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Navigation of Brain Networks Navigation of brain networks Caio Seguina,1, Martijn P. van den Heuvelb,c, and Andrew Zaleskya,d aMelbourne Neuropsychiatry Centre, The University of Melbourne and Melbourne Health, Melbourne, VIC 3010, Australia; bDutch Connectome Lab, Department of Complex Trait Genetics, Center for Neurogenomics and Cognitive Research, Amsterdam Neuroscience, VU University Amsterdam, 1081 HV Amsterdam, The Netherlands; cDepartment of Clinical Genetics, Amsterdam Neuroscience, VU University Medical Center, 1081 HV Amsterdam, The Netherlands; and dDepartment of Biomedical Engineering, Melbourne School of Engineering, The University of Melbourne, Melbourne, VIC 3010, Australia Edited by Edward T. Bullmore, University of Cambridge, Cambridge, United Kingdom, and accepted by Editorial Board Member Michael S. Gazzaniga May 7, 2018 (received for review January 24, 2018) Understanding the mechanisms of neural communication in large- Navigation is a network communication strategy that routes scale brain networks remains a major goal in neuroscience. We information based on the distance between network nodes (23). investigated whether navigation is a parsimonious routing model Navigating a network is as simple as progressing to the next node for connectomics. Navigating a network involves progressing to that is closest in distance to a desired target. Navigation is not the next node that is closest in distance to a desired destina- guaranteed to successfully reach a target destination. Moreover, tion. We developed a measure to quantify navigation efficiency targets might be reached using long, inefficient paths. However, and found that connectomes in a range of mammalian species several real-world networks are known to be efficiently naviga- (human, mouse, and macaque) can be successfully navigated with ble, including biological, social, transportation, and technological near-optimal efficiency (>80% of optimal efficiency for typical systems (24–26). Successful navigation depends on certain topo- connection densities). Rewiring network topology or reposition- logical properties such as small worldness (23) and a combination ing network nodes resulted in 45–60% reductions in navigation of high clustering and heterogeneous degree distribution (24), all performance. We found that the human connectome cannot of which are found in the brain networks of several species (9). be progressively randomized or clusterized to result in topolo- Here, we comprehensively investigate the feasibility of nav- gies with substantially improved navigation performance (>5%), igation routing as a model for large-scale neural communica- suggesting a topological balance between regularity and ran- tion. We develop a measure of navigation efficiency and apply domness that is conducive to efficient navigation. Navigation it to publicly available connectomics data acquired from the was also found to (i) promote a resource-efficient distribution of macaque, mouse, and human brain. We find that brain networks NEUROSCIENCE the information traffic load, potentially relieving communication are highly navigable, with connectome topology well poised bottlenecks, and (ii) explain significant variation in functional con- between regularity and randomness to facilitate efficient nav- nectivity. Unlike commonly studied communication strategies in igation. In addition, we characterize the centrality of nodes connectomics, navigation does not mandate assumptions about and connections under navigability and investigate the relation global knowledge of network topology. We conclude that the topology and geometry of brain networks are conducive to efficient decentralized communication. Significance connectome j neural communication j network navigation j We show that the combination of topology and geometry in complex networks mammalian cortical networks allows for near-optimal decen- tralized communication under navigation routing. Following a simple propagation rule based on local knowledge of the ervous systems are networks and one of the key functions distance between cortical regions, we demonstrate that brain Nof a network is to facilitate communication. Complex topo- networks can be successfully navigated with efficiency that logical properties such as small worldness (1, 2), modularity is comparable to shortest paths routing. This finding helps (3), and a core of highly interconnected hubs (4) are univer- to conciliate the major progress achieved over more than sally found across the brain networks of advanced and simple a decade of connectomics research, under the assumption species, including mouse (5, 6), macaque (7, 8), and human of communication via shortest paths, with recent questions connectomes (9). Support for efficient communication between raised