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(16), brain models that ensembles indicate diffusion path studies and (11), 18), dynamics communi- (17, spreading neural communicability as large-scale such decentralized, of are cation, network models systems nervous of alternative that motivating been knowledge basis has the global knowledge on centralized challenged possess for requirement to This topology. systems nervous elements individual of nodes. requires two proposes paths com- between shortest which path identifying network 15), However, shortest the (14, on traverses routing information focused 13). optimal that largely on (12, based has neuroscience munication systems connectomics for date, challenge To pathways open matter an white of remains networks complex main through complex municated the these of of one emergence 11). be (10, the properties organizational human to behind and conjectured advantages 8), adaptive is between (7, populations communication simple macaque efficient neuronal univer- and for 6), Support are advanced (5, (9). of (4) connectomes mouse networks hubs modularity including brain interconnected 2), the species, highly across (1, of found worldness core sally small a and as (3), such properties logical N networks complex to conducive connectome are networks brain the that of communication. conclude decentralized geometry efficient We topology. and network topology about of assumptions knowledge mandate global not in does strategies navigation communication connectomics, studied commonly Unlike nectivity. communication (ii relieving and potentially Navigation bottlenecks, load, navigation. traffic information efficient (i the ran- to to and found conducive also regularity was is between that balance domness topolo- topological in cannot a (>5%), result performance suggesting connectome navigation to improved human substantially clusterized with or gies the randomized that navigation progressively in be found reductions 45–60% We in performance. resulted reposition- nodes typical or network topology for ing network efficiency Rewiring optimal densities). species of connection mammalian (>80% with of navigated efficiency destina- successfully range near-optimal be desired a can macaque) a in and efficiency mouse, to navigation connectomes (human, quantify that distance to found in measure and closest a developed is We that to tion. progressing node involves next network the We a neuroscience. 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(CC Editorial 4.0 access the License by NonCommercial-NoDerivatives invited open editor guest This a is E.T.B. Submission. Direct PNAS Board. a is article This interest. of performed conflict analyzed no A.Z. A.Z. declare authors and paper. and The the C.S. wrote C.S. tools; A.Z. research; and reagents/analytic M.P.v.d.H., C.S., new the and contributed data; designed A.Z. and A.Z. C.S. and research; C.S. contributions: Author gto.I diin ecaatrz h etaiyo nodes of relation centrality the investigate the and navigability characterize nav- under poised we efficient connections and well facilitate addition, to In topology randomness igation. connectome and the with regularity from networks between navigable, brain acquired that highly find data We are brain. apply connectomics human and and available efficiency mouse, communica- macaque, navigation publicly neural of to large-scale measure it a for develop model We a tion. as (9). routing species several igation of networks brain the in found all are (24), which distribution of degree heterogeneous and combination a clustering and high (23) of worldness small topo- as certain such on properties logical depends navigation Successful naviga- (24–26). efficiently technological systems and be transportation, to social, biological, known including However, ble, are paths. networks inefficient not real-world long, is using several Navigation reached Moreover, be target. destination. might desired target targets a a reach to successfully distance to node next guaranteed in the closest to progressing is as (23). simple that as nodes is network network between a distance Navigating the on based information owo orsodnesol eadesd mi:[email protected]. Email: addressed. be should correspondence whom To h motneo h ri’ pta medn,suggest- embedding, geometry, communication. connectome and spatial topology, reiterate between relationship brain’s results three-way the a Our ing routes. of optimal importance involved of the requirements computation unrealistic questions the biologically recent assumption in the with the than by paths, under more shortest raised research, via over communication connectomics achieved helps of of progress finding decade major This a routing. the