories, the ordering of the connectomes from left to right roughly follows the increasing physical volume they occupy. (b) Average topological and geometrical stretch of the GRE greedy paths in each connectome with the error bars showing the 10% and 90% percentiles.

Fig 3. Distribution of the locally outgoing and incoming success rates for the (a) Drosophila1 and (b) Human5 datasets in Euclidean and hyperbolic space.

Results for the 14 datasets for which distances in Euclidean space are known are given on S2 Fig.

Fig 4

(a) The connectome Human5 embedded in the hyperbolic disk. Nodes belonging to the two different hemispheres are shown in blue and red. See Materials and methods for details on this representation. (b) Success rate (SR) of the greedy routing protocol for the hyperbolic embeddings of several connectomes (GRH) as well as for randomized versions of these connectomes generated using the three null models presented in the Materials and Methods section. As in Fig 2, the x-axis is broken to highlight the difference between connectomes at the neuron level from the ones where nodes correspond to areas of the brain. Within these two resolution categories, the ordering of the connectomes from left to right roughly follows the increasing physical volume they occupy. (c) Average stretch of the greedy paths in each connectome with the error bars showing the 10% and 90% percentiles. (d)–(e) A sample of representative neuroanatomical regions from the DK atlas are superimposed over the inferred positions of nodes shown on (a). (f) Distribution of the average normalized angular separation between every pair of nodes belonging to the same neuroanatomical regions defined by the DK atlas.

Fig 5. Superposition of various neuroanatomical regions and functional clusters on the inferred positions of the nodes in the hyperbolic disk for the Human8 dataset.

(a–b) The neuroanatomical regions correspond to the lobes as identified by Ref. [52]. (c–d) The functional clusters correspond to the 17-region parcellation proposed in Ref. [59] using the name introduced in Ref. [60]. The clusters shown were chosen so that many could fit on a single plot without clutering it; the complete set of neuroanatomical regions and functional clusters are shown on Fig. S10 Fig.