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Mapping the community structure of the rat cerebral cortex with weighted stochastic block modeling

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Abstract

The anatomical architecture of the mammalian brain can be modeled as the connectivity between functionally distinct areas of cortex and sub-cortex, which we refer to as the connectome. The community structure of the connectome describes how the network can be parsed into meaningful groups of nodes. This process, called community detection, is commonly carried out to find internally densely connected communities—a modular topology. However, other community structure patterns are possible. Here we employ the weighted stochastic block model (WSBM), which can identify a wide range of topologies, to the rat cerebral cortex connectome, to probe the network for evidence of modular, core, periphery, and disassortative organization. Despite its algorithmic flexibility, the WSBM identifies substantial modular and assortative topology throughout the rat cerebral cortex connectome, significantly aligning to the modular approach in some parts of the network. Significant deviations from modular partitions include the identification of communities that are highly enriched in core (rich club) areas. A comparison of the WSBM and modular models demonstrates that the former, when applied as a generative model, more closely captures several nodal network attributes. An analysis of variation across an ensemble of partitions reveals that certain parts of the network participate in multiple topological regimes. Overall, our findings demonstrate the potential benefits of adopting the WSBM, which can be applied to a single weighted and directed matrix such as the rat cerebral cortex connectome, to identify community structure with a broad definition that transcends the common modular approach.

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References

  • Aicher C, Jacobs AZ, Clauset A (2013) Adapting the stochastic block model to edge-weighted networks. arXiv:1305.5782

  • Aicher C, Jacobs AZ, Clauset A (2014) Learning latent block structure in weighted networks. J Complex Netw 3(2):221–248

    Google Scholar 

  • Airoldi EM, Blei DM, Fienberg SE, Xing EP (2008) Mixed membership stochastic blockmodels. J Mach Learn Res 9:1981–2014

    PubMed  PubMed Central  Google Scholar 

  • Akiki TJ, Abdallah CG (2019) Determining the hierarchical architecture of the human brain using subject-level clustering of functional networks. https://doi.org/10.1101/350462

  • Alexander-Bloch A, Lambiotte R, Roberts B, Giedd J, Gogtay N, Bullmore E (2012) The discovery of population differences in network community structure: new methods and applications to brain functional networks in schizophrenia. NeuroImage 59(4):3889–3900

    PubMed  Google Scholar 

  • Bassett DS, Sporns O (2017) Network neuroscience. Nat Neurosci 20(3):353–364

    CAS  PubMed  PubMed Central  Google Scholar 

  • Bassett DS, Greenfield DL, Meyer-Lindenberg A, Weinberger DR, Moore SW, Bullmore ET (2010) Efficient physical embedding of topologically complex information processing networks in brains and computer circuits. PLoS Comput Biol 6(4):e1000748

    PubMed  PubMed Central  Google Scholar 

  • Battiston F, Guillon J, Chavez M, Latora V, De Vico Fallani F (2018) Multiplex core–periphery organization of the human connectome. J R Soc Interface 15(146):20180514

    PubMed  PubMed Central  Google Scholar 

  • Baum GL, Ciric R, Roalf DR, Betzel RF, Moore TM, Shinohara RT et al (2017) Modular segregation of structural brain networks supports the development of executive function in youth. Curr Biol 27(11):1561–1572

    CAS  PubMed  PubMed Central  Google Scholar 

  • Betzel RF, Bassett DS (2017) Multi-scale brain networks. Neuroimage 160:73–83

    PubMed  Google Scholar 

  • Betzel RF, Avena-Koenigsberger A, Goni J, He Y, de Reus MA, Griffa A, Sporns O (2016) Generative models of the human connectome. Neuroimage 124(Pt A):1054–1064

    PubMed  PubMed Central  Google Scholar 

  • Betzel RF, Medaglia JD, Papadopoulos L, Baum GL, Gur R, Gur R, Bassett DS (2017) The modular organization of human anatomical brain networks: accounting for the cost of wiring. Netw Neurosci 1(1):42–68

    PubMed  PubMed Central  Google Scholar 

  • Betzel RF, Medaglia JD, Bassett DS (2018) Diversity of meso-scale architecture in human and non-human connectomes. Nat Commun 9(1):346

    PubMed  PubMed Central  Google Scholar 

  • Betzel RF, Bertolero MA, Gordon EM, Gratton C, Dosenbach NUF, Bassett DS (2019) The community structure of functional brain networks exhibits scale-specific patterns of inter- and intra-subject variability. NeuroImage 202:115990

