Figure 2.

Full pseudocode description for multi-subject, multi-modal MNM inversion based on spm_eeg_invert. m in SPM8. For details of the ReML algorithm and its precise updates using a Fisher-Scoring ascent on free-energy cost function, see Friston et al. (2008). (Note that an alternative “greedy search” algorithm is used for maximizing the free-energy in the case of MSP inversions.) The two main stages are: (1) optimizing the source component hyperparameters over subjects (by re-referencing each subject's gain matrix and data to an “average” space), and (2) optimizing the sensor and source hyperparameters separately for each subject, using the single optimized source component from stage 1. For a single subject, the first stage has negligible effect. Note also that this example assumes a single set of k trials: the full code allows for different numbers of trials for different numbers of conditions. We have also ignored any filtering or temporal whitening of the data (see Friston et al., 2006, for more details). ⁺ refers to the pseudoinverse; cat_j refers to the vertical concatenation of a vector or matrix along the j-th dimension; tr refers to the trace of a matrix; X = svd(Y, m) refers to a single-value decomposition of the matrix Y in order to produce the matrix X containing the m singular vectors with the highest singular values. Note that these update rules are generic – i.e., apply to all Applications in this paper – all that changed across Applications was the choice of data (Y) and covariance components (Q), as illustrated in the generative models shown in subsequent Figures.