Skip to main content
SpringerLink
Log in

A Python Hands-on Tutorial on Network and Topological Neuroscience

  • Conference paper
  • First Online:
Geometric Science of Information (GSI 2021)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 12829))

Included in the following conference series:

Abstract

Network neuroscience investigates brain functioning through the prism of connectivity, and graph theory has been the main framework to understand brain networks. Recently, an alternative framework has gained attention: topological data analysis. It provides a set of metrics that go beyond pairwise connections and offer improved robustness against noise. Here, our goal is to provide an easy-to-grasp theoretical and computational tutorial to explore neuroimaging data using these frameworks, facilitating their accessibility, data visualisation, and comprehension for newcomers to the field. We provide a concise (and by no means complete) theoretical overview of the two frameworks and a computational guide on the computation of both well-established and newer metrics using a publicly available resting-state functional magnetic resonance imaging dataset. Moreover, we have developed a pipeline for three-dimensional (3-D) visualisation of high order interactions in brain networks.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Notice that the notion of metric in mathematics defines distance between two points in a set [16], which is distinct from what we are using in this work. We denote as metric any quantity that can be computed, i.e., “measured”, in a brain network or simplicial complex.

References

  1. Barbarossa, S., Sardellitti, S.: Topological signal processing over simplicial complexes. IEEE Trans. Sig. Process. 68, 2992–3007 (2020). https://doi.org/10.1109/TSP.2020.2981920

    Article  Google Scholar 

  2. Bassett, D.S., Bullmore, E.T.: Small-world brain networks revisited. Neuroscientist 23(5), 499–516 (2017). https://doi.org/10.1177/1073858416667720

    Article  Google Scholar 

  3. Bassett, D.S., Sporns, O.: Network neuroscience. Nat. Neurosci. 20(3), 353 (2017). https://doi.org/10.1038/nn.4502

    Article  Google Scholar 

  4. Battiston, F., et al.: Networks beyond pairwise interactions: structure and dynamics. Phys. Rep. 874, 1–92 (2020). https://doi.org/10.1016/j.physrep.2020.05.004

    Article  MathSciNet  Google Scholar 

  5. Baudot, P.: The poincare-shannon machine: statistical physics and machine learning aspects of information cohomology. Entropy 21(9), 881 (2019). https://doi.org/10.3390/e21090881

    Article  MathSciNet  Google Scholar 

  6. Baudot, P., Bennequin, D.: The homological nature of entropy. Entropy 17(5), 3253–3318 (2015). https://doi.org/10.3390/e17053253

    Article  MathSciNet  Google Scholar 

  7. Baudot, P., et al.: Topological information data analysis. Entropy 21(9), 869 (2019). https://doi.org/10.3390/e21090869

    Article  MathSciNet  Google Scholar 

  8. Biswal, B.B., et al.: Toward discovery science of human brain function. Proc. Nat. Acad. Sci. USA 107(10), 4734–4739 (2010). https://doi.org/10.1073/pnas.0911855107

    Article  Google Scholar 

  9. Bullmore, E., Sporns, O.: Complex brain networks: graph theoretical analysis of structural and functional systems. Nat. Rev. Neurosci. 10(3), 186–198 (2009). https://doi.org/10.1038/nrn2575

    Article  Google Scholar 

  10. Carlsson, G.: Topological methods for data modelling. Nat. Rev. Phys. 2(12), 697–708 (2019). https://doi.org/10.1038/s42254-020-00249-3

    Article  Google Scholar 

  11. Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255–308 (2009). https://doi.org/10.1090/S0273-0979-09-01249-X

    Article  MathSciNet  MATH  Google Scholar 

  12. Centeno, E., Santos, F.N., MultinetLAB: Notebook for Network and Topological Analysis in Neuroscience (2021). https://doi.org/10.5281/zenodo.4483651

  13. Centeno, E.G.Z., et al.: A hands-on tutorial on network and topological neuroscience. bioRxiv. p. 2021.02.15.431255. https://doi.org/10.1101/2021.02.15.431255

  14. Curto, C.: What can topology tell us about the neural code? Bull. Am. Math. Soc. 54(1), 63–78 (2017). https://doi.org/10.1090/bull/1554

    Article  MathSciNet  MATH  Google Scholar 

  15. Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLOS Comput. Biol. 4(10), e1000205 (2008). https://doi.org/10.1371/journal.pcbi.1000205

