Brazilian Journal of Probability and Statistics

Studying the effective brain connectivity using multiregression dynamic models

Lilia Costa, Thomas Nichols, and Jim Q. Smith

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The Multiregression Dynamic Model (MDM) is a multivariate graphical model for a multidimensional time series that allows the estimation of time-varying effective connectivity. An MDM is a state space model where connection weights reflect the contemporaneous interactions between brain regions. Because the marginal likelihood has a closed form, model selection across a large number of potential connectivity networks is easy to perform. With application of the Integer Programming Algorithm, we can quickly find optimal models that satisfy acyclic graph constraints and, due to a factorisation of the marginal likelihood, the search over all possible directed (acyclic or cyclic) graphical structures is even faster. These methods are illustrated using recent resting-state and steady-state task fMRI data.

Article information

Braz. J. Probab. Stat., Volume 31, Number 4 (2017), 765-800.

Received: September 2016
Accepted: August 2017
First available in Project Euclid: 15 December 2017

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Zentralblatt MATH identifier

Multiregression dynamic model Bayesian network effective connectivity functional magnetic resonance imaging integer programming algorithm


Costa, Lilia; Nichols, Thomas; Smith, Jim Q. Studying the effective brain connectivity using multiregression dynamic models. Braz. J. Probab. Stat. 31 (2017), no. 4, 765--800. doi:10.1214/17-BJPS375.

