Variability Analysis & Its Applications To

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VARIABILITY ANALYSIS & ITS APPLICATIONS TO PHYSIOLOGICAL TIME SERIES DATA

FARHAD KAFFASHI

Submitted in partial fulﬁllment of the requirements For the degree of For the degree of Doctor of Philosophy

Dissertation Advisor: Dr. Kenneth A. Loparo

Department of Electrical Engineering & Computer Science

CASE WESTERN RESERVE UNIVERSITY

August 2007 CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

Farhad Kaﬀashi

candidate for the degree of Doctor of Philosophy *

Committee Chair: Dr. Kenneth A. Loparo Thesis Advisor Professor, Department of Electrical Engineering & Computer Science

Committee: Dr. Mark S. Scher Professor, Department of Pediatric & Neurology

Committee: Dr. Mary Ann Werz Associate Professor, Department of Neurology

Committee: Dr. Vira Chankong Associate Professor, Department of Electrical Engineering & Computer Science

Committee: Dr. M Cenk C¸avu¸so˘glu Assistant Professor, Department of Electrical Engineering & Computer Science

August 2007

*We also certify that written approval has been obtained for any proprietary material contained therein. Table of Contents

Table of Contents ...... iii List of Tables ...... v List of Figures ...... vii Acknowledgement ...... ix Abstract ...... x

1 Introduction 1

2 Variability Analysis Techniques 4 2.1 Detrended Fluctuation Analysis ...... 5 2.2 Characterization of Power-Law Behavior Using Change Point Detection 8 2.2.1 Gradient Detection Algorithm ...... 9 2.2.2 Results ...... 11 2.2.3 Conclusions ...... 18 2.3 Approximate & Sample Entropy ...... 18 2.3.1 Incorporating a Time Delay In The Calculation ...... 20 2.3.2 Parameter Selection ...... 21 2.3.3 Parameter Validation ...... 22 2.3.4 Conclusion ...... 28

3 Complexity Analysis of Neonatal EEG 29 3.1 Data Collection & Description ...... 30 3.2 The Eﬀect of Time Delay on ApEn & SpEn Computation ...... 33 3.3 Results & Discussion ...... 35 3.3.1 Histogram Analysis ...... 37 3.3.2 Histogram Matching Scheme ...... 45 3.3.3 Closeness Test ...... 46 3.4 Conclusions ...... 47

4 Epilepsy 53 4.1 Data description ...... 57

iii 4.2 DFA Limitation ...... 58 4.3 Brain Activity Quantiﬁcation ...... 59 4.4 Data Analysis & Results ...... 60 4.4.1 Brain Activity Index ...... 63 4.5 Discussion & Conclusions ...... 74 4.6 Future work & Recommendations ...... 77

5 Analysis of Respiratory Data 78 5.1 Complexity of network ...... 78 5.1.1 Methods ...... 79 5.1.2 In vitro extracellular recording ...... 79 5.1.3 Parameter selection ...... 80 5.1.4 Results ...... 80 5.2 Cardioventilatory Coupling ...... 83 5.2.1 Methods ...... 83 5.2.2 Protocol ...... 84 5.2.3 Surrogate Data Analysis ...... 85 5.2.4 Data Analysis & Discussion ...... 86

A Variability Analysis Techniques from the Literature 95 A.1 Non-Parametric Change Point Detection ...... 95 A.1.1 Statistical Methods ...... 95 A.1.2 Description of Algorithm ...... 97 A.2 Correlation Dimension ...... 98 A.3 Information Theory Based Entropy ...... 100 A.3.1 Shannon Entropy ...... 100 A.3.2 Interval Entropy & Entropy of Intervals ...... 100 A.3.3 Spectral Entropy & Variance ...... 102 A.4 Surrogate Data Analysis ...... 102 A.5 Hjorth Parameters ...... 103 A.6 Dynamical Similarity Index ...... 104 A.7 Phase Synchronization ...... 105

Bibliography 106

iv List of Tables

2.1 DFA log-log plot gradient for diﬀerent signals ...... 6 2.2 D2 results for intracranial EEG ...... 16 2.3 ApEn & SamEn Results ...... 27 2.4 ApEn & SamEn Results for ﬁltered white noise ...... 27

3.1 Neonate EEG Channels ...... 32 3.2 Pittsburgh Study Group ...... 32 3.3 Unity vs. 1 Seconds Time Delay Analysis Results for EEG Number 96 (Mean±STD) ...... 36 3.4 Time Delay Analysis Results for EEG Number 96 (Mean±STD of STD) 36 3.5 Means of Complexity Results ...... 39 3.6 Means of Active Complexity Results ...... 40 3.7 Means of Quiet Complexity Results ...... 41 3.8 Standard Deviation of Complexity Results ...... 42 3.9 Standard Deviation of Active Complexity Results ...... 43 3.10 Standard Deviation of Quiet Complexity Results ...... 44 3.11 Diﬀerence Between Histograms of Pittsburgh & Pilot Studies . . . . . 46 3.12 Mean Mahalanobis Distance Between Neonates at 40-41 Week . . . . 48 3.13 Means of Complexity Results after Histogram Matching ...... 51 3.14 T-test result between Pittsburgh full term and premature at the age 40-41week...... 52

4.1 Short recording speciﬁcs for each patient ...... 57 4.2 Long recording speciﬁcs for each patient ...... 58 4.3 K-means clustering result ...... 67

5.1 Subject Demographic Data ...... 84 5.2 Interested parameters from cardioventilatory system ...... 88 5.3 Entropy results for normal breathing in healthy subjects ...... 88 5.4 Entropy results for 5 cm H2O of PEEP breathing in healthy subjects 89

v 5.5 Entropy and ranking result for Cardioventilatory intervals during normal breathing in healthy subjects ...... 91 5.6 Entropy and ranking result for Cardioventilatory intervals during PEEP breathing in healthy subjects ...... 92 5.7 Median of Cardioventilatory parameters during normal breathing in healthy subjects ...... 92 5.8 Median of Cardioventilatory parameters during PEEP breathing in healthy subjects ...... 93

vi List of Figures

2.1 DFA Results for diﬀerent frequencies ...... 7 2.2 Intracranial EEG during ictal & non-ictal ...... 12 2.3 Diagnostics for DFA of the Lorenz attractor ...... 13 2.4 Diagnostics for DFA of random noise ...... 13 2.5 Diagnostics for DFA of intracranial ictal EEG ...... 14 2.6 Diagnostics for DFA of non-ictal EEG ...... 14 2.7 Diagnostics for correlation dimension of the Lorenz attractor . . . . . 15 2.8 Diagnostics for correlation dimension of the for intracranial non-ictal EEG (W =1)...... 16 2.9 Diagnostics for correlation dimension of the for intracranial non-ictal EEG (W =10) ...... 16 2.10 Diagnostics for correlation dimension of the for intracranial ictal EEG (W =1)...... 17 2.11 Diagnostics for correlation dimension of the for intracranial ictal EEG (W =14) ...... 17 2.12 Sample Burst Data from an in vitro respiratory slice preparation (neona- talrat)...... 24 2.13 Normalized auto correlation function ...... 25 2.14 Histogram of Approximate Entropy for Lorenz Attractor ...... 26 2.15 Histogram of Sample Entropy for Lorenz Attractor ...... 26

3.1 Electrodes Placement for International 10-20 System ...... 31 3.2 Neonatal EEG Autocorrelation Function for Pittsburgh Study . . . . 34 3.3 Normalized Histogram of Approximate Entropy Results for Pittsburgh Study ...... 48 3.4 Normalized Histogram of Approximate Entropy Results for Pilot Study 49 3.5 Normalized Histogram of Approximate Entropy Results for Pittsburgh Fullterm 40-41 Weeks Study Active vs. Quiet ...... 49 3.6 Normalized Histogram of Approximate Entropy Results for Pilot Pre- mature 40-41 Weeks Study Active vs. Quiet ...... 50

vii 3.7 Normalized Histogram of Approximate Entropy Results of Pilot vs. Pittsburgh Study ...... 50 3.8 Normalized Histogram of Approximate Entropy Results of Pilot vs. Pittsburgh Study After Mean Correction ...... 51

4.1 DFA result for patient S-1 ...... 61 4.2 Dynamical similarity index result for patient S-1 ...... 62 4.3 Phase coherence result for patient S-1 ...... 63 4.4 Hjorth parameters result for patient L-1 ...... 64 4.5 DFA result for patient L-1 ...... 65 4.6 Dynamical similarity index result for patient L-1 ...... 65 4.7 Phase coherence result for patient L-1 ...... 66 4.8 Brain activity index for patient L-1 ...... 68 4.9 Smoothed versus normal brain activity index for patient L-1 . . . . . 69 4.10 Smoothed versus normal brain activity index for patient L-2 . . . . . 70 4.11 Smoothed versus normal brain activity index for patient L-3 . . . . . 71 4.12 Smoothed versus normal brain activity index for patient L-3 . . . . . 72 4.13 Smoothed versus normal brain activity index for patient L-4 . . . . . 73 4.14 Sample EEG before and after seizure as well as seizure onset . . . . . 75 4.15 DFA Result for patient L-4 ...... 76

5.1 Approximate Entropy result for respiratory network bursts at diﬀerent potassium concentration ...... 81 5.2 Sample Entropy result for respiratory network bursts at diﬀerent potassium concentration ...... 82 5.3 Sample ECG & Breathing ...... 87 5.4 Entropy of intervals during normal breathing ...... 89 5.5 Entropy of intervals during PEEP breathing ...... 90 5.6 Entropy of intervals for Cardioventilatory intervals in healthy subjects 93 5.7 Ranking results for cardioventilatory intervals in healthy subjects . . 94

viii ACKNOWLEDGEMENTS

First, I would like to thank my father for encouraging me to pursue my doctoral degree. My deepest gratitude goes to Prof. Kenneth A. Loparo for his guidance and support during this study as my advisor and friend. I also would like to express the words of appreciation to the members of my dissertation committee and our collaborators for their time and suggestion to improve this work. I have furthermore to thank the entire faculty, staﬀ and students in the department for their assistance and kindness, especially to my friends Reza Jamasabi and Evren Gurkan for the good times we shared together.

ix Variability Analysis & Its Applications to Physiological Time Series Data

Abstract

Farhad Kaﬀashi

In this thesis, novel variability analysis techniques are developed and refinements are made to some currently available methods to enhance their use and effectiveness. These variability analysis techniques are applied to physiological time series data to study both health and disease. In particular, the addition of a new parameter, the time delay τ, is proposed to enhance the performance of Approximate and Sample Entropy calculations; a novel technique is developed to estimate the gradient of power law behavior based on non-parametric change-point detection; a novel technique is developed to quantify the coupling between time series data based on surrogate data analysis, and the limitations of Detrended Fluctuation Analysis (DFA) are studied in the context of the detection of self similarities in EEG time series data. The techniques that are developed in the thesis are applied in several areas to evaluate their suitability and effectiveness. In one application, neurodevelopment and maturation of the neonatal brain is studied and the effectiveness of strategies that can improve sleep organization, such as skin-to-skin contact or Kangaroo Care (KC) intervention, are evaluated using Approximate and Sample Entropy. The results show that the KC intervention improves neurodevelopment and maturation. In a study of epilepsy, a novel technique to quantify electrocorticography data using the DFA is presented. The DFA can detect changes in the electrical activity of the brain that

x are associated with different brain states such as seizure (ictal), preictal, postictal as well as arousals that are occurring during sleep. In the analysis of respiratory data, the complexity of in vitro modularly prepared neonatal rat slices, that are capable of generating a spontaneous respiratory related rhythm at different extra cellular K+ levels are quantified and further, the coupling between the respiratory and cardiac networks is investigated using a novel approach based on surrogate data analysis.

xi Chapter 1

Introduction

The study of variability, or how pattern are changing over time, of physiological time series is very complicated because of diverse characteristic of time series data. One of the important issues is non-stationarity of the time series, where many analysis techniques designed to characterize variability requires stationarity or at a minimum the basic statistics such as mean and standard deviation of signal are constant within a recording. This requirement is very difficult to meet because the noise patterns on physiological time series are likely to be influenced by body movement, physiological state or probe connection thus making the analysis more challenging. Therefore the time series are assumed to be at least piece-wise stationary or wide-sense stationary where the mean of the signal is constant and the autocorrelation function is invariant for the segment or epoch of data being analyzed. Further there are artifacts in the data that have a significant effect on the signal. These artifacts characteristics can often easily identified by visual inspection of raw data, however there are artifacts that might not be easily recognized and corrupt the signal under study. Generally, researchers that are interested in quantifying the variability of physiological time series data, do not have sufficient background in physiology and clinicians do not have the necessary understanding of sophisticated signal analysis techniques. In fact, for accurate interpretations of variability analysis results, it is very important

1 to understand the nature of the recorded signals. For example, a standard electroencephalogram (EEG) recording from the scalp of a patient represents the collective response of a large group of neurons which is subsequently filtered through the shall and scalp. It is still an open question as to the exact nature of the filtering of the EEG by skull and scalp. EEG recordings directly from the brain cortex or deep brain electrodes are more precise, less noisy where each electrodes recodes a smaller population of neurons as compare to scalp recordings. Therefore a close collaboration of engineers with clinicians is necessary in the quantification of variability to provide diagnostic, prognostic or any physiological information. The literature related to variability analysis is very extensive and each variability analysis technique is commonly defined for measuring different aspects of time series data such as complexity, dimensionality, regularity or irregularity, randomness, predictability, self similarity, synchrony, etc. In fact, all of the various signal processing techniques have similar objectives, but quantify the signal variability from a different perspective. In particular, there is no agreement that any single technique is the best means of characterizing the variability of specific biological signal; rather investigators agree that multiple techniques should be performed simultaneously to facilitate comparison between methods, techniques and studies. Variability analysis can be done with different approaches. The simplest approach is the application of techniques that do not require any a prior information and after analysis the task is to correlate observed changes to physiology. This method may appear to be easy but it is very time consuming and any changes in variability may not have a corresponding biological reason. On the other hand, researchers generally design an experiment to understand the changes in biology during specific conditions. In this approach, the objective is very clear and the changes in variability during a given condition are compared to the normal condition. For example, to understand breathing variability in an animal model of the lung injury, one can try to extract features from the breathing time series to distinguish the breathing patterns with

2 lung injury from those without lung injury. There are two main contributions at this work: as variability development and/or reﬁnement of techniques for analysis and the applications to real physiological time series. Many of the techniques have been developed for a speciﬁc application but can also be applied to other systems. In particular the study of variability in the following problem areas has been accomplished:

• Epilepsy: Quantiﬁcation of the electrical brain activity of epileptic patients using electrocorticography (EcOG)

• Neonatal EEG: Measuring maturation and neurodevelopment.

• Respiration: Measuring respiratory network complexity and detecting the coupling or synchrony of cardiac and respiratory rhythms.

In fact each of these problem areas constitute a research endeavor on their own, and understanding the complete variability is still an open question. However in this work related literature has been investigated and novel techniques that have not been applied to each of these problem areas has been studied. The remaining chapters of this dissertation are organized as follows. In the second chapter the variability analysis techniques are presented and the methods from the literature are given in the Appendix. In the third, forth and ﬁfth chapters each of the above problems is introduced and the details of collected data are given. Further the results of the variability analysis techniques along with a discussion and conclusion are presented, and each chapter concludes with suggested future work.

3 Chapter 2

Variability Analysis Techniques

Variability or the analysis of how patterns change over time is important in many applications from engineering, the physical and social sciences, to biology and medicine. In biology and medicine analysis of the variability in physiological time series data, e.g. EEG, respiration, breathing and heart rate (HR), and circadian rhythms such as sleep/wake cycles, core body temperature and metabolic processes are of interest in the study of both health and disease phenotypes. Although quantifying variability had its roots in the statistical sciences, recent results from dynamical systems and information theory have had a significant impact on this field. The statistical techniques are primarily directed at understanding variability in the context of linear stochastic signal models using techniques such as regression, correlation analysis, and modeling in the time (AR, ARMA, etc.) and frequency (windowed spectral analysis, PSD, Pe- riodogram, etc.) domains. More recently attention has been directed at developing new measures of variability and a number of these measures have been derived from information (entropy) theory and the analysis of complex (chaotic) systems. From an applications perspective, because most physiological signals (e.g. EEG, HR and respiration) will include both linear stochastic and nonlinear deterministic-features, our approach is to develop modeling and analysis methods that can effectively and efficiently use both types of information in characterizing a phenotype.

4 Fractal analysis [79], Correlation Dimension [20, 19], Approximate and Sample Entropy [52, 53] are typical dynamical system measures. DFA (Detrended Fluctuation Analysis) [51] and noise titration [57, 56] are other techniques that have been proposed to distinguish between intrinsic fluctuations generated by a complex system, e.g. beat- to-beat fluctuations in HR in both health and disease. The most recognized entropy techniques are: Shannon Entropy [72], Approximate Entropy (Pincus [52, 55, 53]), Sample Entropy (Richman and Moorman [65]), Spectral Entropy, Wavelet Entropy (Rosso [66]), Renyi Entropy (Gonzalez [18]), Multiscale Entropy (Costa [12]), and Interval Entropy (Reeke [64]). Seely [71] has classified most of the variability analysis techniques with their advantages and disadvantages. In the following sections the variability analysis techniques that have been used in the analysis of physiological time series data in this dissertation are presented and the methods from the literature are presented in Appendix A.

2.1 Detrended Fluctuation Analysis

Peng et al. [51] introduced a technique referred to as Detrended Fluctuation Anal- ysis (DFA) for quantifying the long-range correlation behavior in non stationary physiological time-series data. This technique has been applied to the analysis of heart interbeat time intervals as follows:

k X y(k) = [B(i) − Bave] (2.1) i=1

th where B(i) is the i interbeat interval and Bave is the average interbeat interval over the signal being studied. The centered signal y(k) is divided into n equal segments and a linear regression line, yn(k), provides a least squares curve ﬁt to each segment.

The integrated interbeat signal, y(k), is then detrended using the local trend yn(k)

5 and the cost function F (n) is calculated:

v u N u 1 X F (n) = t [y(k) − y (k)]2 (2.2) N n k=1

The cost function F (n) is calculated for diﬀerent values of n, and the existence of a power law of the form (2.3) is investigated:

F (n) ∝ nα (2.3)

Normally, a linear relationship in the log − log plot of F (n) versus n indicates the presence of such a scaling law, and the scaling exponent α is calculated by estimating the factor relating log F (n) to log n. In addition, forward and backward calculations of F (n) are used to lower the deviations of the estimate of α on small time scales [28]. DFA requires a large data set and Peng et al. [51] used a minimum of 8 hours of interbeat intervals to quantify heart rate variability. Seely [71] suggests that at least 8000 samples needs to be used in the DFA calculation. Also, the minimum number of samples in each segment is another parameter that needs to be selected properly. Peng et al. [51] chose 20 ≤ n ≤ 1000 in their DFA computation. The typical α values for diﬀerent time series are given in Table 2.1.

Table 2.1: DFA log-log plot gradient for diﬀerent signals Signals α Random noise 0.5 Brownian noise (integrated white noise) 1.5 Sinusoidal wave 2

In this section we are interested in investigating the eﬀect of the number of samples from each cycle of a periodic trend on the calculation of the DFA by considering a sine wave signal of the form:

u(t) = As sin(2πft) (2.4)

6 where As is the amplitude of signal and f is the frequency. It is known that the α value for a sinusoidal signal is 2 and it is independent of As and f [23]. Therefore, we would like to consider sine waves at different frequencies sampled at a fixed rate of 1000 Samples/Sec1. The DFA of 10 seconds (10000 samples) of simulated data were computed and the logarithmic plot of log(n) vs. log(F (n)) for changes in frequency from 1 to 150 Hz are given in figure 2.1. In the DFA calculation the minimum number

DFA 2.5

1.5

0.5

−0.5

Log(F(n)) 1 Hz −1 2.3 Hz 5.3 Hz −1.5 12.2 Hz −2 28.3 Hz 65.1 Hz −2.5 150 Hz −3 0.5 1 1.5 2 2.5 3 Log(n)

Figure 2.1: DFA Results for different frequencies of samples in each segment, n, has been chosen as 5 which corresponds to a scaling of 0.69 = log10(5), and the maximum number of samples in each segment is 1000. The gradient of log-log plots of the linear scaling regions is 2 which is consistent with the literature [23]. However as the frequency gets higher the crossover moves toward the lower values of log(n) (the left of the plot) and the linear scaling region gets smaller and smaller. This makes the detection of this region more difficult and there is insufficient data to quantify it is impossible to quantify the gradient of the linear scaling region for frequencies greater than 150Hz. The problem is further complicated

1 1 The time between samples T = Sampling Rate

7 in applications where noisy time series data are recorded for analysis.

