The Annals of Applied Statistics

A statistical approach to the inverse problem in magnetoencephalography

Zhigang Yao and William F. Eddy

Full-text: Open access


Magnetoencephalography (MEG) is an imaging technique used to measure the magnetic field outside the human head produced by the electrical activity inside the brain. The MEG inverse problem, identifying the location of the electrical sources from the magnetic signal measurements, is ill-posed, that is, there are an infinite number of mathematically correct solutions. Common source localization methods assume the source does not vary with time and do not provide estimates of the variability of the fitted model. Here, we reformulate the MEG inverse problem by considering time-varying locations for the sources and their electrical moments and we model their time evolution using a state space model. Based on our predictive model, we investigate the inverse problem by finding the posterior source distribution given the multiple channels of observations at each time rather than fitting fixed source parameters. Our new model is more realistic than common models and allows us to estimate the variation of the strength, orientation and position. We propose two new Monte Carlo methods based on sequential importance sampling. Unlike the usual MCMC sampling scheme, our new methods work in this situation without needing to tune a high-dimensional transition kernel which has a very high cost. The dimensionality of the unknown parameters is extremely large and the size of the data is even larger. We use Parallel Virtual Machine (PVM) to speed up the computation.

Article information

Ann. Appl. Stat., Volume 8, Number 2 (2014), 1119-1144.

First available in Project Euclid: 1 July 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Ill-posed problem sequential importance sampling state space model parallel computing source localization


Yao, Zhigang; Eddy, William F. A statistical approach to the inverse problem in magnetoencephalography. Ann. Appl. Stat. 8 (2014), no. 2, 1119--1144. doi:10.1214/14-AOAS716.