paths conciliate that shortest efficiency to to with comparable navigated successfully is the be of brain can that knowledge demonstrate networks local we regions, on cortical based between distance rule propagation Following simple routing. a navigation decen- under near-optimal in communication geometry for tralized and allows topology networks of cortical combination mammalian the that show We Significance ee ecmrhnieyivsiaetefaiiiyo nav- of feasibility the investigate comprehensively we Here, routes that strategy communication network a is Navigation a,d mouse.brain-map.org/ n os aawr ahrdfo h olwn pub- following the from gathered were data mouse and core-nets.org/; connectivity.brain-map.org; . b uc oncoeLab, Connectome Dutch www.pnas.org/lookup/suppl/doi:10. raieCmosAttribution- Commons Creative developingmouse.brain-map.org/ NSLts Articles Latest PNAS | f6 of 1 ;

NEUROSCIENCE N ×N between navigation path lengths and functional connectivity Let L ∈ 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 = ∞. Otherwise, Λij = Liu + ... + Lvj , where {u, ..., v} 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 next directly connected node that is closest in distance to the i 6= j target node and stop if the target is reached (Fig. 1). To implement navigation, we defined the distance between pairs to compare navigation with shortest path routing. For any net- ∗ of nodes as the Euclidean distance between node centroids work, E ≥ E and thus 0 ≤ ER ≤ 1. The closer ER is to 1, the (27, 28). Importantly, navigation can fail to identify a path. better navigation is at finding paths that are as efficient as This occurs when a navigation path becomes trapped between shortest paths (Fig. 1). nodes without neighbors closer to the destination than them- bin wei We focus on binary (ER ) and weighted (ER ) naviga- selves (Fig. 1B). The success ratio (SR) measures the proportion tion efficiency ratios, quantifying how efficient navigation paths of node pairs in a network that can be successfully reached are compared with shortest paths computed on binarized and via navigation. weighted connectomes, respectively. In addition, we compute dis ER to determine how close navigation paths are to routes that minimize the sum of physical (Euclidean) connection distances traversed between nodes. A A B A Navigability of the Human Connectome. High-resolution diffusion BCBCMRI data from 75 healthy participants of the Human Connec- D D tome Project (HCP) (30) were used to map structural brain net- works at several spatial resolutions (N = 256, 360, 512, 1,024). EF G EF G Whole-brain tractography was performed for each individual and the number of streamlines interconnecting each pair of nodes H H was enumerated to provide a measure of structural connectivity (SI Appendix, Connectivity Data). A group-level connectome was computed as the average of all individual connectivity matrices Binary SP (22). Connection weights were remapped into binary, weighted, Navigation and distance-based connection lengths to allow for the com- putation of communication path lengths (SI Appendix, Network C A D Analysis). Consistent with previous reports (24, 25), we found that nav- BC D igation can successfully identify paths for the majority of nodes pairs composing the human connectome (SR= 89%, 94%, and 96% for 10%, 15%, and 20% connection density, respectively, EF G N = 360; Fig. 2A). Remarkably, navigation was only marginally wei bin less efficient than shortest paths (e.g., ER = 72%, ER = 83%, H dis and ER = 83%, for N = 360 at 15% connection density; Fig. B D -50 0 50 2 – ), with navigation performance improving as connection density increased. Note that navigation does not use connec- wei Fig. 1. Illustrative examples of navigation (green) and shortest (red) paths tion weights, and thus ER quantifies the extent to which from a source to the target node (circled in orange) in a binary network. navigation can blindly identify weighted shortest paths. Naviga- Grid indicates spatial embedding of the networks. Efficiency ratios (ER(i, j)) tion remained efficient and successfully identified paths for the are the ratio of the number of hops in the navigation path to the number majority of node pairs across various parcellation resolutions (SI of hops in the shortest path. (A) The shortest path between A and H has Appendix, Fig. S1), with moderate decreases in success and effi- three hops (A-B-E-H) while navigation leads to a four-hop path (A-D-F-G-H). ciency ratios as the number of nodes increased (SR= 95%, 91%, Navigation routes information from A to H at 75% of optimal efficiency. (B) bin and 90% and ER = 84%, 80%, and 81% for N = 256, 512, and Navigation fails to find a path from B to F, becoming trapped between E 1,024, respectively, at 15% connection density; Fig. 2E). When and H. (C) Both strategies lead to three-hop paths, and navigation routes stratified by hop count, navigation performance remained high information from G to B at 100% of optimal efficiency. (D) Example of a wei successful navigation path in the human connectome that achieves 75% for long, multihop paths (61% median ER and 79% median dis efficiency. ER benchmarked against five-hop shortest paths; Fig. 2F and