    PubMed  Google Scholar 

  • Blondel VD, Guillaume J-L, Lambiotte R, Lefebvre E (2008) Fast unfolding of communities in large networks. J Stat Mech Theory Exp 10:P10008

    Google Scholar 

  • Bota M, Swanson LW (2007) Online workbenches for neural network connections. J Comp Neurol 500(5):807–814

    PubMed  Google Scholar 

  • Bota M, Sporns O, Swanson LW (2015) Architecture of the cerebral cortical association connectome underlying cognition. Proc Natl Acad Sci 112(16):E2093–E2101

    CAS  PubMed  Google Scholar 

  • Bullmore E, Sporns O (2012) The economy of brain network organization. Nat Rev Neurosci 13(5):336–349

    CAS  PubMed  Google Scholar 

  • Chen M, Nguyen T, Szymanski BK (2013) On measuring the quality of a network community structure. In: 2013 international conference on social computing. IEEE, pp 122–127

  • Contreras JA, Avena-Koenigsberger A, Risacher SL, West JD, Tallman E, McDonald BC et al (2019) Resting state network modularity along the prodromal late onset Alzheimer’s disease continuum. NeuroImage 22:101687

    PubMed  Google Scholar 

  • Crossley NA, Mechelli A, Vértes PE, Winton-Brown TT, Patel AX, Ginestet CE, Bullmore ET (2013) Cognitive relevance of the community structure of the human brain functional coactivation network. Proc Natl Acad Sci 110(28):11583–11588

    CAS  PubMed  Google Scholar 

  • Faskowitz J, Yan X, Zuo X-N, Sporns O (2018) Weighted stochastic block models of the human connectome across the life span. Sci Rep 8(1):12997

    PubMed  PubMed Central  Google Scholar 

  • Fortunato S, Barthelemy M (2007) Resolution limit in community detection. Proc Natl Acad Sci 104(1):36–41

    CAS  PubMed  Google Scholar 

  • Fortunato S, Hric D (2016) Community detection in networks: a user guide. Phys Rep-Rev Sect Phys Lett 659:1–44

    Google Scholar 

  • Good BH, de Montjoye Y-A, Clauset A (2010) Performance of modularity maximization in practical contexts. Phys Rev E 81(4):046106

    Google Scholar 

  • Harriger L, van den Heuvel MP, Sporns O (2012) Rich club organization of macaque cerebral cortex and its role in network communication. PLoS One 7(9):e46497

    CAS  PubMed  PubMed Central  Google Scholar 

  • He Y, Lim S, Fortunato S, Sporns O, Zhang L, Qiu J, Zuo XN (2018) Reconfiguration of cortical networks in MDD uncovered by multiscale community detection with fMRI. Cereb Cortex 28(4):1383–1395

    PubMed  Google Scholar 

  • Holland PW, Laskey KB, Leinhardt S (1983) Stochastic blockmodels: first steps. Soc Netw 5(2):109–137

    Google Scholar 

  • Hric D, Peixoto TP, Fortunato S (2016) Network structure, metadata, and the prediction of missing nodes and annotations. Phys Rev X 6(3):031038

    Google Scholar 

  • Jeub LG, Sporns O, Fortunato S (2018) Multiresolution consensus clustering in networks. Sci Rep 8(1):3259

    PubMed  PubMed Central  Google Scholar 

  • Karrer B, Newman ME (2011) Stochastic blockmodels and community structure in networks. Phys Rev E 83(1 Pt 2):016107

    Google Scholar 

  • Kurmukov A, Ananyeva M, Dodonova Y, Gutman B, Faskowitz J, Jahanshad N, Zhukov L (2017) Classifying phenotypes based on the community structure of human brain networks. In: Graphs in biomedical image analysis, computational anatomy and imaging genetics. Springer, Cham, pp 3–11

    Google Scholar 

  • Kurmukov A, Musabaeva A, Denisova Y, Moyer D, Gutman B (2018) Connectivity-driven brain parcellation via consensus clustering. In: Connectomics in neuroimaging. Springer, Cham, pp 117–126

    Google Scholar 

  • Kwak H, Choi Y, Eom Y-H, Jeong H, Moon S (2009) Mining communities in networks: a solution for consistency and its evaluation. In: Proceedings of the 9th ACM SIGCOMM conference on Internet measurement conference. ACM, pp 301–314