2.2 Characterization of Power-Law Behavior Us- ing Change Point Detection

Various methods that quantify power-law scaling in time, space, and distribution, specifically the Correlation Dimension and Detrended Fluctuation Analysis (DFA), show promise for characterizing time series phenomena in the pure and applied sciences. If the system in question follows a global power law, a log − log representation of the characteristic vs. scale will be linear, and the gradient can be easily computed by linear regression. For more complex processes where scaling laws vary and cross-over phenomena [51] may occur, a major challenge to the implementation of automatic scale analysis is the lack of a single easily identifiable linear region. In this section, an innovative algorithm for the automatic detection of linear regions in a log-log plot is presented. A three-point Lagrangian approximation of the derivative is used to generate a diagnostic sequence for isolating the most statistically invariant segments of the log-log plot based on Non-parametric Change-Point Detection, from which the scaling law can be determined. Power law scaling behavior has been observed in the dynamics of many phenomena in the sciences and physiology, including the analysis of heart rate fluctuations, inter- breath intervals, earthquakes, solar flares and stock market fluctuations [71, 21, 39]. Techniques for the automatic quantification of these phenomena are therefore of interest for both engineering and clinical applications. The main premise is the presence of locally scale-independent power law behavior, which can be described by

f(x) = αxβ, (2.5) where α and β are constants. Taking the logarithm of both sides yields an aﬃne relationship between log f(x) and log x, speciﬁcally an equation of the form:

log f(x) = log α + β log x (2.6)

8 The theoretical advantage of scale-independent analysis [71] is due to the fact that if the variable x is replaced by Ax, where A is a constant, the fundamental power law relation β is invariant. This property is called self-similarity, and a process f(x) is called self-similar with index β if for some A > 0

f(Ax) ∼ Aβf(x), (2.7) where “∼” may indicate similarity in structure or distribution. The objective of power-law analysis is to find the scale parameter β for a given data set, usually a time series. This is normally accomplished by measuring some characteristic f(x) over several time scales x, then fitting a linear regression line to the most clearly linear region of the log − log plot of f(x) vs. x. Observations of real measurement data from a majority of applications are in general not globally self-similar, and can exhibit different scaling behaviors over varying scales and are also often corrupted by noise that further makes the determination of the scaling region difficult. Regions with different scaling characteristic are marked by transitions, or ‘crossovers’, in the log-log plot of the characteristic in question, where β can change abruptly. The objective of the technique developed in this section is to automatically isolate such crossovers, identify the scaling regions where the specific power-law behavior is invariant, and quantify the scaling exponent β for each such segment. In the following sections we present an innovative gradient-detection technique based on change point detection for the automatic detection and quantification of power-law scaling behaviors from time-series data. A brief review of the non-parametric change point detection algorithm is given in Appendix A.

2.2.1 Gradient Detection Algorithm

To identify the linear scaling regions of a log − log plot where the derivative of the function is statistically invariant, we apply the following procedure:

9 1. The point-wise derivatives of the log − log plot are computed using the following three point Lagrangian approximation. That is, given x(t) we approximate dx(t) x˙ = dt at the point tk by:

x(t ) − x(t ) x˙(t ) = k+1 k−1 ; h = t − t = t − t (2.8) k 2h k+1 k k k−1

2. The change-point detection algorithm is then used to estimate the locations where statistically signiﬁcant changes in the sequence of estimated point-wise derivatives occurs, and the gradient in each of the identiﬁed regions is obtained using linear regression.

3. Using an application-dependent criterion the appropriate linear region is selected from the identiﬁed regions and an estimate of the slope in the selected region is computed.

Remark-1 For the correlation dimension D2, the mean square error between the actual data points in each region and the linear regression line is used to select the appropriate linear region. In addition, because the correlation dimension is defined for small values of r, the linear regions with smaller r-values have a higher weight in the cost function. Therefore the problem of selecting the most linear region is defined as an optimization problem over identified segments using change point detection with the objective function:

m k 2 X log(ri)h i min log C(ri) − Yk(ri) (2.9) k m2 i=1 k

where for each segment, k, mk is the length of the segment and Yk is the the linear regression line ﬁt to the data.

Remark-2 In computing an estimate of the scaling parameter using the DFA algorithm, the smallest linear region is chosen, unless it is very short (n < 5 points2),

2The total number of points to characterize the log-log plot of DFA has been considered as 40.

10 in which case the second region is selected. This is done to mitigate the eﬀects of “local” correlations in characterizing the global behavior of the time-series.

2.2.2 Results

The improved estimation of the power-law scaling parameters based on selection of the appropriate linear region is demonstrated below using simulated data from the Lorenz system, uniformly distributed random noise, as well as intracranial electroencephalographic (EEG) data collected from a patient with severe epilepsy.

Simulated Data

The Lorenz attractor, a well known dynamical system often used to validate techniques for quantifying chaotic behavior and complexity, is used to generate a one- dimensional time series, chosen as the x-component of the Lorenz dynamics. The Lorenz system is given by the system of diﬀerential equations:

x˙ = σ(y − x) y˙ = −xz + ρx − y (2.10) z˙ = xy − βz where σ, ρ, and β are constant parameters that determine the dynamical behavior of the system, which may be periodic or chaotic. For chaotic dynamics σ, ρ, and β have been chosen as 10, 28, and 8/3 respectively. These equations have been simulated in

MATLAB Simulink with a ﬁxed integration step size of Ts = 0.01, and the ﬁrst 10000 samples of the simulation have been ignored, so that the steady state dynamics on the attractor are analyzed.

EEG data

To demonstrate the applicability of the proposed method to experimental data, an intracranial EEG time-series from the Epilepsy Unit of University Hospitals of Cleveland has been analyzed. The data were recorded using a Nihon Kohden data

11 Intracranial EEG During Ictal Intracranial EEG During Non−Ictal

0 1 2 3 4 0 1 2 3 4 Time(Seconds) Time(Seconds)

Figure 2.2: Intracranial EEG during ictal & non-ictal collection system at a 1000 Hz sampling rate. A sample of ictal (during a seizure) EEG and non-ictal EEG are shown in Figure 2.2. Data analysis techniques that examine power-law behavior are very sensitive to the number of samples in the time-series, particularly for the computation of correlation dimension. Therefore time series containing 10000 samples were used in the analysis.

Application of DFA

The log − log plot of the DFA characteristic for a Lorenz attractor time-series and the diagnostics for gradient detection are shown in Figure 2.3. From this ﬁgure it is evident that one crossover has been detected and the gradients within the two linear regions have been estimated as 1.45±0.11 and 0.80±0.06 respectively for 10 simulated data sets starting from diﬀerent initial conditions. In addition, the results of the same analysis applied to uniformly distributed random noise are shown in Figure 2.4. The gradient was found to be 0.50±0.03 for 10 randomly generated time series, which is in agreement with the theoretical value for random noise [51]. DFA analysis is also applied to ictal EEG as well as non-ictal EEG data collected from intracranial (ECoG) recordings. The results of the gradient detection algorithm for the chosen time-series are shown in Figures 2.5 and 2.6. Two crossovers have been detected

12 DFA 3

0 Pointwise Gradient Detected Gradient Estimated Gradient −1 1.5 2 2.5 3 Log(n)

3 Slope−1 2.5 Slope−2 Actual Data 2 Log(Fn) 1.5

1 1.5 2 2.5 3 Log(n)

Figure 2.3: Diagnostics for DFA of the Lorenz attractor

DFA 1.5 Pointwise Slopes 1 Detected Slopes

0.5

Estimated Gradient −0.5 1.5 2 2.5 3 Log(n)

0.4 Slope−1 0.2 Actual Data

0 Log(Fn) −0.2

−0.4 1.5 2 2.5 3 Log(n)

Figure 2.4: Diagnostics for DFA of random noise

13 for the ictal ECoG time series and the scaling exponents within the corresponding invariant regions are 1.69, 1.32, and 0.44 respectively and 0.95 has been estimated for the non-ictal ECoG time series segment.

DFA 3 Pointwise Gradient Detected Gradient 2

Estimated Gradient 0 1.5 2 2.5 3 Log(n)

Slope−1

Log(Fn) 3 Slope−2 Slope−3 Actual Data 2 1.5 2 2.5 3 Log(n)

Figure 2.5: Diagnostics for DFA of intracranial ictal EEG

DFA 2 Pointwise Gradient 1.5 Detected Gradient

0.5

Estimated Gradient 0 1.5 2 2.5 3 Log(n)

3.5

3 Log(Fn) 2.5 Slope−1 Actual Data 2 1.5 2 2.5 3 Log(n)

Figure 2.6: Diagnostics for DFA of non-ictal EEG

Application of Correlation Dimension

Prior to calculation of the correlation integral, the attractor is embedded in the phase-space as speciﬁed in section A.2. The time-delay τ is determined by choosing

14 the first minimum or zero-crossing of the autocorrelation function or the mutual information [14]. After finding the proper delay τ, the appropriate embedding dimension is determined by the false nearest neighborhood method [31]. In addition, in order to reduce the effects of short range time-correlations, the Theiler window [79, 78] is used. In our computations we use an embedding dimension of m = 6 and a time-delay of τ = 0.13 time units for the x variable of the Lorenz attractor, for which the correlation integral and diagnostics are shown in Figure 2.7. The correlation dimension has been computed from the appropriate linear regions of the log − log plots of 10 Lorenz attractor time-series with different initial conditions and we obtained D2 = 2.0 ± 0.1, which is consistent with previously published values [19, 77].

Correlation Dimension 4 Pointwise Gradient 3 Detected Gradient

Derivative of Log(Fn) 0 −1 0 1 2 3 4 5 6

0 Actual Data Selected Linear Plateau −5

−10 Log(C(r))

−15 −1 0 1 2 3 4 5 6 Log(r)

Figure 2.7: Diagnostics for correlation dimension of the Lorenz attractor

Automatic estimation of the correlation dimension of the ECoG time series was also examined. For a sample epoch of ictal intracranial EEG, the appropriate delay and embedding dimension have been chosen as τ = 0.014 seconds (14 samples) and m = 6 respectively. For the non-ictal intracranial EEG epoch m = 8 and τ = 0.01 seconds (10 samples). The results of the correlation integral analysis and the gradient detection algorithm for non-ictal intracranial EEG with W = 1 and W = τ is shown in Figures 2.8 and 2.9. For the ictal intracranial EEG time series, the results are shown in Figures 2.10 and 2.11. Our technique detects the linear segment of the log − log

15 plot and evaluates the corresponding gradient, the estimated correlation dimensions are given in Table 2.2.

Table 2.2: D2 results for intracranial EEG W = 1 W = τ Non-Ictal EEG 5.77 7.08 Ictal EEG 3.82 4.69

Correlation Dimension 8 Pointwise Gradient 6 Detected Gradient

Derivative of Log(Fn) 0 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

−5

−10 Log(C(r)) −15 Actual Data Selected Linear Plateau −20 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 q Log(r) Figure 2.8: Diagnostics for correlation dimension of the for intracranial non-ictal EEG (W = 1)

Correlation Dimension 15 Pointwise Gradient Detected Gradient 10

Derivative of Log(Fn) 0 3 4 5 6 7 8

−5

−10

−15 Log(C(r)) −20 Actual Data Selected Linear Plateau −25 3 4 5 6 7 8 Log(r)

Figure 2.9: Diagnostics for correlation dimension of the for intracranial non-ictal EEG (W = 10)

16 Correlation Dimension 5 Pointwise Gradient 4 Detected Gradient 3

Derivative of Log(Fn) 0 3 4 5 6 7 8 9 10

−5

−10 Log(C(r)) −15 Actual Data Selected Linear Plateau −20 3 4 5 6 7 8 9 10 Log(r)

Figure 2.10: Diagnostics for correlation dimension of the for intracranial ictal EEG (W = 1)

Correlation Dimension 8 Pointwise Gradient 6 Detected Gradient

Derivative of Log(Fn) 0 3 4 5 6 7 8 9 10

−5

−10 Log(C(r)) −15 Actual Data Selected Linear Plateau −20 3 4 5 6 7 8 9 10 Log(r)

Figure 2.11: Diagnostics for correlation dimension of the for intracranial ictal EEG (W = 14)

17 2.2.3 Conclusions

Time-series data from real applications can exhibit complex scaling behavior with marked changes across a variety of scales. We have demonstrated the use of a non- parametric statistical change-point detection technique to automatically identify such transitions, or crossovers, in a scaling characteristic, and to quantify self-similarity in scaling regions where that characteristic is statistically invariant. We propose that this approach to characterizing power-law phenomena from time series data is important, especially for automatic analysis. The computation time for this additional step is not signiﬁcant and is recommended for any method that requires linear regression applied to a log − log plot to examine power-law scaling exponents. The proposed approach can improve the consistency and accuracy of methods intended to automatically extract scaling information from large data sets.

2.3 Approximate & Sample Entropy

Approximate Entropy is a scale independent measure of complexity (predictability) that was introduced by Pincus [52]. The motivation for this approach is the classiﬁcation of complex systems including both deterministic and stochastic processes from time series with a limited number of data points when compared to other measures like correlation dimension [20, 19]. This measure has been used in the analysis of heart rate variability [33, 53, 55], EEG (electroencephalogram) [3, 6, 63, 54], and the study of respiratory neural networks [11]. Approximate Entropy is deﬁned for a given one-dimensional time series of length n (x1, x2, ··· , xn) as:

ApEn(m, r, n, τ) = Φm(r) − Φm+1(r) n−(m−1)τ m −1 X m (2.11) Φ (r) = [n − (m − 1)τ] ln Ci (r) i=1 where B Cm(r) = i (2.12) i n − (m − 1)τ

18 Bi = number of j such that d|Xi,Xj| ≤ r (2.13)

In equation 2.13, (Xi,Xj) are m-dimensional embedding vectors, whose components are time-delayed versions of the elements in the original time series with delay τ, a multiple of the sampling time, that is:

m Xi = (xi, xi+τ , xi+2τ , ··· , xi+(m−1)τ ) Xi ∈ R (2.14)

m Xj = (xj, xj+τ , xj+2τ , ··· , xj+(m−1)τ ) Xj ∈ R (2.15) and d|Xi,Xj| is a measure of the distance between Xi and Xj. In deﬁnition of Ap- proximate Entropy in literature [52] and the subsequent applications of the technique the time delay has been chosen as τ = 1 and such a τ is not included in computation. However, we have determine a need to include a non-unity time delay in the computation of Approximate and Sample Entropy and this will be discussed in the next section. The main idea of Approximate and Sample Entropy measures is to calculate the “conditional probability” that a given data set of length n, with a given number of m-dimensional embedding vectors within a tolerance r will have a similar number of m + 1-dimensional vectors within the same tolerance r. Here the time delay τ chosen for both embedding is the same. For large values of n the Approximate Entropy is given by: n−mτ −1 X ApEn(m, r, n) = (n − mτ) [− ln(Ai/Bi)] (2.16) i=1 where Ai is the number of Xi within tolerance r of Xj for the m + 1-dimensional embedding and Bi is the number of Xi in tolerance r of Xj in the m-dimensional embedding. Sample Entropy is another measure of complexity [65] which is very similar to Approximate Entropy. The main diﬀerence between these two measures is how self counting is handled in the computation. In Approximate Entropy self counting is required at each iteration to prevent computing the natural logarithm of zero. How- ever, in Sample Entropy, the natural logarithm is computed once and self counting is

19 excluded by requiring that i 6= j in equation 2.13. Sample Entropy is computed by modifying the Approximate Entropy formula given in equation 2.16 to:

n−mτ X Ai A i=1 SamEn(m, r, n) = − ln = − ln n−mτ (2.17) B X Bi i=1

2.3.1 Incorporating a Time Delay In The Calculation

In typical applications where Approximate Entropy has been used to quantify the complexity or predictability, the autocorrelation of the time series data generally de- cays very quickly. Therefore, in these applications, a unity delay is an acceptable choice for generating the embedding vector and for studying the (nonlinear) complexity in the signal. However for those signals that have long range correlation, the use of a unity time delay may result in artificially low complexity values simply due to the (linear) autocorrelation effects that are present in the signal. In Chen et al. [11], in order to obtain higher values of complexity for respiratory time series, the investigators down-sampled the data by factors of 2, 4, and 8. In fact, down-sampling the data reduces the correlation between consecutive samples and, as a result, the complexity as measured by Approximate Entropy increases. In this section, we are interested in developing a systematic technique for the computation of Approximate Entropy that will allow meaningful comparisons across data sets. We propose the use of a (non-unity) time-delay embedding technique in the computation of Approx- imate Entropy to reduce the effects of autocorrelation.To motivate the development, we begin by investigating the effects of autocorrelation of a time series signal on the computation of the Approximate and Sample Entropy measures in some detail. The effects of autocorrelation were previously studied for other nonlinear anal- yses such as correlation dimension [20, 19]. Theiler [79] found that the long range correlation property of a signal can add a shoulder to the logarithmic plot of the cor-

20 relation integral and this can lead to inaccurate and spurious dimension estimation. To address this issue the Theiler window was introduced in the computation of the correlation integral [78]. We hypothesize that the addition of an appropriately chosen time delay, τ, will also improve the accuracy and the consistency of entropy-based complexity measures. To test this hypothesis, we examine the performance of the entropy-based complexity measures in two cases: with unity delay and with a delay equal to the ﬁrst zero crossing or minimum of the autocorrelation function.

2.3.2 Parameter Selection

In the calculation of Approximate or Sample Entropy typically the time delay τ is chosen to be unity [52, 55, 53, 33, 65]. This value is appropriate for dynamics that have a rapidly decaying autocorrelation function such as the Henon or Logistic maps [52]. However, for dynamic systems that have long range correlation, the choice of different delays can have a significant impact on the calculation of both the Approximate and Sample Entropy measures. If the delay is too small, the effect of autocorrelation results in a much higher conditional probability that an m + 1-dimensional embedding is within tolerance r given that the m-dimensional embedding vector is within tolerance r. So if this effect is to be minimized, the choice of the autocorrelation window is of critical importance, that is the delay needs to be chosen appropriately, for example to minimize the autocorrelation effects in the computation. If this is not done properly then instead of measuring the actual nonlinear properties of the signal, we end up measuring the autocorrelation within the data of interest and the dynamics can appear to have lower complexity than actually present in the data. Furthermore, selection of an appropriate time delay allows a more complete un- folding of the attractor, reduces the autocorrelation effects and provides a more accurate and consistent measures of the true complexity of the system. There are two common techniques for determining the time delay for use in a time-delay embedding

21 scheme; the first is to choose a delay that corresponds to the first minimum or zero crossing of the autocorrelation function, and the second is to use mutual information [14] instead of the autocorrelation function. Because we are primarily concerned with excluding linear stochastic (autocorrelation) effects in the Approximate and Sample Entropy calculations, the time delay has been chosen based on the first minimum or zero crossing of the autocorrelation function. In addition, the embedding dimension, m, and the tolerance value, r, are the other parameters that need to be selected. In phase space reconstruction, the embedding dimension is usually determined using the false nearest neighbor algorithm [31]. However, the objective of the Approximate or Sample Entropy algorithms is not to unfold the attractor, but rather to have a better estimate of the conditional probability function. To accomplish this, the idea is to maximize Bi (in equation

2.13) for a given m. We observe that Bi decrease as m increases. So in our analysis m is set at 2 which is consistent with previously suggested values from the literature [52, 55, 53]. The typical tolerance value r is between 0.1 and 0.2 of the standard deviation (SD) of the signal. In our work r has been chosen as 0.2 of the SD.

2.3.3 Parameter Validation Data

In an effort to better understand the effects of the time-delay parameter on the computation of Approximate and Sample Entropy measures, we simulated data from well-known dynamical systems such as the Lorenz, Rössler, Logistic and Henon systems. The Lorenz equations are given in Section 2.2.2. For the Rössler system:

x˙ = −y − z y˙ = x + ay (2.18) z˙ = b + z(x − c)

22 where a, b, and c are constant parameters that have been chosen as 0.2, 0.2, and 5.7 respectively. The simulation step size is 0.02 unit. The Logistic map is given by:

xi+1 = Rxi(1 − xi) (2.19) where R has been chosen as 3.6 to generate chaotic dynamics. A parameterized version of Henon map is given by:

x = Ry + 1 − 1.4x2 i+1 i i (2.20) yi+1 = 0.3Rxi where R = 1. In addition to these well known dynamical systems, we also use real biological data obtained from in vitro respiratory slice preparation experiments [76]. Brieﬂy, a 300 µm slice of a rat brainstem was transected from the central nervous system of a two day old rat and placed in a recording chamber containing artiﬁcial cerebrospinal

fluid oxygenated with 95% O2 and 5% CO2. Fictive inspiratory neural drive was recorded from the hypoglossal cranial rootlet using suction electrodes and filtered to have frequency content in the range at 10–1000 Hz. The neural activity was digitized using a Digidata 1322 analog-to-digital converter (8× oversampled). Bursts of inspiratory drive were isolated and extracted from the recorded data file using custom developed algorithms implemented in Matlab. We then analyzed 10 bursts of hypoglossal nerve discharge data which represents fictive inspiratory activity. A typical sample burst time series is given in figure 2.12.

