A Probabilistic Framework to Infer Brain Functional Connectivity from Anatomical Connections

  • Fani Deligianni
  • Gael Varoquaux
  • Bertrand Thirion
  • Emma Robinson
  • David J. Sharp
  • A. David Edwards
  • Daniel Rueckert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6801)


We present a novel probabilistic framework to learn across several subjects a mapping from brain anatomical connectivity to functional connectivity, i.e. the covariance structure of brain activity. This prediction problem must be formulated as a structured-output learning task, as the predicted parameters are strongly correlated. We introduce a model selection framework based on cross-validation with a parametrization-independent loss function suitable to the manifold of covariance matrices. Our model is based on constraining the conditional independence structure of functional activity by the anatomical connectivity. Subsequently, we learn a linear predictor of a stationary multivariate autoregressive model. This natural parameterization of functional connectivity also enforces the positive-definiteness of the predicted covariance and thus matches the structure of the output space. Our results show that functional connectivity can be explained by anatomical connectivity on a rigorous statistical basis, and that a proper model of functional connectivity is essential to assess this link.


Functional Connectivity Default Mode Network Structural Connectivity Probabilistic Framework Sparsity Pattern 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aljabar, P., Heckemann, R., Hammers, A., Hajnal, J., Rueckert, D.: Multi-atlas based segmentation of brain images: atlas selection and its effect on accuracy. Neuroimage 46(3), 726–738 (2009)CrossRefGoogle Scholar
  2. 2.
    Behrens, T., Woolrich, M., Jenkinson, M., Johansen-Berg, H., Nunes, R., Clare, S., Matthews, P., Brady, J., Smith, S.: Characterization and propagation of uncertainty in diffusion-weighted mr imaging. Magnet Reson Med. 50(5), 1077–1088 (2003)CrossRefGoogle Scholar
  3. 3.
    Burns, J.: An evolutionary theory of schizophrenia: Cortical connectivity, metarepresentation, and the social brain. Behavioral and Brain Sciences 27(6), 831 (2004)Google Scholar
  4. 4.
    Damoiseaux, J., Greicius, M.: Greater than the sum of its parts: a review of studies combining structural connectivity and resting-state functional connectivity. Brain Struct. Funct. 213(6), 525–533 (2009)CrossRefGoogle Scholar
  5. 5.
    Deligianni, F., Robinson, E., Beckmann, C., Sharp, D., Edwards, A., Rueckert, D.: Inference of functional connectivity from structural brain connectivity. In: ISBI, pp. 1113–1116 (2010)Google Scholar
  6. 6.
    Donoho, D.: For most large underdetermined systems of linear equations the minimal l1-norm solution is also the sparsest solution. Comm. Pure Appl. Math. 59(6), 797–829 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Efron, B., Hastie, T., Johnstone, I., Tibshirani, R.: Least angle regression. Ann. Stat. 32(2), 407–499 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ferrarelli, F., Massimini, M., Sarasso, S., Casali, A., Riedner, B., Angelini, G., Tononi, G., Pearce, R.: Breakdown in cortical effective connectivity during midazolam-induced loss of consciousness. P. Natl. Acad. Sci. Usa 107(6), 2681–2686 (2010)CrossRefGoogle Scholar
  9. 9.
    Förstner, W., Moonen, B.: A metric for covariance matrices. Qua. Vadis Geodesia, 113–128 (1999)Google Scholar
  10. 10.
    Fransson, P., Marrelec, G.: The precuneus/posterior cingulate cortex plays a pivotal role in the default mode network: Evidence from a partial correlation network analysis. Neuroimage 42, 1178–1184 (2008)CrossRefGoogle Scholar
  11. 11.
    Friston, K.: Statistical parametric mapping: the analysis of functional brain images. Academic Press, London (2007)CrossRefGoogle Scholar
  12. 12.
    Greicius, M., Supekar, K., Menon, V., Dougherty, R.: Resting-state functional connectivity reflects structural connectivity in the default mode network. Cereb. Cortex (2008)Google Scholar
  13. 13.
    van den Heuvel, M.P., Mandl, R.C.W., Kahn, R.S., Pol, H.E.H.: Functionally linked resting-state networks reflect the underlying structural connectivity architecture of the human brain. Human Brain Mapping 30(10), 3127–3141 (2009)CrossRefGoogle Scholar