2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1801351115 Seguin et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 i.S n Table and S7 Fig. Appendix, (SI S1 age with the deteriorated sig- connectome performance nificantly to navigation group-averaged Furthermore, S6). comparable Fig. the Appendix, (SI was for connectomes determined performance individual Appendix, of (SI formance normalizations count Analyses Supplementary streamline and measures performance. navigation empirical of extent accounted large embeddings these a spatial for of realistic topo- to restricting either randomization while connectome’s of logical navigation, the Randomization impeded both markedly geometry. attributes by its facilitated suggest and jointly results topology is less these the navigation together, than Taken that performance models. navigation null greater constrained showed S3), Fig. and the than (P navigable connectome less human significantly although in that, resulted networks randomization null topological cost-preserving 2 contrast, Fig. density; connection 15% akdyrdcdi oprsnwt h miia networks empirical E was in the decrease networks 60% with null and comparison topological (46% and in spatial reduced Nav- the markedly Analysis ). both Network structurally Appendix, of between (SI igability 28) distances cost, (10, Euclidean nodes connectome network of connected con- sum total (iii) the (ii) preserving or as shuffled; while defined (31); was randomized nodes randomized was in topology of was networks position topology null spatial nectome random the (i) of was which ensembles performance connectome’s against Navigation the embedding. benchmarked spatial by its facilitated determine or is to topology navigation sought next efficient we whether navigated, efficiently be and can cessfully that pairs node hops. few or a pairs only node in not navigated connected is efficiency directly navigation to that due suggesting indicating S2), colors Fig. with Appendix, Elsevier. count, from SI hop permission with given 9, a ref. for from paths reproduced shortest diagrams of Brain number paths. navigation the (red) show failed plots and Bar (green) count. successful hop of matching proportion the with paths shortest against (E benchmarked intervals. (S confidence networks 95% (F structural indicates networks. mammalian human Shading of networks. resolutions null parcellation (blue) different E repositioned across spatially shown and metrics (green) formance rewired 1,000 with compared were (E ratio ciency i.S8 Fig. 2. Fig. eune al. et Seguin R dis R wei hs nig eerbs ovrain nnd distance node in variations to robust were findings These aigetbihdta h ua oncoecnb suc- be can connectome human the that established Having Weighted Efficiency Ratio Success Ratio IAppendix , (SI females and males between differ not did but ), 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 nprl) rage eoennua pce hl ice eoehmndt.Dse ie eoetesm oncindniy(5)ars all across (15%) density connection same the denote lines Dashed data. human denote circles while species nonhuman denote Triangles purple). in Connection Density Connection Density 1 1 E o oooia n pta admzto,rsetvl,at respectively, randomization, spatial and topological for , C AB R wei . . 0.5 0.3 0.1 0.5 0.3 0.1 ). ucs ai (S ratio Success (A–D) connectomes. mammalian of Navigability epciey t1%cneto density; connection 15% at respectively, , Empirical R dis o ua oncoe (N connectomes human for ) aiaiiysrtfidb o on ( count hop by stratified Navigability ) Rewired <

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NEUROSCIENCE regular networks with a high clustering coefficient to disordered A Navigation (NC) B Weightet shortest paths (BC) and costly random networks (SI Appendix, Network Analysis). Clusterization was found to progressively decrease navigation efficiency, whereas slight randomization of connectome topology yielded networks with marginal increases in navigation perfor- wei mance (Fig. 3A). Specifically, peak ER was, on average, 0.8%, 1.4%, and 4.7% more efficient than the connectome after 4.0%, 4 3.3%, and 4.8% of connections were randomly swapped, for 3 N = 256, 360, 512, respectively. Further randomization beyond 2 C D these peaks resulted in deterioration of navigation efficiency. 151 1 1 bin 4 4 Similar results were found for ER (SI Appendix, Fig. S10). Node Centrality (log) 0 0.8 BC Next, we investigated the effects of connectome rewiring 3 3 NC 0.6 Degree aimed to explicitly increase navigability. To this end, we progres- 76 sively performed connection swaps between randomly chosen 2 2 0.4 Degree node pairs that (i) preserved connection density and degree 1 1 0.2 distribution, (ii) led to an increase of a measure of naviga- NC BC 0 4