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  • Achterberg, T. (2007). Constraint integer programming. PhD thesis, TU Berlin.
  • Baba, K., Shibata, R. and Sibuya, M. (2004). Partial correlation and conditional correlation as measures of conditional independence. Australian and New Zealand Journal of Statistics 46, 4, 657–664.
  • Bartlett, M. and Cussens, J. (2013). Advances in Bayesian network learning using integer programming. arXiv preprint. Available at arXiv:1309.6825.
  • Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society, Series B 57, 1, 289–300.
  • Chang, C., Thomason, M. E. and Glover, G. H. (2008). Mapping and correction of vascular hemodynamic latency in the BOLD signal. NeuroImage 43, 90–102.
  • Costa, L., Smith, J., Nichols, T., Cussens, J., Duff, E. P. and Makin, T. R. (2015). Searching multiregression dynamic models of resting-state fMRI networks using integer programming. Bayesian Analysis 10, 441–478.
  • Cowell, R. G. (2013). A simple greedy algorithm for reconstructing pedigrees. Theoretical Population Biology 83, 55–63.
  • Cussens, J. (2010). SMaximum likelihood pedigree reconstruction using integer programming. WCB@ ICLP 8–19.
  • Cussens, J. (2012). Bayesian network learning with cutting planes. arXiv preprint. Available at arXiv:1202.3713.
  • David, O., Guillemain, I., Saillet, S., Reyt, S., Deransart, C., Segebarth, C. and Depaulis, A. (2008). Identifying neural drivers with functional MRI: An electrophysiological validation. PLoS Biology 6, 2683–2697.
  • Duff, E., Tamar, M., Smith, S. M. and Woolrich, M. W. (2017). Disambiguating brain functional connectivity. bioRxiv.
  • Friston, K. J. (2011). Functional and Effective Connectivity: a review. Brain Connectivity 1, 1, 13–36.
  • Friston, K. J., Harrison, L. and Penny, W. (2003). Dynamic causal modelling. NeuroImage 19, 1273–1302.
  • Goldenberg, A., Zheng, A. X., Fienberg, S. E., Airoldi, E. M. and others (2010). A survey of statistical network models. Foundations and Trends® in Machine Learning 2, 2, 129–233.
  • Granger, C. W. J. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424–438.
  • Griffanti, L., Salimi-Khorshidi, G., Beckmann, C. F., Auerbach, E. J., Douaud, G., Sexton, C. E., Zsoldos, E., Ebmeier, K. P., Filippini, N., Mackay, C. E. and Moeller, S. (2014). ICA-based artefact removal and accelerated fMRI acquisition for improved resting state network imaging. NeuroImage 95, 232–247.
  • Havlicek, M., Jan, J., Brazdil, M. and Calhoun, V. D. (2010). Dynamic Granger causality based on Kalman filter for evaluation of functional network connectivity in fMRI data. NeuroImage 53, 65–77.
  • Heckerman, D. (1998). A tutorial on learning with Bayesian networks. Nato Asi Series D Behavioural And Social Sciences 89, 301–354.
  • Jeffreys, H. (1961). Theory of Probability, 3rd ed. London: Oxford University Press.
  • Jenkinson, M., Beckmann, C. F., Behrens, T. E., Woolrich, M. W. and Smith, S. M. (2012). FSL. NeuroImage 62, 782–790.
  • Koster, J. T. (1996). Markov properties of nonrecursive causal models. The Annals of Statistics 2148–2177.
  • Marrelec, G., Krainik, A., Duffau, H., Pélégrini-Issac, M., Lehéricy, S., Doyon, J. and Benali, H. (2006). Partial correlation for functional brain interactivity investigation in functional MRI. NeuroImage 62, 228–237.
  • Pearl, J. (2000). Causality: Models, Reasoning, and Inference. Cambridge: Cambridge University Press.
  • Penny, W., Ghahramani, Z. and Friston, K. (2005). Bilinear dynamical systems. Philosophical Transactions of the Royal Society of London Series B, Biological Sciences 360, 983–993.
  • Petris, G., Petrone, S. and Campagnoli, P. (2009). Dynamic Linear Models with R. New York: Springer.
  • Poldrack, R. A., Mumford, J. A. and Nichols, T. E. (2011). Handbook of fMRI Data Analysis. Cambridge University Press.
  • Queen, C. M. and Albers, C. J. (2008). Forecast covariances in the linear multiregression dynamic model. J. Forecast. 27, 175–191.
  • Queen, C. M. and Albers, C. J. (2009). Intervention and causality: Forecasting traffic flows using a dynamic Bayesian network. Journal of the American Statistical Association 104, 669–681.
  • Queen, C. M. and Smith, J. Q. (1993). Multiregression dynamic models. Journal of the Royal Statistical Society, Series B 55, 849–870.
  • R Core Team (2016). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.
  • Raichle, M. E. (2010). Two views of brain function. Trends in Cognitive Sciences 14, 180–190.
  • Ramsey, J. D., Hanson, S. J., Hanson, C., Halchenko, Y. O., Poldrack, R. A. and Glymour, C. (2010). Six problems for causal inference from fMRI. NeuroImage 49, 1545–1558.
  • Ridgway, G., Leite, A. B., Penny, W. and Friston, K. (2013). Stochastic DCM of the DMN using resting-state fMRI: test-retest reliability. figshare.
  • Ryali, S., Supekar, K., Chen, T. and Menon, V. (2011). Multivariate dynamical systems models for estimating causal interactions in fMRI. NeuroImage 54, 807–823.
  • Salimi-Khorshidi, G., Douaud, G., Beckmann, C. F., Glasser, M. F., Griffanti, L. and Smith, S. M. (2014). Automatic denoising of functional MRI data: Combining independent component analysis and hierarchical fusion of classifiers. NeuroImage 90, 449–468.
  • Schwab, S., Harbord, R., Costa, L. and Nichols, T. E. (2017). multdyn: A package for Multiregression Dynamic Models (MDM). Available at
  • Shehzad, Z., Kelly, A. C., Reiss, P. T., Gee, D. G., Gotimer, K., Uddin, L. Q., Lee, S. H., Margulies, D. S., Roy, A. K., Biswal, B. B. and Petkova, E. (2009). The resting brain: Unconstrained yet reliable. Cerebral Cortex 19, 2209–2229.
  • Sloane, N. J. A. and Plouffe, S. (1995). The Encyclopedia of Integer Sequences. Academic Press.
  • Smith, J. F., Pillai, A., Chen, K. and Horwitz, B. (2010). Identification and validation of effective connectivity networks in functional magnetic resonance imaging using switching linear dynamic systems. NeuroImage 52, 1027–1040.
  • Smith, J. F., Pillai, A., Chen, K. and Horwitz, B. (2011). Effective connectivity modeling for fMRI: Six issues and possible solutions using linear dynamic systems. Frontiers in Systems Neuroscience 5, 104.
  • Smith, J. Q. and Croft, J. (2003). Bayesian networks for discrete multivariate data: An algebraic approach to inference. Journal of Multivariate Analysis 84, 387–402.
  • Smith, S. M., Bandettini, P. A., Miller, K. L., Behrens, T. E. J., Friston, K. J., David, O., Liue, T., Woolricha, M. W. and Nichols, T. E. (2012). The danger of systematic bias in group-level FMRI-lag-based causality estimation. NeuroImage 59, 1228–1229.
  • Smith, S. M., Fox, P. T., Miller, K. L., Glahn, D. C., Fox, P. M., Mackay, C. E., Filippini, N., Watkins, K. E., Toro, R., Laird, A. R. and Beckmann, C. F. (2009). Correspondence of the brain’s functional architecture during activation and rest. Proceedings of the National Academy of Sciences of the United States of America 106, 13040–13045.
  • Smith, S. M., Jenkinson, M., Woolrich, M. W., Beckmann, C. F., Behrens, T. E. J., Johansen-Berg, H., Bannister, P. R., De Luca, M., Drobnjak, I., Flitney, D. E., Niazy, R., Saunders, J., Vickers, J., Zhang, Y., De Stefano, N., Brady, J. M. and Matthews, P. M. (2004). Advances in functional and structural MR image analysis and implementation as FSL. NeuroImage 23, 208–219.
  • Spirtes, P. (1995). Directed cyclic graphical representations of feedback models. In Uncertainty in Artificial Intelligence 11 (P. Besnard and S. Hanks, eds.) 491–498. Morgan Kaufmann.
  • Spirtes, P., Glymour, C. N. and Scheines, R. (2000). Causation, Prediction, and Search, 2nd ed. Cambridge, MA: MIT Press.
  • Sporns, O. (2010). Networks of the Brain, 1st ed. MIT Press.
  • Stephan, K. E., Kasper, L., Harrison, L. M., Daunizeau, J., den Ouden, H. E., Breakspear, M. and Friston, K. J. (2008). Nonlinear dynamic causal models for fMRI. NeuroImage 42, 649–662.
  • Valdés-Sosa, P. A., Roebroeck, A., Daunizeau, J. and Friston, K. (2011). Effective connectivity: Influence, causality and biophysical modeling. NeuroImage 58, 339–361.
  • West, M. and Harrison, P. J. (1997). Bayesian Forecasting and Dynamic Models, 2nd ed. New York: Springer.
  • Williams, H. P. (2009). Logic and Integer Programming. Springer. ISBN 978-0-387-92279-9.
  • Woolrich, M. W., Jbabdi, S., Patenaude, B., Chappell, M., Makni, S., Behrens, T., Beckmann, C., Jenkinson, M. and Smith, S. M. (2009). Bayesian analysis of neuroimaging data in FSL. NeuroImage 45, S173–S186.