Paths between Hypnotic States

This section defines a simplified state space for the ’ultrahigh-dimensional dynamical system’ of the human brain. The aim is to describe essential properties to map the dynamical regimes that are of interest for this study, namely the states of consciousness that correspond to different depths of anesthesia.

I derived the methodology used here from an approach that proved to be useful for the cou- pled map system in chapter 4. A symbolic representation was mapped into a low-dimensional description space using the distribution of the algebraic order. This results in a trajectory that represents the changes in the state of consciousness over the course of the experiment.

In an analogy to the coupled map lattice in chapter 4, I defined a neighborhood-relation between the EEG electrodes (see fig. 5.6), specifying between 2 and 6 neighbors for each elec- trode. Later, I will draw topograms, assigning the data corresponding to a pair of electrodes to the center of their connecting line. In order not to have different pairs assigned to the same po- sition, I avoided crossing edges, i. e. the neighborhood relation corresponds to a triangulation. For each pair of neighboring electrodes, the transfer symbol distribution (n = 5, τ = 6 ms) was calculated.

Fig. 5.5: Transfer symbol entropy (n= 5, τ = 6 ms) versus electrode distance for pairs of electrodes within different regions of the skull. a) Definition of the regions b) Comparison of front and back region c) Comparison of left and right region. The lighter curves mark the standard deviation (across probands) of the fitted curves, based on a bootstrap estimation. Continuous lines mark the baseline and dashed lines the unconsciousness regime, respectively.

Fig. 5.6: A neighbor relation was defined on the electrode grid, avoiding crossing edges.

In order to allow a visualization as a trajectory in a low-dimensional space, the algebraic order distribution of the transfer symbols was averaged over all neighboring pairs of electrodes. The symbols of sequence lengthn= 5 can have an algebraic order from 1 to 6. Since the relative frequencies add up to 1, the order distribution has 5 degrees of freedom. The description space is the 5-dimensionalaffine space spanned by the orders 1, . . . ,6 (however, the set of normalized order distributions is the convex hull of the orders). See section 4.4.1, especially fig. 4.18, for the corresponding description of the CML.

Since the following investigations used data from all regimes (see fig. 5.1), only the data from 10 probands could be used (inclusion criterion: 5 min of continuously recorded data in the baseline, unconscious and awake regime, respectively). Since the recovery process can be individual, the recording was stopped soon after the return of consciousness for certain probands. The investigations done so far only used data from the baseline and unconsciousness regimes, which is available for all 15 probands.

Fig. 5.7 shows 2-dimensional projections of the trajectory in the description space, based on overlapping sliding windows of 30 sec length. The regimes are color coded.

The baseline, unconsciousness and awake regimes are expected to be approximately station- ary (near steady state conditions). In the description space, they can be identified as compact clusters. The descent and recovery regimes were defined as transition between the supposably stationary states. Those transitions would resemble smooth (or at least monotonic) paths with added noise, which is in a reasonable agreement with fig. 5.7. The trajectory follows a corridor between the awake regime and the unconsciousness regime. The individuals differ mainly in the positions of the cluster centers.

The baseline regime was not always consistent with the expectations. Even though it was generally related to the more awake states, the connection from the baseline sample to the rest of the trajectory was not as smooth as expected. In contrast to the assumption of an approximately stationary state, the transition from the baseline to the descent regime was clearly discontinuous in some cases. The variation in the location of the baseline cluster between individuals was larger than for the other regimes, including a single case (#6) whose baseline regime had the same order distribution as it is typical for the unconsciousness regime. The observed discrepancy between the baseline and the rest of the data needs some ex- planation. It should be mentioned that the data was not continuously recorded at this point (the baseline data was separate). There was possibly a short time gap between the data of the baseline regime and the rest of the data (in the order of several minutes). However, when

Fig. 5.7:EEG trajectory in terms of the algebraic order distribution (for single probands, over all pairs of neighboring electrodes). The regimes are color coded, the values are based on overlapping sliding windows of 30 s length. a) Order 2 versus 1 b) Order 5 versus 3.

considering the variation over time, a gap in the data is not likely to fully explain the observed discrepancy. This observation suggests that the probands behaved more individually before the medication and more commonly under the influence of Propofol.

Some individuals differ fundamentally from the bulk of the group. For proband #10 for instance, the separation line between unconsciousness (green) and awake (red) in fig. 5.7a had a completely different direction, because the unconsciousness regime had more order 2 and less order 1 symbols than usual. Another example is the aforementioned proband #6, where baseline and unconsciousness were not distinguished at all.

In summary, I conclude that the proposed description space provides a useful mapping of the main features that distinguish the regimes addressed by this study.