tion performance, and (iii) preserved total network cost. We 1 180 360 1 180 360 % of communication paths 0 50 100 6 performed a total of 2 × 10 connection swap attempts, with Centrality distributions Top % highest degree nodes rejection of swaps that did not simultaneously meet all three conditions. Direct optimization of network navigability led to Fig. 4. Comparison between navigation (NC) and weighted betweenness wei (BC) node centralities for N = 360 at 15% connection density. Centrality val- an 18–20% increase in ER , with 16–18% of the improve- 6 ues are logarithmically scaled. (A and B) NC (A) and BC (B) projected onto ment taking place in the first 10 swap attempts. Therefore, the cortical surface. (C) NC (Left) and BC (Right) sorted from highest to increasing connectome navigability rapidly became more diffi- lowest values. (D) Relationship between the cumulative sum of centrality cult as a function of swap attempts, suggesting that the observed measures and degree. The horizontal axis is a percentage ranking of nodes improvements converge to an asymptote. Comparable improve- from highest to lowest degree (e.g., for N = 360, the 10% most connected ments between 5% and 22% were found for other measures nodes are the 36 nodes with highest degree). Solid curves (left-hand verti- of navigation performance (SI Appendix, Fig. S11). Collec- cal axis) represent the cumulative sum of BC (red) and NC (green) over all tively, these results indicate that the human connectome is well nodes ordered from most to least connected, divided by the total number of poised between randomness and regularity to facilitate efficient communication paths in the network, indicating the fraction of communi- cation paths mediated by nodes. Blue dots (right-hand vertical axis) show navigation, while explicit optimization by connectome rewiring the original degree associated with each percentage of most-connected converges to 5–22% improvements in navigability. nodes.

Navigation Centrality. The number of shortest paths that traverse a node defines its betweenness centrality (BC), a measure that lated (Pearson correlation coefficient r = 0.54). Several regions finds utility in identifying connectome hub nodes (12) and nodes were central to both shortest paths (BC) and navigation (NC), mediating the bulk of neural communication (7). We defined a including portions of the left and right superior frontal gyrus, new path-based centrality measure called navigation centrality insula, central gyri, and precuneus (Fig. 4 A and B). However, (NC), which quantifies the number of successful navigation paths NC was more uniformly distributed across nodes compared with that traverse each node (SI Appendix, Network Analysis). BC, suggesting that navigation uses network resources more We computed NC and BC for the human connectome (group homogeneously (Fig. 4C). High values of BC were found only average, N = 360, at 15% connection density), with BC based in a small group of high-degree nodes (r = 0.86 between log- on weighted shortest paths. We found that both NC and BC arithm of BC and degree), which mediated most of the net- spanned four orders of magnitude and were positively corre- work’s communication routes. For instance, 99.3% of all shortest paths traveled exclusively through the top 50% most connected nodes (Fig. 4D). In contrast, although high-degree nodes showed high NC (r = 0.61 between logarithm of NC and degree), A B medium- and low-degree regions were responsible for mediat- 1.1 1.25 256 360 512 Connectome ing a share of navigation paths, with the 50% least-connected wei R 1.05 wei R 1.2 nodes responsible for 26% of navigation paths. Greater diver- 1 sity in paths may lead to fewer communication bottlenecks 1.1 and less signal congestion (32), as well as stronger resilience 0.95 against failure of network elements (16, 33). Similar results Normalized E Normalized E 0.9 1 were obtained for BC computed in binarized connectomes (SI 4000 2000 0 2000 4000 6000 0 5 1 1.5 2 Appendix, Fig. S12) and for edge-centric definitions of BCs and 6 Connection swaps Connection swap attempts 10 NCs (SI Appendix, Fig. S13). All correlation coefficients (r) were (Clusterize Connectome Randomize) (Connectome Improve navigability) significant (P < 10−8). wei Fig. 3. Navigation performance (ER ) of progressively rewired connectome wei topologies (at 15% connection density). The ER of rewired topologies Navigation and Functional Connectivity. Finally, we tested whether was normalized by the empirical value found for the human connectome. navigation path lengths can explain variation in functional con- Curves indicate the mean values (inner line) and 95% confidence intervals nectivity across nodes pairs of the human connectome. The (outer shadow) obtained from several runs of the rewiring routines. (A) strength of functional connectivity between node pairs that are Normalized Ewei of clusterized and randomized networks for 100 runs of R not directly connected can be attributed to signal propagation the randomization–clusterizing procedure and different parcellation reso- along multisynaptic (multihop) paths (22, 34). Therefore, if mul- lutions. Dashed lines show performance peaks (vertical axis) and number of connection swaps (horizontal axis), with red indicating the values obtained tihop neural communication is indeed facilitated by navigation, wei we hypothesized that navigation path lengths should be inversely for the empirical brain. (B) Normalized ER obtained from direct optimiza- wei correlated with functional connectivity strength. Resting-state tion of the connectome’s empirical ER , as a function of connection swap attempts, for 50 (N = 256), 50 (N = 360), and 30 (N = 512) independent functional MRI data from the same 75 participants of the HCP rewiring runs. were used to map functional brain networks. A group-averaged

4 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1801351115 Seguin et al. 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linked been has navigation Successful IApni,SplmnayAnalyses Supplementary Appendix, SI NSLts Articles Latest PNAS odt,the date, To o further for Several | f6 of 5

NEUROSCIENCE hyperbolic space) may provide further insight into the nav- the definition of L. The matrix of navigation path lengths Λ was computed igability of nervous systems across species (40). In parallel, by navigating every node pair. Note that Λ is asymmetric, requiring N2 − N brain stimulation techniques could be used to evaluate evi- navigation path computations. Using different connection length measures (Lbin, Lwei, Ldis), we computed binary (Λbin), weighted (Λwei), and distance- dence for competing communication strategies by means of dis bin wei based (Λ ) navigation path lengths. Navigation efficiency ratios (ER , ER , electrophysiological tracking of local perturbations (41, 42). dis ER ) were computed by comparing navigation path lengths to shortest path bin∗ wei∗ dis∗ Materials and Methods lengths (Λ , Λ , Λ ) using Eq. 1. Details on the acquisition and preprocessing of network datasets are Data Sharing. Human, macaque, and mouse datasets are publicly available. described in SI Appendix, Connectivity Data. SI Appendix, Network Analysis Our implementation of navigation routing is available at https://github.com/ provides details on network modeling. Supporting and replication analyses caioseguin/connectomics/. are presented in SI Appendix, Supplementary Analyses. ACKNOWLEDGMENTS. We thank Mikail Rubinov for providing connectivity Navigation Implementation. For a network with N nodes, navigation routing data for the mouse. Human data were provided by the Human Connec- from node i to j was implemented as follows. Determine which of i’s neigh- tome Project, WU–Minn Consortium (1U54MH091657; Principal Investiga- bors is closest (shortest Euclidean distance) to j and progress to it. Repeat tors David Van Essen and Kamil Ugurbil) funded by the 16 National Institutes this process for each new node until j is reached—constituting a success- of Health (NIH) institutes and centers that support the NIH Blueprint for ful navigation path—or a node is revisited—constituting a failed naviga- Neuroscience Research, and by the McDonnell Center for Systems Neuro- science at Washington University. C.S. is funded by a Melbourne Research tion path. Scholarship. M.P.v.d.H. was funded by an ALW (Earth and Life Sciences) open Navigation paths are identified based on network topology and the (ALWOP.179) and VIDI (452-16-015) grant from the Netherlands Organiza- spatial positioning of nodes and thus independent from how connection tion for Scientific Research (NWO) and a Fellowship of MQ. A.Z. is supported lengths are defined. Navigation path lengths, however, are the sum of con- by the Australian National Health and Medical Research Council (NHMRC) nection lengths composed in navigation paths and will vary depending on Senior Research Fellowship B (1136649).

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