The model

In blind source separation, the original independent sources are assumed to be unknown, and we only have access to their weighted sum. In this model, the signals recorded in an MEG study are noted as x_k(i) (i ranging from 1 to L, the number of sensors used, and k denoting discrete time); see Fig. . Each x_k(i) is expressed as the weighted sum of M independent signals s_k(j), following the vector expression:   

$${\bf x}_k = \sum_{j=1}^M {\bf a}(j)s_k(j) = {\bf As}_k,$$ (1)

where ${\bf x}_k = [x_k(1), \dots, x_k(L)]^T$ is an L-dimensional data vector, made up of the L mixtures at discrete time k. The $s_k(1), \dots, s_k(M)$ are the M zero mean independent source signals, and ${\bf A} = [{\bf a}(1), \dots, {\bf a}(M)]$ is a mixing matrix independent of time whose elements a_ij are the unknown coefficients of the mixtures. In order to perform ICA, it is necessary to have at least as many mixtures as there are independent sources ($L \geq M$). When this relation is not fully guaranteed, and the dimensionality of the problem is high enough, we should expect the first independent components to present clearly the most strongly independent signals, while the last components still consist of mixtures of the remaining signals. In our study, we did expect that the artifacts, being clearly independent from the brain activity, should come out in the first independent components. The remaining of the brain activity (e.g. $\alpha$ and $\mu$ rhythms) may need some further processing.

The mixing matrix ${\bf A}$ is a function of the geometry of the sources and the electrical conductivities of the brain, cerebrospinal fluid, skull and scalp. Although this matrix is unknown, we assume it to be constant, or slowly changing (to preserve some local constancy).

The problem is now to estimate the independent signals s_k (j) from their mixtures, or the equivalent problem of finding the separating matrix ${\bf B}$ that satisfies (see Eq. )  

$${\bf \hat{s}}_k = {\bf B}{\bf x}_k.$$ (2)

In our algorithm, the solution uses the statistical definition of fourth-order cumulant or kurtosis that, for the ith source signal, is defined as

$$kurt(s(i)) = E\{s(i)^4\} - 3[E\{s(i)^2\}]^2,$$

where E(s) denotes the mathematical expectation of s.