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- https://ufdc.ufl.edu/UFE0022411/00001
## Material Information- Title:
- Brain Dynamics, System Control and Optimization Techniques with Applications in Epilepsy
- Creator:
- Liu, Chang-Chia
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2008
- Language:
- english
- Physical Description:
- 1 online resource (169 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Biomedical Engineering
- Committee Chair:
- Pardalos, Panagote M.
- Committee Members:
- Uthman, Basim M.
Carney, Paul R. Van Oostrom, Johannes H. Roper, Steven N. - Graduation Date:
- 8/9/2008
## Subjects- Subjects / Keywords:
- Brain ( jstor )
Connectivity ( jstor ) Electrodes ( jstor ) Electroencephalography ( jstor ) Epilepsy ( jstor ) Neurons ( jstor ) Nonlinearity ( jstor ) Seizures ( jstor ) Signals ( jstor ) Time series ( jstor ) Biomedical Engineering -- Dissertations, Academic -- UF clustering, dynamics, epilepsy, nonlinear, statistics, svm - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Biomedical Engineering thesis, Ph.D.
## Notes- Abstract:
- The basic mechanisms of epileptogenesis remain unclear and investigators agree that no single mechanism underlies the epileptiform activity. Different forms of epilepsy are probably initiated by different mechanisms. The quantification for preictal dynamic changes among different brain cortical regions have been shown to yield important information in understanding the spatio-temporal epileptogenic phenomena in both humans and animal models. In the first part of this study, methods developed from nonlinear dynamics are used for detecting the preictal transitions. Dynamical changes of the brain, from complex to less complex spatio-temporal states, during preictal transitions were detected in intracranial electroencephalogram (EEG) recordings acquired from patients with intractable mesial temporal lobe epilepsy (MTLE). The detection performance was further enhanced by the dynamics support vector machine (D-SVM) and a maximum clique clustering framework. These methods were developed from optimization theory and data mining techniques by utilizing dynamic features of EEG. The quantitative complexity analysis in multi-channel intracranial EEG recordings is also presented. The findings suggest that it is possible to distinguish epilepsy patients with independent bi-temporal seizure onset zones (BTSOZ) from those with unilateral seizure onset zone (ULSOZ). Furthermore, for the ULSOZ patients, it is also possible to identify the location of the seizure onset zone in the brain. Improving clinician?s certainty in identifying the epileptogenic focus will increase the chances for better outcome of epilepsy surgery in patient with intractable MTLE. Recent advances in nonlinear dynamics performed on EEG recordings have shown the ability to characterize changes in synchronization structure and nonlinear interdependence among different brain cortical regions. Although these changes in cortical networks are rapid and often subtle, they may convey new and valuable information that are related to the state of the brain and the effect of therapeutic interventions. Traditionally, clinical observations evaluating the number of seizures during a given period of time have been gold standard for estimating the efficacy of medical treatment in epilepsy. EEG recordings are only used as a supplemental tool in clinical evaluations. In the later part of this study, a connectivity support vector machine (C-SVM) is developed for differentiating patients with epilepsy that are seizure free from those that are not. To that end, a quantitative outcome measure using EEG recordings acquired before and after anti-epileptic drug treatment is introduced. Our results indicate that connectivity and synchronization between different cortical regions at higher order EEG properties change with drug therapy. These changes could provide a new insight for developing a novel surrogate outcome measure for patients with epilepsy when clinical observations could potentially fail to detect a significant difference. ( en )
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- In the series University of Florida Digital Collections.
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- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2008.
- Local:
- Adviser: Pardalos, Panagote M.
