Abstract
Functional connectivity has emerged as a promising approach to study the functional organisation of the brain and to define features for prediction of brain state. The most widely used method for inferring functional connectivity is Pearson-s correlation, but it cannot differentiate direct and indirect effects. This disadvantage is often avoided by computing the partial correlation between two regions controlling all other regions, but this method suffers from Berkson-s paradox. Some advanced methods, such as regularised inverse covariance, have been applied. However, these methods usually depend on some parameters. Here we propose use of minimum partial correlation as a parameter-free measure for the skeleton of functional connectivity in functional magnetic resonance imaging (fMRI). The minimum partial correlation between two regions is the minimum of absolute values of partial correlations by controlling all possible subsets of other regions. Theoretically, there is a direct effect between two regions if and only if their minimum partial correlation is non-zero under faithfulness and Gaussian assumptions. The elastic PC-algorithm is designed to efficiently approximate minimum partial correlation within a computational time budget. The simulation study shows that the proposed method outperforms others in most cases and its application is illustrated using a resting-state fMRI dataset from the human connectome project.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K. J. Friston. Functional and effective connectivity: A review. Brain Connectivity, vol. 1, no. 1, pp. 13–36, 2011.
R. L. Buckner, F. M. Krienen, B. T. T. Yeo. Opportunities and limitations of intrinsic functional connectivity MRI. Nature Neuroscience, vol. 16, no. 7, pp. 832–837, 2013.
R. C. Craddock, S. Jbabdi, C. G. Yan, J. T. Vogelstein, F. X. Castellanos, A. D. Martino, C. Kelly, K. Heberlein, S. Colcombe, M. P. Milham. Imaging human connectomes at the macroscale. Nature Methods, vol. 10, no. 6, pp. 524–539, 2013.
D. J. Hawellek, J. F. Hipp, C. M. Lewis, M. Corbetta, A. K. Engel. Increased functional connectivity indicates the severity of cognitive impairment in multiple sclerosis. Proceedings of the National Academy of Sciences of the United States of America, vol. 108, no. 47, pp. 19066–19071, 2011.
W. R. Shirer, S. Ryali, E. Rykhlevskaia, V. Menon, M. D. Greicius. Decoding subject-driven cognitive states with whole-brain connectivity patterns. Cerebral Cortex, vol. 22, no. 1, pp. 158–165, 2012.
S. M. Smith, K. L. Miller, G. Salimi-Khorshidi, M. Webster, C. F. Beckmann, T. E. Nichols, J. D. Ramsey, M.W. Woolrich. Network modelling methods for fMRI. NeuroImage, vol. 54, no. 2, pp. 875–891, 2011.
K. J. Friston, L. Harrison, W. Penny. Dynamic causal modelling. NeuroImage, vol. 19, no. 4, pp. 1273–1302, 2003.
A. M. Hermundstad, D. S. Bassett, K. S. Brown, E. M. Aminoff, D. Clewett, S. Freeman, A. Frithsen, A. Johnson, C. M. Tipper, M. B. Miller, S. T. Grafton, J. M. Carlson. Structural foundations of resting-state and task-based functional connectivity in the human brain. Proceedings of the National Academy of Sciences of the United States of America, vol. 110, no. 15, pp. 6169–6174, 2013.
N. B. Turk-Browne. Functional interactions as big data in the human brain. Science, vol. 342, no. 6158, pp. 580–584, 2013.
G. Marrelec, A. Krainik, H. Duffau, M. Pélégrini-Issac, S. Lehéricy, J. Doyon, H. Benali. Partial correlation for functional brain interactivity investigation in functional MRI. NeuroImage, vol. 32, no. 1, pp. 228–237, 2006.
H. Lee, D. S. Lee, H. Kang, B. N. Kim, M. K. Chung. Sparse brain network recovery under compressed sensing. IEEE Transactions on Medical Imaging, vol. 30, no. 5, pp. 1154–1165, 2011.
S. Ryali, T. W. Chen, K. Supekar, V. Menon. Estimation of functional connectivity in fMRI data using stability selection-based sparse partial correlation with elastic net penalty. NeuroImage, vol. 59, no. 4, pp. 3852–3861, 2012.
S. M. Smith, D. Vidaurre, C. F. Beckmann, M. F. Glasser, M. Jenkinson, K. L. Miller, T. E. Nichols, E. C. Robinson, G. Salimi-Khorshidi, M. W. Woolrich, D. M. Barch, K. Uğurbil, D. C. Van Essen. Functional connectomics from resting-state fMRI. Trends in Cognitive Sciences, vol. 17, no. 12, pp. 666–682, 2013.
