Symbolic Recurrence Structure Analysis Features

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Incision Effect on EEG

Several works have studied the effect of incision or any other nociceptive stimulus on an anesthetized patients or animals using EEG. It is of primary interest for anesthesia research, as it might lead to well established methods to monitor the patient pain during anesthesia together with good marker of unconsciousness, which would lead to a general improvement of the surgery conditions for patients and savings in anesthetic and analgesic drugs [120].
Classification of patient reaction on noxious stimuli during a surgery is still not clear as the underlying mechanism are not well understood. There is still no consensus for reliable markers of pain during anesthesia as it seems to depend a lot on the patient: age, condition and natural variability of metabolism [62].
Nevertheless two different kind of responses have been observed by experimental setups with both patients and animals. In [84] 46 patients were put under isoflurane anesthesia and monitored with 17 scalp electrodes according to the 10–20 location system. The recording started 6 min before incision and stopped 14 min after anesthesia. To study the influence of the level of the anesthesia on the response, the patients were separated in two groups one with 0.6% of isoflurane and one with 1.2% of isoflurane. In each group half of the patients were incised and the other half were not. The results are showing a strong increase in -power and a decrease in -power. The increase of -power is stronger with low concentration group.
Theses results are interesting because they are in contradiction with what was traditionally found in nociceptive stimuli influence on EEG during anesthesia. Indeed, previous studies re-ported arousal pattern during sensory stimulation similar to the emergence from anesthesia: an increase in the -power and lower amplitude characteristic of EEG resynchronization. This study show an opposite effect, called reverse arousal or paradoxical arousal which looks like a deeper anesthesia state with slow waves of high amplitude.
These finding have been reproduced on sheep, cats and humans [113] and with different kinds of anesthetic [133]. It seems that these two patterns depend on the level of anesthesia but are still present in deep anesthesia.
In [83] the effect of extradural analgesia on the EEG response to incision was studied. The signals were recorded during 30 minutes after incision. They have shown that if analgesia is strong enough, the hemodynamic variables as well as the EEG does not vary during incision. Without analgesia, EEG is displaying the paradoxical arousal pattern and arterial pressure is rising. These two responses are predominant in the frontal lobe, it is thus expected that single or bi-electrode monitoring on the forehead would be able to capture it.
In [63], the EEG bicoherence was shown to be a good indicator of the analgesia of the patient. The data were recorded before incision, 5min after incision, and after administration of fentanyl intravenously (opioid). There were two groups: one with isoflurane and one with sevoflurane. Two index derived from bicoherence were shown to be good markers of incision with both anesthetic. The pBIC-low is the maximal value of bicoherence between 2 to 6 Hz and pBIC-high, from 7 to 13 Hz. Both these markers are decreasing during incision and return to pre-incisional values after the fentanyl administration.
On the contrary, the BIS index and SEF (spectral edge frequency) were not able to separate the classes due to the difference between arousal and paradoxical arousal patterns. Hagihira has shown similar results with isoflurane and even reported some patients displaying a mixture of paradoxical arousal and arousal [62].
With the same data set as the present work, Sleigh et al. [133] were able to design a marker of post-incision with the spindle amplitudes in the alpha range. They use 1=f regression on power spectrum, a method that we will discuss in the following chapters. Other markers like BIS, -peak and others were not able to separate pre- and post-incision with significance.