Export citation


  • Andrieu, C. and Thoms, J. (2008). A tutorial on adaptive MCMC. Stat. Comput. 18 343–373.
  • Bai, X. and He, B. (2006). Estimation of number of independent brain electric sources from the scalp EEGs. IEEE Trans. Biomed. Eng. 53 1883–1892.
  • Barkley, G. L. and Baumgartner, C. (2003). MEG and EEG in epilepsy. J. Clin. Neurophysiol. 20 163–178.
  • Bertrand, C., Ohmi, M., Suzuki, R. and Kado, H. (2001). A probabilistic solution to the MEG inverse problem via MCMC methods: The reversible jump and parallel tempering algorithms. IEEE Trans. Biomed. Eng. 48 533–542.
  • Berzuini, C., Best, N. G., Gilks, W. R. and Larizza, C. (1997). Dynamic conditional independence models and Markov chain Monte Carlo methods. J. Amer. Statist. Assoc. 92 1403–1412.
  • Campi, C., Pascarella, A., Sorrentino, A. and Piana, M. (2008). A Rao–Blackwellized particle filter for magnetoencephalography. Inverse Problems 24 25023–25037.
  • Campi, C., Pascarella, A., Sorrentino, A. and Piana, M. (2011). Highly automated dipole estimation. Computational Intelligence and Neuroscience 2011 Article ID 982185, 11 pp.
  • Carlin, B. P., Polson, N. G. and Stoffer, D. S. (1992). A Monte Carlo approach to nonnormal and nonlinear state-space modeling. J. Amer. Statist. Assoc. 87 493–500.
  • Carter, C. K. and Kohn, R. (1994). On Gibbs sampling for state space models. Biometrika 81 541–553.
  • Cohen, D. (1968). Magnetoencephalography: Evidence of magnetic fields produced by alpha-rhythm currents. Science 161 784–786.
  • Del Moral, P., Doucet, A. and Jasra, A. (2006). Sequential Monte Carlo samplers. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 411–436.
  • Fearnhead, P. (2008). Computational methods for complex stochastic systems: A review of some alternatives to MCMC. Stat. Comput. 18 151–171.
  • Gamerman, D. (1998). Markov chain Monte Carlo for dynamic generalised linear models. Biometrika 85 215–227.
  • Geist, A., Beguelin, A., Dongarra, J., Jiang, W., Manchek, R. and Sunderam, V. S. (1994). PVM: Parallel Virtual Machine: A Users’ Guide and Tutorial for Network Parallel Computing (Scientific and Engineering Computation). MIT Press, Cambridge, MA.
  • Gelman, A., Roberts, G. O. and Gilks, W. R. (1996). Efficient Metropolis jumping rules. In Bayesian Statistics 599–607. Oxford Univ. Press, New York.
  • Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statist. Sci. 7 457–511.
  • Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In Bayesian Statistics 169–193. Oxford Univ. Press, New York.
  • Gordon, N. J., Salmond, D. J. and Smith, A. F. M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings F (Radar and Signal Processing) 140 107–113.
  • Griffiths, D. J. (1999). Introduction to Electrodynamics. Prentice Hall, New York.
  • Hämäläinen, M. S. and Ilmoniemi, R. J. (1994). Interpreting magnetic fields of the brain: Minimum norm estimates. Med. Biol. Eng. Comput. 32 35–42.
  • Hämäläinen, M. S., Hari, R., Ilmoniemi, R. J., Knuutila, J. and Lounasmaa, O. V. (1993). Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of signal processing in the human brain. Rev. Modern Phys. 65 413–497.
  • Heidelberger, P. and Welch, P. D. (1983). Simulation run length control in the presence of an initial transient. Oper. Res. 31 1109–1144.
  • Jun, S. C., George, J. S., Paré-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 28 84–98.
  • Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. J. Comput. Graph. Statist. 5 1–25.
  • Knorr-Held, L. (1999). Conditional prior proposals in dynamic models. Scand. J. Stat. 26 129–144.
  • Kristeva-Feige, R., Rossi, S., Feige, B., Mergner, Th., Lucking, C. H. and Rossini, P. M. (1997). The bereitschaftspotential paradigm in investigating voluntary movement organization in humans using magnetoencephalography (MEG). Brain Res. Protoc. 1 13–22.
  • Kybic, J., Clerc, M., Faugeras, O., Keriven, R. and Papadopoulo, T. (2006). Generalized head models for MEG/EEG: Boundary element method beyond nested volumes. Phys. Med. Biol. 51 1333–1346.
  • Liu, J. S. (1996). Metropolized independent sampling with comparisons to rejection sampling and importance sampling. Statist. Comput. 6 113–119.
  • Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc. 93 1032–1044.
  • Mattout, J., Phillips, C., Penny, W. D., Rugg, M. D. and Friston, K. J. (2006). MEG source localization under multiple constraints: An extended Bayesian framework. NeuroImage 30 753–767.
  • Miao, L., Michael, S., Kovvali, N., Chakrabarti, C. and Papandreou-Suppappola, A. (2013). Multi-source neural activity estimation and sensor scheduling: Algorithms and hardware implementation. Journal of Signal Processing Systems 70 145–162.
  • 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.
  • Neal, R. M. (2001). Annealed importance sampling. Stat. Comput. 11 125–139.
  • Okada, Y., Lähteenmäki, A. and Xu, C. (1999). Comparison of MEG and EEG on the basis of somatic evoked responses elicited by stimulation of the snout in the juvenile swine. Clin. Neurophysiol. 110 214–229.
  • Ou, W., Hämäläinen, M. S. and Golland, P. (2009). A distributed spatio-temporal EEG/MEG inverse solver. NeuroImage 44 932–946.
  • Raftery, A. E. and Lewis, S. M. (1992). One long run with diagnostics: Implementation strategies for Markov chain Monte Carlo. Statist. Sci. 7 493–497.
  • Roberts, G. O. and Rosenthal, J. S. (2009). Examples of adaptive MCMC. J. Comput. Graph. Statist. 18 349–367.
  • Sarvas, J. (1984). Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. Phys. Med. Biol. 32 11–22.
  • Schmidt, D. M., George, J. S. and Wood, C. C. (1999). Bayesian inference applied to the electromagnet inverse problem. Hum. Brain Mapp. 7 195–212.
  • Shephard, N. and Pitt, M. K. (1997). Likelihood analysis of non-Gaussian measurement time series. Biometrika 84 653–667.
  • Somersalo, E., Voutilainen, A. and Kaipio, J. P. (2003). Non-stationary magnetoencephalography by Bayesian filtering of dipole models. Inverse Problems 19 1047–1063.
  • 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.
  • Sorrentino, A., Johansen, A. M., Aston, J. A. D., Nichols, T. E. and Kendall, W. S. (2013). Dynamic filtering of static dipoles in magnetoencephalography. Ann. Appl. Stat. 7 955–988.
  • Srinivasan, R. (2002). Importance Sampling: Applications in Communications and Detection. Springer, Berlin.
  • Uutela, K., Hämäläinen, M. S. and Somersalo, E. (1999). Visualization of magnetoencephalographic data using minimum current estimates. NeuroImage 10 173–180.
  • Van Veen, B., Joseph, J. and Hecox, K. (1992). Localization of intra-cerebral sources of electrical activity via linearly constrained minimum variance spatial filtering. In Proc. IEEE 6th SP Workshop on Statistical Signal and Array Processing 526–529. Victoria, BC.
  • Waldorp, L. J., Huizenga, H. M., Nehorai, A., Grasman, R. P. P. P. and Molenaar, P. C. M. (2005). Model selection in spatio-temporal electromagnetic source analysis. IEEE Trans. Biomed. Eng. 52 414–420.
  • Wang, W., Sudre, G. P., Xu, Y., Kass, R. E., Collinger, J. L., Degenhart, A. D., Bagic, A. I. and Weber, D. J. (2010). Decoding and cortical source localization for intended movement direction with MEG. J. Neurophysiol. 104 2451–2461.
  • Yao, Z. and Eddy, W. F. (2012). Statistical approaches to estimating the number of signal sources in magnetoencephalography. Unpublished manuscript.