S. M. Smith, T. E. Nichols, D. Vidaurre, A. M. Winkler, T. E. J. Behrens, M. F. Glasser, K. Ugurbil, D. M. Barch, D. C. Van Essen, K. L. Miller. A positive-negative mode of population covariation links brain connectivity, demographics and behavior. Nature Neuroscience, vol. 18, no. 11, pp. 1565–1567, 2015.
J. Berkson. Limitations of the application of fourfold table analysis to hospital data. Biometrics Bulletin, vol. 2, no. 3, pp. 47–53, 1946.
J. Friedman, T. Hastie, R. Tibshirani. Sparse inverse covariance estimation with the graphical lasso. Biostatistics, vol. 9, no. 3, pp. 432–441, 2008.
G. Varoquaux, A. Gramfort, J. B. Poline, B. Thirion. Brain covariance selection: Better individual functional connectivity models using population prior. In Proceedings of Neural Information Processing Systems, NIPS, Vancouver, Canada, pp. 2334–2342, 2010.
M. Hinne, L. Ambrogioni, R. J. Janssen, T. Heskes, M. A. J. van Gerven. Structurally-informed Bayesian functional connectivity analysis. NeuroImage, vol. 86, pp. 294–305, 2014.
K. P. Murphy. Machine Learning: A Probabilistic Perspective, Cambridge, USA: MIT Press, 2012.
S. Huang, J. Li, L. Sun, J. P. Ye, A. Fleisher, T. Wu, K. W. Chen, E. Reiman. Learning brain connectivity of Alzheimers disease by sparse inverse covariance estimation. NeuroImage, vol. 50, no. 3, pp. 935–949, 2010.
M. G. G’Sell, J. Taylor, R. Tibshirani. Adaptive testing for the graphical lasso, [Online], Available: https://arxiv.org/abs/1307.4765, 2013.
R. Lockhart, J. Taylor, R. J. Tibshirani, R. Tibshirani. A significance test for the lasso. The Annals of Statistics, vol. 42, no. 2, pp. 413–468, 2014.
P. Spirtes, C. Glymour, R. Scheines. Causation, Prediction, and Search, 2nd ed., Cambridge, USA: MIT Press, 2000.
S. M. Smith, C. F. Beckmann, J. Andersson, E. J. Auerbach, J. Bijsterbosch, G. Douaud, E. Duff, D. A. Feinberg, L. Griffanti, M. P. Harms, M. Kelly, T. Laumann, K. L. Miller, S. Moeller, S. Petersen, J. Power, G. Salimi-Khorshidi, A. Z. Snyder, A. T. Vu, M. W. Woolrich, J. Q. Xu, E. Yacoub, K. Ugŭrbil, D. C. Van Essen, M. F. Glasser. Resting-state fMRI in the Human Connectome Project. NeuroImage, vol. 80, pp. 144–168, 2013.
L. Nie, X. Yang, P. M. Matthews, Z. X. Xu, Y. K. Guo. Minimum partial correlation: An accurate and parameterfree measure of functional connectivity in fMRI. In Proceedings of International Conference on Brain Informatics and Health, Springer, Cham, Switzerland, pp. 125–134, 2015.
J. Pearl. Causality: Models, Reasoning and Inference, Cambridge, UK: Cambridge University Press, 2000.
J. A. Mumford, J. D. Ramsey. Bayesian networks for fMRI: A primer. NeuroImage, vol. 86, pp. 573–582, 2014.
C. Bielza, P. Larra˜naga. Bayesian networks in neuroscience: A survey. Frontiers in Computational Neuroscience, vol.8, Article number 131, 2014.
S. L. Lauritzen. Graphical Models, Oxford, UK: Oxford University Press, 1996.
R. A. Fisher. The distribution of the partial correlation coefficient. Metron, vol. 3, pp. 329–332, 1924.
J. Cheng, R. Greinera, J. Kelly, D. Bell, W. R. Lius. Learning Bayesian networks from data: An information-theory based approach. Artificial Intelligence, vol. 137, no. 1–2, pp. 43–90, 2002.
I. Tsamardinos, L. E. Brown, C. F. Aliferis. The maxmin hill-climbing Bayesian network structure learning algorithm. Machine Learning, vol. 65, no. 1, pp. 31–78, 2006.
Z. X. Wang, L. W. Chan. Learning Bayesian networks from Markov random fields: An efficient algorithm for linear models. ACM Transactions on Knowledge Discovery from Data (TKDD), vol. 6, no. 3, Article number 10, 2012.
S. P. Iyer, I. Shafran, D. Grayson, K. Gates, J. T. Nigg, D. A. Fair. Inferring functional connectivity in MRI using Bayesian network structure learning with a modified PCalgorithm. NeuroImage, vol. 75, no. 4, pp. 165–175, 2013.