Symbolic Recurrence Structure Analysis

The next step in our work is to analyze non-stationary behavior in the data. For this purpose we chose recurrence-based methods. More precisely, we utilize symbolic recurrence structure anal-ysis (SRSA), which allows to identify recurrent and stationary domains in system’s dynamics and rewrite the signal as a sequence of symbols. Symbolic representation then allows for analysis with information-based methods, such as complexity measure.
Recurrence is a fundamental property of nonlinear dynamical systems, which was first formu-lated by Poincaré in [118]. It was further illustrated in recurrence plot (RP) technique proposed by Eckmann et al. [41]. This relatively simple method allows to visualize multidimensional phase trajectories of a nonlinear system on a two-dimensional plane. The RP can be obtained by plotting the recurrence matrix: Rij = ( » jj xi xjjj) ;i; j = 1; 2; : : : ; N : (2.16).
Here xi 2 Rd is the state of the complex system in the phase space of dimension d at a time instance i, jj jj denotes a metric, is the Heaviside step function, and  » is a threshold distance. It can be seen from Eq. (2.16), that if two points in the phase space are relatively close, the corresponding element of the recurrence matrix Rij = 1 and will be represented by a black dot on the RP, in contrast, white dots on the RP represent the absence of recurrence, i.e., Rij = 0, an example of a recurrence plot can be seen on Fig. 2.6c. By visual inspection of RPs an expert can draw conclusions on the studied system’s dynamics and behavior. For example, isolated black dots may indicate rare or unique event, which is not repeated in time or it fluctuates; parallel diagonal lines may indicate periodical behavior; horizontal or vertical lines mean that system is evolving very slowly or not evolving at all; dots uniformly distributed over the RP may be a sign of a stochastic stationary process.
Visual analysis of RPs may be challenging and require a lot of expertise, the search for more quantitative and formal approach gave rise to Recurrence Quantification Analysis (RQA) developed by Zbilut and Webber [143, 148] and then further extended by Marwan et al. in [107]. RQA allows to identify the nature of a system under study by calculating a set of single-valued measures. It can be adopted to time-dependent study by using sliding analysis window. This analysis was adopted in such domains as biology [3, 10, 97], physics [31, 87] and economics [5, 108].
RQA is a powerful tool, which permits to identify non-stationarity, drifts, transitions to chaos and to order. However, we are interested in specific types of activity, which can be extracted from RPs. This activity can be partitioned into three broad groups: (i) stable states, meaning that the system evolves close to a single state within a long time interval, (ii) recurrent states, which are the states to which trajectory returns after some time and stays there for a short intervals of time, and (iii) transient states, which evolve fast, are unique for the system and do not repeat within the time of observation. The first two states have a subtle difference between each other and both are denoted on a recurrence plot with a black-colored dot, i.e., Rij = 1, however, stable states occupy a region of the RP, which does not necessarily repeat, on another hand, the second type of activity is always present on the RP more than once. The last type of activity can be seen as non-recurrent regimes which occurs during system’s transitions from one stable state to another.
To identify the different kinds of behavior described above, we use symbolic recurrence plots. By transforming original signal to a symbolic sequence, we can assign a particular symbol to each type of dynamics and afterwards quantify it using information theory. Previous studies showed successful application of different implementations of this approach. For instance, order patterns recurrence plots [59, 128], where recurrence plots were obtained from order patterns, were used to provide a measure of coupling strength between two systems robust against artifacts and successfully applied to EEG analysis. Another variation of the approach, that consist of constructing RPs from words in symbolic sequences, found applications in such domains as performance analysis [39] or behavior analysis [44]. The symbol-based RPs provide advantage of coarse-graining, i.e., focusing on larger time scales, in addition, such RPs are transitive, which means if Rij = 1 and Rjk = 1, then Rik = 1.
In this study, we deviate from previously described methods and use approach proposed by beim Graben and Hutt, which generates symbolic representation of a signal from the recurrence plots. Previous studies [21–23] have shown successful application of the method to the detection of transient and stable states in synthetic and event-related brain potential.
The main idea of the method is to partition phase trajectory into a set of balls with diameter  » around sampling point xi. And then merge intersecting balls into a new set by examining the system’s RP: if two balls i and j are intersecting, then Rij = 1. By continuing merging intersecting balls, we partition phase trajectory into n disjoint sets Sk. Let P = fSk Rdj1 6 k 6 ng be the final partitioning of the phase trajectory, then desired sequence s = si represents states visited by the system at each discrete time, i.e., si = k when xi 2 Sk. To obtain aforementioned partitioning, we start by mapping a phase trajectory x to an initial symbolic sequence, where each symbol correspond to a time index, si = i. If two states at times i and j are recurrent, i.e., Rij = 1, and i > j, we can rewrite the symbol at time i with the symbol sj. We now see that the recurrence matrix can be interpreted as a grammar in rewriting process, where higher indices are replaced by the lower ones if the states at these discrete times are recurrent. In addition, this method resolves cases when three states are recurrent, i.e., Rij = 1 and Rik = 1. The described rewriting rules can be formulated as follows: i ! j if i > j and Rij = 1 (2.17). j ! k if i > j > k and Rij = 1; R ik = 1 : i ! k.

