The Annals of Applied Statistics

A two-way regularization method for MEG source reconstruction

Tian Siva Tian, Jianhua Z. Huang, Haipeng Shen, and Zhimin Li

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The MEG inverse problem refers to the reconstruction of the neural activity of the brain from magnetoencephalography (MEG) measurements. We propose a two-way regularization (TWR) method to solve the MEG inverse problem under the assumptions that only a small number of locations in space are responsible for the measured signals (focality), and each source time course is smooth in time (smoothness). The focality and smoothness of the reconstructed signals are ensured respectively by imposing a sparsity-inducing penalty and a roughness penalty in the data fitting criterion. A two-stage algorithm is developed for fast computation, where a raw estimate of the source time course is obtained in the first stage and then refined in the second stage by the two-way regularization. The proposed method is shown to be effective on both synthetic and real-world examples.

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Ann. Appl. Stat., Volume 6, Number 3 (2012), 1021-1046.

First available in Project Euclid: 31 August 2012

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Inverse problem MEG two-way regularization spatio-temporal


Tian, Tian Siva; Huang, Jianhua Z.; Shen, Haipeng; Li, Zhimin. A two-way regularization method for MEG source reconstruction. Ann. Appl. Stat. 6 (2012), no. 3, 1021--1046. doi:10.1214/11-AOAS531.