Test Design

Each of the dynamical systems described in 2.3.3 with the given constants have been simulated ten times from diﬀerent initial conditions and 3000 samples from the steady state portion of the simulation have been analyzed. In addition, for each of the corresponding time series data, we generated surrogate data and calculated the Approximate and Sample Entropy measures using τ = 1, m = 2, and r = 0.2. We

23 na : nucleus ambiguus XII : hypoglossal motor nucleus Raw XII : integrated output from XII nerve preBötC: preBötzinger complex (region of rhythm generating/auto-rhythmic neurons) Figure 2.12: Sample Burst Data from an in vitro respiratory slice preparation (neonatal rat) also calculated the Approximate and Sample Entropy measures for dynamical systems that do not have a fast decaying autocorrelation function using τ equal to the first minimum, or zero crossing, of the autocorrelation function and with the same m and r values. The autocorrelation functions of the dynamic systems are shown in Figure 2.13 and the means and standard deviations of the Approximate and Sample Entropy measures are given in Table 2.3. The results of surrogate data analysis for the Henon and Logistic maps are considerably different from the original data and, thereby, provide strong evidence that the calculated measures capture and quantify the nonlinear properties of the time series data. Conversely, the results from both the Lorenz and Rösslersystems for unity delay are very similar. Because the surrogate data generation does not change the distribution and linear second-order properties of the signal, this suggests that with τ = 1 Approximate and Sample Entropy are measuring this linear statistical behavior and not measuring the nonlinear characteristics of the time series. By modifying the delay to be the first minimum or zero crossing of the autocorrelation function for the Lorenz and Rössler systems, the Approxi- mate and Sample Entropy measures for the surrogate and original data are no longer

24 similar. This demonstrates that, by selecting the proper delay, complexity resulting from signal nonlinearity can be measured and appropriately quantified. Further, the Lorenz equations have been simulated from 100 different initial conditions and the histogram of calculated Approximate and Sample Entropy for original and surrogate data sets with unity and non-unity time delay are given in figure 2.14 and figure 2.15 respectively. The histograms of the original and surrogate results for unity time delay overlap considerably and are not distinguishable. However the histograms with non- unity time delay (τ = 77) are completely separable and this further validates that the Approximate and Sample Entropy computations are quantifying nonlinear behavior in time series data. For the burst samples, the results of the Approximate and Sample Entropy measures compared to ten surrogate data provide similar conclusion to those of the Lorenz and Rösslersystems. In order to further investigate the effect of time delay selection in the computation of the Approximate and Sample Entropy measures, we analyzed ten time series of white noise each consisting of 3000 samples with zero mean and unity variance. After

Original Surrogate 1 1

0.9 0.9 Logistic 0.8 0.8 0 5 10 0 5 10 1 1

0 0 Henon −1 −1 0 5 10 0 5 10 1 1

0.5 0.5 Rossler 0 0 0 50 100 0 50 100 1 1

0 0 Lorenz −1 −1 0 50 100 0 50 100 Time delay Time delay

Figure 2.13: Normalized auto correlation function

25 τ = 1 40

10 Number of Samples 0 0.16 0.18 0.2 0.22 0.24 0.26 Surrogate Data τ = 77 Original Data 30

10 Number of Samples 0 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Figure 2.14: Histogram of Approximate Entropy for Lorenz Attractor

τ = 1 50

10 Number of Samples 0 0.14 0.16 0.18 0.2 0.22 Surrogate Data τ = 77 Original Data 30

10 Number of Samples 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Figure 2.15: Histogram of Sample Entropy for Lorenz Attractor

26 Table 2.3: ApEn & SamEn Results Original Data Surrogate Data ApEn SamEn ApEn SamEn Logistic map (τ = 1) 0.2107±0.0019 0.2057±0.0017 0.8898±0.0094 0.5127±0.0672 Henon map x-axis (τ = 1) 0.4751±0.0050 0.4597±0.0073 1.9592±0.0079 1.9925±0.0124 Henon map y-axis (τ = 1) 0.4751±0.0050 0.4597±0.0072 1.9594±0.0087 1.9916±0.0121 Lorenz x-axis (τ = 1) 0.1969±0.0057 0.1509±0.0067 0.2207±0.0087 0.1944±0.0074 Lorenz x-axis (τ = 77) 1.0339±0.0732 0.9698±0.1140 1.3724±0.0322 1.4793±0.0495 Lorenz y-axis (τ = 1) 0.2518±0.0045 0.1500±0.0077 0.3013±0.0063 0.2359±0.0121 Lorenz y-axis (τ = 69) 1.0484±0.0629 0.9659±0.0826 1.4980±0.0309 1.6706±0.0368 Lorenz z-axis (τ = 1) 0.3134±0.0099 0.2483±0.0119 0.3773±0.0076 0.3386±0.0143 Lorenz z-axis (τ = 38) 0.9863±0.0341 0.9082±0.0465 1.5241±0.0358 1.5621±0.0742 Rösslerx-axis (τ = 1) 0.0706±0.0020 0.0676±0.0020 0.1006±0.0131 0.0930±0.0105 Rösslerx-axis (τ = 73) 0.3249±0.0699 0.3261±0.0665 0.7822±0.0568 0.8205±0.0583 Rösslery-axis (τ = 1) 0.0639±0.0005 0.0644±0.0014 0.0721±0.0063 0.0705±0.0053 Rösslery-axis (τ = 79) 0.3169±0.0851 0.3326±0.0737 0.6674±0.0834 0.7183±0.1026 Rösslerz-axis (τ = 1) 0.0493±0.0168 0.0063±0.0014 0.0660±0.0224 0.0102±0.0031 Rösslerz-axis (τ = 75) 0.6900±0.1008 0.4991±0.1656 0.6551±0.1107 0.4460±0.1405 Burst Samples (τ = 1) 0.6373±0.0121 0.4537±0.0139 0.6813±0.0133 0.5483±0.0213 Burst Samples (τ = 10) 1.5589±0.0314 1.2078±0.0279 1.8061±0.0401 1.6869±0.0709

Table 2.4: ApEn & SamEn Results for filtered white noise White Noise ApEn SamEn Original (τ = 1) 2.0100±0.0121 2.1911±0.0189 1. filtering (τ = 1) 1.8868±0.0111 1.9915±0.0180 1. filtering (τ = 2) 2.0121±0.0128 2.1926±0.0181 2. filtering (τ = 1) 1.7237±0.0159 1.7622±0.0280 2. filtering (τ = 3) 1.9984±0.0118 2.1762±0.0166 3. filtering (τ = 1) 1.5759±0.0122 1.5786±0.0162 3. filtering (τ = 4) 2.0053±0.0141 2.1857±0.0198 4. filtering (τ = 1) 1.4459±0.0188 1.4369±0.0226 4. filtering (τ = 5) 2.0041±0.0134 2.1798±0.0149 data generation, to add linear correlation to the signal, a first order moving average filter (given in equation 2.21) has been applied to the white noise signal:

y[n] = 0.5x[n] + 0.5x[n − 1] (2.21)

The results of the Approximate and Sample Entropy computation for the filtered white noise signals with several iterations of filtering to make the resultant signals more correlated are shown in Table 2.4. The filtering does not add any nonlinear properties to the signal and, thus, the results of the Approximate and Sample Entropy calculations should be unchanged. However, in each step of the filtering process the entropy results with unity delay are

27 lower and the resulting signal appears to be less complex. This decrease in complexity is due to the autocorrelation eﬀect and by selecting a time delay equal to the ﬁrst zero crossing of the autocorrelation function, more consistent results are obtained.

2.3.4 Conclusion

Approximate and Sample Entropy are intended to quantify signal complexity and predictability, and can measure both linear and nonlinear properties of a time series signal that exhibits short or long range correlation. If the autocorrelation function is decaying rapidly, unity delay (τ = 1) may be sufficient to provide an accurate measure of signal complexity resulting from the nonlinear feature in the signal. However, for signals with a slowly decaying autocorrelation function, the story can be very different. Hence the linear signal properties are measured with a unity time-delay because the signal analysis is focused within the autocorrelation region. If τ is chosen as the first zero crossing or minimum of the autocorrelation function, this is a more appropriate choice if the objective is to quantify the complex aspects of the signal that are the inherent result of nonlinearity within the system. We are currently applying these methods to analyze biological signals that exhibit both linear and nonlinear properties. The results of this analysis are similar to those obtained by Theiler with regard to the computation of correlation dimension [78]. We have applied the modified Approximate and Sample Entropy methods to biological systems with a relatively simple neural network architecture and these measures are used to quantify the changes that occur as the experimental conditions are varied over a physiologically meaningful range.

28 Chapter 3

Complexity Analysis of Neonatal EEG

Neonatal electroencephalographic-polysomnographic studies have been performed for over half a century. From the earliest days of the development of the neonatal intensive care unit, EEG sleep studies have been proposed to assess for brain organization and maturation, assessment of the severity and persistence of neonatal encephalopathy, the detection of neonatal seizures, and correlation with other examination and imaging studies. In this study we examined maturation and neurodevelopment of the neonatal brain by quantification of their brain signals (EEG). We were especially interested to understand the effect of strategies designed to improve sleep organization. One of these strategies is skin-to-skin contact or Kangaroo Care (KC). For this purpose two data set have been analyzed. The first neonatal EEG data set is from Pittsburgh which consists of full term and premature neonates who did not receive KC intervention. The second data set is from Cleveland in a controlled trial of the KC intervention of premature infants [37]. In quantification of the EEG, Approximate Entropy and Sample Entropy which are scale independent measures of complexity or regularity have been used. These measures have been used in the analysis of physiological time series data such as heart rate variability [33, 53, 55]. In addition, the effect of the time delay on the

29 computation of Approximate Entropy and Sample Entropy of neonatal EEG has been investigated in detail. In the data analysis, the first objective was to calculate the complexity of the EEG during sleep of premature neonates at the post menstrual ages of 31-32 weeks and 40-41 weeks given that they were born at a PMA of 31- 32 weeks, and full term neonates at the post menstrual age of 40-41 weeks from the Pittsburgh study. Further, the complexity of the EEG acquired from different brain regions during sleep of neonates from the KC intervention study (Cleveland) were computed and compared to the results from the control study (Pittsburgh) at the same PMA. It has been established that the KC intervention alters sleep organization of premature neonates [2, 13], however, these changes have not previously been quantified in terms of complexity measures such as Approximate Entropy and Sample Entropy. In this chapter, we show that the complexity of premature neonates is lower than the complexity of full term neonates at the same PMA using data from the Pittsburgh study as controls, which is consistent with our previous work [68] and leads to the hypothesis that EEG-complexity increases with brain maturation and neurodevelopment. We further show using discriminant analysis methods that the complexity of the KC intervention cohort is closer in a statistical sense to the complexity of the full term cohort from the Pittsburgh study than to the complexity of the premature cohort from the Pittsburgh study.

3.1 Data Collection & Description

The design, method and recording procedure as well as the neonates selected for the KC intervention study have been given in [37]. The EEG data for the KC intervention study has been collected at a sampling rate of 240 Samples/Sec using a 16 bit ADC (Analog Digital Converter) with a single-ended (referential) measurement technique. A diﬀerential measurement technique has been used in the collection of the Pittsburgh study study using a 12 bit ADC at a sampling rate of 64 Samples/Sec. The KC data was transformed to a diﬀerential format to facilitate the comparison of the data and

30 FP1 FP2

F7 F8 F3 FZ F4

T3 T4 A1 C3CZ C4 A2

PZ T5 P3 P4 T6

O1 O2

Figure 3.1: Electrodes Placement for International 10-20 System analysis results with the Pittsburgh study data. High pass and low pass filtering of the analog EEG data were employed prior to acquisition in both studies. The high-pass filter time constant1 for the Pittsburgh study data collection system was 0.3 seconds 1 and T is inversely proportional to the cutoff frequency f = 2πT which is equal to 0.531Hz. In addition to prevent aliasing on each channel of data a 35Hz anti-aliasing analog low pass filter has been used. This setting of the analog anti-aliasing filter, did in fact result in some aliasing of the data. In particular, frequencies in the range 32-35 Hz will be aliased into the frequency range of 0-2.5 Hz. Further filtering of the data before analysis was used to remove this corrupted data from the collected EEG data for the Pittsburgh cohort. The EEG channels that were recorded in each study and electrode placement are shown in Table 3.1 and Figure 3.1. The KC data set for neonates born at the PMA of 31-32 weeks does not include the channels Fp1 and Fp2. Therefore, in a comparison study we are not able to use any channels that involve Fp1 and Fp2 in the analysis of the 31-32 week data.

1The time constant (T ) is the time that it takes for the response of a system to a sudden impulse to decay from 100% to 37%.

31 Table 3.1: Neonate EEG Channels

Pittsburgh Study KC Intervention Study 31-32 Weeks 40-41 Weeks Fp1 - T3 FP1 T3 - O1 FP2 Fp2 - T4 C3 C3 T4 - O2 C4 C4 Fp1 - C3 O1 O1 C3 - O1 O2 O2 Fp2 - C4 T3 T3 C4 - O2 T4 T4 T3 - C3 Cz Cz C3 - Cz Cz - C4 C4 - T4

Table 3.2: Pittsburgh Study Group

Preterm Full term EEG Number PMA EEG Number PMA EEG Number PMA 34 31 37 41 55 41.43 44 31 39 40 61 41 46 31 81 40 63 41 73 31 82 41 79 40.43 96 31 84 40 88 40 114 32 112 40 100 41.43 PMA(Postmenstrual Age)

Two EEG data sets were collected for each subject in the KC intervention study. The ﬁrst set was at a PMA of 31-32 weeks and the second was at the PMA of 40-41 weeks. Therefore neonate data from the Pittsburgh study was chosen to be comparable with this group, i.e. premature neonates that had EEG studies at PMAs of 31-32 weeks and 40-41 weeks. Furthermore, full term subjects with a PMA of 40-41 weeks deﬁne the normal group. The subjects in the Pittsburgh study are shown in Table 3.2.

32 3.2 The Eﬀect of Time Delay on ApEn & SpEn Computation

The effect of the time delay τ on the Approximate Entropy and Sample Entropy measures has been investigated in detail in [27]. In certain sleep states the neonatal EEG signal has long-range correlation (memory). Therefore, based on our previous results [27], selecting a unity time delay in the embedding process can be misleading in the computation of ApEn and SpEn depending on what signal qualities are of interest in the study. For example, for a signal with long-range correlation a unity delay will result in ApEn and SpEn measures that primarily quantify the linear stochastic features in the data. However, if one is interested in quantifying the complexity in the signal as a result of nonlinearity, a time delay equal to the first zero crossing or minimum of the autocorrelation function is more appropriate and preferred. A rep- resentative plot of the autocorrelation function of various EEG channels of a sample epoch (one minute) from the Pittsburgh study is given in Figure 3.2. The first zero crossing or minimum of the autocorrelation function is different for different channels and it also changes for different epochs of the EEG data reflecting the non-stationary characteristics and sleep stage dependence in the data. In our data analysis in order to improve consistency in the computations and facilitate comparisons among channels and various epochs in the two data sets being analyzed, we would like to select a constant time delay τ which may not be exactly the first minimum or zero crossing of the autocorrelation function for each epoch of data, but rather represents a satisfac- tory choice of time delay for all channels and epochs to be analyzed. Selecting such a time delay across the channels, across epochs, and across studies in a consistent way will allow us to make more effective comparisons. Therefore, the effect of choosing different time delays that are close to the first minimum or zero crossing has been investigated. In order to quantify the variability of the time delay in each epoch of data for this study we have chosen 2 hours (120 epochs) from the Pittsburgh data set (study number 96). Further to validate the necessity of choosing a non-unity time de-

33 Fp1− T3 T3 − O1 Fp2− T4 T4 − O2 1 1 1 1

0.5 0.5 0.5 0.5

0 0 0 0

−0.5 −0.5 −0.5 −0.5 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 Fp1− C3 C3 − O1 Fp2− C4 C4 − O2 1 1 1 1

0.5 0.5 0.5 0.5

0 0 0 0

−0.5 −0.5 −0.5 −0.5 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3

Normalized Autocorrelation T3 − C3 C3 − Cz Cz − C4 C4 − T4 1 1 1 1

0.5 0.5 0.5 0.5

0 0 0 0

−0.5 −0.5 −0.5 −0.5 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 Time delay (Sec) Time delay (Sec) Time delay (Sec) Time delay (Sec)

Figure 3.2: Neonatal EEG Autocorrelation Function for Pittsburgh Study

34 lay, the complexity measures for the time delay equal to both unity (one sample) and 1 second (64 samples) with r = 0.2 and m = 2 have been calculated and the means and standard deviations of the entropy measures using these parameters are given in Table 3.3. The results for a unity time delay are very low and provides compelling evidence of the need of selecting a time delay to quantify the nonlinear properties of the time series data. Next, for each epoch of data ApEn and SpEn measures with a time delay ranging from 0.8 seconds (51 samples) to 1.2 seconds (77 samples) with resolution of 1 sample and with the same values of m and r have been analyzed. Be- cause the complexity measures for various epochs of data will be diﬀerent due to the variability in the data, the STD (Standard Deviation) of the complexity measures for each epoch with diﬀerent time delays have also been calculated. The results of this analysis demonstrate the stability of the Approximate Entropy and Sample Entropy measures in quantifying the complexity in each epoch of the neonatal EEG, when the time delay is suitably chosen based on the autocorrelation properties of the signal. The mean and STD of the calculated STD for each epoch of data over the entire data set (120 epochs) is presented in Table 3.4. The results show that the Approximate and Sample Entropy measures are very stable when calculated using a suitable time delay.

3.3 Results & Discussion

In the computation of Approximate and Sample Entropy, there are three parameters that need to be chosen: embedding dimension, m; tolerance, r; and time delay , τ. The most important parameter is the time delay and it needs to be selected properly. In the previous section we showed that the variability in the complexity measure due to a time delay that is properly selected, i.e. is close to the ﬁrst minimum or zero crossing of autocorrelation function, is negligible. In the data analysis that follows, we have considered three constant time delays of 0.8 seconds, 1 second, and 1.2 seconds. Furthermore, the embedding dimension , m, and tolerance, r, have

35 Table 3.3: Unity vs. 1 Seconds Time Delay Analysis Results for EEG Number 96 (Mean±STD)

Approximate Entropy FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 τ = 1 0.71±0.33 0.53±0.19 0.68±0.29 0.48±0.07 0.81±0.28 0.46±0.05 τ = 64 1.60±0.16 1.54±0.12 1.63±0.15 1.58±0.09 1.74±0.12 1.55±0.08 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 τ = 1 0.83±0.3 0.49±0.06 0.66±0.21 0.66±0.05 0.67±0.06 0.56±0.09 τ = 64 1.73±0.14 1.59±0.06 1.63±0.13 1.72±0.05 1.70±0.07 1.61±0.10

Sample Entropy FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 τ = 1 0.48±0.29 0.34±0.17 0.46±0.27 0.32±0.08 0.56±0.25 0.30±0.05 τ = 64 1.55±0.25 1.50±0.22 1.6±0.26 1.62±0.18 1.75±0.21 1.58±0.15 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 τ = 1 0.59±0.30 0.34±0.04 0.45±0.20 0.47±0.06 0.46±0.06 0.38±0.10 τ = 64 1.74±0.24 1.66±0.10 1.65±0.21 1.77±0.11 1.74±0.13 1.63±0.19

Table 3.4: Time Delay Analysis Results for EEG Number 96 (Mean±STD of STD)

Approximate Entropy FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 0.016±0.006 0.017±0.007 0.016±0.007 0.016±0.006 0.014±0.005 0.016±0.006 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 0.015±0.006 0.016±0.005 0.015±0.006 0.014±0.005 0.015±0.005 0.016±0.007

Sample Entropy FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 0.029±0.014 0.030±0.017 0.031±0.014 0.027±0.014 0.024±0.015 0.027±0.013 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 0.024±0.012 0.025±0.011 0.026±0.013 0.024±0.010 0.024±0.011 0.027±0.015

36 been chosen based on suggested values from the literature [27] as m = 2 and r = 0.2 of STD (Standard Deviation) of the signal. In addition, the epoch length has been chosen based on the number of samples suggested in Pincus [52] where 3000 samples was sufficient to get a good estimate of the complexity of a signal. Therefore, in the analysis of the Pittsburgh data the epoch length has been chosen as one minute which corresponds to 3840 samples. In the KC intervention study, the epoch length was chosen as 15 seconds which corresponds to 3600 samples. Because of this difference in the time window of the data that is analyzed, and the effect of non-stationarity on the computation of the complexity of the data, we expect the ApEn and SpEn of the Pittsburgh data to be greater than a comparable study from the KC intervention study. The means and standard deviations of the analysis of the data are given in Tables 3.5-3.8. In addition, the EEG data has also been analyzed according to their sleep states, i.e. active or quiet sleep, except for the 31-32 week neonates from the Pittsburgh study where sleep state scoring information was not available. Therefore, the results of the complexity measures independent of sleep state are also presented in Table 3.5. The results for the active sleep states are shown in Tables 3.6 and 3.9, and the results for the quiet sleep states are given in Tables 3.7 and 3.10. Furthermore, to investigate the separation of the distributions of the complexity measures between preterm and full term neonates at a PMA of 40-41 weeks from the Pittsburgh study, a T-test has been used. The T-test results are given in Table 3.14 and show that for the majority of channels the Approximate Entropy measures between the preterm and fullterm groups of Pittsburgh study are statistically significantly different.