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Phase Space Reconstruction

The discussed technique of recurrence plots and recurrence-based symbolic dynamics usually deals with dynamical systems, whose phase space is multidimensional. Each dimension of the phase space is a certain property of a system, it can, for instance, be obtained from a system of ordinary differential equations, which defines system’s evolution law or it can be presented with multivariate measurements of an observable (e.g., several EEG channels). The system’s trajectory in the phase space correspond to the system’s states at particular times.
In certain cases only discrete univariate measurements of an observable are available, in this situation a phase space should be reconstructed according to Takens’s theorem [135], which states that phase space presented with a d-dimensional manifold can be mapped into 2d+1-dimensional Euclidean space preserving dynamics of the system. Several method of phase space reconstruction exist: delayed embeddings [135], numerical derivatives [114] and others (see for instance [88]).
In this work we propose a new heuristic method of phase space reconstruction based on the time-frequency representation (TFR) of the signal, a distribution of the signal’s power over time and frequency. Here, the power in each frequency band contributes to a dimension of the reconstructed phase space. This approach is well-adapted for non-stationary and, especially, for oscillatory data. The current section presents different phase space reconstruction methods, starting with delayed embeddings and continuing by describing different TFRs: spectrogram, reassigned spectrogram and scalograms (based on continuous wavelet transform). Below we argue that the proposed reconstruction with the signal’s TFR is more advantageous for SRSA analysis of oscillatory signals.

Delayed Embeddings

Assume, we have a time series which represents scalar measurements of a system’s observable in discrete time: xn = x(n t); n = 1; : : : ; N ; (2.20).
Then reconstruction of the phase space is done in the following way: sn = xn; xn+ ; xn+2 ; : : : ; xn+(m 1) ; n = 1; : : : ; N (m 1) ; (2.21).
where m is the embedding dimension and is the time delay. These parameters play an important role in correct reconstruction and should be estimated correctly.
Optimal time delay should ensure that reconstructed delay vectors from Eq. (2.21) are sufficiently independent. Several approaches to estimate this parameter exist. One can visually inspect obtained phase portraits; for data with periodic component, a good initial estimate of is a quarter of the period [79]. However, these are empirical approaches. Amongst more formal methods, the two leading estimates are based on autocorrelation function [1, 79] or average mutual information (AMI) [49, 98]. The autocorrelation approach considers linear independence between delayed coordinates as a sufficient criterion for optimal , while AMI also takes non-linear relations between the coordinates into account. Owing to this reason we chose to use AMI to estimate the proper value of the time delay .
Mutual information is the amount of information that one can learn about one system by observing another system. Let us call these two systems X and Y with possible observations xi and yj. If we denote the probability of observing xi by P (x), probability of observing yj by P (y) and the joint probability of xi and yj by P (x; y), then average mutual information between the two systems is given in [55] as: I(X;Y) = x;y P (x; y) log2 P (x)P (y) : (2.22).

Table of contents :

Chapter 1 Analgesia Effect of Anesthesia 
1.1 Mechanisms of Anesthesia
1.2 Analgesia
1.2.1 The Pathways of Pain
1.2.2 Analgesia During Anesthesia
1.3 Anesthesia and EEG
1.3.1 Anesthesia Monitoring
1.3.2 Drugs Effect on EEG
1.3.3 Incision Effect on EEG
1.4 Conclusion
Chapter 2 Feature construction 
2.1 Data Description
2.1.1 Experimental Data
2.1.2 Synthetic data
2.2 Power Spectral Analysis
2.3 Symbolic Recurrence Structure Analysis
2.4 Phase Space Reconstruction
2.4.1 Delayed Embeddings
2.4.2 Time-Frequency Embedding
2.5 Complexity Measure
2.6 Results
2.6.1 Power Spectral Analysis
2.6.2 Symbolic Recurrence Structure Analysis
2.7 Discussion
Chapter 3 Classification 
3.1 Linear Discriminant Analysis
3.2 Support Vector Machines
3.2.1 Margin Maximization
3.2.2 Use of Kernels in SVM
3.3 Cross-Validation
3.4 Feature Ranking
3.5 Results
3.5.1 Power Spectral Analysis Features
3.5.2 Symbolic Recurrence Structure Analysis Features
3.5.3 Full Feature Set
3.5.4 Feature Ranking
3.6 Discussion
General Conclusions
Bibliography 

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