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Harmonic Holes as the Submodules of Brain Network and Network Dissimilarity

  • Hyekyoung LeeEmail author
  • Moo K. Chung
  • Hongyoon Choi
  • Hyejin Kang
  • Seunggyun Ha
  • Yu Kyeong Kim
  • Dong Soo Lee
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11382)

Abstract

Persistent homology has been applied to brain network analysis for finding the shape of brain networks across multiple thresholds. In the persistent homology, the shape of networks is often quantified by the sequence of k-dimensional holes and Betti numbers. The Betti numbers are more widely used than holes themselves in topological brain network analysis. However, the holes show the local connectivity of networks, and they can be very informative features in analysis. In this study, we propose a new method of measuring network differences based on the dissimilarity measure of harmonic holes (HHs). The HHs, which represent the substructure of brain networks, are extracted by the Hodge Laplacian of brain networks. We also find the most contributed HHs to the network difference based on the HH dissimilarity. We applied our proposed method to clustering the networks of 4 groups, normal controls (NC), stable and progressive mild cognitive impairment (sMCI and pMCI), and Alzheimer’s disease (AD). The results showed that the clustering performance of the proposed method was better than that of network distances based on only the global change of topology.

Keywords

Topological data analysis Brain network Alzheimer’s disease Harmonic holes Hodge Laplacian 

Notes

Acknowledgements

Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at http://adni.loni.usc.edu. This work is supported by Basic Science Research Program through the National Research Foundation (NRF) (No. 2013R1A1A2064593 and No. 2016R1D1A1B03935463), NRF Grant funded by MSIP of Korea (No. 2015M3C7A1028926 and No. 2017M3C7A1048079), NRF grant funded by the Korean Government (No. 2016R1D1A1A02937497, No. 2017R1A5A1015626, and No. 2011-0030815), and NIH grant EB022856.

