Skip to main content
SpringerLink
Log in
Book cover

START UP RESEARCH

START UP RESEARCH 2017: Studies in Neural Data Science pp 111–130Cite as

Hierarchical Spatio-Temporal Modeling of Resting State fMRI Data

  • Conference paper
  • First Online:

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 257))

Abstract

In recent years, state of the art brain imaging techniques like Functional Magnetic Resonance Imaging (fMRI), have raised new challenges to the statistical community, which is asked to provide new frameworks for modeling and data analysis. Here, motivated by resting state fMRI data, which can be seen as a collection of spatially dependent functional observations among brain regions, we propose a parsimonious but flexible representation of their dependence structure leveraging a Bayesian time-dependent latent factor model. Adopting an assumption of separability of the covariance structure in space and time, we are able to substantially reduce the computational cost and, at the same time, provide interpretable results. Theoretical properties of the model along with identifiability conditions are discussed. For model fitting, we propose a mcmc algorithm to enable posterior inference. We illustrate our work through an application to a dataset coming from the enkirs project, discussing the estimated covariance structure and also performing model selection along with network analysis. Our modeling is preliminary but offers ideas for developing fully Bayesian fMRI models, incorporating a plausible space and time dependence structure.

This is a preview of subscription content, 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
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Notes

  1. 1.

    Richer modeling might allow heterogeneity in variances, e.g., across regions but we do not consider that here.

References

  1. Biswal, B., Yetkin, F.Z., V.M., H., Hyde, J.: Functional connectivity in the motor cortex of resting human brain using echo-planar MRI. Magn. Reson. Med. 34(4), 537–541 (1995). https://doi.org/10.1002/mrm.1910340409

    Article  Google Scholar 

  2. Lee, M.H., Smyser, C.D., Shimony, J.S.: Resting state fMRI: a review of methods and clinical applications. Am. J. Neuroradiol. 34(10), 1866–1872 (2013). https://doi.org/10.3174/ajnr.A3263

    Article  Google Scholar 

  3. Poldrack, R.A., Mumford, J.A., Nichols, T.E.: Handbook of Functional MRI Data Analysis. Cambridge University Press, Cambridge (2011)

    Google Scholar 

  4. Smitha, K., Akhil Raja, K., Arun, K., Rajesh, P., Thomas, B., Kapilamoorthy, T., Kesavadas, C.: Resting state fMRI: A review on methods in resting state connectivity analysis and resting state networks. Neuroradiol. J. 30(4), 305–317 (2017). https://doi.org/10.1177/1971400917697342

    Article  Google Scholar 

  5. Bowman, F.D., Caffo, B., Bassett, S.S., Kilts, C.: A Bayesian hierarchical framework for spatial modeling of fMRI data. NeuroImage 39 (2008). https://doi.org/10.1016/j.neuroimage.2007.08.012

    Article  Google Scholar 

  6. Hartvig, N.G.: A stochastic geometry model for functional magnetic resonance images. Scand. J. Stat. 29(3), 333–353 (2002). https://doi.org/10.1111/1467-9469.00294

    Article  MathSciNet  MATH  Google Scholar 

  7. Quirós, A., Diez, R.M., Gamerman, D.: Bayesian spatiotemporal model of fMRI data. NeuroImage 49(1), 442–456 (2010). https://doi.org/10.1016/j.neuroimage.2009.07.047

    Article  Google Scholar 

  8. Stephan, K.E., Kasper, L., Harrison, L.M., Daunizeau, J., den Ouden, H.E., Breakspear, M., Friston, K.J.: Nonlinear dynamic causal models for fMRI. NeuroImage 42(2), 649–662 (2008). https://doi.org/10.1016/j.neuroimage.2008.04.262

    Article  Google Scholar 

  9. Zhang, L., Guindani, M., Versace, F., Engelmann, J.M., Vannucci, M.: A spatiotemporal nonparametric Bayesian model of multi-subject fMRI data. Ann. Appl. Stat. 10(2), 638–666 (2016). https://doi.org/10.1214/16-AOAS926

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhang, L., Guindani, M., Versace, F., Vannucci, M.: A spatio-temporal nonparametric Bayesian variable selection model of fMRI data for clustering correlated time courses. NeuroImage 95, 162–175 (2014). https://doi.org/10.1016/j.neuroimage.2014.03.024