3.3.1 Histogram Analysis

A histogram, h, is deﬁned as a mapping that counts the number of observations that fall into various disjoint categories (known as bins). Given a data set where the total

37 number of observations=N and total number of bins=n; we have:

n X N = h(k) (3.1) k=1 where k is an index over the bins. In our computation, in order to be able to compare histograms from diﬀerent studies that do not have the same number of observations, the histograms are normalized. As a result, the histograms are similar to a probability density function with the total area (sum of histogram) is equal to unity. The histograms of the Approximate Entropy for the Pittsburgh and KC intervention studies are given in Figures 3.3 and 3.4, and the histogram of the Approximate Entropy measures of active vs. quiet sleep states for the Pittsburgh full term and KC intervention preterm studies at a PMA of 40-41 weeks are shown in Figures 3.5 and 3.6 respectively.

38 Table 3.5: Means of Complexity Results Approximate Entropy Channel Name FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 Pitts. Fullterm 40-41 Weeks τ = 0.8 Sec 1.722 1.756 1.634 1.669 1.732 1.754 1.623 1.636 1.747 1.778 1.687 1.604 Pitts. Fullterm 40-41 Weeks τ = 1 Sec 1.730 1.760 1.643 1.674 1.738 1.758 1.631 1.642 1.753 1.781 1.693 1.611 Pitts. Fullterm 40-41 Weeks τ = 1.2 Sec 1.735 1.764 1.649 1.679 1.742 1.761 1.637 1.646 1.758 1.785 1.697 1.618 Pitts. Premature 40-41 Weeks τ = 0.8 Sec 1.696 1.682 1.600 1.628 1.697 1.744 1.594 1.647 1.732 1.678 1.573 1.604 Pitts. Premature 40-41 Weeks τ = 1 Sec 1.703 1.688 1.610 1.634 1.704 1.749 1.601 1.652 1.737 1.685 1.580 1.611 Pitts. Premature 40-41 Weeks τ = 1.2 Sec 1.710 1.691 1.616 1.638 1.710 1.751 1.606 1.656 1.741 1.690 1.587 1.616 Pitts. Premature 31-32 Weeks τ = 0.8 Sec 1.513 1.491 1.470 1.481 1.604 1.524 1.589 1.558 1.549 1.631 1.614 1.515 Pitts. Premature 31-32 Weeks τ = 1 Sec 1.533 1.506 1.490 1.496 1.618 1.536 1.602 1.569 1.565 1.643 1.626 1.531 Pitts. Premature 31-32 Weeks τ = 1.2 Sec 1.548 1.519 1.507 1.509 1.629 1.549 1.615 1.577 1.579 1.652 1.637 1.546 Pilot Premature 40-41 Weeks τ = 0.8 Sec 0.940 0.855 0.890 0.881 0.930 0.858 0.899 0.901 0.919 0.974 0.966 0.926 Pilot Premature 40-41 Weeks τ = 1 Sec 0.923 0.837 0.873 0.863 0.914 0.840 0.882 0.882 0.904 0.958 0.949 0.910 Pilot Premature 40-41 Weeks τ = 1.2 Sec 0.906 0.820 0.856 0.847 0.898 0.822 0.865 0.865 0.887 0.942 0.931 0.892 Pilot Premature 31-32 Weeks τ = 0.8 Sec 0.762 0.735 0.781 0.772 0.783 0.818 0.793 0.767

39 Pilot Premature 31-32 Weeks τ = 1 Sec 0.754 0.728 0.772 0.761 0.774 0.803 0.780 0.757 Pilot Premature 31-32 Weeks τ = 1.2 Sec 0.744 0.718 0.760 0.749 0.765 0.791 0.765 0.748

Sample Entropy Channel Name FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 Pitts. Fullterm 40-41 Weeks τ = 0.8 Sec 1.770 1.829 1.649 1.716 1.800 1.838 1.649 1.688 1.806 1.853 1.736 1.616 Pitts. Fullterm 40-41 Weeks τ = 1 Sec 1.791 1.844 1.673 1.732 1.817 1.852 1.670 1.705 1.821 1.866 1.751 1.633 Pitts. Fullterm 40-41 Weeks τ = 1.2 Sec 1.806 1.857 1.693 1.747 1.831 1.864 1.687 1.717 1.836 1.877 1.762 1.650 Pitts. Premature 40-41 Weeks τ = 0.8 Sec 1.752 1.749 1.649 1.692 1.756 1.834 1.627 1.708 1.803 1.742 1.603 1.645 Pitts. Premature 40-41 Weeks τ = 1 Sec 1.772 1.768 1.672 1.707 1.775 1.848 1.647 1.722 1.817 1.760 1.621 1.664 Pitts. Premature 40-41 Weeks τ = 1.2 Sec 1.791 1.782 1.689 1.722 1.791 1.860 1.664 1.736 1.831 1.772 1.638 1.679 Pitts. Premature 31-32 Weeks τ = 0.8 Sec 1.374 1.380 1.343 1.396 1.517 1.480 1.509 1.541 1.463 1.598 1.583 1.435 Pitts. Premature 31-32 Weeks τ = 1 Sec 1.423 1.427 1.391 1.441 1.557 1.517 1.547 1.575 1.504 1.633 1.620 1.479 Pitts. Premature 31-32 Weeks τ = 1.2 Sec 1.466 1.469 1.436 1.481 1.592 1.553 1.584 1.604 1.541 1.662 1.651 1.517 Pilot Premature 40-41 Weeks τ = 0.8 Sec 1.073 0.984 1.022 1.014 1.061 0.987 1.032 1.036 1.051 1.106 1.101 1.061 Pilot Premature 40-41 Weeks τ = 1 Sec 1.076 0.988 1.018 1.012 1.065 0.989 1.021 1.033 1.056 1.101 1.093 1.060 Pilot Premature 40-41 Weeks τ = 1.2 Sec 1.080 0.987 1.030 1.017 1.070 0.989 1.037 1.037 1.059 1.115 1.106 1.067 Pilot Premature 31-32 Weeks τ = 0.8 Sec 0.872 0.842 0.900 0.893 0.885 0.939 0.909 0.872 Pilot Premature 31-32 Weeks τ = 1 Sec 0.885 0.859 0.912 0.902 0.899 0.944 0.917 0.884 Pilot Premature 31-32 Weeks τ = 1.2 Sec 0.896 0.871 0.920 0.911 0.912 0.952 0.921 0.897 Table 3.6: Means of Active Complexity Results Approximate Entropy Channel Name FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 Pitts. Fullterm 40-41 Weeks τ = 0.8 Sec 1.788 1.811 1.696 1.744 1.812 1.838 1.722 1.749 1.836 1.877 1.789 1.699 Pitts. Fullterm 40-41 Weeks τ = 1 Sec 1.797 1.818 1.707 1.751 1.819 1.843 1.731 1.757 1.842 1.882 1.793 1.707 Pitts. Fullterm 40-41 Weeks τ = 1.2 Sec 1.804 1.822 1.716 1.756 1.824 1.846 1.737 1.760 1.848 1.885 1.797 1.714 Pitts. Premature 40-41 Weeks τ = 0.8 Sec 1.758 1.756 1.691 1.733 1.787 1.819 1.734 1.779 1.806 1.812 1.772 1.750 Pitts. Premature 40-41 Weeks τ = 1 Sec 1.765 1.761 1.700 1.737 1.793 1.825 1.740 1.782 1.811 1.817 1.776 1.757 Pitts. Premature 40-41 Weeks τ = 1.2 Sec 1.771 1.766 1.706 1.740 1.800 1.828 1.745 1.787 1.814 1.822 1.782 1.762 Pitts. Premature 31-32 Weeks τ = 0.8 Sec 1.513 1.491 1.469 1.481 1.604 1.523 1.589 1.558 1.549 1.630 1.614 1.515 Pitts. Premature 31-32 Weeks τ = 1 Sec 1.532 1.506 1.490 1.495 1.617 1.536 1.602 1.569 1.564 1.643 1.626 1.531 Pitts. Premature 31-32 Weeks τ = 1.2 Sec 1.548 1.519 1.507 1.508 1.629 1.548 1.614 1.577 1.578 1.651 1.636 1.546 Pilot Premature 40-41 Weeks τ = 0.8 Sec 0.998 0.878 0.919 0.933 0.995 0.872 0.932 0.940 10000 1.082 1.042 0.988 Pilot Premature 40-41 Weeks τ = 1 Sec 0.978 0.857 0.898 0.910 0.975 0.853 0.910 0.917 0.980 1.063 1.021 0.970 Pilot Premature 40-41 Weeks τ = 1.2 Sec 0.957 0.836 0.879 0.890 0.958 0.833 0.892 0.895 0.961 1.047 1.004 0.951 Pilot Premature 31-32 Weeks τ = 0.8 Sec 0.787 0.762 0.810 0.805 0.817 0.883 0.857 0.811

40 Pilot Premature 31-32 Weeks τ = 1 Sec 0.778 0.753 0.801 0.794 0.805 0.867 0.843 0.798 Pilot Premature 31-32 Weeks τ = 1.2 Sec 0.765 0.742 0.788 0.783 0.793 0.852 0.825 0.785

Sample Entropy Channel Name FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 Pitts. Fullterm 40-41 Weeks τ = 0.8 Sec 1.850 1.904 1.714 1.818 1.900 1.939 1.764 1.830 1.917 1.972 1.859 1.739 Pitts. Fullterm 40-41 Weeks τ = 1 Sec 1.873 1.926 1.747 1.843 1.921 1.957 1.788 1.852 1.937 1.987 1.874 1.761 Pitts. Fullterm 40-41 Weeks τ = 1.2 Sec 1.896 1.940 1.776 1.858 1.938 1.969 1.809 1.864 1.955 2.001 1.887 1.782 Pitts. Premature 40-41 Weeks τ = 0.8 Sec 1.830 1.848 1.762 1.826 1.866 1.923 1.809 1.879 1.899 1.903 1.860 1.834 Pitts. Premature 40-41 Weeks τ = 1 Sec 1.855 1.869 1.787 1.841 1.888 1.941 1.831 1.893 1.916 1.920 1.873 1.854 Pitts. Premature 40-41 Weeks τ = 1.2 Sec 1.875 1.884 1.805 1.855 1.908 1.957 1.848 1.909 1.931 1.935 1.889 1.868 Pitts. Premature 31-32 Weeks τ = 0.8 Sec 1.373 1.379 1.342 1.395 1.516 1.479 1.508 1.541 1.462 1.597 1.582 1.434 Pitts. Premature 31-32 Weeks τ = 1 Sec 1.422 1.426 1.390 1.440 1.556 1.517 1.546 1.575 1.504 1.633 1.620 1.478 Pitts. Premature 31-32 Weeks τ = 1.2 Sec 1.465 1.468 1.435 1.481 1.591 1.553 1.584 1.604 1.541 1.661 1.650 1.517 Pilot Premature 40-41 Weeks τ = 0.8 Sec 1.141 1.011 1.055 1.068 1.133 1.009 1.075 1.081 1.141 1.228 1.187 1.126 Pilot Premature 40-41 Weeks τ = 1 Sec 1.140 1.008 1.054 1.064 1.134 1.011 1.071 1.075 1.140 1.231 1.186 1.128 Pilot Premature 40-41 Weeks τ = 1.2 Sec 1.136 1.004 1.051 1.061 1.135 1.008 1.072 1.070 1.139 1.232 1.188 1.127 Pilot Premature 31-32 Weeks τ = 0.8 Sec 0.911 0.884 0.937 0.934 0.936 1.017 0.988 0.928 Pilot Premature 31-32 Weeks τ = 1 Sec 0.921 0.893 0.947 0.943 0.945 1.017 0.992 0.936 Pilot Premature 31-32 Weeks τ = 1.2 Sec 0.927 0.900 0.954 0.951 0.952 1.021 0.991 0.944 Table 3.7: Means of Quiet Complexity Results Approximate Entropy Channel Name FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 Pitts. Fullterm 40-41 Weeks τ = 0.8 Sec 1.676 1.742 1.568 1.623 1.686 1.730 1.554 1.572 1.682 1.714 1.598 1.491 Pitts. Fullterm 40-41 Weeks τ = 1 Sec 1.683 1.744 1.577 1.626 1.692 1.733 1.561 1.577 1.687 1.717 1.604 1.499 Pitts. Fullterm 40-41 Weeks τ = 1.2 Sec 1.686 1.746 1.582 1.630 1.694 1.736 1.566 1.581 1.693 1.721 1.609 1.506 Pitts. Premature 40-41 Weeks τ = 0.8 Sec 1.640 1.625 1.520 1.543 1.619 1.695 1.468 1.537 1.672 1.567 1.401 1.481 Pitts. Premature 40-41 Weeks τ = 1 Sec 1.647 1.631 1.530 1.550 1.625 1.699 1.477 1.543 1.678 1.576 1.411 1.490 Pitts. Premature 40-41 Weeks τ = 1.2 Sec 1.656 1.634 1.536 1.556 1.631 1.701 1.483 1.547 1.683 1.581 1.419 1.496 Pitts. Premature 31-32 Weeks τ = 0.8 Sec 1.513 1.491 1.469 1.481 1.604 1.523 1.589 1.558 1.549 1.630 1.614 1.515 Pitts. Premature 31-32 Weeks τ = 1 Sec 1.532 1.506 1.490 1.495 1.617 1.536 1.602 1.569 1.564 1.643 1.626 1.531 Pitts. Premature 31-32 Weeks τ = 1.2 Sec 1.548 1.519 1.507 1.508 1.629 1.548 1.614 1.577 1.578 1.651 1.636 1.546 Pilot Premature 40-41 Weeks τ = 0.8 Sec 1.047 0.961 0.987 0.990 1.035 0.980 1.011 1.033 1.040 1.151 1.133 1.044 Pilot Premature 40-41 Weeks τ = 1 Sec 1.035 0.946 0.974 0.976 1.022 0.964 0.996 1.017 1.025 1.133 1.116 1.030 Pilot Premature 40-41 Weeks τ = 1.2 Sec 1.020 0.929 0.960 0.960 1.007 0.947 0.981 1.001 1.008 1.117 1.098 1.016 Pilot Premature 31-32 Weeks τ = 0.8 Sec 0.774 0.760 0.812 0.822 0.778 0.892 0.863 0.796

41 Pilot Premature 31-32 Weeks τ = 1 Sec 0.777 0.764 0.807 0.817 0.781 0.884 0.857 0.798 Pilot Premature 31-32 Weeks τ = 1.2 Sec 0.774 0.763 0.800 0.809 0.784 0.879 0.847 0.797

Sample Entropy Channel Name FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 Pitts. Fullterm 40-41 Weeks τ = 0.8 Sec 1.723 1.817 1.583 1.661 1.750 1.811 1.569 1.609 1.728 1.774 1.626 1.476 Pitts. Fullterm 40-41 Weeks τ = 1 Sec 1.741 1.827 1.602 1.672 1.764 1.824 1.586 1.622 1.741 1.785 1.639 1.493 Pitts. Fullterm 40-41 Weeks τ = 1.2 Sec 1.751 1.836 1.617 1.686 1.774 1.833 1.600 1.635 1.754 1.794 1.651 1.507 Pitts. Premature 40-41 Weeks τ = 0.8 Sec 1.684 1.674 1.552 1.584 1.663 1.775 1.470 1.570 1.726 1.604 1.384 1.492 Pitts. Premature 40-41 Weeks τ = 1 Sec 1.701 1.691 1.574 1.600 1.681 1.785 1.489 1.583 1.740 1.623 1.407 1.510 Pitts. Premature 40-41 Weeks τ = 1.2 Sec 1.720 1.706 1.590 1.615 1.691 1.796 1.505 1.596 1.752 1.635 1.425 1.525 Pitts. Premature 31-32 Weeks τ = 0.8 Sec 1.373 1.379 1.342 1.395 1.516 1.479 1.508 1.541 1.462 1.597 1.582 1.434 Pitts. Premature 31-32 Weeks τ = 1 Sec 1.422 1.426 1.390 1.440 1.556 1.517 1.546 1.575 1.504 1.633 1.620 1.478 Pitts. Premature 31-32 Weeks τ = 1.2 Sec 1.465 1.468 1.435 1.481 1.591 1.553 1.584 1.604 1.541 1.661 1.650 1.517 Pilot Premature 40-41 Weeks τ = 0.8 Sec 1.185 1.102 1.129 1.132 1.176 1.122 1.150 1.178 1.184 1.298 1.278 1.189 Pilot Premature 40-41 Weeks τ = 1 Sec 1.194 1.106 1.137 1.137 1.183 1.124 1.156 1.178 1.190 1.300 1.280 1.196 Pilot Premature 40-41 Weeks τ = 1.2 Sec 1.203 1.108 1.144 1.142 1.190 1.127 1.162 1.180 1.193 1.303 1.282 1.203 Pilot Premature 31-32 Weeks τ = 0.8 Sec 0.845 0.837 0.911 0.924 0.839 0.994 0.955 0.864 Pilot Premature 31-32 Weeks τ = 1 Sec 0.881 0.873 0.937 0.945 0.872 1.015 0.978 0.898 Pilot Premature 31-32 Weeks τ = 1.2 Sec 0.909 0.900 0.954 0.964 0.906 1.036 0.992 0.925 Table 3.8: Standard Deviation of Complexity Results Approximate Entropy Channel Name FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 Pitts. Fullterm 40-41 Weeks τ = 0.8 Sec 0.227 0.233 0.279 0.290 0.249 0.252 0.318 0.326 0.280 0.273 0.351 0.375 Pitts. Fullterm 40-41 Weeks τ = 1 Sec 0.223 0.231 0.276 0.288 0.246 0.249 0.315 0.324 0.277 0.270 0.346 0.371 Pitts. Fullterm 40-41 Weeks τ = 1.2 Sec 0.221 0.228 0.273 0.286 0.244 0.245 0.311 0.321 0.274 0.267 0.342 0.367 Pitts. Premature 40-41 Weeks τ = 0.8 Sec 0.240 0.242 0.259 0.276 0.257 0.218 0.315 0.310 0.240 0.297 0.370 0.295 Pitts. Premature 40-41 Weeks τ = 1 Sec 0.238 0.240 0.255 0.272 0.254 0.215 0.312 0.306 0.238 0.292 0.365 0.292 Pitts. Premature 40-41 Weeks τ = 1.2 Sec 0.233 0.238 0.251 0.270 0.250 0.214 0.308 0.305 0.235 0.288 0.361 0.290 Pitts. Premature 31-32 Weeks τ = 0.8 Sec 0.214 0.175 0.218 0.161 0.197 0.145 0.187 0.106 0.188 0.154 0.151 0.178 Pitts. Premature 31-32 Weeks τ = 1 Sec 0.204 0.168 0.209 0.154 0.188 0.140 0.179 0.102 0.179 0.147 0.144 0.170 Pitts. Premature 31-32 Weeks τ = 1.2 Sec 0.196 0.163 0.200 0.149 0.181 0.138 0.172 0.100 0.172 0.143 0.140 0.163 Pilot Premature 40-41 Weeks τ = 0.8 Sec 0.306 0.286 0.304 0.280 0.331 0.305 0.323 0.273 0.344 0.369 0.345 0.318 Pilot Premature 40-41 Weeks τ = 1 Sec 0.312 0.288 0.305 0.282 0.332 0.311 0.324 0.275 0.347 0.371 0.344 0.316 Pilot Premature 40-41 Weeks τ = 1.2 Sec 0.309 0.286 0.305 0.282 0.329 0.307 0.323 0.274 0.341 0.367 0.345 0.320 Pilot Premature 31-32 Weeks τ = 0.8 Sec 0.143 0.138 0.156 0.152 0.162 0.197 0.194 0.158

42 Pilot Premature 31-32 Weeks τ = 1 Sec 0.148 0.142 0.159 0.156 0.164 0.199 0.198 0.162 Pilot Premature 31-32 Weeks τ = 1.2 Sec 0.150 0.145 0.163 0.160 0.167 0.202 0.200 0.165

