Lempel-Ziv complexity is also key for the perturbational complexity index (PCI), a measure that quantifies the diversity of the brain response activity to magnetic perturbation. PCI is colloquially known as "zap and zip", since first the brain is magnetically "zapped" then the response activity’s diversity assessed by Lempel-Ziv compressibility, i.e. "zipped".
PCI’s most robust indication of levels of consciousness (Section 1.4.9 listed global states of consciousness successfully indexed by PCI) while at the same time being interpreted to capture brain activity patterns that reflect phenomenological features of "differentiation"
and "integration" made us investigate multidimensional spontaneous signal diversity. The comparison of spontaneous and perturbational signal diversity as "explanatory"3correlates of consciousness is a central question of this thesis. The following technical details about PCI further apply for our computation of PCI for a Stuart-Landau oscillator model in Chapter 6.
2.2.1 Definition and methods
In order to quantify the diversity of the response activity to perturbation, first a perturba-tion must be applied and the response activity in the EEG across cortical regions defined.
The transcanial magnetic perturbation was a short magnetic pulse, applied to a cortical region, say the visual cortex4, and the EEG activity across the whole scalp (60 sensors) up to 300ms after the pulse was considered to reflect the response to this perturbation sufficiently. The signals were transformed from sensor space into source space, using the 3-spheres BERG method and the inverse problem was solved by the Weighted Minimum Norm constraint applied to an "empirical" Bayesian approach. This procedure turned the 60 EEG sensor channels into 3000 source channels (see supplement of [53] for more details).
Perturbing the brain about 100 times per subject (see [53] for varying trial numbers across different experiments) allowed an average response waveform across these trials to be found for each source. This average waveform was then binarised by a threshold T obtained from the pre-perturbation activity (from −500 to −1ms) across all trials. T was chosen as the 99th percentile of the distribution of the maximum absolute values of pre-perturbation activity sampled from all trials (perturbations), using the
bootstrap-2The variation in LZc for 50 different shufflings of the same input string of 25000 binary digits was under 0.002% of the mean result across those 50, showing that the data matrices we analysed - typically 10 channels times 2500 observations - were sufficiently large such that there was negligible variance arising from basing the normalisation on just a single random shuffling.
3Explanatory correlates of consciousness are brain activity patterns that can be directly mapped to phenomenological properties, characterizing all conscious experiences [202].
4The results did not vary qualitatively when other cortical target locations for the TMS stimulation were used [53].
based statistical procedure described here [151]. With this threshold, a binary response matrix SS(x, t) ("significant sources") with 3000 rows (one for each source x) and 109 columns (one for each observation t, being 300ms sampled at 362.5Hz) was filled, setting SS(x, t) = 1 if the activity of source x at time t was above the threshold T and SS(x, t) = 0 otherwise. Thus for one subject in one state and a chosen target region for the magnetic perturbation, one binary response matrix SS was obtained from approximately 100 trials (TMS stimulations), representing the brain response to the magnetic perturbation.
A Lempel-Ziv algorithm is now used to determine the number of new binary words in each column of the SS matrix, while keeping track of binary words found already in previous columns as described in Fig. 2.2. The resulting Lempel-Ziv complexity score, i.e.
the length of the dictionary of binary words, denoted as c(SS), depends on the size L of the SS matrix, L being the product of SS’s number of rows and number of columns. The values of c(SS) are large in practice and it strongly depends on the source entropy H(L) of SS, defined as
H(L) = −p log2(p) − (1 − p) log2(1 − p) (2.1) where p the fraction of ones across all entries of the SS matrix. Thus c(SS) is normalised by its asymptotic value for a random binary string of length L [126], i.e. as L 7→ ∞, c(SS) 7→ LH(L)/ log2(L). Thus the perturbational complexity index is defined as
P CI = c(SS) log2(L)
LH(L) . (2.2)
2.2.2 Differentiation and integration
Fig. 2.3 shows Casali et al.’s illustration of the difference of the SS between wakeful rest and deep sleep (NREM). The large black frames contain the SS matrices for the same subject, one for each state of consciousness, rows being source channels and columns observations (dimensions of the SS matrices are 3000 sources times approx. 100 observations, repre-senting 300ms). The corresponding PCI values are 0.51 for wake and 0.23 for deep sleep, indicating higher algorithmic complexity for spatio-temporal response to TMS during wake than deep sleep, for this representative subject. These PCI differences are clearly in line with visual inspection of continuous response waveforms between the two states. For wake, the continuous waveforms show greater diversity across channels but also, for each chan-nel, across observations. This diversity in the response to perturbation is a clear marker of differentiation, one of the theoretically proposed neural hallmarks of consciousness.
In order to argue for integration, note that the strength of the TMS stimulation in both states, WR and NREM, was the same, causing an electric field strength at the tar-get of 90V /m, but for deep sleep, far fewer sources showed an above-threshold response (Fig. 2.3A,B). This is seen as a sign that the effective connectivity between macroscopic brain regions is lower in deep sleep than wake. Effective connectivity in this sense can be seen as a measure of integration, the other of the theoretically proposed neural hallmarks of consciousness, proposedly reflecting the phenomenological property of "wholeness" of
Figure 2.2: Ziv algorithm applied to SS matrix Flow-diagram of the Lempel-Ziv algorithm (based on LZ-76) applied by Casali et al. to estimate the algorithmic com-plexity of the binary response matrix SS to transcanial magnetic perturbation. x and t are discrete indices of the spatial and temporal dimensions (x = 1, ..., L1 and t = 1, ..., L2), l(t) is the number of spatiotemporal samples of SS(x, t) up to time t. r, q, a are indices to keep track of location and length of found binary patterns. The input ("Data") is the binary SS matrix, having L1 rows (row i for the ith source, i = 0, .., L1) and L2 columns (column j for the jth observation, j = 0, .., L2). Patterns of length k are strings of k bits, Data(i : k, j) = SS(i + 1, j)SS(i + 2, j)...SS(i + k, j), with 0 < i + k ≤ L1. The algorithm outputs the non-normalised Lempel-Ziv complexity c of the SS matrix, say c(SS). Copied from Fig.S3 of [53], reprinted with permission from AAAS.
every experience, binding together features of each experience into a consciously perceived integrated whole.
Yet a high fraction of channels with significant response is not sufficient to indicate elevated effective connectivity and thus integration. In addition, diversity in the response
Figure 2.3: PCI
activity is needed. This can be seen in Fig. 2.3C, showing the binary response matrix (SS) for NREM sleep again but this time with the amplitude of the magnetic perturbation nearly doubled (inducing an electric field of 160V /m at target) and thus forcing significant response activity upon most channels. Yet PCI scores are still comparably low as for NREM with stimulation strength at 90V /m, in line with its interpretation to measure integration
(effective connectivity) as the fraction of channels that show significant and diverse response activity. That PCI captures exactly this was further confirmed with computer simulations where PCI increases monotonically with the fraction of non-zero random channels in SS.
In summary, PCI captures integration by the fraction of channels with strong enough and diverse response (i.e. how "far" the response signal to TMS perturbation spreads across sources) and differentiation in terms of response signal diversity across sources and observations.