    Article  Google Scholar 

  11. Woolrich, M.W., Jbabdi, S., Patenaude, B., Chappell, M., Makni, S., Behrens, T., Beckmann, C., Jenkinson, M., Smith, S.M.: Bayesian analysis of neuroimaging data in FSL. NeuroImage 45 (2009). https://doi.org/10.1016/j.neuroimage.2008.10.055

    Article  Google Scholar 

  12. Zhang, L., Guindani, M., Vannucci, M.: Bayesian models for functional magnetic resonance imaging data analysis. Comput. Stat. 7(1), 21–41 (2015). https://doi.org/10.1002/wics.1339. Wiley Interdisciplinary Reviews

    Article  MathSciNet  Google Scholar 

  13. Friston, K.J., Holmes, A.P., Worsley, K.J., Poline, J.P., Frith, C.D., Frackowiak, R.S.J.: Statistical parametric maps in functional imaging: a general linear approach. Hum. Brain Mapp. 2(4), 189–210 (1994). https://doi.org/10.1002/hbm.460020402

    Article  Google Scholar 

  14. Craddock, R.C., Jbabdi, S., Yan, C., Vogelstein, J.T., Castellanos, F.X., Di Martino, A., Kelly, C., Heberlein, K., Colcombe, S., Milham, M.P.: Imaging human connectomes at the macroscale. Nat. Methods 6, 524–539 (2013). https://doi.org/10.1038/nmeth.2482

    Article  Google Scholar 

  15. Erhardt, E.B., Allen, E.A., Wei, Y., Eichele, T., Calhoun, V.D.: Simtb, a simulation toolbox for fmri data under a model of spatiotemporal separability. NeuroImage 59(4), 4160–4167 (2012). https://doi.org/10.1016/j.neuroimage.2011.11.088

    Article  Google Scholar 

  16. Desikan, R.S., Segonne, F., Fischl, B., Quinn, B.T., Dickerson, B.C., Blacker, D., Buckner, R.L., Dale, A.M., Maguire, R.P., Hyman, B.T., Albert, M.S., Killiany, R.J.: An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest. NeuroImage 31, 968–980 (2006). https://doi.org/10.1016/j.neuroimage.2006.01.021

    Article  Google Scholar 

  17. Cressie, N., Wikle, C.K.: Statistics for Spatio-Temporal Data. Wiley, New York (2011)

    Google Scholar 

  18. Banerjee, S., Carlin, B.P., Gelfand, A.E.: Hierarchical Modeling and Analysis for Spatial Data, 2nd edn. CRC Press, Boca Raton (2014)

    Google Scholar 

  19. Quick, H., Banerjee, S., Carlin, B.P.: Modeling temporal gradients in regionally aggregated California asthma hospitalization data. Ann. Appl. Stat. 7(1), 154–176 (2013). https://doi.org/10.1214/12-AOAS600

    Article  MathSciNet  MATH  Google Scholar 

  20. Ramsay, J., Silverman, B.W.: Functional Data Analysis. Springer, New York (2005)

    Google Scholar 

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

    Article  Google Scholar 

  22. van den Heuvel, M., Stam, C.J., Boersma, M., Hulshoff Pol, H.E.: Small-world and scale-free organization of voxel-based resting-state functional connectivity in the human brain. NeuroImage 43, 528–539 (2008). https://doi.org/10.1016/j.neuroimage.2008.08.010

    Article  Google Scholar 

  23. Alexander-Bloch, A.F., Vértes, P., Stidd, R., Lalonde, F., Clasen, L., Rapoport, J., Giedd, J., Bullmore, E.T., Gogtay, N.: The anatomical distance of functional connections predicts brain network topology in health and schizophrenia. Cereb. Cortex 23(1), 127–138 (2013). https://doi.org/10.1093/cercor/bhr388

    Article  Google Scholar 

  24. Salvador, R., Suckling, J., Coleman, M., Pickard, J., Menon, D., Bullmore, E.: Neurophysiological architecture of functional magnetic resonance images of human brain. Cereb. Cortex 15, 1332–1342 (2005). https://doi.org/10.1093/cercor/bhi016