    Article  MathSciNet  Google Scholar 

  16. Do Carmo, M.P.: Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition. Courier Dover Publications, New York (2016)

    Google Scholar 

  17. Edelsbrunner, H., Harer, J.: Computational topology: an introduction, vol. 69. 1st edn. American Mathematical Society, Providence, USA (2010). https://doi.org/10.1090/mbk/069

  18. Expert, P., et al.: Editorial: topological neuroscience. Network Neurosci. 3(3), 653–655 (2019). https://doi.org/10.1162/netne00096

    Article  Google Scholar 

  19. Farahani, F.V., Karwowski, W., Lighthall, N.R.: Application of graph theory for identifying connectivity patterns in human brain networks: a systematic review. Front. Neurosci. 13, 585 (2019). https://doi.org/10.3389/fnins.2019.00585

    Article  Google Scholar 

  20. Farooq, H., et al.: Network curvature as a hallmark of brain structural connectivity. Nat. Commun. 10(1), 4937 (2019). https://doi.org/10.1038/s41467-019-12915-x

    Article  MathSciNet  Google Scholar 

  21. Fornito, A., Zalesky, A., Bullmore, E.: Fundamentals of Brain Network Analysis, 1st edn. Academic Press, San Diego (2016)

    Google Scholar 

  22. Gatica, M., et al.: High-order interdependencies in the aging brain. bioRxiv (2020)

    Google Scholar 

  23. Giusti, C., Pastalkova, E., Curto, C., Itskov, V.: Clique topology reveals intrinsic geometric structure in neural correlations. Proc. Nat. Acad. Sci. USA 112(44), 13455–13460 (2015). https://doi.org/10.1073/pnas.1506407112

    Article  MathSciNet  MATH  Google Scholar 

  24. Gross, J.L., Yellen, J.: Handbook of Graph Theory, 1st edn. CRC Press, Boca Raton (2003)

    Book  Google Scholar 

  25. Hagberg, A., Swart, P., S Chult, D.: Exploring network structure, dynamics, and function using networkx. In: Varoquaux, G.T., Vaught, J.M. (ed.) Proceedings of the 7th Python in Science Conference (SciPy 2008), pp. 11–15 (2008)

    Google Scholar 

  26. Hallquist, M.N., Hillary, F.G.: Graph theory approaches to functional network organization in brain disorders: a critique for a brave new small-world. Network Neurosci. 3(1), 1–26 (2018). https://doi.org/10.1162/netna00054

    Article  Google Scholar 

  27. Jalili, M.: Functional brain networks: does the choice of dependency estimator and binarization method matter? Sci. Rep. 6, 29780 (2016). https://doi.org/10.1038/srep29780

    Article  Google Scholar 

  28. Kartun-Giles, A.P., Bianconi, G.: Beyond the clustering coefficient: a topological analysis of node neighbourhoods in complex networks. Chaos Solitons Fract. X 1, 100004 (2019). https://doi.org/10.1016/j.csfx.2019.100004

    Article  Google Scholar 

  29. Lambiotte, R., Rosvall, M., Scholtes, I.: From networks to optimal higher-order models of complex systems. Nat. Phys. 15(4), 313–320 (2019). https://doi.org/10.1038/s41567-019-0459-y

    Article  Google Scholar 

  30. Liu, T.T.: Noise contributions to the FMRI signal: an overview. Neuroimage 143, 141–151 (2016). https://doi.org/10.1016/j.neuroimage.2016.09.008

    Article  Google Scholar 

  31. Maletić, S., Rajković, M., Vasiljević, D.: Simplicial complexes of networks and their statistical properties. In: Bubak, M., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds.) ICCS 2008. LNCS, vol. 5102, pp. 568–575. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-69387-1_65

    Chapter  Google Scholar 

  32. Maria, C., Boissonnat, J.-D., Glisse, M., Yvinec, M.: The Gudhi library: simplicial complexes and persistent homology. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 167–174. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44199-2_28

    Chapter  Google Scholar 

  33. Najman, L., Romon, P. (eds.): Modern Approaches to Discrete Curvature. LNM, vol. 2184. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-58002-9

    Book  MATH  Google Scholar 

  34. Pardalos, P.M., Xue, J.: The maximum clique problem. J. Glob. Optim. 4(3), 301–328 (1994)

    Article  MathSciNet  Google Scholar 

  35. Petri, G., et al.: Homological scaffolds of brain functional networks. J. R. Soc. Interface 11(101), 20140873 (2014). https://doi.org/10.1098/rsif.2014.0873

