## Brazilian Journal of Probability and Statistics

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

### Studying the effective brain connectivity using multiregression dynamic models

Lilia Costa, Thomas Nichols, and Jim Q. Smith

**Full-text: Open access**

#### Abstract

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

**Source**

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

**Dates**

Received: September 2016

Accepted: August 2017

First available in Project Euclid: 15 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1513328767

**Digital Object Identifier**

doi:10.1214/17-BJPS375

**Mathematical Reviews number (MathSciNet)**

MR3738178

**Zentralblatt MATH identifier**

1385.92013

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.bjps/1513328767

#### References

- 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.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.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. http://biorxiv.org/content/early/2017/01/25/103002.
- 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. https://www.R-project.org/. - 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. https://doi.org/10.6084/m9.figshare.866771.v1.
- 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 https://github.com/schw4b/multdyn.
- 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.

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