- Statement of Responsibility:
- by Chang-Chia Liu.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Copyright Liu, Chang-Chia. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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detect the effect of therapeutic interventions. This quantification would also help us identify the regions that actively participate during epileptic seizures. In the later chapter, I investigate possibilities for identify the regions in the brain that are actively participate prior to epileptic seizures and the effect of therapeutic interventions on EEG recordings. Although it is known that the fluctuations in EEG frequency and voltage arise from spontaneous interactions between excitatory and inhibitory neurons in circuit loops, the cause of neuron discharge remains unclear. The synchronization of neuron activity is considered to be important for information processing in the developing brain. There is now clear evidence that there are distinct differences between the immature and mature brain in the pathophysiology and consequences of seizures, abnormal synchronization activity in the developing brain can result in irreversible alterations in neuronal connectivity [108]. In cognitive task studies, In 1980, Freeman found "more rc, l oi spatiotemporal activities in EEG for a brief period of time when the animal inhaled a familiar odor until the animal exhaled [109; 96]. Some earlier studies also indicated the significant role of synchronization for physiological systems in humans; the detectable alters in synchronization phenomena have been associated to a number of chronic, acute diseases or the normality of brain. [110; 111]. In the field of epilepsy research, several authors have -,i--.- -1. 1 direct relationship between alters in synchronization phenomena and onset of the epileptic seizures using EEG recordings. For example, Iasemidis et al., (1996) reported from intracranial EEG that the entrainment in the largest Lyapunov exponents from critical cortical regions is a necessary condition for onset of seizures for patients with temporal lobe epilepsy [112; 61]; Le Van Quyen et al., showed epileptic seizure can be anticipated by nonlinear analysis of dynamical similarity between recordings [35]. Mormann et al., showed the preictal state can be detected based on a decrease in synchronization on intracranial EEG recordings [89; 43]. The highly complex behavior on the EEG recordings is considered to normality of brain state, while transitions into a lower complexity brain state are regarded as a pathological Dependency Matrix (Before Treatment PI) Dependency Matrix (After Treatment PI) 5 1.5 5 1.5 10 0 I 10 0 1 1 U 0.5 15 0.5 O 0 5 10 15 5 10 15 Dependency Matrix (Before Treatment P2) Dependency Matrix (After Treatment P2) 2 2 5 1.5 5 -. 1.5 10 1 10 15 m 0.5 15 U 0.5 0 0 5 10 15 5 10 15 Dependency Matrix (Before Treatment P3) Dependency Matrix (After Treatment P3) 2 2 5 1.5 5 1.5 10 I 10 15 0.5 15 0.5 0 0 5 10 15 5 10 15 Dependency Matrix (Before Treatment P4) Dependency Matrix (After Treatment P4) .5 1.5 5 1.5 :- *0 5L5 15 N .5 15 OE 0.5 5 10 15 5 10 15 Figure 8-2. Pairwise mutual information between for all electrodes- before v.s. after treatment 3.3 Proper Time Delay We used the mutual information function to estimate the proper time d. 1i, between successive components in delay vectors. In theory, the time delay used for time delay vector reconstruction is not the subject of the embedding theorem. Since the data are assumed to have infinite precision, from mathematical point of view delay time can be chosen arbitrary. On the other hand, it is essential to have a good estimation for proper time delay when dealing with none artificial data. For none artificial data, the time delay parameter can affect the dynamical properties under studying, if time delay is very large, the different coordinates may be almost uncorrelated. In this case, the attractor may become very complicated, even if the underlying true attractor is simple. If delay is too small, there is almost no difference between the different components between delay vectors, such that all points are accumulated around the bisectrix in the embedding space. Therefore, it is slr.