    Article  Google Scholar 

  36. Rosas, F.E., et al.: Quantifying high-order interdependencies via multivariate extensions of the mutual information. Phys. Rev. E 100(3), 032305 (2019). https://doi.org/10.1103/PhysRevE.100.032305

    Article  Google Scholar 

  37. Saggar, M., et al.: Towards a new approach to reveal dynamical organization of the brain using topological data analysis. Nat. Commun. 9(1), 1399 (2018). https://doi.org/10.1038/s41467-018-03664-4

    Article  Google Scholar 

  38. Santos, F.A.N., et al.: Topological phase transitions in functional brain networks. Phys. Rev. E 100(3–1), 032414 (2019). https://doi.org/10.1103/PhysRevE.100.032414

    Article  MathSciNet  Google Scholar 

  39. Singh, G., et al.: Topological analysis of population activity in visual cortex. J. Vis. 8(8), 11 (2008). https://doi.org/10.1167/8.8.11

    Article  Google Scholar 

  40. Sizemore, A.E., Giusti, C., Kahn, A., Vettel, J.M., Betzel, R.F., Bassett, D.S.: Cliques and cavities in the human connectome. J. Comput. Neurosci. 44(1), 115–145 (2017). https://doi.org/10.1007/s10827-017-0672-6

    Article  MathSciNet  MATH  Google Scholar 

  41. Sizemore Blevins, A., Bassett, D.S.: Reorderability of node-filtered order complexes. Phys. Rev. E 101(5–1), 052311 (2020). https://doi.org/10.1103/PhysRevE.101.052311

    Article  MathSciNet  Google Scholar 

  42. Sporns, O.: Graph theory methods: applications in brain networks. Dialogues Clin. Neurosci. 20(2), 111–121 (2018)

    Article  Google Scholar 

  43. Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440 (1998). https://doi.org/10.1038/30918

    Article  MATH  Google Scholar 

  44. Weber, M., et al.: Curvature-based methods for brain network analysis. arXiv preprint arXiv:1707.00180 (2017)

  45. Wu, Z., et al.: Emergent complex network geometry. Sci. Rep. 5, 10073 (2015). https://doi.org/10.1038/srep10073

    Article  Google Scholar 

  46. Zalesky, A., Fornito, A., Bullmore, E.: On the use of correlation as a measure of network connectivity. Neuroimage 60(4), 2096–2106 (2012). https://doi.org/10.1016/j.neuroimage.2012.02.001

    Article  Google Scholar 

  47. Zomorodian, A.J.: Topology for Computing, vol. 16, 1st edn. Cambridge University Press, New York (2005)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Amsterdam UMC, Vrije Universiteit Amsterdam, Anatomy and Neurosciences, Amsterdam Neuroscience, De Boelelaan 1117, Amsterdam, Netherlands

    Eduarda Gervini Zampieri Centeno, Giulia Moreni, Chris Vriend, Linda Douw & Fernando Antônio Nóbrega Santos

  2. Bordeaux Neurocampus, Université de Bordeaux, Institut des Maladies Neurodégénératives, CNRS UMR 5293, 146 Rue Léo Saignat, Bordeaux, France

    Eduarda Gervini Zampieri Centeno

  3. Amsterdam UMC, Vrije Universiteit Amsterdam, Psychiatry, Amsterdam Neuroscience, De Boelelaan 1117, Amsterdam, Netherlands

    Chris Vriend

  4. Institute of Advanced Studies, University of Amsterdam, Oude Turfmarkt 147, 1012, Amsterdam, GC, Netherlands

    Fernando Antônio Nóbrega Santos

Authors
  1. Eduarda Gervini Zampieri Centeno
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giulia Moreni
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chris Vriend
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Linda Douw
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Fernando Antônio Nóbrega Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eduarda Gervini Zampieri Centeno .

Editor information

Editors and Affiliations

  1. Sony Computer Science Laboratories Inc., Tokyo, Japan

    Frank Nielsen

  2. THALES Land & Air Systems, Meudon, France

    Frédéric Barbaresco

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Centeno, E.G.Z., Moreni, G., Vriend, C., Douw, L., Santos, F.A.N. (2021). A Python Hands-on Tutorial on Network and Topological Neuroscience. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_71

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-80209-7_71

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-80208-0

  • Online ISBN: 978-3-030-80209-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation