Multi-dimensional connectivity: a conceptual and mathematical review
Open accessAbstract
The estimation of functional connectivity between regions of the brain, for example based on statistical dependencies between the time series of activity in each region, has become increasingly important in neuroimaging. Typically, multiple time series (e.g. from each voxel in fMRI data) are first reduced to a single time series that summarises the activity in a region of interest, e.g. by averaging across voxels or by taking the first principal component; an approach we call one-dimensional connectivity. However, this summary approach ignores potential multi-dimensional connectivity between two regions, and a number of recent methods have been proposed to capture such complex dependencies. Here we review the most common multi-dimensional connectivity methods, from an intuitive perspective, from a formal (mathematical) point of view, and through a number of simulated and real (fMRI and MEG) data examples that illustrate the strengths and weaknesses of each method. The paper is accompanied with both functions and scripts, which implement each method and reproduce all the examples.
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A rising approach of EEG data handling is to aggregate electrode data into physiologically pre-defined brain regions (ROIs).
“A rising approach of EEG data handling is to aggregate the electrode data into physiologically pre-defined brain regions (or region of interest, ROI) (Basti et al., 2020).”
From Frontal Electroencephalography Asymmetry and Desynchronized Functional Connectivity Associated with Long-Term and Short-Term Breathing Training (2026) · cites this paper as Basti et al., 2020
Section 5.2 describes projecting MEG sensor data into anatomically defined cortical ROIs and discusses ROIs in source-reconstructed MEG/EEG data. It does not describe aggregating EEG electrode data into ROIs or establish that this is a rising approach; it also cautions that simple averaging of source-reconstructed MEG/EEG signals is inappropriate. (LLM, conf=0.82)
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A multi-modal parcellation of human cerebral cortex
Glasser, M. F., Coalson, T. S., Robinson, E. C., Hacker, C. D., Harwell, J., Yacoub, E., Ugurbil, K., Andersson, J., Beckmann, C. F., Jenkinson, M., et al. (2016) · Nature · doi:10.1038/nature18933 · Cited by 5,111 · see citing works
Exploring the brain network: A review on resting-state fMRI functional connectivity
van den Heuvel, M. P., & Hulshoff Pol, H. E. (2010) · European Neuropsychopharmacology · doi:10.1016/j.euroneuro.2010.03.008 · Cited by 3,326 · see citing works
Measuring and testing dependence by correlation of distances
Székely, G. J., Rizzo, M. L., & Bakirov, N. K. (2007) · The Annals of Statistics · doi:10.1214/009053607000000505 · Cited by 2,684 · see citing works
Large-scale brain networks in cognition: emerging methods and principles
Bressler, S. L., & Menon, V. (2010) · Trends in Cognitive Sciences · doi:10.1016/j.tics.2010.04.004 · Cited by 2,348 · see citing works
Recurrence is required to capture the representational dynamics of the human visual system
Kietzmann, T. C., Spoerer, C. J., Sörensen, L. K. A., Cichy, R. M., Hauk, O., & Kriegeskorte, N. (2019) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1905544116 · Cited by 406 · see citing works
Self-citation
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Research works printed but not linked: 33
- Anzellotti, S. , Fedorenko, E. , Kell, A.J. , Caramazza, A. , Saxe, R. , 2017a. Measuring and modeling nonlinear interactions between brain regions with fMRI. bioRxiv, 074856 .
- Anzellotti, S. , Caramazza, A. , Saxe, R. , 2017b. Multivariate pattern dependence. PLoS Comput. Biol. 13 (11), e1005799 .
- Anzellotti, S. , Coutanche, M.N. , 2018. Beyond functional connectivity: investigating net- works of multivariate representations. Trends Cogn. Sci. (Regul. Ed.) 22 (3), 258-269 .
- Baccalá, L.A. , Sameshima, K. , 2001. Partial directed coherence: a new concept in neural structure determination. Biol. Cybern. 84 (6), 463-474 .
- Barrett, A.B. , Barnett, L. , Seth, A.K. , 2010. Multivariate Granger causality and generalized variance. Phys. Rev. E 81 (4), 041907 .
- Biswal, B. , Zerrin Yetkin, F. , Haughton, V.M. , Hyde, J.S. , 1995. Functional connectivity in the motor cortex of resting human brain using echo-planar MRI. Magn. Reson. Med. 34 (4), 537-541 .
- Bressler, S.L. , Seth, A.K. , 2011. Wiener-Granger causality: a well established methodology. Neuroimage 58 (2), 323-329 .
- Craddock, R.C. , Tungaraza, R.L. , Milham, M.P. , 2015. Connectomics and new approaches for analyzing human brain functional connectivity. Gigascience 4 (1), s13742 -015 .
- Engel, A.K. , Gerloff, C. , Hilgetag, C.C. , Nolte, G. , 2013. Intrinsic coupling modes: multi- scale interactions in ongoing brain activity. Neuron 80 (4), 867-886 .
- Ewald, A. , Marzetti, L. , Zappasodi, F. , Meinecke, F.C. , Nolte, G. , 2012. Estimating true brain connectivity from EEG/MEG data invariant to linear and static transformations in sensor space. Neuroimage 60 (1), 476-488 .
- Friston, K.J. , Rotshtein, P. , Geng, J.J. , Sterzer, P. , Henson, R.N. , 2006. A critique of func- tional localisers. Neuroimage 30 (4), 1077-1087 .
- Geerligs, L. , Tsvetanov, K.A. , Henson, R.N. , 2017. Challenges in measuring individual differences in functional connectivity using fMRI: the case of healthy aging. Hum. Brain Mapp. 38 (8), 4125-4156 .
- Golub, G.H. , Heath, M. , Wahba, G. , 1979. Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21 (2), 215-223 .
- Harrison, L. , Penny, W.D. , Friston, K. , 2003. Multivariate autoregressive modeling of fMRI time series. Neuroimage 19 (4), 1477-1491 .
- Hipp, J.F. , Hawellek, D.J. , Corbetta, M. , Siegel, M. , Engel, A.K. , 2012. Large-scale cortical correlation structure of spontaneous oscillatory activity. Nat. Neurosci. 15 (6), 884 .
- Hotelling, H., 1936. Relations between two sets of variates. Biometrika 28 (3/4), 321-377. doi: 10.2307/2333955 . · DOI
- Kherif, F. , Poline, J.B. , Flandin, G. , Benali, H. , Simon, O. , Dehaene, S. , Worsley, K.J. , 2002. Multivariate model specification for fMRI data. Neuroimage 16 (4), 1068-1083 .
- Lachaux, J.P. , Rodriguez, E. , Martinerie, J. , Varela, F.J. , 1999. Measuring phase synchrony in brain signals. Hum. Brain Mapp. 8 (4), 194-208 .
- Nolte, G. , Bai, O. , Wheaton, L. , Mari, Z. , Vorbach, S. , Hallett, M. , 2004. Identifying true brain interaction from EEG data using the imaginary part of coherency. Clin. Neuro- physiol. 115 (10), 2292-2307 .
- Palva, S. , Palva, J.M. , 2012. Discovering oscillatory interaction networks with M/EEG: challenges and breakthroughs. Trends Cogn. Sci. (Regul. Ed.) 16 (4), 219-230 .
- Pascual-Marqui, R.D. (2007a). Coherence and phase synchronization: generalization to pairs of multivariate time series, and removal of zero-lag contributions. arXiv preprint, 0706.1776 . · arXiv
- Pascual-Marqui, R.D. (2007b). Instantaneous and lagged measurements of linear and non- linear dependence between groups of multivariate time series: frequency decomposi- tion. arXiv preprint, 0711.1455 . · arXiv
- Ranganath, C. , Ritchey, M. , 2012. Two cortical systems for memory-guided behaviour. Nat. Rev. Neurosci. 13 (10), 713-726 .
- Rangaprakash, D. , Wu, G.R. , Marinazzo, D. , Hu, X. , Deshpande, G. , 2018. Hemodynamic response function (HRF) variability confounds resting-state fMRI functional connec- tivity. Magn. Reson. Med. 80 (4), 1697-1713 .
- Sanchez-Romero, R. , Cole, M.W. , 2019. Combining multiple functional connectivity meth- ods to improve causal inferences. bioRxiv .
- Stam, C.J. , Nolte, G. , Daffertshofer, A. , 2007. Phase lag index: assessment of functional connectivity from multi channel EEG and MEG with diminished bias from common sources. Hum. Brain Mapp. 28 (11), 1178-1193 .