Sample Entropy Channel Name FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 Pitts. Fullterm 40-41 Weeks τ = 0.8 Sec 0.289 0.291 0.357 0.367 0.305 0.303 0.406 0.402 0.351 0.337 0.445 0.478 Pitts. Fullterm 40-41 Weeks τ = 1 Sec 0.285 0.286 0.355 0.365 0.300 0.298 0.402 0.399 0.348 0.334 0.440 0.474 Pitts. Fullterm 40-41 Weeks τ = 1.2 Sec 0.282 0.283 0.353 0.362 0.297 0.292 0.398 0.395 0.344 0.329 0.434 0.470 Pitts. Premature 40-41 Weeks τ = 0.8 Sec 0.303 0.301 0.334 0.343 0.323 0.267 0.410 0.394 0.301 0.364 0.475 0.383 Pitts. Premature 40-41 Weeks τ = 1 Sec 0.299 0.296 0.328 0.338 0.320 0.263 0.406 0.390 0.297 0.359 0.467 0.380 Pitts. Premature 40-41 Weeks τ = 1.2 Sec 0.295 0.291 0.322 0.334 0.317 0.260 0.401 0.386 0.293 0.354 0.462 0.377 Pitts. Premature 31-32 Weeks τ = 0.8 Sec 0.340 0.295 0.346 0.282 0.332 0.235 0.311 0.188 0.304 0.259 0.239 0.286 Pitts. Premature 31-32 Weeks τ = 1 Sec 0.331 0.285 0.338 0.271 0.318 0.226 0.298 0.179 0.295 0.248 0.228 0.277 Pitts. Premature 31-32 Weeks τ = 1.2 Sec 0.322 0.277 0.329 0.262 0.307 0.221 0.285 0.172 0.285 0.239 0.217 0.267 Pilot Premature 40-41 Weeks τ = 0.8 Sec 0.332 0.312 0.329 0.304 0.358 0.331 0.349 0.297 0.374 0.403 0.373 0.344 Pilot Premature 40-41 Weeks τ = 1 Sec 0.334 0.311 0.327 0.302 0.356 0.334 0.348 0.296 0.374 0.402 0.369 0.338 Pilot Premature 40-41 Weeks τ = 1.2 Sec 0.334 0.313 0.330 0.304 0.355 0.332 0.349 0.296 0.369 0.398 0.371 0.345 Pilot Premature 31-32 Weeks τ = 0.8 Sec 0.169 0.166 0.180 0.177 0.191 0.221 0.218 0.185 Pilot Premature 31-32 Weeks τ = 1 Sec 0.173 0.167 0.182 0.181 0.191 0.223 0.221 0.187 Pilot Premature 31-32 Weeks τ = 1.2 Sec 0.175 0.171 0.185 0.186 0.193 0.226 0.223 0.191 Table 3.9: Standard Deviation of Active Complexity Results Approximate Entropy Channel Name FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 Pitts. Fullterm 40-41 Weeks τ = 0.8 Sec 0.190 0.162 0.270 0.230 0.188 0.166 0.283 0.241 0.164 0.157 0.282 0.316 Pitts. Fullterm 40-41 Weeks τ = 1 Sec 0.185 0.159 0.266 0.228 0.184 0.161 0.280 0.238 0.163 0.157 0.277 0.311 Pitts. Fullterm 40-41 Weeks τ = 1.2 Sec 0.181 0.155 0.262 0.225 0.180 0.160 0.276 0.236 0.160 0.153 0.273 0.308 Pitts. Premature 40-41 Weeks τ = 0.8 Sec 0.194 0.178 0.216 0.215 0.185 0.131 0.245 0.208 0.164 0.214 0.273 0.197 Pitts. Premature 40-41 Weeks τ = 1 Sec 0.191 0.177 0.213 0.212 0.182 0.128 0.241 0.205 0.159 0.210 0.268 0.194 Pitts. Premature 40-41 Weeks τ = 1.2 Sec 0.186 0.174 0.208 0.210 0.178 0.128 0.237 0.203 0.156 0.205 0.263 0.193 Pitts. Premature 31-32 Weeks τ = 0.8 Sec 0.214 0.175 0.218 0.161 0.197 0.145 0.187 0.106 0.188 0.154 0.151 0.178 Pitts. Premature 31-32 Weeks τ = 1 Sec 0.204 0.168 0.209 0.154 0.188 0.140 0.179 0.102 0.179 0.147 0.144 0.170 Pitts. Premature 31-32 Weeks τ = 1.2 Sec 0.196 0.163 0.200 0.149 0.181 0.138 0.172 0.100 0.172 0.143 0.140 0.163 Pilot Premature 40-41 Weeks τ = 0.8 Sec 0.288 0.293 0.292 0.254 0.311 0.303 0.319 0.246 0.332 0.323 0.306 0.304 Pilot Premature 40-41 Weeks τ = 1 Sec 0.290 0.291 0.291 0.256 0.309 0.304 0.320 0.247 0.332 0.323 0.308 0.303 Pilot Premature 40-41 Weeks τ = 1.2 Sec 0.290 0.292 0.291 0.253 0.309 0.306 0.319 0.246 0.333 0.322 0.307 0.302 Pilot Premature 31-32 Weeks τ = 0.8 Sec 0.109 0.107 0.116 0.114 0.113 0.134 0.123 0.113

43 Pilot Premature 31-32 Weeks τ = 1 Sec 0.113 0.110 0.116 0.116 0.117 0.135 0.125 0.117 Pilot Premature 31-32 Weeks τ = 1.2 Sec 0.115 0.114 0.120 0.120 0.120 0.137 0.130 0.118

Sample Entropy Channel Name FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 Pitts. Fullterm 40-41 Weeks τ = 0.8 Sec 0.241 0.193 0.344 0.283 0.224 0.197 0.361 0.288 0.211 0.201 0.360 0.396 Pitts. Fullterm 40-41 Weeks τ = 1 Sec 0.234 0.185 0.339 0.279 0.217 0.189 0.357 0.287 0.207 0.200 0.355 0.391 Pitts. Fullterm 40-41 Weeks τ = 1.2 Sec 0.228 0.179 0.335 0.275 0.213 0.186 0.352 0.284 0.205 0.192 0.348 0.390 Pitts. Premature 40-41 Weeks τ = 0.8 Sec 0.241 0.215 0.276 0.258 0.234 0.160 0.305 0.247 0.202 0.262 0.332 0.250 Pitts. Premature 40-41 Weeks τ = 1 Sec 0.234 0.206 0.268 0.253 0.229 0.152 0.298 0.242 0.192 0.257 0.323 0.245 Pitts. Premature 40-41 Weeks τ = 1.2 Sec 0.227 0.199 0.258 0.248 0.222 0.149 0.292 0.237 0.185 0.250 0.317 0.243 Pitts. Premature 31-32 Weeks τ = 0.8 Sec 0.340 0.295 0.346 0.282 0.332 0.235 0.311 0.188 0.304 0.259 0.239 0.286 Pitts. Premature 31-32 Weeks τ = 1 Sec 0.331 0.285 0.338 0.271 0.318 0.226 0.298 0.179 0.295 0.248 0.228 0.277 Pitts. Premature 31-32 Weeks τ = 1.2 Sec 0.322 0.277 0.329 0.262 0.307 0.221 0.285 0.172 0.285 0.239 0.217 0.267 Pilot Premature 40-41 Weeks τ = 0.8 Sec 0.309 0.311 0.310 0.272 0.334 0.321 0.341 0.263 0.355 0.348 0.327 0.324 Pilot Premature 40-41 Weeks τ = 1 Sec 0.310 0.309 0.309 0.273 0.329 0.321 0.341 0.262 0.353 0.345 0.328 0.323 Pilot Premature 40-41 Weeks τ = 1.2 Sec 0.310 0.309 0.311 0.270 0.330 0.323 0.340 0.261 0.352 0.342 0.326 0.320 Pilot Premature 31-32 Weeks τ = 0.8 Sec 0.132 0.134 0.140 0.142 0.140 0.157 0.143 0.139 Pilot Premature 31-32 Weeks τ = 1 Sec 0.135 0.136 0.141 0.143 0.144 0.160 0.143 0.144 Pilot Premature 31-32 Weeks τ = 1.2 Sec 0.138 0.141 0.145 0.149 0.149 0.163 0.151 0.145 Table 3.10: Standard Deviation of Quiet Complexity Results Approximate Entropy Channel Name FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 Pitts. Fullterm 40-41 Weeks τ = 0.8 Sec 0.254 0.253 0.294 0.322 0.287 0.272 0.340 0.357 0.334 0.329 0.395 0.412 Pitts. Fullterm 40-41 Weeks τ = 1 Sec 0.251 0.250 0.290 0.318 0.283 0.269 0.336 0.353 0.329 0.324 0.391 0.408 Pitts. Fullterm 40-41 Weeks τ = 1.2 Sec 0.248 0.248 0.287 0.316 0.280 0.265 0.332 0.349 0.325 0.319 0.386 0.402 Pitts. Premature 40-41 Weeks τ = 0.8 Sec 0.262 0.280 0.274 0.303 0.281 0.254 0.329 0.348 0.281 0.314 0.368 0.320 Pitts. Premature 40-41 Weeks τ = 1 Sec 0.260 0.278 0.267 0.298 0.278 0.250 0.325 0.343 0.278 0.308 0.365 0.316 Pitts. Premature 40-41 Weeks τ = 1.2 Sec 0.256 0.277 0.264 0.296 0.273 0.246 0.322 0.342 0.275 0.303 0.361 0.314 Pitts. Premature 31-32 Weeks τ = 0.8 Sec 0.214 0.175 0.218 0.161 0.197 0.145 0.187 0.106 0.188 0.154 0.151 0.178 Pitts. Premature 31-32 Weeks τ = 1 Sec 0.204 0.168 0.209 0.154 0.188 0.140 0.179 0.102 0.179 0.147 0.144 0.170 Pitts. Premature 31-32 Weeks τ = 1.2 Sec 0.196 0.163 0.200 0.149 0.181 0.138 0.172 0.100 0.172 0.143 0.140 0.163 Pilot Premature 40-41 Weeks τ = 0.8 Sec 0.227 0.228 0.259 0.213 0.273 0.245 0.278 0.171 0.270 0.261 0.254 0.245 Pilot Premature 40-41 Weeks τ = 1 Sec 0.233 0.231 0.262 0.215 0.274 0.248 0.281 0.175 0.271 0.263 0.258 0.248 Pilot Premature 40-41 Weeks τ = 1.2 Sec 0.236 0.236 0.265 0.218 0.275 0.252 0.284 0.178 0.271 0.265 0.260 0.251 Pilot Premature 31-32 Weeks τ = 0.8 Sec 0.153 0.153 0.146 0.141 0.166 0.165 0.169 0.161

44 Pilot Premature 31-32 Weeks τ = 1 Sec 0.157 0.155 0.149 0.148 0.168 0.166 0.170 0.161 Pilot Premature 31-32 Weeks τ = 1.2 Sec 0.161 0.158 0.155 0.149 0.171 0.166 0.171 0.166

Sample Entropy Channel Name FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 Pitts. Fullterm 40-41 Weeks τ = 0.8 Sec 0.325 0.312 0.381 0.405 0.352 0.326 0.441 0.445 0.416 0.403 0.503 0.528 Pitts. Fullterm 40-41 Weeks τ = 1 Sec 0.321 0.307 0.377 0.400 0.346 0.322 0.436 0.439 0.410 0.400 0.497 0.523 Pitts. Fullterm 40-41 Weeks τ = 1.2 Sec 0.317 0.305 0.373 0.398 0.342 0.316 0.432 0.435 0.405 0.392 0.492 0.517 Pitts. Premature 40-41 Weeks τ = 0.8 Sec 0.331 0.344 0.356 0.375 0.352 0.311 0.433 0.445 0.353 0.389 0.479 0.418 Pitts. Premature 40-41 Weeks τ = 1 Sec 0.327 0.342 0.348 0.370 0.348 0.305 0.428 0.440 0.347 0.383 0.476 0.412 Pitts. Premature 40-41 Weeks τ = 1.2 Sec 0.324 0.337 0.343 0.367 0.344 0.300 0.422 0.435 0.344 0.377 0.470 0.409 Pitts. Premature 31-32 Weeks τ = 0.8 Sec 0.340 0.295 0.346 0.282 0.332 0.235 0.311 0.188 0.304 0.259 0.239 0.286 Pitts. Premature 31-32 Weeks τ = 1 Sec 0.331 0.285 0.338 0.271 0.318 0.226 0.298 0.179 0.295 0.248 0.228 0.277 Pitts. Premature 31-32 Weeks τ = 1.2 Sec 0.322 0.277 0.329 0.262 0.307 0.221 0.285 0.172 0.285 0.239 0.217 0.267 Pilot Premature 40-41 Weeks τ = 0.8 Sec 0.248 0.245 0.280 0.231 0.292 0.264 0.297 0.187 0.292 0.283 0.271 0.265 Pilot Premature 40-41 Weeks τ = 1 Sec 0.254 0.249 0.283 0.232 0.292 0.266 0.299 0.190 0.291 0.285 0.275 0.268 Pilot Premature 40-41 Weeks τ = 1.2 Sec 0.258 0.257 0.289 0.235 0.295 0.271 0.305 0.196 0.292 0.286 0.278 0.271 Pilot Premature 31-32 Weeks τ = 0.8 Sec 0.200 0.195 0.182 0.176 0.212 0.206 0.206 0.205 Pilot Premature 31-32 Weeks τ = 1 Sec 0.198 0.191 0.180 0.178 0.207 0.203 0.207 0.200 Pilot Premature 31-32 Weeks τ = 1.2 Sec 0.196 0.189 0.181 0.176 0.207 0.200 0.206 0.199 3.3.2 Histogram Matching Scheme

The quality of the Pittsburgh EEG data is very different from that of the KC intervention study and as a result, the complexity measures from these two data sets are not directly comparable on the same scale. Therefore in order to be able to compare the results across the studies a histogram adjustment scheme is required. The underlying hypothesis is that the preterm neonates at 31-32 weeks from the Pittsburgh and KC intervention studies are statistically the same. That is, the distribution of the complexity measures for these two group should be the same. Hence the baseline for this adjustment procedure is to align these two preterm 31-32 week histograms. This is reasonable because these are two groups of neonates at the same PMA. The alignment procedure is based on defining a linear transformation that transform the histograms from the Pittsburgh to be comparable with the histograms form the KC intervention study for preterm 31-32 week neonates. We observe that the support of the two histogram is about the same, so the transformation is a linear shift in the data where matching is then defined as an optimization problem for each EEG channel where the objective is:

n X h i2 min hPitts.(k) − hKC(k − d) (3.2) d k=1 where d is the shift. The histograms of the Pittsburgh vs. KC intervention studies for the 31-32 week preterm neonates are shown in Figure 3.7 and then for each channel the optimal shift is given in Table 3.11 and the adjusted histograms are shown in Figure 3.8. Because we assume that the diﬀerence in the complexity distributions between the two study groups is a result of data collection and data quality, we then apply the same adjustment procedure as computed using the 31-32 week data to the Pittsburgh preterm and full term neonates at a PMA 40-41 weeks. The results of adjusting the complexity measures for the Pittsburgh study along with the results for the KC intervention study are given in Table 3.13. The results show that ﬁve brain regions

45 Table 3.11: Diﬀerence Between Histograms of Pittsburgh & Pilot Studies Approximate Entropy T3-O1 T4-O2 C3-O1 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 0.79 0.79 0.78 0.80 0.81 0.80 0.82 0.80

Sample Entropy T3-O1 T4-O2 C3-O1 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 0.64 0.71 0.65 0.71 0.70 0.73 0.72 0.69 had greater complexity in the KC intervention study when compared to neonates at the same PMA without the KC intervention. These included speciﬁc brain regions in both the right hemisphere as well as left and right parasagittal regions corresponding to the electrodes C4-T4, T4-02, C4-CZ, C4-O2, and C3-CZ. There were two brain regions that showed greater complexity for the KC intervention study. The regions involved were exclusively in the right hemisphere C4-CZ, C4-T4. Less complexity was noted in the channels corresponding with the posterior quadrant of the left hemisphere in the KC intervention group, consisting of T3-01, C3-O1, and T3-C3 channels.

3.3.3 Closeness Test

From an analysis of individual studies from the Pittsburgh and KC intervention study, it is evident that along with brain maturation and neurodevelopment comes an increased in complexity of the EEG. But a question that remains is how the KC intervention inﬂuences brain maturation and neurodevelopment. Therefore our objective is to measure the distance between the complexities of preterm neonates at 40-41 week PMA from the Pittsburgh and KC intervention cohorts after complexity results have been adjusted to bring them to the same scale. The hypothesis is that the KC intervention improves the brain maturation process, and we will investigate the validity of this using the Pittsburgh full term cohort as the normative group. In order to understand the distance between these groups the Mahalanobis distance between the adjusted complexity measures will be used.

46 Mahalanobis Distance

Mahalanobis distance is a statistical distance measure based on correlation between variables through which diﬀerent patterns can be identiﬁed and analyzed. Let X and Y be two points in a N-dimensional feature space selected from the same distribution.

Then if the covariance matrix is C, then the Mahalanobis DM distance is given by:

p T −1 DM = (X − Y ) C (X − Y ) (3.3)

The calculation of Mahalanobis distance was done using built-in the Matlab function mahal.m. This function computes a multivariate measure of the separation of a data set from a given point in a N-dimensional space. Therefore, the distance between the complexity results of each epoch of each individual neonate with KC intervention at 40-41 week PMA and the adjusted distributions of the Pittsburgh preterm and full term neonates at the same PMA is computed. In this distance calculation the channels from the areas of the brain where we have shown that KC intervention has the most noticeable eﬀect on neurodevelopment have been chosen and the selected channels are T4-O2, C4-O2, and Cz-C4. The average distance of each neonate with KC intervention from the Pittsburgh preterm and full term studies are given in Table 3.12. The results show that all of the neonates with KC intervention are closer to the full term neonate cohort than to the preterm neonate cohort at 40-41 week PMA.

3.4 Conclusions

We have shown that as the brain matures the complexity of the EEG signals during sleep as measured by Approximate and Sample Entropy increases. In particular the full term neonate cohort statistically (T-test) shows greater complexity than the premature neonate cohort from the Pittsburgh study at the same PMA (40-41 week) without any intervention. The complexity of KC intervention cohort at a PMA of 40-41 weeks is between the complexity of the adjusted full term and preterm neonates

47 Table 3.12: Mean Mahalanobis Distance Between Neonates at 40-41 Week Approximate Entropy Sample Entropy KC Intervention Pittsburgh Pittsburgh Neonate Number Full term Preterm Full term Preterm 1 4.33 5.47 3.07 4.20 2 2.90 3.27 2.12 2.57 3 2.22 3.45 1.71 2.75 4 2.73 4.50 2.13 3.65 5 2.85 4.71 2.27 3.94 6 2.64 3.53 2.10 2.90 7 2.49 4.64 1.95 3.72 8 2.96 4.67 2.23 3.81

Pitts Premature 31−32 Weeks Pitts Premature 40−41 Weeks Pitts Fullterm 40−41 Weeks

Fp1− T3 T3 − O1 Fp2− T4 5 5 4

0 0 0 1 1.5 2 1 1.5 2 1 1.5 2 T4 − O2 Fp1− C3 C3 − O1 4 5 10

2 5

0 0 0 1 1.5 2 1 1.5 2 1 1.5 2 Fp2− C4 C4 − O2 T3 − C3 4 10 10

2 5 5

0 0 0 1 1.5 2 1 1.5 2 1 1.5 2 C3 − Cz Cz − C4 C4 − T4 10 10 5

5 5

0 0 0 1 1.5 2 1 1.5 2 1 1.5 2

Figure 3.3: Normalized Histogram of Approximate Entropy Results for Pittsburgh Study

48 Pilot Premature 31−32 Weeks

Pilot Premature 40−41 Weeks

T3 − O1 T4 − O2 4 4

2 2

0 0 0 0.5 1 1.5 0 0.5 1 1.5 C3 − O1 C4 − O2 4 4

2 2

0 0 0 0.5 1 1.5 0 0.5 1 1.5 T3 − C3 C3 − Cz 4 4

2 2

0 0 0 0.5 1 1.5 0 0.5 1 1.5 Cz − C4 C4 − T4 4 4

2 2

0 0 0 0.5 1 1.5 0 0.5 1 1.5

Figure 3.4: Normalized Histogram of Approximate Entropy Results for Pilot Study

Pitts Fullterm 40−41 Weeks Active Sleep Pitts Fullterm 40−41 Weeks Quiet Sleep

Fp1− T3 T3 − O1 Fp2− T4 10 10 5

5 5

0 0 0 1.6 1.8 2 1.6 1.8 2 1.6 1.8 2 T4 − O2 Fp1− C3 C3 − O1 10 10 10

5 5 5

0 0 0 1.6 1.8 2 1.6 1.8 2 1.6 1.8 2 Fp2− C4 C4 − O2 T3 − C3 10 10 10

5 5 5

0 0 0 1.6 1.8 2 1.6 1.8 2 1.6 1.8 2 C3 − Cz Cz − C4 C4 − T4 20 20 10

10 10 5

0 0 0 1.6 1.8 2 1.6 1.8 2 1.6 1.8 2

Figure 3.5: Normalized Histogram of Approximate Entropy Results for Pittsburgh Fullterm 40-41 Weeks Study Active vs. Quiet

49 Pilot Premature 40−41 Weeks Active Sleep Pilot Premature 40−41 Weeks Quiet Sleep

Fp1− T3 T3 − O1 Fp2− T4 5 4 4

2 2

0 0 0 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 T4 − O2 Fp1− C3 C3 − O1 5 4 4

2 2

0 0 0 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 Fp2− C4 C4 − O2 T3 − C3 4 5 4

2 2

0 0 0 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 C3 − Cz Cz − C4 C4 − T4 4 4 4

2 2 2

0 0 0 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5

Figure 3.6: Normalized Histogram of Approximate Entropy Results for Pilot Prema- ture 40-41 Weeks Study Active vs. Quiet

Pilot Premature 31−32 Weeks Pitts. Premature 31−32 Weeks T3 − O1 T4 − O2 4 4

2 2

0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 C3 − O1 C4 − O2 5 10

0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 T3 − C3 C3 − Cz 5 4

0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Cz − C4 C4 − T4 4 4

2 2

0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2

Figure 3.7: Normalized Histogram of Approximate Entropy Results of Pilot vs. Pitts- burgh Study

50 Pilot Premature 31−32 Weeks Pitts. Premature 31−32 Weeks

T3 − O1 T4 − O2 4 4

2 2

0 0 0 0.5 1 1.5 0 0.5 1 1.5 C3 − O1 C4 − O2 5 10

0 0 0 0.5 1 1.5 0 0.5 1 1.5 T3 − C3 C3 − Cz 5 4

0 0 0 0.5 1 1.5 0 0.5 1 1.5 Cz − C4 C4 − T4 4 4

2 2

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Figure 3.8: Normalized Histogram of Approximate Entropy Results of Pilot vs. Pitts- burgh Study After Mean Correction

Table 3.13: Means of Complexity Results after Histogram Matching Approximate Entropy Channel Name T3-O1 T4-O2 C3-O1 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 Pitts. Full term 40-41 Weeks 0.970 0.884 0.977 0.842 0.942 0.981 0.872 0.811 Pitts. Premature 40-41 Weeks 0.897 0.843 0.968 0.851 0.926 0.884 0.760 0.811 Pilot Premature 40-41 Weeks 0.838 0.860 0.839 0.879 0.901 0.948 0.937 0.904

Sample Entropy Channel Name T3-O1 T4-O2 C3-O1 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 Pitts. Full term 40-41 Weeks 1.204 1.021 1.202 0.994 1.121 1.135 1.031 0.943 Pitts. Premature 40-41 Weeks 1.127 0.997 1.197 1.011 1.117 1.030 0.900 0.974 Pilot Premature 40-41 Weeks 0.987 1.012 0.988 1.033 1.055 1.100 1.092 1.059

51 Table 3.14: T-test result between Pittsburgh full term and premature at the age 40-41 week

Approximate Entropy Channel Name FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 T-test Result 1 1 1 1 1 0 p-value 99.4 100 99.6 99.9 99.8 64.44 Channel Name FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 T-test Result 1 0 0 1 1 0 p-value 99.4 53.1 85.2 100 100 0.6

Sample Entropy Channel Name FP1-T3 T3-O1 FP2-T4 T4-O2 FP1-C3 C3-O1 T-test Result 0 1 0 0 1 0 p-value 86.9 99.9 4.4 89.9 99.8 31.6 Channel Name FP2-C4 C4-O2 T3-C3 C3-Cz Cz-C4 C4-T4 T-test Result 0 0 0 1 1 0 p-value 82.3 70.1 22.4 99.9 99.9 90.8 that did not have any KC intervention at the PMA. In addition, the KC intervention cohort is closer to the full term cohort as measured by the Mahalanobis Distance than it is to the premature cohort. This provides a preliminary validation of our hypothesis that KC intervention can improve the brain maturation in neonates in addition to the eﬀects we observed previously on the organization of the neonatal sleep [37].