    Article  Google Scholar 

  25. Durante, D., Scarpa, B., Dunson, D.B.: Locally adaptive factor processes for multivariate time series. J. Mach. Learn. Res. 15, 1493–1522 (2014). http://jmlr.org/papers/v15/durante14a.html

  26. Fox, E., Dunson, D.: Bayesian nonparametric covariance regression. J. Mach. Learn. Res. 16, 2501–2542 (2015). http://jmlr.org/papers/v16/fox15a.html

  27. Prado, R., West, M.: Time Series. Modeling, Computation and Inference. CRC Press (2010)

    Google Scholar 

  28. Carvalho, C.M., Chang, J., Lucas, J.E., Nevins, J.R., Wang, Q., West, M.: High-dimensional sparse factor modeling: applications in gene expression genomics. J. Am. Stat. Assoc. 103(484), 1438–1456 (2008). https://doi.org/10.1198/016214508000000869

    Article  MathSciNet  MATH  Google Scholar 

  29. West, M.: Bayesian factor regression models in the large p, small n paradigm. In: J. Bernardo, M.J. Bayarri, J.O. Berger, A.P. Dawid, D. Heckerman, A.F.M. Smith, M. West (eds.) Bayesian Statistics, vol. 7, pp. 733–742. Oxford University Press (2003)

    Google Scholar 

  30. Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. The MIT Press (2006)

    Google Scholar 

  31. Allen, E.A., Damaraju, E., Plis, S.M., Erhardt, E.B., Eichele, T., Calhoun, V.D.: Tracking whole-brain connectivity dynamics in the resting state. Cereb. Cortex 24, 663–676 (2014). https://doi.org/10.1093/cercor/bhs352

    Article  Google Scholar 

  32. Dawid, A.P.: Some matrix-variate distribution theory: notational considerations and a Bayesian application. Biometrika 68(1), 265–274 (1981). https://doi.org/10.2307/2335827

    Article  MathSciNet  Google Scholar 

  33. Papp, T.K.: klin: Linear Equations with Kronecker Structure R package version 2007-02-05 (2012). https://CRAN.R-project.org/package=klin.

  34. Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., Rubin, D.B.: Bayesian Data Analysis, 3rd edn. CRC Press (2014)

    Google Scholar 

  35. Spiegelhalter, D.J., Best, N.G., Carlin, B.P., Van Der Linde, A.: Bayesian measures of model complexity and fit. J. R. Stat. Society. Ser. B: Stat. Methodol.64(4), 583–616 (2002). 10.1111/1467-9868.00353

    Article  MathSciNet  Google Scholar 

  36. Breiman, L.: Random forests. Mach. Learn. 45(1), 5–32 (2001). https://doi.org/10.1023/A:1010933404324

    Article  MATH  Google Scholar 

  37. Kolaczyk, E.D.: Statistical Analysis of Network Data. Springer, New York (2009). https://doi.org/10.1007/978-0-387-88146-1

    Book  Google Scholar 

  38. Ghosh, J., Dunson, D.B.: Default prior distributions and efficient posterior computation in Bayesian factor analysis. J. Comput. Graph. Stat. 18(2), 306–320 (2009). https://doi.org/10.1198/jcgs.2009.07145

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We are grateful to Greg Kiar and Eric Bridgeford from NeuroData at Johns Hopkins University, who graciously pre-processed the raw DTI and R-fMRI imaging data available at http://fcon_1000.projects.nitrc.org/indi/CoRR/html/nki_1.html, using the pipelines ndmg and c-pac. The authors are also thankful to the organizers of StartUp Research for coordinating such a stimulating event.

Author information

Authors and Affiliations

  1. Department of Statistical Sciences, Sapienza University of Rome, Rome, Italy

    Alessia Caponera

  2. Department of Statistics and Quantitative Methods, University of Milano-Bicocca, Milan, Italy

    Francesco Denti

  3. Department of Decision Sciences, Bocconi University, Milan, Italy

    Tommaso Rigon

  4. Department of Statistical Sciences, University of Padova, Padua, Italy

    Andrea Sottosanti

  5. Department of Statistical Science, Duke University, Durham, North Carolina, USA

    Alan Gelfand

Authors
  1. Alessia Caponera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesco Denti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tommaso Rigon
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Andrea Sottosanti
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Alan Gelfand
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tommaso Rigon .

