Variational Bayesian Framework for Estimating Parameters of Integrated E/MEG and fMRI Model

The integrated analysis of the Electroencephalography (EEG), Magnetoencephalography (MEG), and functional magnetic resonance imaging (fMRI) are instrumental for functional neuroimaging of the brain. A bottom-up integrated E/MEG and fMRI model based on physiology as well as a method for estimating its parameters are keys to the integrated analysis. We propose the variational Bayesian expectation maximization (VBEM) method to estimate parameters of our proposed integrated model. VBEM method iteratively optimizes a lower bound on the marginal likelihood. An iteration of the VBEM consists of two steps: a variational Bayesian expectation step implemented using the extended Kalman smoother (EKS) and the posterior probability of the parameters in the previous step, and a variational Bayesian maximization step to estimate the posterior distributions of the parameters. For a given external stimulus, a variety of multi-area models can be considered in which the number of areas and the configuration and strength of connections between the areas are different. The proposed VBEM method can be used to select an optimal model as well as estimate its parameters. The efficiency of the proposed VBEM method is illustrated using simulation and real datasets. The proposed VBEM method can be used to estimate parameters of other non-linear dynamical systems. This study proposes an effective method to integrate E/MEG and fMRI and plans to use these techniques in functional neuroimaging.