52 Chapter 4

Epilepsy

Epilepsy is a brain disorder where nerve cells, or neurons, in the brain some- times behave abnormally and this condition affects about one to five percent of the total population of the world. Epilepsy may develop because of an abnormality in brain wiring, an imbalance of nerve signaling chemicals called neurotransmitters, or some combination of these factors. The understanding brain electrical activity in an epileptic brain, one of the most challenging dynamical system problems in nature, is a topic of great interest in clinical and basic neuroscience communities. Epilepsy is also one of the most costly neurological disorders because it can be lifelong. Most of the epilepsy patients, 70 to 80 percent, can be treated with anti epileptic medication and about 5 to 10 percent of the patients that do not respond to any medication are candidates for epilepsy surgery. If the area of the brain where the seizure encounters (the local region) from can be identified and surgically resected, the epilepsy can be treated. However the remaining 10-15 percent of epilepsy patients generally have non-focal seizures that involve multiple areas of brain, for which there is currently no treatment available. Finding better treatments to control or even prevent seizures requires developing a fundamental understanding of how the disease begins and develops. Although, these are still unanswered questions, but researchers believe that a device similar to

53 heart pacemaker could be an alternative treatment for the disease. The concept is that the brain pacemaker would use implanted electrodes to acquire and monitor brain electrical activity as well as providing the data that is necessary for an early seizure prediction algorithm. Once the pre-seizure activity is detected, the device could initiate an electrical stimulation of the brain or activate an implanted mini- pump for anti epileptic drug delivery [48]. There are studies that show that electrical stimulation can reduce the severity of ecliptic seizures [49, 47], however the results available to date from seizure prediction algorithms are not sufficient for clinical applications [38, 41]. Seizure detection or prediction consists of two main steps: feature extraction, extracting relevant information from the EEG data, and classification, determine the current state of the brain. Lopes da Silva [74] has provided a summary of most of the EEG analysis techniques. Although, in the literature there have been discussion regarding various features extraction techniques for seizure prediction, there is not sufficient evidence that changes in these extracted features before seizure onset are sufficient classification for clinical application. Whenever changes in features are obvious as the state of patient changes, classification is trivial. Therefore feature extraction is the most critical part of the any seizure detection or prediction algorithm. Currently, most algorithms do classification based on threshold crossing, i.e. an alarm is indicated whenever extracted feature value cross a specified threshold. In the last decade, in order to predict the onset of epileptic seizures, several linear or nonlinear time series analysis techniques have been applied to intracranial EEG recording from either deep brain or cortical electrodes. In the early 1990 to characterize seizure dynamics, Iasemidis et al. used the largest Lyapunov exponent which is an indictor of chaotic behavior [24] and in 2003 they developed an adaptive seizure prediction technique with 82% sensitivity and 0.16/h false prediction rate focused on the computation of the short-term maximum Lyapunov exponent [25]. Lehnertz and Elger [35] used the computation of the correlation dimension for seizure

54 prediction. Lehnertz et al.[34] stressed the importance of nonlinear EEG analysis in the study of epilepsy especially from the perspective of chaotic dynamics and synchronization. However, Harrison et al. [21] have reported correlation dimension can not be used effectively in the prediction seizures. At 1999, La Van Quyen et al. introduced the dynamical similarity index [60]. This nonlinear measure has been designed to detect changes in dynamic properties of electrical brain activity that anticipate epileptic seizures and it is one of the most computationally efficient nonlinear techniques available [61, 59]. The basis for this algorithm is the analysis of the time intervals between consecutive positive zero crossings of the EEG. A review of this method is given in Appendix A. Mean phase coherence, a statistical measure of phase synchronization, is the latest promising seizure prediction technique which was introduced by Mormann et al. in 2000 [46, 44, 45, 42, 40] and has also been used by other researchers [58, 62, 10, 67]. This method does not require the reference intervals but evaluates synchrony as a relative measure of phase synchronization between two channels. Mormann et al. measured synchrony using adjacent channels of EEG data collected from depth electrodes. Schiff et al. [26] have claimed that there is insufficient statistical evidence that seizure can be discriminated from interictal dynamics based on multivariate synchrony measures, and further that the dynamics of a pre-seizure state can be identified. Lopes da Silva et al. define epilepsy as a dynamical disease of the brain system with at least two distinct states: the interictal state, the normal state, and the ictal state being associated with seizures [73]. In principle, there are two different scenarios of how a seizure can occur. It can be a gradual or sudden transition from the normal state to the ictal state and they have studied these transition using the mathematics of nonlinear system [75]. The future development of seizure prediction algorithms may depend critically on understanding of these dynamical states changes as well as how confounding variables such as sleep or awake may influence the computation of

55 seizure detection and prediction measures [67]. It has been reported that seizure prediction algorithms can predict epileptic seizures from minutes up to 1.5 hours prior to onset [41, 38]. However the prediction performance of none of these has been evaluated on the same basis. Winterhalder et al. [82] have proposed a general framework to assess and compare such prediction algorithms. In this work the dynamical similarity index has been used to analyze 582 hours of intracranial EEG data, including 88 seizures. In addition, two surrogate data [70] methods have been proposed: Seizure Time Surrogate [1] and Measure Pro- file Surrogate [32] for validating the performance of seizure prediction algorithms. In the seizure time surrogate technique the seizure-onset times of the original EEG recordings are replaced by surrogate onset times. If the performance of a prediction method on the original data is improved when compared to the performance on surrogate seizure onset times, then the performance of this measure can be considered significantly better than a random prediction. Chaovalitwongse et al. [7] used the seizure time surrogate method to validate the performance of the short-term maximum Lyapunov exponent as a seizure prediction method, however they didn’t apply this technique properly and there were lots of discussion in literature regrading their algorithm evaluation [43, 8, 9, 83]. In this chapter we examine the application of Deterended Fluctuation Analysis as a novel technique to quantify the electrical activity in the brain. Changes in activity can be sudden or gradual from the pre seizure to seizure state and in particular the time intervals between changes in the brain activity can be statistically significantly different within and across patient recordings. Next, we present and evaluate the performance of DFA along with the automated gradient detection method which is given in Chapter 2. In addition the results of other promising techniques such as Phase Synchronization, Dynamical Similarity Index, and Hjorth parameters, given in Appendix A, are presented for comparison purpose. Further some computational issues and limitations of the DFA algorithm in quantifying brain electrical activity

56 have been investigated and presented.

4.1 Data description

In this dissertation, the intracranial EEG data recorded from implanted grids and strips from patients in the Epilepsy Unit at University Hospitals of Cleveland have been used. These data have been collected to evaluate the epilepsy patients especially for localization of epileptic area of the brain in preparation of epilepsy surgery. The data have been visually analyzed and annotated by a neurologist. Normally patients are monitored for 5-6 continuous days at a sampling rate for data collection of 1000 Samples/Sec which is higher than the sampling rate of most other studies [36]. The collected data for each patient including the acquired video during the monitoring is around 150-200 GB for study which makes storing, archiving, and handling the data diﬃcult. Therefore after epilepsy surgery, most of the data are erased and only the short recordings (1-3 hours) before and after epileptic seizures are kept. In the preliminary stages of our data analysis, these short recorded data sets have been considered and analyzed. Recording speciﬁcs for each patient of the short data sets are given in Table 4.1.

Table 4.1: Short recording speciﬁcs for each patient # of Recorded Grid & Strip # of # of Total Time of Patient ID Channels Size Seizures Files Recording (Minutes) S-1 98 8x8 2(1x4) 2(1x6) 1x10 4 3 359.8 S-2 72 6x8 8(1x4) 2(1x6) 4 2 119.8 S-3 134 6x8 5(1x4) 9(1x6) 1x8 6 4 480.0 S-4 58 2(1x4) 3(1x6) 2(1x10) 2 2 121.9 S-5 64 1x4 6(1x6) 2(1x10) 6 9 387.3 S-6 90 8x8 3(1x4) 1x10 3 4 205.2 S-7 98 6x8 4(1x4) 5(1x6) 15 1 179.9 S-8 92 8x8 2(1x4) 1x6 1x10 4 4 253.7

In these short data sets, there are between 20 to 60 minutes of EEG data prior to seizure onset and it is observed that the proposed measure, DFA, detects some activity in advance of seizure onset. Because that data before the seizure was limited to 20 to 60 minutes, we were not sure how these changes in activity were evolving

57 into a seizure and disappearing afterward. Therefore keeping the entire recorded data obtained from monitoring sessions was necessary and analysis of the entire recordings was initiated. The speciﬁcations of these long data are given in Table 4.2.

Table 4.2: Long recording speciﬁcs for each patient # of Recorded Grid & Strip # of # of Total Time of Patient ID Channels Size Seizures Files Recording (Hours) L-1 102 8x8 4(1x4) 1x6 1x10 4 19 37.1 L-2 100 8x8 6(1x4) 1x10 5 63 140.9 L-3 90 . 6x8 5(1x4) 3(1x6) 10 52 143.5 L-4 93 8x8 2(1x10) 16 51 137.9

The intracranial EEG data have been collected using a Nihon Kohden data collection system at a sampling rate of 1000 Samples/Sec using a 16 bit ADC (Analog Digital Converter) with referential measurement technique. Prior to data collection to prevent aliasing a 200Hz anti-aliasing ﬁlter has been used. In scalp EEG data recordings, the reference is commonly connected to the shoulder of patient. However in intracranial EEG, the reference is chosen from the grids and strip electrodes that are far from the suspected center of epileptic area. Therefore, any change in potential of reference electrode causes a change in the potential measured at other EEG electrode sites. In the Nihon Kohden data acquisition system, in order to reduce the ﬂuctuations due to the reference two channels far away from the center of epileptic area are selected and the median of the potential measures at these sites is used as the reference. The reference channels used in our recordings are the third and the fourth channels of recorded data.

4.2 DFA Limitation

In chapter 2 DFA was introduced and the eﬀect of the number of samples were studied. In the DFA computation, as the number of samples per cycle becomes smaller, the scaling region gets shorter. The typical sampling rate of acquired intracranial EEG data is around 200 Samples/Sec. If the sampling rate were 1000 Samples/Sec,

58 the maximum frequency that can be detected in the scaling region is around 70 Hz. Therefore, DFA is only sensitive to frequencies up to 14 Hz for EEG data recorded at 200 Samples/Sec and DFA may not be able to properly quantify the self similarity properties of the EEG data in this case. The sampling rate of the EEG used in this study data is 1000 Samples/Sec and it is sufficient to quantify self similarities up to 70Hz oscillations. According to recent studies, there are high frequency (over 100Hz) activities prior to seizure onset and one of our objectives is to understand the high frequency oscillations in EEG time series. Therefore approaches to increase the sensitivity of DFA to high frequencies have been investigated. In fact, the problem is not information content in the signal, because there is sufficient high frequency information in the data with 200Hz anti aliasing analog low pass filter. The problem is just the result of the number of samples per cycle. One approach to overcoming this problem is to up sample the time series. The basis for this procedure is that up sampling of a bandlimited time series data does not add any additional information to signal. Therefore, if the problem is just the number of samples per cycle, DFA should be able to detect the underlying self similarity. Up sampling excludes the lower scaling region of the log-log plot of DFA and the DFA is able to quantify the up sampled signal. Generally, up sampling requires interpolation and as a result, any bandlimited signal after sufficient up sampling will be smooth in interpolated regions and the gradient computed for the lower scaling region of log-log plot of DFA will approach to 2. Hence no useful information can be obtained with this approach.

4.3 Brain Activity Quantiﬁcation

In the following sections, we will discuss the features of brain activity that facilitate the comparison between the techniques being studied. DFA measures self similarity and we interpret an increased in self similarity with a decrease in the complexity of the brain activity. The range of this measure is between 0.5 and 2 (Chapter 2). From our analysis, the range between 1-1.2 has been accepted as ”normal” brain activity

59 and greater than 1.45 is the characteristic of a state of abnormal brain activity. For the dynamical similarity index, the signal dynamics are being compared to a segment of a data refereed to as the reference dynamics prior to seizure onset. The more similar the signal is to the reference, the more complex the brain activity. In phase synchronization, the synchrony of other channels to a reference channel, selected from epileptogenic region, is computed and as the synchrony increases the complexity of brain activity decreases. Hence in this work, decrease in complexity are synonymous with an increase in brain activity.

4.4 Data Analysis & Results

In the DFA calculation, the raw EEG data was directly used without any pre- processing or normalization and the epoch size was selected as 10 seconds (10000 samples) without any overlapping between selected data window. The minimum number of samples in each segment, n, was chosen as 33 while the maximum is chosen as 1000 samples. Further, the logarithmic plot of DFA corresponding to log10(33) and log10(1000) with 40 points has been calculated and the statistically linear segments with corresponding gradients has been estimated using the automatic gradient detection algorithm (Chapter 2). The details of the automatic gradient estimation of the log-log plot of DFA for sample ictal and non-ictal EEG data are also given in Chapter 2. The results of DFA applied to a 4 hour recording for patient S-1 is given in Figure 4.1. This plot has 3 seizures at 25 minutes, 93 minutes, and 180 minutes. The occurrence of the first seizure is sudden and there was not appear to be any obvious brain activity before seizure onset. However, after the first and the second seizures, there appears to be quantifiable changes in brain activity that may be useful in the prediction of the onset of future seizure events. In the dynamical similarity index analysis, 5 minutes of EEG data taken 20 minutes prior to the first seizure onset has been chosen as the reference dynamics. Further the EEG signals have been bandpass filtered to remove the low frequency trends and

60 high frequency oscillations prior to detection of the positive zero crossings. The bandpass region is 1-58Hz and the embedding dimension m = 16 was used in time delay embedding to compute the cross correlation and auto correlation sum before calculation of the similarity index (Appendix A). The results of this analysis for 30 second of non overlapping epochs of patient S-1 are given in Figure 4.2. The results depends signiﬁcantly on the selection of reference. For instance, in this patient two cables have been used to transfer the collected EEG data from grids and strips to the computer. Each of the cable sets has a diﬀerent noise level and as a result the noise level on the reference channels are not the same. It is obvious from Figure 4.2 that there are two regions in the plot and the lower channels of the plot, which have a higher noise level, seem to be more similar to the reference dynamics. Therefore the result of this analysis shows that the higher channels are mostly coming from the epileptogenic area which is in contradiction with the results of visual analysis of EEG by neurologists and results evident from DFA. In the phase coherence computation, the synchrony needs to be measured with

DFA (Starting Time 2:39AM)

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Figure 4.1: DFA result for patient S-1

61 respect to reference channel which has been chosen based on the DFA analysis result. The epoch length has been selected as 10 seconds and the details of algorithm are given in Appendix A. In the literature this algorithm has been used to measure the synchrony between adjacent channels from depth electrode recordings [46]. In our analysis phase coherence for all channels are measured with respect to a reference channel chosen from the center of the epileptogenic area. The reference channel for patient S-1 has been chosen as GR2-Org and the result of phase coherence analysis is given in Figure 4.3. In addition to the above analysis Hjorth parameters (Appendix A) for 10 second non-overlapping epochs have been calculated and the result of the Hjorth parameters analysis as well as DFA, dynamical similarity index and phase coherence for 7 hours of EEG data for patient L-1 are given in Figures 4.4, 4.5, 4.6 and 4.7 respectively. In these plots, there is a seizure at 119 minutes and DFA, dynamical similarity index and phase synchronization detect changes in brain activity very similar to patient S-1. However Hjorth parameters can hardly detect the seizures and their performance in

Dynamic Similarity Index (Starting Time 2:39AM)

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Figure 4.2: Dynamical similarity index result for patient S-1

62 Phase Coherence (Starting Time 2:39AM)

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Figure 4.3: Phase coherence result for patient S-1 comparison to DFA, and the dynamical similarity index is poor.

4.4.1 Brain Activity Index

In order to evaluate the performance of the DFA algorithm and to develop a brain activity index, clustering of the DFA result has been considered. K-means clustering one of the simplest unsupervised learning algorithms that can be used for clustering has been used. This clustering technique has been implemented using the kmeans.m function of Matlab. The main idea of using k-means clustering is to ﬁnd the thresholds or centroids of the brain activity level from the DFA analysis that can be used to classify the brain state. Therefore the results of brain activity as measured by DFA from seizure and non-seizure states have been chosen as input to the k-means clustering algorithm. This data has been selected from patient L-1 for which the results of DFA analysis for a 7 hour sample is given in Figure 4.5. Further, before clustering the DFA results have been inspected carefully to remove time segments that are corresponding to artifact

63 Channels Channels Channels 0 0 0 50 50 50 iue44 jrhprmtr eutfrptetL-1 patient for result parameters Hjorth 4.4: Figure 100 100 100 Activity (StartingTime0:27AM) 150 150 150 Complexity Time (Min) 200 200 200 Mobility 64 250 250 250 300 300 300 350 350 350 400 400 400 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 DFA (Starting Time 0:27AM)

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Figure 4.5: DFA result for patient L-1

Dynamical Similarity Index (Starting Time 0:27AM)

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Figure 4.6: Dynamical similarity index result for patient L-1

65 as well as bad channels resulting from poor electrode connections. After data selection for the k-means algorithm, the number of clusters need to be chosen. In our analysis, 4 clusters have been used. However, 2 and 3 state clustering have also been calculated and presented for comparison purposes. The thresholds and centroids for the 2, 3 and 4 state clustering analysis are given in Table 4.3. These thresholds then have been used to find the states of the entire DFA analysis results. For instance when 2 state clustering is used, whenever the DFA result is exceed than 1.362 the state has been assigned as 2 else if is assigned as 1. In the same way 3 and 4 state clustering has been done. Further to measure overall brain activity, an index which is the mean of the assigned states for all channels for a given epoch has been considered. The result of the brain activity index for patient L-1 is given in Figure 4.8. Further to reduce the noise of the brain activity index smoothing has been considered. In the smoothing process a fourth-order Butterworth low pass filter with zero phase shift (forward and backward filtering) has been used. The result of the smoothed brain activity index versus normal brain activity index for patients L-1,

Phase Coherence (Starting Time 0:27AM)

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Figure 4.7: Phase coherence result for patient L-1

66 L-2, L-3, and L-4 are given in Figures 4.9, 4.10, 4.11, and 4.13 respectively.

Table 4.3: K-means clustering result 2 Clusters 3 Clusters 4 Clusters Centroid Threshold Centroid Threshold Centroid Threshold 1.214 1.362 1.118 1.238 1.628 1.534 1.501 1.358 1.471 1.440 1.360 1.584 1.280 1.174 1.069

67 2 State Clustering 2

1.5 Mean States

1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

3 State Clustering 3

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4 State Clustering 4

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Figure 4.8: Brain activity index for patient L-1

68 Brain Activity Index 4

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Figure 4.9: Smoothed versus normal brain activity index for patient L-1

69 Brain Activity Index 4

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Figure 4.10: Smoothed versus normal brain activity index for patient L-2

70 Brain Activity Index 4

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Figure 4.11: Smoothed versus normal brain activity index for patient L-3

71 Brain Activity Index 4

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Figure 4.12: Smoothed versus normal brain activity index for patient L-3

72 Brain Activity Index 4

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Figure 4.13: Smoothed versus normal brain activity index for patient L-4

73 4.5 Discussion & Conclusions

The DFA has been computed for all EEG channels independent of the visual scores assigned to the data by a neurologist. The changes in the DFA results have been used for the detection of interictal, preictal, ictal and post ictal states. Changes in patient state corresponding to annotation of signals by a neurologist have also been investigated. The DFA performance was different for patients that had global seizures (seizures that start from the center of epileptogenic area of the brain and spreads to almost all of the channels) when compared to patients that had short (10-15 seconds) and localized seizures. While the change from the non-seizure state to the seizure state for patients that have global seizures occurred suddenly, there were some seizures in which increases in the brain activity as quantified by the slope of log-log DFA plot started several minutes in advance of clinically identified EEG seizure onset. Moreover, the brain activity as measured by the DFA can remain high up to 3-4 hours after the seizure onset and after the end of the seizure event can be visually identified. In particular DFA can detect changes between the preictal and postictal EEG both of which are identified as non seizure states by the neurologists. For instance, there is seizure at the 25 minutes in the result of DFA analysis of S- 1 patient is given in Figure (4.1) and the results of the DFA before and after this seizure onset are different. In fact the EEG signals are not similar, and further they are different from the EEG at seizure onset and during the seizure. A sample of the EEG at seizure onset and during the seizure, as well as before and after the seizure for patient S-1 is shown in Figure 4.14. For patients such as L-4 that have short and localized seizures, the situation is quite different. The result of 3 hour EEG analysis starting from 1:31AM for patient L-4 is shown in Figure 4.15. According to the previous results of DFA analysis, this patient is expected to have multiple seizures during this 3 hour period. However, the result of visual scoring of the EEG and analysis of the video recording do not verify the occurrence of these seizures. Therefore there is a conflict between the visual

74 Before seizure (20 minutes) 1000

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Figure 4.14: Sample EEG before and after seizure as well as seizure onset

75 scoring and the DFA analysis, analysis of the video recordings during data collection revealed that the observed changes in DFA were the result of changes in brain activity due to sleep state changes. In particular, the sharp transition between hot zone and yellow zone are mostly arousals that are occurring during sleep.