References

  1. 1.
    Batagelj, V., Mrvar, A.: Pajek - analysis and visualization of large networks. In: Jünger, M., Mutzel, P. (eds.) Graph Drawing Software. Mathematics and Visualization, pp. 77–103. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-642-18638-7_4CrossRefGoogle Scholar
  2. 2.
    Carlsson, G., Collins, A., Guibas, L.J.: Persistence barcodes for shapes. Int. J. Shape Model. 11, 149–187 (2005)CrossRefGoogle Scholar
  3. 3.
    Carlsson, G., Ishkhanov, T., de Silva, V., Zomorodian, A.: On the local behavior of spaces of natural images. Int. J. Comput. Vis. 76(1), 1–12 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Choi, H., Jin, K.H.: Predicting cognitive decline with deep learning of brain metabolism and amyloid imaging. Behav. Brain Res. 344, 103–109 (2018).  https://doi.org/10.1016/j.bbr.2018.02.017. https://www.sciencedirect.com/science/article/pii/S0166432818301013CrossRefGoogle Scholar
  5. 5.
    Chung, M.K., Bubenik, P., Kim, P.T.: Persistence diagrams of cortical surface data. In: Prince, J.L., Pham, D.L., Myers, K.J. (eds.) IPMI 2009. LNCS, vol. 5636, pp. 386–397. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-02498-6_32CrossRefGoogle Scholar
  6. 6.
    Chung, M.K., Villalta-Gil, V., Lee, H., Rathouz, P.J., Lahey, B.B., Zald, D.H.: Exact topological inference for paired brain networks via persistent homology. In: Niethammer, M., et al. (eds.) IPMI 2017. LNCS, vol. 10265, pp. 299–310. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-59050-9_24CrossRefGoogle Scholar
  7. 7.
    Chung, M.K., et al.: Topological brain network distances. arXiv:1809.03878 [stat.AP] (2018). https://arxiv.org/abs/1809.03878
  8. 8.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37, 103–120 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Coifman, R.R., Lafon, S., Lee, A.B., Maggioni, M., Warner, F., Zucker, S.: Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. In: Proceedings of the National Academy of Sciences, pp. 7426–7431 (2005)Google Scholar
  10. 10.
    Edelsbrunner, H., Harer, J.: Persistent homology - a survey. Contemp. Math. 453, 257–282 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society Press, New York (2009)CrossRefGoogle Scholar
  12. 12.
    Friedman, J.: Computing Betti numbers via combinatorial Laplacians. In: Proceedings of the 28th Annual ACM Symposium on the Theory of Computing, pp. 386–391 (1996)Google Scholar
  13. 13.
    Horak, D., Jost, J.: Spectra of combinatorial Laplace operators on simplicial complexes. Adv. Math. 244, 303–336 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kim, Y.-J., Kook, W.: Harmonic cycles for graphs. Linear Multilinear Algebra, 1–11 (2018).  https://doi.org/10.1080/03081087.2018.1440519
  15. 15.
    Lee, H., Chung, M.K., Kang, H., Choi, H., Kim, Y.K., Lee, D.S.: Abnormal hole detection in brain connectivity by kernel density of persistence diagram and Hodge Laplacian. In: 2018 IEEE 15th International Symposium on Biomedical Imaging (ISBI 2018), pp. 20–23, April 2018.  https://doi.org/10.1109/ISBI.2018.8363514
  16. 16.
    Lee, H., Chung, M.K., Kang, H., Kim, B.N., Lee, D.S.: Persistent brain network homology from the perspective of dendrogram. IEEE Trans. Med. Imaging 31, 2267–2277 (2012)CrossRefGoogle Scholar
  17. 17.
    Lee, H., Chung, M.K., Kang, H., Lee, D.S.: Hole detection in metabolic connectivity of alzheimer’s disease using k–laplacian. In: Golland, P., Hata, N., Barillot, C., Hornegger, J., Howe, R. (eds.) MICCAI 2014, LNCS, vol. 8675, pp. 297–304. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-10443-0_38Google Scholar
  18. 18.
    Lim, L.H.: Hodge Laplacians on graphs. Geometry and topology in statistical inference. In: Proceedings of Symposia in Applied Mathematics, vol. 73 (2015)Google Scholar
  19. 19.
    Petri, G., et al.: Homological scaffolds of brain functional networks. J. Roy. Soc. Interface 11(101), 20140873 (2014).  https://doi.org/10.1098/rsif.2014.0873CrossRefGoogle Scholar
  20. 20.
    Reininghaus, J., Huber, S., Bauer, U., Kwitt, R.: A stable multi-scale kernel for topological machine learning. In: The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 4741–4748, June 2015Google Scholar
  21. 21.
    Rolls, E.T., Joliot, M., Tzourio-Mazoyer, N.: Implementation of a new parcellation of the orbitofrontal cortex in the automated anatomical labeling atlas. Neuroimage 122, 1–5 (2015)CrossRefGoogle Scholar
  22. 22.
    Sanabria-Diaz, G., Martìnez-Montes, E., Melie-Garcia, L., Alzheimer’s Disease Neuroimaging Initiative: Glucose metabolism during resting state reveals abnormal brain networks organization in the Alzheimer’s disease and mild cognitive impairment. PLOS ONE 8(7), 1–25 (2013).  https://doi.org/10.1371/journal.pone.0068860CrossRefGoogle Scholar
  23. 23.
    Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D.L.: Topological analysis of population activity in visual cortex. J. Vis. 8, 1–18 (2008)CrossRefGoogle Scholar
  24. 24.
    Sizemore, A., Giusti, C., Kahn, A., Vettel, J., Betzel, R., Bassett, D.: Cliques and cavities in the human connectome. J. Comput. Neurosci. 44, 115–145 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Solo, V., et al.: Connectivity in fMRI: blind spots and breakthroughs. IEEE Trans. Med. Imaging 37(7), 1537–1550 (2018).  https://doi.org/10.1109/TMI.2018.2831261CrossRefGoogle Scholar
  26. 26.
    Sporns, O., Tononi, G., Edelman, G.: Theoretical neuroanatomy: relating anatomical and functional connectivity in graphs and cortical connection matrices. Cereb. Cortex 10(2), 127–141 (2000).  https://doi.org/10.1093/cercor/10.2.127CrossRefGoogle Scholar
  27. 27.
    Sporns, O., Betzel, R.F.: Modular brain networks. Ann. Rev. Psychol. 67, 19.1–19.28 (2016)CrossRefGoogle Scholar
  28. 28.
    Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33, 249–274 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Hyekyoung Lee
    • 1
    Email author
  • Moo K. Chung
    • 4
  • Hongyoon Choi
    • 1
  • Hyejin Kang
    • 2
  • Seunggyun Ha
    • 1
  • Yu Kyeong Kim
    • 3
  • Dong Soo Lee
    • 1
    • 2
  1. 1.Seoul National University HospitalSeoulRepublic of Korea
  2. 2.Seoul National UniversitySeoulRepublic of Korea
  3. 3.SMG-SNU Boramae Medical CenterSeoulRepublic of Korea
  4. 4.University of Wisconsin-MadisonMadisonUSA

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