  • Lancichinetti A, Fortunato S (2012) Consensus clustering in complex networks. Sci Rep 2:336

    PubMed  PubMed Central  Google Scholar 

  • Meilă M (2007) Comparing clusterings—an information based distance. J Multivar Anal 98(5):873–895

    Google Scholar 

  • Meunier D, Achard S, Morcom A, Bullmore E (2009) Age-related changes in modular organization of human brain functional networks. Neuroimage 44(3):715–723

    PubMed  Google Scholar 

  • Moyer D, Gutman B, Prasad G, Faskowitz J, Ver Steeg G, Thompson P (2015a) Blockmodels for connectome analysis. 11th International Symposium on Medical Information Processing and Analysis (SIPAIM 2015): 96810A-96810A-96819

  • Moyer D, Gutman B, Prasad G, Ver Steeg G, Thompson P (2015b) Mixed membership stochastic blockmodels for the human connectome. In: MICCAI–workshop on bayesian and graphical models for biomedical imaging

  • Munkres J (1957) Algorithms for the assignment and transportation problems. J Soc Ind Appl Math 5(1):32–38

    Google Scholar 

  • Najafi M, McMenamin BW, Simon JZ, Pessoa L (2016) Overlapping communities reveal rich structure in large-scale brain networks during rest and task conditions. NeuroImage 135:92–106

    PubMed  PubMed Central  Google Scholar 

  • Newman ME (2006) Modularity and community structure in networks. Proc Natl Acad Sci 103(23):8577–8582

    CAS  PubMed  Google Scholar 

  • Noori HR, Schöttler J, Ercsey-Ravasz M, Cosa-Linan A, Varga M, Toroczkai Z, Spanagel R (2017) A multiscale cerebral neurochemical connectome of the rat brain. PLoS Biol 15(7):e2002612

    PubMed  PubMed Central  Google Scholar 

  • Park H-J, Friston K (2013) Structural and functional brain networks: from connections to cognition. Science 342(6158):1238411

    PubMed  Google Scholar 

  • Pavlovic DM, Vertes PE, Bullmore ET, Schafer WR, Nichols TE (2014) Stochastic blockmodeling of the modules and core of the Caenorhabditis elegans connectome. PLoS One 9(7):e97584

    PubMed  PubMed Central  Google Scholar 

  • Peel L, Larremore DB, Clauset A (2017) The ground truth about metadata and community detection in networks. Sci Adv 3(5):e1602548

    PubMed  PubMed Central  Google Scholar 

  • Peixoto TP (2018) Nonparametric weighted stochastic block models. Phys Rev E 97(1–1):012306

    CAS  PubMed  Google Scholar 

  • Rubinov M, Sporns O (2010) Complex network measures of brain connectivity: uses and interpretations. Neuroimage 52(3):1059–1069

    Google Scholar 

  • Rubinov M, Ypma RJ, Watson C, Bullmore ET (2015) Wiring cost and topological participation of the mouse brain connectome. Proc Natl Acad Sci 112(32):10032–10037

    CAS  PubMed  Google Scholar 

  • Schaub MT, Delvenne J-C, Rosvall M, Lambiotte R (2017) The many facets of community detection in complex networks. Appl Netw Sci 2(1):4

    PubMed  PubMed Central  Google Scholar 

  • Schmitt O, Eipert P (2012) neuroVIISAS: approaching multiscale simulation of the rat connectome. Neuroinformatics 10(3):243–267

    PubMed  Google Scholar 

  • Shih C-T, Sporns O, Yuan S-L, Su T-S, Lin Y-J, Chuang C-C, Chiang A-S (2015) Connectomics-based analysis of information flow in the drosophila brain. Curr Biol 25(10):1249–1258

    CAS  PubMed  Google Scholar 

  • Shinn M, Romero-Garcia R, Seidlitz J, Váša F, Vértes PE, Bullmore E (2017) Versatility of nodal affiliation to communities. Sci Rep 7(1):4273

    PubMed  PubMed Central  Google Scholar 

  • Sohn Y, Choi MK, Ahn YY, Lee J, Jeong J (2011) Topological cluster analysis reveals the systemic organization of the Caenorhabditis elegans connectome. PLoS Comput Biol 7(5):e1001139