--- -1. I to look for the first minimum of the time d, li1 mutual information [75]. The concept of mutual information is given as below Mutual information is originated from information theory and it has been used for measuring interdependence between two series of variables. Let us denote the time series of two observable variables as X = {x}N 1 and Y = {yj}N where N is the length of the series and the time between consecutive observations (i.e. iii,,,,J, .irate) is fixed. The mutual information between observations xi and yi is defined as: Slo PY(Si, qj) where Px,y(xi, yj) is the joint probability density of x and y evaluated at (xi, yj) and Px(xi),Py(yj) are the marginal probability densities of x and y evaluated at xi and yj respectively. The unit of mutual information is in bit, when based 2 logarithm is taken. If x and y are completely independent, the joint probability density Px,y(xi, yj) equals to the product of its two marginal probabilities and the mutual information between [40] C.E. Elger. Future trends in epileptology. Current Opinion in Neur 4l.i , 14(2):185-186, 2001. [41] F. Mormann, R.G. Ai,11.. i 1. C. E. Elger, and K. Lehnertz. Seizure prediction: the long and winding road. Brain, 130(2):314-333, 2006. [42] S. Gigola, F. Ortiz, C. E. DAttellis, W. Silva, and S. Kochen. Prediction of epileptic seizures using accumulated energy in a multiresolution framework. Journal of Neuroscience Methods, 138:107-111, 2004. [43] F. Mormann, R.G. Ail1i.. i.1: T. Kreuz, C. Rieke, P. David, C.E. Elger, and K. Lehnertz. Automated detection of a preseizure state based on a decrease in synchronization in intracranial electroencephalogram recordings from epilepsy patients. Phys. Rev. E, 67(2):021912, 2003. [44] 0. Rossler. An equation for continuous chaos. Phys. Lett., 35A:397-398, 1976. [45] E.N. Lorenz. Deterministic nonperiodic flow. Journal of Atmospheric Sciences, 20:130-141, 1963. [46] P. Grassberger and I. Procaccia. Measuring the strangeness of strange attractors. P,;;-,..: D: Nonlinear Phenomena, 9:189-208, 1983. [47] A. Babloi-,-nt: and A. Destexhe. Low dimensional chaos in an instance of epilepsy. Proc. Natl. Acad. Sci. USA, 83:3513-3517, 1986. [48] J.P.M. Pijn, D.N. Velis, M.J. van der Heyden, J. DeGoede, C.W.M. van Veelen, and F.H. Lopes da Silva. Nonlinear dynamics of epileptic seizures on basis of intracranial eeg recordings. Brain T(',' 'i,,''l i, ; 9:249-270, 1997. [49] K. Lehnertz and C.E. Elger. Spatio-temporal dynamics of the primary epileptogenic area in temporal lobe epilepsy characterized by neuronal complexity loss. Electroen- ,, ,,l,, i. Clin. N,-ur.ph .l 95:108-117, 1995. [50] U.R. Acharya, O. Fausta, N. Kannathala, T. Chuaa, and S. Laxminarayan. Non-linear analysis of eeg signals at various sleep stages. Computer Methods and Pi.. 'gn,,.- in Biomedicine, 80(1):37-45, 2005. [51] H. Whitney. Differentiable manifolds. The Annals of Mathematics, Second Series, 37(3):645-680, 1936. [52] F. Takens. Detecting strange attractors in turbulence. In D.A. Rand and L.S. Young, editors, D;,l,; ,,i.. ,i1 Sl' 1/. ,- and Turbulence, Lecture Notes in Mathematics, volume 898, pages 366-381. Springer-Verlag, 1981. [53] J.-P. Eckmann and D. Ruelle. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys., 57:617-656, 1985. yiai =0 (5-16) i 1 0 < ai < C, i = 1,...n (5-17) The solution of the primal problem is given by w = u* '':' ,: where w is the vector that is perpendicular to the separating hyperplane. The free coefficient b can be found from ai(yi(w xi + b) 1) = 0, for any i such that ci is not zero. D-SVM map a given EEG data set of binary labeled training data into a high dimensional feature space and separate the two classes of data linearly with a maximum margin hyperplane in the dynamical feature space. In the case of nonlinear separability, each data point x in the input space is mapped into a different space using some nonlinear mapping function p. A nonlinear kernel function, k(x, f), can be used to substitute the dot product < p(x), p(') >. This kernel function allows the D-SVM to operate efficiently in a nonlinear high-dimensional feature space without being adversely affected by dimensionality of that space. 5.6 Connectivity Support Vector Machine In this subsection, we describe the framework of C-SVM. Instead of modeling the deterministic evolution of the physiological state from time, we now model the evolution of an ensemble of possible states by implementing EEG representation as a "information path , :r;1 or "Brain C .,,,... /.;.