Starting Time = 1:31AM

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Figure 4.15: DFA Result for patient L-4

The results show that the proposed technique (DFA) is very sensitive to changes in brain activity but these changes may not be related to seizure onset. In conclusion, DFA is a robust, sensitive, and non reference-based analysis technique that can be used for the analysis of cortical EEG. However, DFA or any other proposed seizure prediction technique by itself may not be able to detect sudden and apparently un- predictable characteristic of the EEG during seizures [41]. Further, confounding by the sleep/awake states is an important limitation that requires more investigation. It is evident, a combination of techniques for assessing the state of patient such as sleep or awake [67] is going to be required to the development of algorithms for the detection or prediction of epileptic seizures.

76 4.6 Future work & Recommendations

In fact, in the DFA analysis the minimum number of samples in each segment, n = 33, does not provide the information that is necessary in the minimum scaling region of the log − log plot. Therefore in further applications of this technique to intracranial EEG, a lower minimum number of samples per segment such as n = 5 or 7 that extends the log − log plot and improve the slope detection of scaling region is recommended. EEG is very complex and during each of the states such as ictal, sleep and awake the properties of time series are changing. Approximate and Sample Entropy are two of the measures that can quantify signal complexity and using these techniques properly is very important in quantiﬁcation of nonlinear behavior in a signal. The most important parameter of the Approximate and Sample Entropy calculation is the time delay which depends on the long range correlation in the signal. Therefore prior to the application of these entropy measures, investigating the long range correlation of the EEG during the various brain activity states to have a good understanding of how the time delay parameter should be chosen. A feature set that includes Approximate and Sample Entropy, together with DFA may provide a unique opportunity to better understand the topological characteristic of brain activity due to changes in state.

77 Chapter 5

Analysis of Respiratory Data

5.1 Complexity of network

The in vitro medullary slice preparation is capable of generating spontaneous respiratory rhythm as long as it contains the preBötzinger complex. This preparation allows for recording of fictive inspiratory motor output from the hypoglossal (XII) nerve. Traditionally, time-series measures such as coefficient of variation, spectral measures and regularity score have been applied to quantify the variability of respiratory bursts from this in vitro model. However these previous techniques are unable to provide clear and reproducible results. Therefore, in this chapter the focus is on the application of Approximate and Sample Entropy to quantify the complexity of this reduced respiratory network. So, we will demonstrate that Approximate and Sample Entropy provide sensitive measures of respiratory network complexity and can provide a better understanding of network state compared to current methods. In this study collected discharge data from the hypoglossal (XII) nerve in an in vitro rat preparation at different levels of network excitability (by altering extracellular + potassium concentration, [K ]o in a controlled and reproducible manner) has been collected. A sample of the respiratory burst that has been collected was given in Figure 2.12. Changes in these entropy measures across different levels of imposed

78 network excitability that provide realistic physiologic perturbations of excitability are used to assess the complexity of the network state.

5.1.1 Methods Animal Preparation

Animal protocols were approved by the IACUC of Case Western Reserve Univer- sity. All experiments were performed in Dr. Christopher Wilson Laboratory using Sprague-Dawley rat pups age 0-5 days old obtained from Zivic Miller Laboratories (Pittsburgh, PA) or Charles River Laboratories (Wilmington, MA). Rat pups were deeply anesthetized with isofluorane then decapitated. This was followed by dissection of the brainstem at the pontomedullary border 95% O2, 5% CO2 (carbogen)-gassed artificial cerebrospinal fluid (ACSF). Transverse medullary slices were made by sectioning the brainstem on a sliding vibratome (Vibratome Instruments, St. Louis, MO) while continuously superfusing the tissue with chilled (3-6 ◦C) ACSF. Facial nucleus, compact formation of nucleus ambiguous, glossopharyngeal rootlets and obex served as guides to ensure the slice contained the preBötzinger complex (pBc), XII premotor and motor neurons, resulting in sufficient neural circuitry for generation of fictive in vitro inspiratory drive.

5.1.2 In vitro extracellular recording

Immediately following sectioning, the brainstem slice was placed caudal face up in a polycarbonate chamber (26GLP, Warner Instruments, Hamden, CT) retained with a platinum ring and overlying nylon threads in the recording chamber. The slice was perfused with ACSF in a range of [K+] (3-9mM) bubbled with carbogen gas at 27 ◦C ± 0.5 continuously for up to 12 hours. A glass suction pipette was used to record whole nerve fictive inspiratory activity from the XII rootlet, which was visually identified by an upright Leica DM/LFS (Leica Microsystems GmbH, Wetzlar, Germany) microscope. The neural XII activity was recorded and bandpass filtered

79 between 10-1000Hz. This activity was digitized at 8k Samples/Sec using a Digidata 1322 analog-to-digital converter. After recording individual bursts were isolated and extracted for further analysis.

5.1.3 Parameter selection

Approximate and Sample Entropy which were described in Chapter 2 were used to analyze each of the individual excised bursts. For this analysis, the appropriate embedding dimension, m, and tolerance value, r, were chosen to be 2 and 0.2*STD (standard deviation of the signal) respectively and these chosen are consistent with the literature [52]. The time delay, τ has been chosen by detecting the first minimum or zero crossing of the autocorrelation function. Because the variability in each epoch of data is different, it reasonable to expect that the same minimum or zero crossing of the autocorrelation function for each epoch will be different. However in this analysis we are interested in choosing a suitable, constant, time delay value that can provide consistent results across all epochs. Previous results from neonatal EEG analysis show that as long as the time delay value is not selected from the region where the signal is highly correlated and is chosen to be close to the first minimum or zero crossing of autocorrelation function, the nonlinear properties of the time series signal being analyzed will be accurately quantified. Therefore in this analysis the time delay, τ, has been chosen as 10 samples which corresponds to 0.125 seconds.

5.1.4 Results

In order to evaluate the changes in network complexity, whole network excitability is increased by varying [K+] and the Approximate and Sample Entropy were calculated for 50-100 individual bursts per [K+] concentration level. This number of bursts is suﬃcient for a statistical estimate of complexity at each excitability level. The results for Approximate and Sample Entropy are shown in Figures 5.1 and 5.2 respectively and show a consistent pattern for how both types are changing as a

80 function of [K+]. Most notable is the decline as [K+] is increased from 3 to 5mM, followed by a large increase in complexity from 5 to 7mM. If [K+] is increased even further, the complexity falls, however, not to the levels observed at 3mM. This change in complexity demonstrates a potential change in the state of the network, reﬂected in changes in burst dynamics, and the choice of [K+] can inﬂuence the complexity of the signal.

1.9

1.8

1.7

1.6

1.5

Approximate Entropy 1.4

1.3

1.2

1.1 3mM 5mM 7mM 8mM 9mM Extracellular potassium concentration

Figure 5.1: Approximate Entropy result for respiratory network bursts at diﬀerent potassium concentration

Our results suggest that as [K+] is elevated, the complexity of the network changes signiﬁcantly and the region from 5mMM to 7mM may contain a bifurcation point for the network activity. Therefore, in vitro recordings at various [K+] may be repre- senting a change in the state of this reduced respiratory network.

81 2.2

1.8

1.6

1.4

Sample Entropy 1.2

0.8

3mM 5mM 7mM 8mM 9mM Extracellular potassium concentration

Figure 5.2: Sample Entropy result for respiratory network bursts at diﬀerent potassium concentration

82 5.2 Cardioventilatory Coupling

Cardioventilatory (C-V) coupling is known as temporal synchronization between the cardiac and respiratory rhythms. In fact, ventilatory behavior (breath timing and tidal volume) and cardiac rhythm are produced by brainstem neural systems which interact at the physiological level. The C-V coupling has been observed in seated subjects [15], spontaneously breathing anesthetized subjects, and in conscious resting human subjects [81]. The objective of this study is to characterize the cardioventilatory (C-V) coupling in resting humans. Tzeng et. al. [81] studied the coupling by considering the time intervals between the R-wave and the next inspiratory onset event using Shannon Entropy [72] and compared the entropy value to a baseline to validate the coupling. Here, baseline estimation is based on randomly generated time intervals between the R-waves and next inspiratory onset event and if the entropy value of C-V turns out to be less than the baseline, it provide statistical evidence of the C-V coupling. In our work, we have considered all possible combinations of intervals between R-waves and inspiratory or expiratory onset events. In addition surrogate data analysis has been used to investigate the coupling. Galletly et. al. [15, 16, 17] have conducted a series of studies, investigating the synchronization of heart rate and breathing. Tzeng et. al. [81] found C-V coupling in 67% of healthy subjects at rest and noted variability between subjects and within the same subject over time. They also observed that C-V coupling was more likely to be associated with high degrees of heart rate variability and with lower breathing frequency.

5.2.1 Methods Subjects

Subjects in good health were studied in a seated position and during wakefulness. Exclusion criteria included chronic medical conditions, chronic prescribed medications, recent surgery, previously diagnosed acute diseases of the heart or lungs and prior diagnosis of a sleep disorder. None of the subjects were currently on chronic

83 Table 5.1: Subject Demographic Data Subject Age Sex BMI Smoker A 34 F 18.5 N B 35 M 40.3 N C 28 M 23.7 Y D 31 F 20.2 N E 21 F 30.5 N F 36 M 24.2 N Scuba G 39 M 23.4 N Athlete H 33 M 25.8 N I 26 F 25.7 N J 21 F 20.2 N K 22 F 23.5 N Athlete One subject was a smoker & One subject was a marathon runner prescribed medications, with the exception of some female subjects taking oral con- traceptives. Subjects were asked to abstain from consuming caffeinated beverages for at least 2 hours before testing. Subjects were asked to complete a questionnaire regarding general health, including: age, height, weight, time of last meal, consumption of caffeinated drinks , chronic medical problems, prior diagnosis of heart or lung disease, prescription medications, OTC medications, current and past smoking history, allergies, difficulty breathing through nose and mouth, surgery within the last 6 months, or a diagnosis of a sleep disorder. Measurements of ventilation were made through a mouthpiece with the subject wearing nose clip, using a pneumotachograph on the inspiratory side of a one-way valve. Heart activation (HA) and R-R interval were measured by electrocardiogram from a modified V5 lead placement. After becoming accustomed to the testing situation, subjects breathed for a 10 minute period. The demographic information of subjects is given in Table 5.1.

5.2.2 Protocol

The subjects were seated in a comfortable chair with reclined head at an angle of 30 degrees in a quiet room at approximately 70 ◦F. Subjects were allowed to become

84 accustomed to the mouth piece and nose clip before data recording was started. There were three steps during data collection as follows:

Calibration: During the ﬁrst 120 seconds of the experiment, a one liter syringe was used to manually deliver a series of three, one liter calibration volumes through the pneumotachograph. Care was taken to deliver these volumes slowly and smoothly.

Normal Breathing: After calibration, subjects were asked to breathe normally through the mouthpiece with a nose clip preventing nose breathing for 10 minutes. During data collection, subjects breathed normally through a mouthpiece which was connected to a one-way-valve that allows inspiration through a pneumotachograph and directs expiration through either a free path to room air or through a valve providing a PEEP (Positive End Expiratory Pressure) of 5 cm

H2O.

PEEP Breathing: After normal breathing, with the setup identical to normal breath-

ing except that the adjustable valve is set to create a PEEP of 5 cm H2O, the subject breathed normally for an additional 10 minutes. This level of PEEP is thought to provide a degree of lung stretch approximately equal to one extra breath.

5.2.3 Surrogate Data Analysis

Surrogate data analysis has been used to validate the presence of nonlinear coupling between heart rate (R-wave) and the onset of inspiration and expiration. For this purpose, 19 surrogate heart rate time series are generated based on the cumulative sum of the surrogate R-R interval time series to meet the desired level of statistical signiﬁcance required in analysis. The new surrogate heart rate time series has the same linear R-R properties as the original heart rate time series, but any exciting relationship between the original R-waves and the respiratory network have been

85 eliminated. The hypothesis is that if the coupling between the original heart rate and respiratory network is random, the entropy of the intervals being analyzed should not be statistically different from those derived with the surrogate heart rate time series. If the entropy of the intervals between the original heart rate and respiratory network are consistent than those for the surrogate heart rate, the null hypothesis is rejected at a given significance level and this validates the existence of non random coupling between heart rate and respiration. In this test, the statistical significance is obtained by counting or ranking the number of surrogate data sets that have higher entropy of the intervals values than the original data set.

5.2.4 Data Analysis & Discussion

The first step of data analysis is the detection of the R-waves in the ECG signal as well as detection of the inspiration events from the breathing air flow time series. A sample of the ECG and breathing waveforms is given in Figure 5.3. The detection of R-waves and inspiration events has been done based on threshold crossing and because there are no expiration events in the breathing time series, the starting of expiration is at the end of inspiration and the start of the next inspiration signal the end of expiration. The parameters of interest have been defined in Table 5.2 and the computation of information theory-based entropy and the entropy of intervals are given in Appendix A. The calculated entropy values for normal and the PEEP breathing are presented in Tables 5.3 and 5.4 respectively. Also, plots of the entropy of intervals are shown in Figures 5.4 and 5.5. The box plots for all of the entropy of intervals calculations during PEEP breathing have lower quartile than normal breathing. The entropy of the R-R intervals during normal breathing has a lower median than during PEEP breathing. This means that the heart rate is more regular or less complex during normal breathing and there are no other significant differences between the medians of the other parameters for normal and PEEP breathing. The results of the entropy analysis for time intervals defined between heart activation (R-wave) and the

86 ECG

Breathing

Figure 5.3: Sample ECG & Breathing onset of the inspiration or expiration events along with the rank ordering from surrogate data analysis to identify the coupling for normal and PEEP breathing are given in Tables 5.5 and 5.6 respectively and the plots of the entropy and ranking results are shown in Figures 5.6 and 5.7. Since the variability of entropy results during normal and PEEP breathing is very similar, it is very diﬃcult to conclude anything about the relative regularity between these two sequences. However, there is obviously evidence of coupling between heart rate and respiratory events. In particular, there is a stronger coupling between heart activation preceding the start of expiration during PEEP breathing.

87 Table 5.2: Interested parameters from cardioventilatory system Parameter Deﬁnition RR Time intervals of R-R. InsVol Inspiratory volume taken as time-series TotTe Time intervals from start of expiration until beginning of next inspiration TotTi Time intervals from start of inspiration until beginning of next expiration Ti Time intervals from start of inspiration until the start of the next inspiration Te Time intervals from start of expiration until the start of the next expiration HrTi Time intervals of heartbeat initiation until the next inspiration. HrTe Time intervals of heartbeat initiation until the next expiration. TiHr Time intervals of start of Inspiration until the next heartbeat initiation. TeHr Time intervals of start of expiration until the next heartbeat initiation

Table 5.3: Entropy results for normal breathing in healthy subjects Subject R-R InsVol TotTe TotTi Ti Te A, Day one -1.3509 -7.6478 3.2663 3.1750 1.0784 2.9959 A, Day two -1.3509 -7.5277 3.4187 3.4263 1.2530 3.1843 B, Day one -2.7618 -6.9643 3.4393 3.5534 2.0925 2.9959 B, Day two -1.3453 -6.8682 2.7010 2.5872 1.7932 2.0817 C, Day one -3.1362 -7.6934 3.0191 2.4137 2.1849 1.9237 C, Day two -3.0729 -6.9629 2.4860 2.2805 1.5419 1.6113 F, Day one -3.0146 -10.5260 1.0068 1.0164 -1.0822 0.9512 F, Day two -3.4656 -10.4485 1.1395 1.1958 -1.0233 0.9616 G, Day one -1.5241 -7.5473 3.1076 3.0609 1.8518 2.4719 G, Day two -1.3469 -6.9086 3.1679 3.1809 2.2829 2.4042 H, Day one -3.9111 -8.9977 0.2651 -0.010 -0.5736 -0.2385 H, Day two -3.6110 -9.9860 0.1543 0.1613 -0.9388 -0.1701 K, Day one -1.5688 -9.1516 1.4150 1.3484 -0.5322 1.3426 K, Day two -1.4676 -9.3185 1.6287 1.7361 -0.6944 1.4717

88 Table 5.4: Entropy results for 5 cm H2O of PEEP breathing in healthy subjects Subject R-R InsVol TotTe TotTi Ti Te A, Day one -0.5851 -7.2435 3.4146 3.3601 1.2269 3.3715 A, Day two -0.8468 -7.4391 2.9539 2.9545 1.1618 2.6833 B, Day one -2.1256 -6.8549 3.0511 3.0415 2.0136 2.4121 B, Day two -1.8507 -7.1046 2.5822 2.6067 1.5051 2.0228 C, Day one -2.7434 -7.4887 2.9931 2.6736 2.3302 1.4883 C, Day two -2.6864 -7.3850 2.8268 2.6704 1.9269 1.8234 F, Day one -2.7413 -10.1887 1.2334 1.2349 -0.4750 1.0288 F, Day two -1.9900 -8.9283 1.9148 1.9776 0.0860 1.7714 G, Day one -1.4642 -7.1703 2.8736 2.8033 1.2867 2.6720 G, Day two -1.2582 -6.5703 3.1487 3.0904 1.6025 2.7145 H, Day one -3.5297 -8.5837 0.5182 0.5862 0.3263 0.1703 H, Day two -3.2941 -9.1450 1.2773 1.2356 0.0946 0.7942 K, Day one -1.6741 -8.5401 1.6848 1.6206 -0.4065 1.5302 K, Day two -1.1901 -9.6269 0.4658 0.6006 -0.7567 0.4949

Normal Breathing 4

−2

−4

−6 Entropy of Intervals (Bits) −8

−10

−12 R−R InsVol TotTe TotTi Ti Te Selected Intervals

Figure 5.4: Entropy of intervals during normal breathing

89 PEEP Breathing 4

−2

−4

−6 Entropy of Intervals (Bits) −8

−10

−12 R−R InsVol TotTe TotTi Ti Te Selected Intervals

Figure 5.5: Entropy of intervals during PEEP breathing

90 Table 5.5: Entropy and ranking result for Cardioventilatory intervals during normal breathing in healthy subjects Subject HrTi HrTe TiHr TeHr A, Day one -0.0104 -0.3176 -0.1521 0.0063 Ranking 0 85 15 10 A, Day two -0.3105 -0.2650 -0.1861 -0.3840 Ranking 85 85 60 95 B, Day one -0.2668 -0.661 -0.4817 -0.5554 Ranking 80 95 95 95 B, Day two -0.0112 -0.0413 0.09293 -0.1623 Ranking 30 35 35 70 C, Day one -1.1463 -0.8257 -0.9482 -0.7613 Ranking 95 95 95 65 C, Day two -0.6788 -0.7625 -0.7523 -0.7540 Ranking 35 80 80 80 F, Day one -0.2783 -0.4148 -0.3034 -0.3064 Ranking 60 95 85 70 F, Day two -0.7413 -0.6436 -0.7371 -0.6289 Ranking 95 95 95 95 G, Day one -0.3168 -0.2574 -0.1312 -0.1999 Ranking 95 95 85 95 G, Day two 0.00283 0.03043 -0.2488 -0.0184 Ranking 75 55 95 85 H, Day one -0.8765 -0.6882 -0.7854 -0.6629 Ranking 95 95 95 95 H, Day two -0.5695 -0.4874 -0.4932 -0.4574 Ranking 95 70 90 40 K, Day one 0.3316 0.2009 0.3096 0.2204 Ranking 15 95 15 85 K, Day two -0.5695 -0.4874 -0.4932 -0.4574 Ranking 95 70 90 40