    CAS  PubMed  PubMed Central  Google Scholar 

  • Sporns O (2011) The human connectome: a complex network. Ann NY Acad Sci 1224(1):109–125

    PubMed  Google Scholar 

  • Sporns O (2013) Making sense of brain network data. Nat Methods 10(6):491–493

    CAS  PubMed  Google Scholar 

  • Sporns O, Betzel RF (2016) Modular Brain Networks. Annu Rev Psychol 67:613–640

    PubMed  Google Scholar 

  • Sporns O, Tononi G, Kötter R (2005) The human connectome: a structural description of the human brain. PLoS Comput Biol 1(4):e42

    PubMed  PubMed Central  Google Scholar 

  • Swanson LW, Sporns O, Hahn JD (2016) Network architecture of the cerebral nuclei (basal ganglia) association and commissural connectome. Proc Natl Acad Sci 113(40):E5972–E5981

    CAS  PubMed  Google Scholar 

  • Swanson LW, Hahn JD, Sporns O (2017) Organizing principles for the cerebral cortex network of commissural and association connections. Proc Natl Acad Sci 114(45):E9692–E9701

    CAS  PubMed  Google Scholar 

  • Swanson LW, Hahn JD, Jeub LGS, Fortunato S, Sporns O (2018) Subsystem organization of axonal connections within and between the right and left cerebral cortex and cerebral nuclei (endbrain). Proc Natl Acad Sci 115(29):E6910–E6919

    PubMed  Google Scholar 

  • Swanson LW, Sporns O, Hahn JD (2019) The network organization of rat intrathalamic macroconnections and a comparison with other forebrain divisions. Proc Natl Acad Sci ​ 116(27):13661–13669

    CAS  PubMed  Google Scholar 

  • Tunç B, Verma R (2015) Unifying inference of meso-scale structures in networks. PLoS One 10(11):e0143133

    PubMed  PubMed Central  Google Scholar 

  • van den Heuvel MP, Scholtens LH, de Reus MA (2016) Topological organization of connectivity strength in the rat connectome. Brain Struct Funct 221(3):1719–1736

    PubMed  Google Scholar 

  • Xiaoran Y, Cosma S, Jacob EJ, Florent K, Cristopher M, Lenka Z, Yaojia Z (2014) Model selection for degree-corrected block models. J Stat Mech 2014(5):P05007

    Google Scholar 

  • Yeh F-C, Panesar S, Fernandes D, Meola A, Yoshino M, Fernandez-Miranda JC, Verstynen T (2018) Population-averaged atlas of the macroscale human structural connectome and its network topology. NeuroImage 178:57–68

    PubMed  PubMed Central  Google Scholar 

  • Zhao T, Cao M, Niu H, Zuo X-N, Evans A, He Y, Shu N (2015) Age-related changes in the topological organization of the white matter structural connectome across the human lifespan. Hum Brain Mapp 36(10):3777–3792

    PubMed  Google Scholar 

  • Zingg B, Hintiryan H, Gou L, Song MY, Bay M, Bienkowski MS, Toga AW (2014) Neural networks of the mouse neocortex. Cell 156(5):1096–1111

    CAS  PubMed  PubMed Central  Google Scholar 

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Acknowledgements

O.S. acknowledges funding support by the National Institutes of Health (R01 AT009036-01 and R01 MH122957-01). This material is based on the work supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1342962 (J.F.). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Supercomputing was supported in part by Lilly Endowment, Inc., through its support for the Indiana University Pervasive Technology Institute, and in part by the Indiana METACyt Initiative. The Indiana METACyt Initiative at IU was also supported in part by Lilly Endowment, Inc. We would like to thank Christopher Aicher and Aaron Clauset for implementing and hosting the WSBM code package.

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  1. Program in Neuroscience, Indiana University, Bloomington, IN, USA

    Joshua Faskowitz & Olaf Sporns

  2. Department of Psychological and Brain Sciences, Indiana University, 1101 E. 10th Street, Bloomington, IN, 47405, USA

    Joshua Faskowitz & Olaf Sporns

  3. Indiana University Network Science Institute, Indiana University, Bloomington, IN, USA

    Olaf Sporns

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Correspondence to Joshua Faskowitz.

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Faskowitz, J., Sporns, O. Mapping the community structure of the rat cerebral cortex with weighted stochastic block modeling. Brain Struct Funct 225, 71–84 (2020). https://doi.org/10.1007/s00429-019-01984-9

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