by the biologically unrealistic requirements involved neuronal populations is conjectured to be one of the main in the computation of optimal routes. Our results reiterate adaptive advantages behind the emergence of these complex the importance of the brain’s spatial embedding, suggest- organizational properties (10, 11). ing a three-way relationship between connectome geometry, Understanding how neural information is routed and com- topology, and communication. municated through complex networks of white matter pathways remains an open challenge for systems neuroscience (12, 13). Author contributions: C.S. and A.Z. designed the research; C.S. and A.Z. performed To date, connectomics has largely focused on network com- research; C.S. and A.Z. contributed new reagents/analytic tools; C.S. and A.Z. analyzed munication based on optimal routing (14, 15), which proposes data; and C.S., M.P.v.d.H., and A.Z. wrote the paper. that information traverses the shortest path between two nodes. The authors declare no conflict of interest. However, identifying shortest paths requires individual elements This article is a PNAS Direct Submission. E.T.B. is a guest editor invited by the Editorial of nervous systems to possess global knowledge of network Board. topology. This requirement for centralized knowledge has been This open access article is distributed under Creative Commons Attribution- challenged on the basis that nervous systems are decentralized, NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND). motivating alternative models of large-scale neural communi- Data deposition: All data analyzed in this work are publicly available. Human data are from the Human Connectome Project (https://db.humanconnectome.org/); macaque data cation, such as spreading dynamics (11), path ensembles (16), can be found at core-nets.org/; and mouse data were gathered from the following pub- communicability (17, 18), and diffusion models (19–21). These licly available resources: connectivity.brain-map.org; developingmouse.brain-map.org/; studies indicate that brain networks may support efficient com- and mouse.brain-map.org/. munication without the need for centralized knowledge. For 1 To whom correspondence should be addressed. Email: [email protected]. instance, random walkers can be biased to travel via efficient This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. routes (22) and shortest paths help facilitate fast spreading of 1073/pnas.1801351115/-/DCSupplemental. local stimuli (11). www.pnas.org/cgi/doi/10.1073/pnas.1801351115 PNAS Latest Articles j 1 of 6 Downloaded by guest on September 24, 2021 N ×N between navigation path lengths and functional connectivity Let L 2 R denote a matrix of connection lengths for a (FC) inferred from resting-state functional magnetic resonance network comprising N nodes, where Lij measures the length of imaging (MRI). Compared with shortest path routing, we find the connection from node i to j , and let Λ denote the matrix that navigation uses the brain’s resources more uniformly and of navigation path lengths. If node i cannot navigate to node yields stronger correlations with FC. j , Λij = 1. Otherwise, Λij = Liu + ::: + Lvj , where fu, :::, vg is the sequence of nodes visited during navigation. We define nav- Results 2 P igation efficiency as E = 1=(N − N ) i =6 j 1=Λij . Analogous Navigation Performance Measures. We consider neural communi- ∗ 2 P ∗ to global efficiency (29) [E = 1=(N − N ) i 6= j 1=Λij , where cation from the graph-theoretic standpoint of delineating paths ∗ Λij is the shortest path length from node i to j ], both mea- (routes) in the connectome between pairs of nodes (gray matter sures characterize the efficiency of information exchange in a regions). A routing strategy defines a set of rules for identifying parallel system in which all nodes are capable of concurrently a path from a source node to a target node. Path length refers exchanging information. In the same way that global efficiency to the number of connections that compose a path (hops) or the can incorporate network disconnectedness, navigation efficiency sum across the lengths of these connections. To minimize con- incorporates unsuccessful navigation paths (Eij = 0 if i cannot duction latency, noise introduced by synaptic retransmission, and reach j under navigation). Therefore, E quantifies both the num- metabolic costs, neural communication should take place along ber of failed paths and the efficiency of successful paths. We paths with short path lengths (12, 13). defined the efficiency ratio Navigation is a decentralized communication strategy that is particularly suited to spatially embedded networks (24, 25). Nav- ∗ 1 X Λij igating a network involves following a simple rule: Progress to the ER = 2 [1] N − N Λij
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