R. Han, L. Nie, M. M. Ghanem, Y. K. Guo. Elastic algorithms for guaranteeing quality monotonicity in big data mining. In Proceedings of IEEE International Conference on Big Data, IEEE, Silicon Valley, USA, pp. 45–50, 2013.
D. Colombo, M. H. Maathuis. Order-independent constraint-based causal structure learning. Journal of Machine Learning Research, vol. 15, no. 1, pp. 3741–3782, 2014.
S. Feizi, D. Marbach, M. Médard, M. Kellis. Network deconvolution as a general method to distinguish direct dependencies in networks. Nature Biotechnology, vol. 31, no. 8, pp. 726–733, 2013.
B. Barzel, A. L. Barabási. Network link prediction by global silencing of indirect correlations. Nature Biotechnology, vol. 31, no. 8, pp. 720–725, 2013.
M. Jenkinson, C. F. Beckmann, T. E. J. Behrens, M. W. Woolrich, S. M. Smith. FSL. NeuroImage, vol. 62, no. 2, pp. 782–790, 2012.
A. M. Dale, B. Fischl, M. I. Sereno. Cortical surface-based analysis: I. Segmentation and surface reconstruction. NeuroImage, vol. 9, no. 2, pp. 179–194, 1999.
D. S. Marcus, M. P. Harms, A. Z. Snyder, M. Jenkinson, J. A.Wilson, M. F. Glasser, D.M. Barch, K. A. Archie, G. C. Burgess. Human connectome project informatics: Quality control, database services, and data visualization. NeuroImage, vol. 80, no. 8, pp. 202–219, 2013.
L. Griffanti, G. Salimi-Khorshidi, C. F. Beckmann, E. J. Auerbach, G. Douaud, C. E. Sexton, E. Zsoldos, K. P. Ebmeier, N. Filippin, C. E. Mackay, S. Moeller, J. Q. Xu, E. Yacoub, G. Baselli, K. Ugurbil, K. L. Miller, S. M. Smith. ICA-based artefact removal and accelerated fMRI acquisition for improved resting state network imaging. NeuroImage, vol. 95, no. 4, pp. 232–247, 2014.
G. Salimi-Khorshidi, G. Douaud, C. F. Beckmann, M. F. Glasser, L. Griffanti, S. M. Smith. Automatic denoising of functional MRI data: Combining independent component analysis and hierarchical fusion of classifiers. NeuroImage, vol. 90, pp. 449–468, 2014.
M. F. Glasser, S. N. Sotiropoulos, J. A. Wilson, T. S. Coalson, B. Fischl, J. L. Andersson, J. Q. Xu, S. Jbabd. The minimal preprocessing pipelines for the human connectome project. NeuroImage, vol. 80, no. 3, pp. 105–124, 2013.
N. Tzourio-Mazoyer, B. Landeau, D. Papathanassiou, F. Crivello, O. Etard, N. Delcroix, B. Mazoyer, M. Joliot. Automated anatomical labeling of activations in SPM using a macroscopic anatomical parcellation of the MNI MRI single-subject brain. NeuroImage, vol. 15, no. 1, pp. 273–289, 2002.
M. L. Stanley, M. N. Moussa, B. M. Paolini, R. G. Lyday, J. H. Burdette, P. J. Laurienti. Defining nodes in complex brain networks. Frontiers in Computational Neuroscience, vol. 7, Article number 169, 2013.
S. B. Eickhoff, B. Thirion, G. Varoquaux, D. Bzdok. Connectivity-based parcellation: Critique and implications. Human Brain Mapping, vol. 36, no. 12, pp. 4771–4792, 2015.
S. M. Smith. The future of fMRI connectivity. NeuroImage, vol. 62, no. 2, pp. 1257–1266, 2012.
C. Y.Wee, P. T. Yap, D. Q. Zhang, L. H.Wang, D. G. Shen. Group-constrained sparse fMRI connectivity modeling for mild cognitive impairment identification. Brain Structure and Function, vol. 219, no. 2, pp. 641–656, 2014.
M. R. Xia, J. H. Wang, Y. He. BrainNet Viewer: A network visualization tool for human brain connectomics. PLoS One, vol. 8, no. 7, Article number e68910, 2013.
Z. N. Fu. A Study of Dynamic Functional Brain Connectivity Using Functional Magnetic Resonance Imaging (fMRI): Method and Applications, Ph. D. dissertation, The University of Hong Kong, China, 2016.
E. S. Finn, X. L. Shen, D. Scheinost, M. D. Rosenberg, J. Huang, M. M. Chun, X. Papademetris, R. T. Constable. Functional connectome fingerprinting: Identifying individuals using patterns of brain connectivity. Nature Neuroscience, vol. 18, no. 11, pp. 1664–1671, 2015.
