Structure on Different Scales

In order to apply methods from coupled map lattices to EEG data, a first step is to provide ev- idence of spatial structure, i. e. investigate dependencies between the time series corresponding to different electrodes. This is a test of the premise that there are local synchronization effects, which is important to support the analogy between the coupled map lattice of the preceding chapter and the EEG, and to motivate the application of the methods used there.

For each pair of electrodes, I described the relation between the corresponding time series by a sequence of transfer symbols. The delayτ = 6 ms turned out to give the best distinction between the states of consciousness. This measure mainly describes the alpha, beta, and gamma band: for sequence length n = 5, the time horizon is 24 ms, which corresponds to about 1/4 period at 10 Hz (alpha band), or about a full period at 40 Hz (gamma band). Therefore, I expect the symbols to be mainly sensitive for the alpha,beta and gamma bands. Lower frequencies are probably manifest as trends, which I expect to be largely masked by high-frequency signals. A digital filter for frequencies from 0.5-50 Hz was used.

As a measure for the degree of interrelations between the signals, I use the entropy of the transfer symbols, leading to a scalar value for each pair of electrodes. High entropy suggests randomness and is expected for a pair of independent signals. Low entropy is an indication for (not necessarily linear) correlations.

The electrode positions on the skull were given in spherical coordinates, hence it is natural to measure the distance between two electrodes in degrees. The electrode distances for a certain proband depend on the geometry of the individual skull, which was not recorded. Fig. 5.3 shows the transfer symbol entropy as a function of the electrode distance. The gray line marks the

upper bound of the entropy at log₂(n!)≈6.9 bits. Curves were fitted by a polynomial of degree 4 for each individual.

For all probands, the dependence was highly similar. The scattering of the individuals was mostly due to the constant term of the polynomial, corresponding to an individual offset in entropy. In the unconsciousness regime, the offset was smaller and showed a larger scattering of the individuals. This suggests an individually different level of ’mean-field-synchronization’, but could possibly be related to individual differences in the power spectrum.

At short electrode distances, there were clear signs of structure. I observed a dip at distances around 135◦, most pronounced during unconsciousness, which possibly indicates long-range synchronization effects. The non-monotonic dependence on the distance indicates that this is not explained purely by neighboring electrodes sharing parts of their sources.

Fig. 5.3: Transfer symbol entropy (n= 5, τ = 6 ms) versus electrode distance. a) Baseline b) Uncon- sciousness. The lines show a polynomial fit of order 4 over all electrode pairs, performed for each of the 15 probands separately (the colors correspond to the different probands).

Next, I asked if the observed long-range correlation is related to the symmetry between the left and right hemisphere of the brain. In fig. 5.4, I separated the data into symmetric (the two electrodes are situated at corresponding positions of different hemispheres) and non-symmetric electrode pairs. Note that the fitted curves are based on different sampling points, which are relatively sparse in the symmetric case.

The green curves (non-symmetric pairs) are similar to the curves in fig. 5.3, because they are based on almost the same data. For the unconscious regime, the red curve (symmetric pairs) is monotonic, suggesting that the observed long-range synchronization effects are not directly related to symmetric pairs of electrodes. For the conscious state, the disparity between symmetric and non-symmetric pairs is not as distinct. As before, the states of consciousness mainly differ by an offset in entropy.

In order to study interrelations on a more local level, fig. 5.5 considers the same measure (transfer symbol entropy versus electrode distance), restricted to pairs of electrodes within selected regions of the scalp (as defined in fig. 5.5a). Since only electrodes within the same region are considered, the range of distances is smaller than in the previous figures.

Fig. 5.3b and c compare regions with identical electrode geometry, namely the front and back region, and the left and right region. For these pairs of regions, the polynomial fits are based on the same set of sampling points, hence it seems reasonable to compare them pointwise.

Fig. 5.4: Transfer symbol entropy (n= 5, τ = 6 ms) versus electrode distance. a) Baseline b) Uncon- sciousness. Polynomial fit of order 4 to the data of the symmetric electrode pairs (red) and of all other pairs (green).

I estimated the variation among the probands using a bootstrap approximation: several curves were fitted to samples of 15 probands that were randomly drawn with replacement. I used the curves to estimate the standard deviation of the fit curve, which is shown in the form of error channels. The qualitative behavior of the examined regions was reproduced well between the baseline and unconsciousness regimes. Generally, the entropy was lower for unconsciousness. This may be due to the aforementioned differences in the power spectra of both populations. Between the front and back region, the fit curves indicate a considerable difference in the structure. The front region had a monotonic dependence on the distance. The back region followed a more complex relation, suggesting long-range dependencies. The left and right regions were similar, yet there might be a systematic difference for short electrode distance, suggesting more structure in the symbolic description of the left hemisphere.