Editor information

Editors and Affiliations

  1. Department of Statistical Sciences, University of Padova, Padua, Italy

    Antonio Canale

  2. Department of Decision Sciences, Bocconi University, Milan, Italy

    Daniele Durante

  3. Department of Statistical Sciences, Università Cattolica del Sacro Cuore, Milan, Italy

    Lucia Paci

  4. Department of Statistical Sciences, University of Padova, Padua, Italy

    Bruno Scarpa

7 Computational Details

7 Computational Details

In this appendix we describe a simple Metropolis-Hastings for posterior inference. The algorithm is summarized in Algorithm 1. Additionally, we derive the identity in Eq. (15). Because of orthogonality, we have that \(\varvec{U}_{\varvec{T}}\varvec{U}_{\varvec{T}}^\mathsf {T} = I_{T \times T}\) and \(\varvec{U}_{\varvec{A}}\varvec{U}_{\varvec{A}}^\mathsf {T} = I_{L \times L}\) and recall also that the matrices \(\varvec{\varLambda }_{\varvec{T}}\) and \(\varvec{\varLambda }_{\varvec{A}}\) are diagonal, containing the eigenvalues of \(\varvec{\varSigma }_{\varvec{T}}\) and \(\varvec{\varSigma }_{\varvec{A}}\), respectively. Exploiting the spectral decompositions of \(\varvec{\varSigma }_{\varvec{T}}\) and \(\varvec{\varSigma }_{\varvec{A}}\) and the basic properties of the Kronecker product, we get

$$\begin{aligned} \begin{aligned} \varvec{C}&= (\varvec{U}_{\varvec{T}} \varvec{\varLambda }_{\varvec{T}} \varvec{U}_{\varvec{T}}^\mathsf {T})\otimes (\varvec{U}_{\varvec{A}} \varvec{\varLambda }_{\varvec{A}} \varvec{U}_{\varvec{A}}^\mathsf {T}) +\sigma ^2I_{n\times n} \\&= (\varvec{U}_{\varvec{T}} \otimes \varvec{U}_{\varvec{A}})(\varvec{\varLambda }_{\varvec{T}} \otimes \varvec{\varLambda }_{\varvec{A}})(\varvec{U}_{\varvec{T}} \otimes \varvec{U}_{\varvec{A}})^\mathsf {T} +\sigma ^2I_{n\times n}. \end{aligned} \end{aligned}$$

Then, we can write the identity matrix \(I_{n \times n} = (\varvec{U}_{\varvec{T}} \varvec{U}_{\varvec{T}}^\mathsf {T}) \otimes (\varvec{U}_{\varvec{A}}\varvec{U}_{\varvec{A}}^\mathsf {T}) = (\varvec{U}_{\varvec{T}} \otimes \varvec{U}_{\varvec{A}})(\varvec{U}_{\varvec{T}} \otimes \varvec{U}_{\varvec{A}})^\mathsf {T}\). Rearranging the above equation, we get

figure a
$$\begin{aligned} \varvec{C} = (\varvec{U}_{\varvec{T}} \otimes \varvec{U}_{\varvec{A}})(\varvec{\varLambda }_{\varvec{T}} \otimes \varvec{\varLambda }_{\varvec{A}}+\sigma ^2I_{n\times n})(\varvec{U}_{\varvec{T}} \otimes \varvec{U}_{\varvec{A}})^\mathsf {T}, \end{aligned}$$

from which decomposition of \(\varvec{C}^{-1}\) in Eq. (15) follows directly.

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Caponera, A., Denti, F., Rigon, T., Sottosanti, A., Gelfand, A. (2018). Hierarchical Spatio-Temporal Modeling of Resting State fMRI Data. In: Canale, A., Durante, D., Paci, L., Scarpa, B. (eds) Studies in Neural Data Science. START UP RESEARCH 2017. Springer Proceedings in Mathematics & Statistics, vol 257. Springer, Cham. https://doi.org/10.1007/978-3-030-00039-4_7

Download citation

Publish with us

Policies and ethics

Search

Navigation