  14. 14.
    Honey, C., Sporns, O., Cammoun, L., Gigandet, X., Thiran, J., Meuli, R., Hagmann, P.: Predicting human resting-state functional connectivity from structural connectivity. P. Natl. Acad. Sci. Usa 106(6), 2035–2040 (2009)CrossRefGoogle Scholar
  15. 15.
    Honey, C., Kötter, R., Breakspear, M., Sporns, O.: Network structure of cerebral cortex shapes functional connectivity on multiple time scales. P. Natl. Acad. Sci. Usa 104(24), 10240 (2007)CrossRefGoogle Scholar
  16. 16.
    Lauritzen, S.: Graphical models. Oxford University Press, USA (1996)zbMATHGoogle Scholar
  17. 17.
    LeCun, Y., Chopra, S., Hadsell, R., Ranzato, M., Huang, F.L.: Energy-based models. In: BakIr, G., Hofmann, T., Schölkopf, B. (eds.) Predicting Structured Data, pp. 191–245. MIT Press, Cambridge (2007)Google Scholar
  18. 18.
    Ledoit, O., Wolf, M.: A well-conditioned estimator for large-dimensional covariance matrices. J. Multivar. Anal. 88, 365–411 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lenglet, C., Rousson, M., Deriche, R., Faugeras, O.: Statistics on the manifold of multivariate normal distributions: Theory and application to diffusion tensor MRI processing. J. Math. Imaging Vis. 25, 423–444 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Morcom, A., Fletcher, P.: Does the brain have a baseline? why we should be resisting a rest. Neuroimage 37(4), 1073–1082 (2007)CrossRefGoogle Scholar
  21. 21.
    Müller, R.: The study of autism as a distributed disorder. Ment. Retard. Dev. Disabil. Res. 13(1), 85–95 (2007)CrossRefGoogle Scholar
  22. 22.
    Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vision 66, 41–66 (2006)CrossRefzbMATHGoogle Scholar
  23. 23.
    Pollonini, L., Pophale, S., Situ, N., Wu, M.H., Frye, R., Leon-Carrion, J., Zouridakis, G.: Information communication networks in severe traumatic brain injury. Brain Topogr. 23(2), 221–226 (2010)CrossRefGoogle Scholar
  24. 24.
    Robinson, E., Hammers, A., Ericsson, A., Edwards, A., Rueckert, D.: Identifying population differences in whole-brain structural networks: a machine learning approach. Neuroimage 50(3), 910–919 (2010)CrossRefGoogle Scholar
  25. 25.
    Rueckert, D., Sonoda, L., Hayes, C., Hill, D.: Non-rigid registration using free-form deformations: application to breast mr images. IEEE Trans. Med. Imag. 18, 712–721 (1999)CrossRefGoogle Scholar
  26. 26.
    Smith, S., Jenkinson, M., Woolrich, M., Beckmann, C., Behrens, T., Johansen-Berg, H., Bannister, P., Luca, M.D., Drobnjak, I., Flitney, D., Niazy, R., Saunders, J., Vickers, J., Zhang, Y., Stefano, N.D., Brady, J., Matthews, P.: Advances in functional and structural mr image analysis and implementation as fsl. Neuroimage 23, 208–219 (2004)CrossRefGoogle Scholar
  27. 27.
    Smith, S., Miller, K., Salimi-Khorshidi, G., Webster, M., Beckmann, C., Nichols, T., Ramsey, J., Woolrich, M.: Network modelling methods for fMRI. Neuroimage (2010) (in press)Google Scholar
  28. 28.
    Tibshirani, R.: Regression shrinkage and selection via the lasso. J. Roy. Stat. Soc. B 58(1), 267–288 (1996)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Varoquaux, G., Gramfort, A., Poline, J.B., Thirion, B.: Brain covariance selection: better individual functional connectivity models using population prior. In: NIPS (2010)Google Scholar
  30. 30.
    Venkataraman, A., Rathi, Y., Kubicki, M., Westin, C.F., Golland, P.: Joint generative model for fmri/dwi and its application to population studies. Med. Image Comput. Comput. Assist Interv. 13(pt. 1), 191–199 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Fani Deligianni
    • 1
  • Gael Varoquaux
    • 2
    • 3
  • Bertrand Thirion
    • 3
  • Emma Robinson
    • 1
  • David J. Sharp
    • 4
  • A. David Edwards
    • 5
  • Daniel Rueckert
    • 1
  1. 1.Department of ComputingImperial College LondonUK
  2. 2.INSERM U922, Neurospin, CEASaclayFrance
  3. 3.Parietal project teamINRIASaclayFrance
  4. 4.C3NL, The Division of Experimental MedicineImperial CollegeLondonUK
  5. 5.Institute of Clinical SciencesImperial College LondonUK

Personalised recommendations