We study the distribution of brain source from the most advanced brain imaging technique, Magnetoencephalography (MEG) which measures the magnetic fields outside of the human head produced by the electrical activity inside the brain. Common time-varying source localization methods assume the source current with a time-varying structure and solve the MEG inverse problem by mainly estimating the source moment parameters. These methods use the fact that the magnetic fields linearly depend on the moment parameters of the source and work well under the linear dynamic system. However, magnetic fields are known to be nonlinearly related to the location parameters of the source. The existing work on estimating the time-varying unknown location parameters is limited. We are motivated to investigate the source distribution for the location parameters based on a dynamic framework, where the posterior distribution of the source is computed in a closed form discretely. The new framework allows us not only to directly approximate the posterior distribution of the source current, where sequential sampling methods may suffer from slow convergence due to the large volume of measurement, but also to quantify the source distribution at any time point from the entire set of measurements reflecting the distribution of the source, rather than using only the measurements up to the time point of interest. Both a dynamic procedure and a switch procedure are pro- posed for the new discrete approach, balancing estimation accuracy and computational efficiency when multiple sources are present. In both simulation and real data, we illustrate that the new method is able to provide comprehensive insight into the time evolution of the sources at different stages of the MEG and EEG experiment.
Ann. Appl. Stat.
14(3):
1379-1408
(September 2020).
DOI: 10.1214/19-AOAS1321
Arulampalam, M. S., Maskell, S., Gordon, N. and Clapp, T. (2002). A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Trans. Signal Process. 50 174–188.Arulampalam, M. S., Maskell, S., Gordon, N. and Clapp, T. (2002). A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Trans. Signal Process. 50 174–188.
Baillet, S. and Garnero, L. (1997). A Bayesian approach to introducing anatomo-functional priors in the EEG/MEG inverse problem. IEEE Trans. Biomed. Eng. 44 374–385.Baillet, S. and Garnero, L. (1997). A Bayesian approach to introducing anatomo-functional priors in the EEG/MEG inverse problem. IEEE Trans. Biomed. Eng. 44 374–385.
Boto, E., Holmes, N., Leggett, J., Roberts, G., Shah, V., Meyer, S. S., Muñoz, L. D., Mullinger, K. J., Tierney, T. M. et al. (2018). Moving magnetoencephalography towards real-world applications with a wearable system. Nature 555 657.Boto, E., Holmes, N., Leggett, J., Roberts, G., Shah, V., Meyer, S. S., Muñoz, L. D., Mullinger, K. J., Tierney, T. M. et al. (2018). Moving magnetoencephalography towards real-world applications with a wearable system. Nature 555 657.
Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39 1–38. 0364.62022 10.1111/j.2517-6161.1977.tb01600.xDempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39 1–38. 0364.62022 10.1111/j.2517-6161.1977.tb01600.x
Destexhe, A., Contreras, D. and Steriade, M. (1999). Spatiotemporal analysis of local field potentials and unit discharges in cat cerebral cortex during natural wake and sleep states. J. Neurosci. 19 4595–4608.Destexhe, A., Contreras, D. and Steriade, M. (1999). Spatiotemporal analysis of local field potentials and unit discharges in cat cerebral cortex during natural wake and sleep states. J. Neurosci. 19 4595–4608.
Felzenszwalb, P. F., Huttenlocher, D. P. and Kleinberg, J. M. (2004). Fast algorithms for large-state-space HMMs with applications to web usage analysis. In Advances in Neural Information Processing Systems 409–416.Felzenszwalb, P. F., Huttenlocher, D. P. and Kleinberg, J. M. (2004). Fast algorithms for large-state-space HMMs with applications to web usage analysis. In Advances in Neural Information Processing Systems 409–416.
Fukushima, M., Yamashita, O., Knösche, T. R. and Sato, M. (2015). MEG source reconstruction based on identification of directed source interactions on whole-brain anatomical networks. NeuroImage 105 408–427.Fukushima, M., Yamashita, O., Knösche, T. R. and Sato, M. (2015). MEG source reconstruction based on identification of directed source interactions on whole-brain anatomical networks. NeuroImage 105 408–427.
Hämäläinen, M. S., 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.Hämäläinen, M. S., 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.
Lamus, C., Hämäläinen, M. S., Temereanca, S., Brown, E. N. and Purdon, P. L. (2012). A spatiotemporal dynamic distributed solution to the MEG inverse problem. NeuroImage 63 894–909.Lamus, C., Hämäläinen, M. S., Temereanca, S., Brown, E. N. and Purdon, P. L. (2012). A spatiotemporal dynamic distributed solution to the MEG inverse problem. NeuroImage 63 894–909.
Lin, F.-H., Witzel, T., Ahlfors, S. P., Stufflebeam, S. M., Belliveau, J. W. and Hämäläinen, M. S. (2006). Assessing and improving the spatial accuracy in MEG source localization by depth-weighted minimum-norm estimates. NeuroImage 31 160–171.Lin, F.-H., Witzel, T., Ahlfors, S. P., Stufflebeam, S. M., Belliveau, J. W. and Hämäläinen, M. S. (2006). Assessing and improving the spatial accuracy in MEG source localization by depth-weighted minimum-norm estimates. NeuroImage 31 160–171.
Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc. 93 1032–1044. 1064.65500 10.1080/01621459.1998.10473765Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc. 93 1032–1044. 1064.65500 10.1080/01621459.1998.10473765
Liu, S., Poh, J.-H., Koh, H. L., Ng, K. K., Loke, Y. M., Lim, J. K. W., Chong, J. S. X. and Zhou, J. (2018). Carrying the past to the future: Distinct brain networks underlie individual differences in human spatial working memory capacity. NeuroImage 176 1–10.Liu, S., Poh, J.-H., Koh, H. L., Ng, K. K., Loke, Y. M., Lim, J. K. W., Chong, J. S. X. and Zhou, J. (2018). Carrying the past to the future: Distinct brain networks underlie individual differences in human spatial working memory capacity. NeuroImage 176 1–10.
Long, C. J., Purdon, P. L., Temereanca, S., Desai, N. U., Hämäläinen, M. S. and Brown, E. N. (2011). State-space solutions to the dynamic magnetoencephalography inverse problem using high performance computing. Ann. Appl. Stat. 5 1207–1228. 1223.62160 10.1214/11-AOAS483 euclid.aoas/1310562719Long, C. J., Purdon, P. L., Temereanca, S., Desai, N. U., Hämäläinen, M. S. and Brown, E. N. (2011). State-space solutions to the dynamic magnetoencephalography inverse problem using high performance computing. Ann. Appl. Stat. 5 1207–1228. 1223.62160 10.1214/11-AOAS483 euclid.aoas/1310562719
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.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.
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.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.
Trujillo-Barreto, N. J., Aubert-Vázquez, E. and Penny, W. D. (2008). Bayesian M/EEG source reconstruction with spatio-temporal priors. NeuroImage 39 318–335.Trujillo-Barreto, N. J., Aubert-Vázquez, E. and Penny, W. D. (2008). Bayesian M/EEG source reconstruction with spatio-temporal priors. NeuroImage 39 318–335.
Uutela, K., Hämäläinen, M. and Somersalo, E. (1999). Visualization of magnetoencephalographic data using minimum current estimates. NeuroImage 10 173–180.Uutela, K., Hämäläinen, M. and Somersalo, E. (1999). Visualization of magnetoencephalographic data using minimum current estimates. NeuroImage 10 173–180.
Vaida, F. (2005). Parameter convergence for EM and MM algorithms. Statist. Sinica 15 831–840. 1087.62035Vaida, F. (2005). Parameter convergence for EM and MM algorithms. Statist. Sinica 15 831–840. 1087.62035
Veen, B., Joseph, J. and Hecox, K. (1992). Localization of intra-cerebral sources of electrical activity via linearly constrained minimum variance spatial filtering. In Proceedings of IEEE Workshop on Statistical Signal and Array Processing 1 526–529.Veen, B., Joseph, J. and Hecox, K. (1992). Localization of intra-cerebral sources of electrical activity via linearly constrained minimum variance spatial filtering. In Proceedings of IEEE Workshop on Statistical Signal and Array Processing 1 526–529.
Yao, Z. and Eddy, W. F. (2014). A statistical approach to the inverse problem in magnetoencephalography. Ann. Appl. Stat. 8 1119–1144. 06333790 10.1214/14-AOAS716 euclid.aoas/1404229528Yao, Z. and Eddy, W. F. (2014). A statistical approach to the inverse problem in magnetoencephalography. Ann. Appl. Stat. 8 1119–1144. 06333790 10.1214/14-AOAS716 euclid.aoas/1404229528
Yao, Z., Zhang, Y., Bai, Z. and Eddy, W. F. (2018). Estimating the number of sources in magnetoencephalography using spiked population eigenvalues. J. Amer. Statist. Assoc. 113 505–518. 1398.62353 10.1080/01621459.2017.1341411Yao, Z., Zhang, Y., Bai, Z. and Eddy, W. F. (2018). Estimating the number of sources in magnetoencephalography using spiked population eigenvalues. J. Amer. Statist. Assoc. 113 505–518. 1398.62353 10.1080/01621459.2017.1341411
Yao, Z., Fan, Z., Hayashi, M. and Eddy, W. F. (2020). Supplement to “Quantifying time-varying sources in magnetoencephalography—A discrete approach.” https://doi.org/10.1214/19-AOAS1321SUPPYao, Z., Fan, Z., Hayashi, M. and Eddy, W. F. (2020). Supplement to “Quantifying time-varying sources in magnetoencephalography—A discrete approach.” https://doi.org/10.1214/19-AOAS1321SUPP
Zhang, J. and Liu, C. (2015). On linearly constrained minimum variance beamforming. J. Mach. Learn. Res. 16 2099–2145. 1351.94016Zhang, J. and Liu, C. (2015). On linearly constrained minimum variance beamforming. J. Mach. Learn. Res. 16 2099–2145. 1351.94016