91 Table 5.6: Entropy and ranking result for Cardioventilatory intervals during PEEP breathing in healthy subjects Subject HrTi HrTe TiHr TeHr A, Day one -0.0570 -0.4023 -0.2539 -0.1241 Ranking 20 95 90 25 A, Day two -0.2021 -0.1341 -0.2561 -0.1331 Ranking 80 65 85 70 B, Day one -0.4117 -0.9263 -0.5642 -0.5718 Ranking 65 95 95 95 B, Day two -0.1250 -0.1406 -0.2485 -0.1860 Ranking 25 30 90 65 C, Day one -0.8243 -0.6585 -0.7652 -0.7962 Ranking 95 95 85 95 C, Day two -0.5942 -0.8141 -0.6310 -0.8230 Ranking 20 95 35 95 F, Day one -0.5146 -0.5841 -0.4854 -0.6007 Ranking 95 95 95 95 F, Day two -0.7855 -0.7978 -0.3167 -0.8441 Ranking 95 90 0 95 G, Day one -0.0162 -0.3102 -0.0747 -0.1693 Ranking 10 95 45 85 G, Day two -0.0423 -0.0776 0.0126 -0.2225 Ranking 70 95 30 95 H, Day one -0.5693 -0.4958 -0.6005 -0.5456 Ranking 80 95 95 55 H, Day two -0.5381 -0.5543 -0.4867 -0.6035 Ranking 80 95 35 90 K, Day one 0.0073 -0.1824 -0.0164 -0.0979 Ranking 30 95 35 80 K, Day two 0.2889 0.0884 0.1849 0.2526 Ranking 80 95 90 80

Table 5.7: Median of Cardioventilatory parameters during normal breathing in healthy subjects Subject Resp Rate Ti (Sec) Te (Sec) Heart Rate A, Day one 10.68 1.61 3.90 71.16 A, Day two 12.00 1.53 3.43 72.00 B, Day one 7.63 2.98 4.67 66.43 B, Day two 10.14 2.59 3.29 73.09 C, Day one 9.70 3.28 2.73 96.10 C, Day two 11.78 2.57 2.30 94.90 F, Day one 11.34 1.73 3.12 68.72 F, Day two 12.24 1.67 3.18 79.92 G, Day one 7.92 3.00 4.39 58.92 G, Day two 14.00 2.17 2.07 57.5 H, Day one 17.45 1.72 1.67 80.83 H, Day two 19.52 1.54 1.52 80.07 K, Day one 15.60 1.60 2.21 58.00 K, Day two 15.60 1.47 2.35 49.50

92 Table 5.8: Median of Cardioventilatory parameters during PEEP breathing in healthy subjects Subject Resp Rate Ti (Sec) Te (Sec) Heart Rate A, Day one 7.32 2.08 6.05 74.16 A, Day two 8.61 1.96 4.92 71.12 B, Day one 6.21 3.42 5.94 69.70 B, Day two 9.36 2.68 3.68 69.24 C, Day one 11.45 3.07 2.11 91.74 C, Day two 13.09 2.20 2.30 91.20 F, Day one 9.92 1.79 4.11 70.03 F, Day two 8.88 1.70 4.88 85.92 G, Day one 9.12 1.83 4.57 62.52 G, Day two 10.69 2.04 3.54 60.65 H, Day one 14.72 2.31 1.72 82.90 H, Day two 13.96 1.69 2.57 81.49 K, Day one 12.21 1.52 3.34 61.96 K, Day two 13.44 1.47 2.94 48.72

Normal Breathing 0.5

−0.5

−1

Entropy of Intervals (Bits) −1.5 HrTi HrTe TiHr TeHr

PEEP Breathing 0.5

−0.5

−1

Entropy of Intervals (Bits) −1.5 HrTi HrTe TiHr TeHr Selected Intervals

Figure 5.6: Entropy of intervals for Cardioventilatory intervals in healthy subjects

93 Normal Breathing 100

40 Ranking 20

0 HrTi HrTe TiHr TeHr

PEEP Breathing 100

40 Ranking 20

0 HrTi HrTe TiHr TeHr Selected Intervals

Figure 5.7: Ranking results for cardioventilatory intervals in healthy subjects

94 Appendix A

Variability Analysis Techniques from the Literature

A.1 Non-Parametric Change Point Detection

The goal of change-point detection is the segmentation of a time-series into epochs with stationary or statistically invariant properties. We use a non-parametric statistical approach to change point detection developed by Brodsky [5]. This method is based on the model of the Brownian bridge process. The technique is applied to the problem of identifying appropriate linear regions of diagnostic sequences for the analysis of scaling properties in physiological time-series.

A.1.1 Statistical Methods

N In our approach the diagnostic sequence XN , {xn}n=1 is assumed to be a weakly piecewise stationary random sequence. We suppose that XN can be represented in the PK form of a piecewise function fN (n) = k=1 akuN (n − bk) where uN (a) is the Heaviside function, i.e. uN (a) = 0 for n < a and uN (a) = 1 for n ≥ a, and a centered random N sequence ΞN , {ξn}n=1. The model is Xn = fN (n) + ξn, where we further suppose that ΞN satisfies the invariance principles defined in [50]. Specifically, we assume that

ΞN has a ﬁnite moment generating function in the neighborhood of zero, an integrable

95 correlation function, and the existence of the limit given in equation (A.1)

n !2 −1 X 2 lim n E ξk = σ < ∞. (A.1) n→∞ k=1

Note that for an independent identically distributed (i.i.d.) sequence equation (A.1) is simply a statement of ﬁnite variance of the ξk. Change-points in the sequence XN are selected based on a generalization of the Kolmogorov-Smirnov statistic given by the sequence

" n N # h n n iδ 1 X 1 X Y (n, δ) = 1 − x − x . (A.2) N N N n k N − n k k=1 k=n+1 where δ ∈ [0, 1] and j ∈ [1, ··· ,N − 1]. Each statistic YN (n, δ) corresponds to a point xn in the diagnostic sequence and is the difference between the means of the sequence before and after that given point, weighted by a function of the position n ∗ and the parameter δ. It is clear that if E[xn] = m1 for n ≤ n and E[xn] = m2 6= m1 ∗ ∗ for n ≥ n , then arg maxn |YN (n, δ)| = n . Then for some given threshold C (to be defined later), if |YN (n, δ)| ≥ C then the null hypothesis H0 that there is no change ∗ point is rejected and n is selected as a change-point of XN . It is shown in [5] that a value of δ = 0 minimizes the probability of ignoring an existing change point, that δ = 1 minimizes the probability of presuming a nonexistent change-point, and δ = 0.5 guarantees a minimum error in the estimation of the change-point location. It is also shown in [5] using functional limit theorems for random sequences [50] √ that for a stationary random sequence with no change-points, the statistics N · O YN (bNtc, 1) converge in distribution, as N → ∞, to the random process σWt . Here W O is the Brownian bridge process, which is discussed in [4], and σ is defined by equation (A.1) and the mixing conditions given in [50]. The convergence of the Y-statistics to the Brownian bridge W O is important

96 because W O has the notable property that

∞ O X k 2 2 FKS(C) , P sup |Wt | ≤ C = 1 + 2 (−1) exp{−2k C } (A.3) t k=1 where C > 0 and FKS(·) denotes the Kolmogorov-Smirnov distribution. Therefore the change-point detection threshold C can be chosen based on this model by deﬁning a false alarm probability Pfa and using the properties in equation A.3. This gives

√ Pfa = lim P max N|YN (j, δ)| > C = N→∞ j (A.4) O = P sup |Wt | > C/σ = 1 − FKS(C/σ) t √ −1 which results in C = σFKS(1 − Pfa)/ N, where FKS is the Kolmogorov-Smirnov distribution and N is the length of the test sequence.

A.1.2 Description of Algorithm

The change-point detection algorithm is implemented in three stages: Stage 1: Preliminary Detection The Y-statistics equation (A.2) are computed for the diagnostic sequence XN with δ = 1. The threshold C is then computed using a false alarm probability Pfa = 0.1. We estimate the parameter σ by ﬁrst sepa- rating the diagnostic sequence at the location n∗ that maximize the Y-statistics into two sub-sequences, Z1 and Z2. Next the sequence Z1 and Z2 are then combining to form a new centered sequence Z. Assuming that ξN is i.i.d. we estimate the standard ∗ deviation σ = std(Z). If |YN (n, 1)| ≥ C, then n is recorded as a change-point. We then repeat this process recursively for Z1 and Z2 until no additional change-points

∗ K1 are detected. The result of stage 1 is a set of candidate change-points {nk}k=1.

K1 Stage 2: Rejection XN is now decomposed into sub-sequences {Xk}k=1 each containing one change-point and with the endpoints of each subsequence chosen to ∗ be at the midpoints between two neighboring nk. The Y-statistics are then computed

97 ∗ for each sub-sequence with δ = 0, and if |Yk(nk, 0)| ≤ C for a much smaller false ∗ alarm probability chosen to be Pfa = 0.01, then nk is discarded. The result of the

∗ K2 rejection stage is a reduced set of potential change points {nk}k=1, with K2 ≤ K1

Stage 3: Final Estimation XN is again partitioned as in Stage 2 using the

∗ K2 remaining candidate change points {nk}k=1. Then, the Y-statistics are computed for each sub-sequence with δ = 0.5 and the absolute maximum of the Y-statistics for each segment is accepted as the ﬁnal location of the corresponding change-point ∗ nk, thereby minimizing the estimation error in the determination of the change-point

∗ K2 locations {nk}k=1. This set of change points is the ﬁnal output of the algorithm and these change points are used to recover the piecewise function fN (n), which is also of interest in our application.

A.2 Correlation Dimension

The correlation dimension D2 quantiﬁes the complexity of a structure on a metric space e.g. Rm. It is commonly used to measure the “strangeness” of the attractor of a nonlinear dynamical system [19, 20]. When computing D2 from a time-series, it is assumed that the data are observations of the dynamic system corresponding to points on an attractor. The attractor must be reconstructed from the observation using a technique such as time-delay embedding of the time-series in Rm. To better understand this measure, consider a uniformly sampled one-dimensional time series

{x1, x2, . . . , xn} and m-dimensional embedding vectors (Xi,Xj), whose components are time delayed versions of the elements in the original time series with delay τ, which is assumed to be a multiple of the sampling time:

m Xi = (xi, xi+τ , xi+2τ , ··· , xi+(m−1)τ ) Xi ∈ R (A.5)

m Xj = (xj, xj+τ , xj+2τ , ··· , xj+(m−1)τ ) Xj ∈ R (A.6)

98 where 1 ≤ (i, j) ≤ n − (m − 1)τ. Then the correlation integral is given as:

n 1 X C(r) = lim θ(r − |Xi − Xj|) (A.7) n→∞ n2 i,j=1 i6=j where θ is the Heaviside function. The objective is to investigate the behavior of the correlation integral C(r) consistent with the power-law model given in equation (A.8) for small values of r, i.e. C(r) ∝ rν (A.8)

The scaling exponent ν is closely related to the correlation dimension D2, which is given by: log C(r) D2 = lim (A.9) r→0 log r

In practice, the correlation integral must be calculated from a limited amount of time-series data according to:

1 X C(r) = θ(r − |X − X |) (A.10) n(n − 1) i j i,j=1 i6=j

The correlation dimension is then estimated from the gradient of the linear region of the plot of log C(r) vs. log r for small values of r. There are a variety of computational issues associated with the estimation of D2 [79]. Theiler [78] slightly modiﬁed the standard correlation integral calculation by adding a window (W ) to overcome spurious estimation of the dimension due to autocorrelation. The Theiler modiﬁcation includes the window W as follows:

n n−j 2 X X C(r) = θ(r − |X − X |) (A.11) (n − W )(n + 1 − W ) i i+j j=W i=1

99 A.3 Information Theory Based Entropy

A.3.1 Shannon Entropy

The concept of Shannon [72] Entropy, a central concept in information theory, is a measure of uncertainty and has also been used as a measure of information content. Entropy is calculated as either discrete or continuous (diﬀerential) entropy. PN The entropy of a discrete random variable with probability pi such that n=1 pi = 1 is given by: N X H = pi log2 pi (A.12) i=1 where N is the total number of events. For a continuous random variable x we have Z H(x) = f(x) log2 f(x)dx (A.13) S where S is the support of x with density f(x) > 0. The entropy of a discrete random variable is always nonnegative where the entropy of a continuous random variable can be negative. If the discrete and continuous random variables are related through sampling, then a discrete approximation of the continuous entropy is a biased estimate of discrete entropy, i.e.

H∆(x) = H(x) − log2 ∆t (A.14) where ∆t is the temporal bin width or the length of the sampling (integration) interval. It is important to note that Shannon Entropy is independent of the order of a time series and hence is invariant under surrogate data transformations.

A.3.2 Interval Entropy & Entropy of Intervals

Interval Entropy was developed to estimate the entropy of neural discharge from a recorded spike train where successive interspike intervals are assumed to be uncor- related for a ﬁnite sample size [64]. Normally it is diﬃcult to obtain a good estimate

100 of the probability distribution from the histogram if the number of samples is small. Therefore the distribution of data is assumed to be a well known distribution and a probability distribution is ﬁt to the data. The entropy then is calculated directly from the estimated distribution. Although any type of probability distribution can be selected, the generalized gamma distribution have been used for neural discharge activity:

a−1 (x − s) −(x−s) p(x) = γ(x; a, s, τ) = e τ , (x ≥ s ≥ 0; a; τ > 0) (A.15) τ aΓ(a) where a, s, and τ are the order, time shift and time scaling parameters of the distribution. It is also possible to ﬁt to the convex combination:

p(x) = fγ1(x; a, s, τ) + (1 − f)γ2(x; a, s, τ) (A.16) where f is a constant between zero and one. If there is sufficient data, it will be possible to obtain a good estimate of distribution and the entropy can be directly calculated from the histogram. As a result, there is no need to fit the data to a known distribution because the underlying distribution can be different than best assumed probability distribution. Therefore the computed entropy is not anymore Interval Entropy and it is entropy of distribution of interval or Entropy of Interval. The Entropy of Intervals can be computed in two ways as discrete or continues. In discrete calculation, the range of the histogram is divided to N number of bins and the probability of each bin is computed by dividing the number of observations in each bin by total number of observations and the entropy then is computed using equation A.12. This entropy is always positive and the maximum value of entropy occurs when the observations are uniformly distributed. This means the probability of each histogram bin is equal to 1/N and the maximum entropy is − log(1/N). Since the value of entropy is depends on number of bins, N, the proportional entropy [81] or normalized entropy have been used to map the entropy value to the range of 0-1 by diving the entropy value by maximum entropy. In continues calculation, the range of

101 histogram is again divided to N number of bins. However the probability of each bin PN is computed by including the bin width of histogram, ∆t such that i=1 pi∆t = 1 where pi is the probability of each bin. The continues entropy then is computed as PN − i=1 pi log(pi)∆t. The diﬀerence between continues and discrete entropy comes from including ∆t in computation. In calculation of Entropy of Intervals in this dissertation, the continuous entropy calculation has been used.

A.3.3 Spectral Entropy & Variance

The spectral entropy & variance are computed by converting the spectrum into a probability distribution function by normalizing the power spectrum in each frequency bin as: X x = i (A.17) i PN i=1 Xi

th where Xi represents the energy of the i frequency component of the spectrum. The spectral variance then is the variance of Xi and the spectral entropy is the Shannon

Entropy of xi which is computed using equation A.12.

A.4 Surrogate Data Analysis

Nonlinear measures such as correlation dimension, Lyapunov Exponents, Approx- imate Entropy and Sample Entropy are often applied to time series data to identify the presence of nonlinear behavior. Surrogate data analysis is a powerful validation tool, introduced by Theiler [80]. Surrogate methods are designed to determine if the nonlinear characteristics in the time series are from deterministic or stochastic (autocorrelation) eﬀects. Theiler proposed generating a surrogate data series by randomizing the phase of the original time series. Then, the PSD (Power Spectral Density) of the signal is preserved and the original and surrogate data have the same autocorrelation function. If the results of analyzing the surrogate data turn out to be diﬀerent than the original time series data, the nonlinearity behavior is believed

102 to be real. Schreiber [69, 30, 70] noticed that surrogate data generation by just randomizing the phase alters the probability distribution of the time series and this can be a problem in the analysis. So, in order to not change the distribution of data, it was recommended that surrogate data be generated by randomizing the order of the time series with a constraint on the autocorrelation function. In this way, the distribution of time series is preserved. A statistical test then is used to determine the signiﬁcance of the test for the existence of nonlinearity. Usually, a rank order test [80, 69] is used to obtain the false rejections probability which is adjustable by the number of surrogate. To obtain 95% signiﬁcance 19 (1/0.05 - 1) surrogate data sets need to be generated. Surrogate data analysis can also be used to evaluate the performance of any detection or prediction algorithm. Kreuz et al. [32] used the surrogate method to validate the performance of seizure prediction techniques. Noise titration [56, 57] has a similar objective to surrogate data analysis but surrogate data analysis is a preferred method of analysis.

A.5 Hjorth Parameters

Hjorth [22] defined activity, mobility and complexity as a set of parameters intended as a clinically useful tool for the quantitative description of an EEG”. All parameters can be calculated both in the time and in the frequency domain. The activity is proportional to the variance of a signal and the mobility is defined as the variance of the slopes of the EEG normalized by the variance of the amplitude distribution of the time series. The complexity is the variance of the rate of slope changes with reference to an ideal sine curve. The Hjorth parameters are defined as:

activity, A = a0 (A.18)

1/2 mobility, M = [a2/a0] (A.19)

1/2 complexity, C = [(a4/a2) − (a2/a0)] (A.20)

103 where a0, a2, and a4 are zero, second and forth order spectral moments. In frequency domain, the mobility and complexity can be estimated from the second and fourth statistical moment of the power spectrum. In time domain, a0 is the variance of the signal, a2 is the variance of the signal’s ﬁrst derivative and a4 is the variance of the signal’s second derivative.

A.6 Dynamical Similarity Index

Le Van Quyen et al. [60, 59] introduced the dynamical similarity index between a running test window and a reference period, usually selected from the beginning of a recording. This technique is based on time interval between successive zero crossing from negative to positive amplitude. In fact interval or period analysis is an alternative method of EEG analysis based on measuring the distribution of time interval between either zero or other level crossing, or between maxima and minima [74]. So, the first step is the transformation from time domain into the interval domain: In = Tn+1 − Tn where Tn is the time intervals between zero crossing. From this sequence of intervals, delay vectors are formed An = (In,In−1,...,In−m+1) using time delay embedding. To reduce the noise then singular value decomposition (SVD) is used X = A.V , where A is the trajectory matrix of original signal and X is the trajectory matrix projected onto the basis V defined by eigenvectors of the covariance matrix AT A. This SVD transformation is applied to both reference period r and test window t and dynamical similarity index between them is defined as:

Crt γt = √ (A.21) CrrCtt where Crt is the cross correlation [29] sum given by:

Nt Nr 1 X X Crt = Θ k xt,k − xr,i k − (A.22) NrNt k=1 i=1

104 and Crr and Ctt are the auto-correlation sum of the reference and test window:

Nt Nt 1 X X Ctt = Θ k xt,k − xt,i k − (A.23) NtNt k=1 i=1

Nr Nr 1 X X Crr = Θ k xr,k − xr,i k − (A.24) NrNr k=1 i=1

A.7 Phase Synchronization

Phase synchronization is deﬁned as phase locking [φx(t) − φy(t) = const] or, in the case of noisy and/or chaotic systems as phase entrainment [φx(t) − φy(t) < const], with φx(t) and φy(t) denoting the phase variables of two oscillating signals x(t) and y(t). The most common measure of phase synchronization that has been proposed for intracranial EEG data analysis is the mean phase coherence [46]

N 1 X R = ei[φx(tj )−φy(tj )] (A.25) N j=1

This measure is conﬁned to the interval [0, 1] where high values indicate a high degree of phase synchronization and low values correspond to unsynchronized signals. In order to measure changes in phase synchronization of two signals x(t) and y(t) over time, their phases φx(t) and φy(t) need to be determined and it can be achieved using instantaneous phase s˜(t) φ(t) = arctan (A.26) s(t) for an arbitrary signal s(t) using Hilbert Transform

Z +∞ 0 1 s(t ) 0 s˜(t) = p.v. 0 dt (A.27) π −∞ t − t where p.v. is the Cauchy principle value.

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