- Stramaglia, S. , Cortes, J.M. , Marinazzo, D. , 2014. Synergy and redundancy in the Granger causal analysis of dynamical networks. New. J. Phys. 16 (10), 105003 .
- Taylor, J.R., Williams, N., Cusack, R., Auer, T., Shafto, M.A., Dixon, M., Tyler, L.K., Cam-CAN, Henson, R.N., 2017. The Cambridge Centre for Ageing and Neuro- science (Cam-CAN) data repository: structural and functional MRI, MEG, and cog- nitive data from a cross-sectional adult lifespan sample. Neuroimage 144, 262-269. doi: 10.1016/j.neuroimage.2015.09.018 . · DOI
- Theodoridis, S. , Koutroumbas, K. , 1999. Pattern recognition and neural networks. In: Ad- vanced Course on Artificial Intelligence. Springer, Berlin, Heidelberg, pp. 169-195 .
- Tibshirani, R. , 1996. Regression shrinkage and selection via the lasso. J. R. Stat. Soc.: Ser. B (Methodol.) 58 (1), 267-288 .
- Uurtio, V. , Monteiro, J.M. , Kandola, J. , Shawe-Taylor, J. , Fernandez-Reyes, D. , Rousu, J. , 2017. A tutorial on canonical correlation methods. ACM Comput. Surv. (CSUR) 50 (6), 1-33 .
- Vidaurre, D. , Hunt, L.T. , Quinn, A.J. , Hunt, B.A. , Brookes, M.J. , Nobre, A.C. , Wool- rich, M.W. , 2018. Spontaneous cortical activity transiently organises into frequency specific phase-coupling networks. Nat. Commun. 9 (1), 1-13 .
- Vinck, M. , Oostenveld, R. , Van Wingerden, M. , Battaglia, F. , Pennartz, C.M. , 2011. An improved index of phase-synchronization for electrophysiological data in the presence of volume-conduction, noise and sample-size bias. Neuroimage 55 (4), 1548-1565 .
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Ng, H.-Y. H., Hsu, A.-L., Wu, C. W., Huang, C.-M., Chao, Y.-P., Jung, T.-P., & Chuang, C.-H. (2026) · Mindfulness · doi:10.1007/s12671-025-02734-1
- Authors
- Basti, A., Nili, H., Hauk, O., Marzetti, L., & Henson, R. N.
- Affiliations
- University of Chieti-Pescara (IT)
- University of Oxford (GB)
- MRC Cognition and Brain Sciences Unit (GB)
- Corresponding author
- Basti, Alessio
- Nili, Hamed
- Venue
- NeuroImage
- Publisher
- Elsevier BV
- Published
- 2020-11
- Volume
- 221
- Pages
- 117179
- ISSN
- 1053-8119 (ISSN-L)
- OpenAlex
- W3043376887
- Cited by
- 48
- References
- 5
- Citations by year
- 2026: 3 · 2025: 3 · 2024: 8 · 2023: 10 · 2022: 7 · 2021: 15 · 2020: 2
- FWCI
- 1.15
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- NeuroImage (publishedVersion, cc-by-nc-nd)
- PubMed (publishedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- Apollo (University of Cambridge) (submittedVersion, cc-by-nc-nd)
- INFM-OAR (INFN Catania) (submittedVersion, cc-by)
- Apollo
- Funders
- Ministero dell’Istruzione, dell’Università e della Ricerca
- Medical Research Council (SUAG/058 G101400)
- Horizon 2020 (810377)
- European Commission
- European Research Council
- Topics
- Life Sciences › Neuroscience › Cognitive Neuroscience › Functional Brain Connectivity Studies
- Health Sciences › Medicine › Radiology, Nuclear Medicine and Imaging › Advanced Neuroimaging Techniques and Applications
- Social Sciences › Psychology › Experimental and Cognitive Psychology › Mental Health Research Topics
- Keywords
- functional connectivity, brain regions, statistical dependence, time series, functional MRI, MEG, dimensionality reduction
- MeSH terms
- Brain, Humans, Magnetic Resonance Imaging, Models, Theoretical, Magnetoencephalography, Connectome
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