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  • Auranen, T., Nummenmaa, A., Hämäläinen, M. S., Jääskeläinen, I. P., Lampinen, J., Vehtari, A. and Sams, M. (2005). Bayesian analysis of the neuromagnetic inverse problem with lp-norm priors. NeuroImage 26 870–884.
  • Baillet, S. and Garnero, L. (1997). A Bayesian approach to introducing anatomo-functional prior in the EEG/MEG inverse problem. IEEE Transactions on Biomedical Engineering 44 374–385.
  • Baillet, S., Riera, J. J., Marin, G., Mangin, J. F., Aubert, J. and Garnero, L. (2001). Evaluation of inverse methods and head models for EEG source localization using a human skull phantom. Phys. Med. Biol. 46 77–96.
  • Barlow, H. B. (1994). What is the computational goal of the neocortex? In Large-Scale Neuronal Theories of the Brain (C. Koch and J. L. Davis, eds.) 1–22. MIT press, Cambridge, MA.
  • Bolstad, A., Veen, B. V. and Nowak, R. (2009). Space-time event sparse penalization for magneto-/electroencephalography. NeuroImage 46 1066–1081.
  • Brent, R. P. (1973). Algorithms for Minimization Without Derivatives. Prentice-Hall Inc., Englewood Cliffs, NJ.
  • Darvas, F., Pantazis, D., Kucukaltun-Yildirim, E. and Leahy, R. M. (2004). Mapping human brain function with MEG and EEG: Methods and validation. NeoroImage 23 289–299.
  • Daunizeau, J., Mattout, J., Clonda, D., Goulard, B., Benali, H. and Lina, J.-M. (2006). Bayesian spatio-temporal approach for EEG source reconstruction: Conciliating ECD and distributed models. IEEE Trans. Biomed. Eng. 53 503–516.
  • Ding, L. and He, B. (2008). Sparse source imaging in electroencephalography with accurate field modeling. Hum. Brain Mapp 29 1053–1067.
  • Dogandžić, A. and Nehorai, A. (2000). Estimating evoked dipole responses in unknown spatially correlated noise with EEG/MEG arrays. IEEE Trans. Signal Process. 48 13–25.
  • Friston, K., Harrison, L., Daunizeau, J., Kiebel, S., Phillips, C., Trujillo-Barreto, N., Henson, R., Flandin, G. and Mattout, J. (2008). Multiple sparse priors for the M/EEG inverse problem. NeuroImage 39 1104–1120.
  • Gorodnitsky, I. F. and Rao, B. D. (1997). Sparse signal reconstruction from limited data using FOCUSS: A re-weighted minimum norm algorithm. IEEE Transactions on Signal Processing 45 600–616.
  • Hämäläinen, M. and Ilmoniemi, R. J. (1994). Interpreting measured magnetic fields of the brain: Estimates of current distribution. Technical Report TKK-F-A599, Helsinki Univ. Technology.
  • Hämäläinen, M., Hari, R., Ilmoniemi, R. J., Knuutila, J. and Lounasmaa, O. V. (1993). Magnetoencephalography-theory, instrumentation, and applications to noninvasive studies of the working human brain. Rev. Modern Phys. 65 413–497.
  • Huang, J. Z., Shen, H. and Buja, A. (2008). Functional principal components analysis via penalized rank one approximation. Electron. J. Stat. 2 678–695.
  • Huang, J. Z., Shen, H. and Buja, A. (2009). The analysis of two-way functional data using two-way regularized singular value decompositions. J. Amer. Statist. Assoc. 104 1609–1620.
  • Jeffs, B., Leahy, R. and Singh, M. (1987). An evaluation of methods for neuromagnetic image reconstruction. IEEE Trans. Biomed. Eng. 34 713–723.
  • Jun, S. C., George, J. S., Paŕe-Blagoev, J., Plis, S. M., Ranken, D. M., Schmidt, D. M. and Wood, C. C. (2005). Spatiotemporal Bayesian inference dipole analysis for MEG neuroimaging data. NeuroImage 29 84–98.
  • Lee, M., Shen, H., Huang, J. Z. and Marron, J. S. (2010). Biclustering via sparse singular value decomposition. Biometrics 66 1087–1095.
  • Lin, F.-H., Belliveau, J. W., Dale, A. M. and Hämäläinen, M. S. (2006). Distributed current estimates using cortical orientation constraints. Hum. Brain Mapp 27 1–13.
  • Lu, Z. and Kaufman, L. (2003). Magnetic Source Imaging of the Human Brain. awrence Erlbaum Associates, Inc., Manwah, New Jersey.
  • Matsuura, K. and Okabe, Y. (1995). Selective minimum-norm solution of the biomagnetic inverse problem. IEEE Trans. Biomed. Eng. 42 608–615.
  • Mosher, J. C., Leahy, R. M. and Lewis, P. S. (1999). EEG and MEG: Forward solutions for inverse methods. IEEE Trans. Biomed. Eng. 46 245–259.
  • Mosher, J. C., Lewis, P. S. and Leahy, R. M. (1992). Multiple dipole modeling and localization from spatio-temporal MEG data. IEEE Trans. Biomed. Eng. 39 541–557.
  • Nummenmaa, A., Auranen, T., Hämäläinen, M. S., Jääskeläinen, I. P., Lampinen, J., Sams, M. and Vehtari, A. (2007a). Hierarchical Bayesian estimates of distributed MEG sources: Theoretical aspects and comparison of variational and MCMC methods. Neuroimage 35 669–685.
  • Nummenmaa, A., Auranen, T., Vanni, S., Hämäläinen, M. S., Jääskeläinen, I. P., Lampinen, J., Vehtari, A. and Sams, M. (2007b). Sparse MEG inverse solutions via hierarchical Bayesian modeling: Evaluation with a parallel fMRI study Technical Report B65, Laboratory of Computational Engineering, Helsinki Univ. Technology, Helsinki, Finland.
  • Nunez, P. L. (1981). Electric Fields of the Brain: The Neurophysics of EEG. Oxford Univ. Press, New York, NY.
  • Ou, W., Hämäläinen, M. S. and Golland, P. (2009). A distributed spatio-temporal EEG/MEG inverse solver. Neuroimage 44 932–946.
  • Papanicolaou, A. C. (1995). An introduction to magnetoencephalography with some applications. Brain Cogn. 27 331–352.
  • Pascual-Marqui, R. D. (2002). Standardized low-resolution brain electromagnetic tomography (sLORETA): Technical details. Methods & Findings in Experimental & Clinical Pharmacology 24 5–12.
  • Pascual-Marqui, R. D., Michel, C. M. and Lehmann, D. (1994). Low resolution electromagnetic tomography: A new method for localizing electrical activity in the brain. Int. J. Psychophysiol. 18 49–65.
  • Sarvas, J. (1987). Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. Phys. Med. Biol. 32 11–22.
  • Scherg, M. and Von Cramon, D. (1986). Evoked dipole source potentials of the human auditory cortex. Electroencephalogr. Clin. Neurophysiol. 65 344–360.
  • Schmidt, R. O. (1986). Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas and Propagation 43 276–280.
  • Shen, H. and Huang, J. Z. (2008). Sparse principal component analysis via regularized low rank matrix approximation. J. Multivariate Anal. 99 1015–1034.
  • Sorrentino, A., Parkkonen, L., Pascarella, A., Campi, C. and Piana, M. (2009). Dynamical MEG source modeling with multi-target Bayesian filtering. Hum. Brain Mapp 30 1911–1921.
  • Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 267–288.
  • Uutela, K., Hämäläinen, M. and Somersalo, E. (1999). Visualization of magnetoencephalographic data using minimum current estimates. Neuroimage 10 173–180.
  • VanVeen, B., van Drongelen, W., Yuchtman, M. and Suzuki, A. (1997). Localization of brain electrical activity via linearly constrained minimum variance spatial filtering. IEEE Transactions on Biomedical Engineering 44 867–880.
  • Veen, B. D. V. and Buckley, K. M. (1988). Beamforming: A versatile approach to spatial filtering. IEEE ASSP Magazine 5 4–24.
  • von Helmholtz, H. (1853). Ueber einige Gesetze der Vertheilung elektrischer Ströme in körperlichen Leitern mit Anwendung auf die thierisch-elektrischen Versuche. Ann. Phys. 165 211–233.
  • Witten, D. M., Tibshirani, R. and Hastie, T. (2009). A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics 10 515–534.
  • Yamazaki, T., Kamijo, K., Kenmochi, A., Fukuzumi, S., Kiyuna, T., Takaki, Y. and Kuroiwa, Y. (2000). Multiple equivalent current dipole source localization of visual event-related potentials during oddball paradigm with motor response. Brain Topogr. 12 159–175.