Identifying interactions in mixed and noisy complex systems

We present a technique that identifies truly interacting subsystems of a complex system from multichannel data if the recordings are an unknown linear and instantaneous mixture of the true sources. The method is valid for arbitrary noise structure. For this, a blind source separation technique is proposed that diagonalizes antisymmetrized cross-correlation or cross-spectral matrices. The resulting decomposition finds truly interacting subsystems blindly and suppresses any spurious interaction stemming from the mixture. The usefulness of this interacting source analysis is demonstrated in simulations and for real electroencephalography data.

Figure 1

(Color online) Each row contains results for two different simulations shown as full and dashed lines, respectively. Top: coupled Rössler system for the noisy vs noise-free case; middle: coupled vs uncoupled Rössler system with noise added; bottom: same as middle row using now normalized cross-spectra. The panels in the left column show power averaged over four channels. The panels in the middle (right) column show the corresponding diagonals of the first (second) ISA component after complex diagonalization for each data set.

Figure 2

(Color online) Power as a function of electrode location for two frequencies, and as function of frequency for two selected channels as indicated.

Figure 3

(Color online) Left: Two basis fields (top and middle) and diagonal spectrum of component corresponding to occipital $\alpha$. Right: same for central mu rhythm.

Figure 4

(Color online) Spatial patterns of $\mu$ components calculated from the first half of the data (top and middle row) and $\text{spectra}\pm \text{twice}$ the SEM indicated in red calculated from the second half of the data (bottom row). The peaks at $12\mathrm{Hz}$ differ from zero by 9.42 and 11.43 standard errors of the mean, respectively. The corresponding $p$ values are below ${10}^{-15}$.