L.Y. Chen, J. Yang, G. G. Xu, Y. Q. Liu, J. T. Li, C. S.Xu. Biomarker identification of rat liver regeneration via adaptive logistic regression. International Journal of Automation and Computing, vol. 13, no. 2, pp. 191–198, 2016.
Acknowledgement
Data were provided by the human connectome project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil, 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research, and by the McDonnell Center for Systems Neuroscience at Washington University. Paul M. Matthews gratefully acknowledges support from the Imperial College NIHR Biomedical Research Centre and personal support from the Edmond Safra Foundation and Lily Safra.
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Associate Editor Hong Qiao
Lei Nie received the B. Sc. degree in information and computing science from Sichuan University, China in 2009, and received the Ph.D. degree in computer science and technology from University of the Chinese Academy of Sciences, China in 2015. He was a visiting Ph.D. student in the Department of Computing, Imperial College London, UK from 2012 to 2014. He is currently a research associate in the Department of Computing, Imperial College London, UK.
His research interests include neuroimaging and machine learning.
Xian Yang received the B. Eng. degree in electronic information engineering from Huazhong University of Science and Technology, China in 2008, received the M. Sc. degree in digital communication from University of Bath, UK in 2009, and received the Ph. D. degree in computing from Imperial College London, UK in 2016. She is currently a research associate in the Department of Computing, Imperial College London, UK.
Her research interests include bioinformatics, system biology, neuroimaging, data mining and health informatics.
Paul M. Matthews received the B.A. degree in chemistry from University of Oxford, UK in 1978, the D.Phil degree in biochemistry from University of Oxford, UK in 1982, and the M.D. degree from Stanford University School of Medicine, USA in 1987. He is the Edmond and Lily Safra Chair and head of the Division of Brain Sciences at Imperial College London, UK. Amongst other external activities, he is chair of the Imaging Enhancement Working Group and a member of the steering group for UK Biobank (https://www.ukbiobank.ac.uk/), which has initiated a programme to image the brain, heart, carotids, bones and body of 100 000 people to understand disease risk in later life. He was the founding director of the Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB) (http://www.fmrib.ox.ac.uk/) and of the GSK Clinical Imaging Centre at the Hammersmith Hospital, for which he was a lead in spinning out Imanova Ltd., which is run as a public-private partnership between Imperial College, UCL, Kings College and the Medical Research Council (http://www.imanova.co.uk/).
His research interests include innovative translational applications of clinical imaging for the neurosciences. His broad area of research interest has been in molecular and functional neuroimaging and in neurological therapeutics development. A particular focus of his work has involved close collaboration with colleagues in computing and engineering to encourage effective translation of advanced imaging data to information.
Zhi-Wei Xu received the B. Sc. degree from University of Electronic Science and Technology of China, in 1982, received the M. Sc. degree from Purdue University, USA in 1984, and received the Ph.D. degree from the University of Southern California, USA in 1987. He is a professor and CTO of the Institute of Computing Technology (ICT) of the Chinese Academy of Sciences (CAS). His prior industrial experience included chief engineer of Dawning Corporation (now Sugon as listed in Shanghai Stock Exchange), a leading high-performance computer vendor in China. He currently leads “Cloud-Sea Computing Systems”, a strategic priority research project of the Chinese Academy of Sciences that aims at developing billion-thread computers with elastic processors.
His research interests include high-performance computer architecture and network computing science.
Yi-Ke Guo received the B. Sc. degree, the M. Sc. degree in computer science and technology from Tsinghua University, China in 1985, and the Ph.D. degree in computational logic from Imperial College London, UK in 1993. He is the founding director of the Data Science Institute, Imperial College London, UK. He is also a professor in the Department of Computing, Imperial College London, UK. During last 15 years, he has been leading a data science group to carry out many research projects, including discovery net on grid based data analysis for scientific discovery, MESSAGE on wireless mobile sensor network for environment monitoring, BAIR on system biology for diabetes study, iHealth on modern informatics infrastructure for healthcare decision making, UBIOPRED on large informatics platform for translational medicine research, digital city exchange on sensor information-based urban dynamics modelling, IC Cloud system for large scale collaborative scientific research. He is now the principal investigator of the eTRIKS project, a 23M Euro project in building a cloud-based translational informatics platform for global medical research.
His research interests include data mining, machine learning and bioinformatics
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Nie, L., Yang, X., Matthews, P.M. et al. Inferring functional connectivity in fMRI using minimum partial correlation. Int. J. Autom. Comput. 14, 371–385 (2017). https://doi.org/10.1007/s11633-017-1084-9
Received:
Accepted:
Published:
Issue date:
DOI: https://doi.org/10.1007/s11633-017-1084-9