-,'.I The brain connectivity can be formulated as follows. Let G be an undirected graph with vertices V1,... V, where Vi represents electrode i. There is an edge (link) with the weight i, for every pair of nodes Vi and Vj corresponding to the connectivity of the brain dynamics between these two electrodes. The connectivity or synchronization can be viewed as been activated by interactions between neurons in the local circuitry underlying the recording electrodes. Figure 5-4 represents a hypothetical brain graph in which each connected path denotes the underlying connectivity. With this graph model, the attributes of C-SVM inputs are the pair-wised relation between two time series profiles rather than time stamps of a time series profile. In this context, the known that the orbitofrontal areas communicate with each other more than other parts of the brain. This has led us to the conclusion that the brain areas that are selected to be in the maximum clique are the vulnerable brain areas, rather than the epileptogenic areas. In other words, the brain areas) that are highly synchronized could be governed or manipulated by the epileptogenic areas so that they continuously show strong neuronal communication through the synchronization of EEG signals (measured by cross-mutual information). RL2 - RE I I II I1 kP3 I.I RLI I 1 IIIII ii ini LF23 Seizure Onset- 2F -i I ImIIziaions e tt RT3 - RT2 - RTI LT4 I s ai II oIll b - LT3 - LT2 LTI RD2- J- .D3 0 20 4- 60 SO 100 120 140 160 180 Figure 7-7. Electrode selection using the maximum clique algorithm for Case 1 7.4.4 Implications of the Results In normal brain functions, the orbitofrontal areas (both left and right) of the brain are highly synchronized active most of the time as it is considered to be the brain's executive function, and the temporal lobe areas are separated into left and right cortical hemispheres that work independently from each other. We hypothesized that this operation in the brain should be applied to the epileptic brains, even in the pre-seizure period. As we predicted, from the spectral partitioning results, both left and right orbitofrontal areas were also highly synchronized and active as well as right and left temporal lobe areas during the pre-seizure state. This -ir-'-i -;that the epileptogenetic Fpl Fp2 F F3 F4 F8 T3 C3 C4 T4 P3 P4 T5 T6 Oz r18 Channels Figure 5-1. Scalp electrode placement 5.2 EEG Data Information In this study, the dataset consists of continuous short-term (about 60 minutes) multi-channel scalp EEG recordings of 10 epileptic patients, 5 with medically intractable epilepsy (TLE) and 5 were seizure after treatment. The 19-32 channels scalp EEG recordings were obtained using standard 10-20 system, Nicolet BMSI 6000. Figure 5-1 shows the location of the electrodes on the scalp. Table 8-1 shows the EEG description from 10 subjects, EEG signals were recorded at sampling rate 250Hz. For consistency, we analyze and investigate EEG time series using bipolar electrodes only from 18 standard channels for every patient.EEG recordings from each subjects were inspected by certificated electroencephalographers. We randomly and uniformly sample two 30-second EEG epochs from each subject. Since EEG recordings were digitized at the sampling rate of 250 Hz, the length of each EEG epoch 7,500 points. where m is given as an integer and ry is a positive real number. The value of N is the length of compared subsequences in S, and rd specifies a tolerance level. d(xi,xj) max Si+k Sj+k (2-32) O d(xi, xj) represents the maximum distance between vectors xi and xj in their respective scalar components. n-m+l C7 rf (rf)= In (2-33) n-m- i=1 Finally the approximate entropy is given by: ApEn(m, rf, N) m(rf)- )m+l(rf). (2-34) The parameter rf corresponds to an a priori fixed distance between neighboring trajectory and r, is chosen according to the standard deviation estimated from data. Hence, rf can be viewed as a filtering level and the parameter m is the embedding dimension determining the dimension of the phase space. Heuristically, ApEn quantifies the likelihood that subsequences in S of patterns that are close and will remain close on the next increment. The lower ApEn value indicates that the given time series is more regular and correlated, and larger ApEn value means that it is more complex and independent. 2.8 Dynamical Support Vector Machine (D-SVM) The underlying dynamics of preictal transitions is changing from case to case, this requires analytical tools which is capable for identifying the changes in brain dynamics when preictal transitions take place. The detection performance is further improved by the dynamics support vector machine (D-SVM), a method developed from optimization theory and data mining techniques by utilizing dynamic features of EEG. D-SVM performs classification by constructing an N-dimensional hyper plane that separates the data into two different classes. The maximal margin classifier rule is used to construct the D-SVM. The objective of maximal margin D-SVM is to minimize the bond types of connection between two or more sources, and quantifying the synchronization between different brain areas (measured by different electrodes) is crucial to a greater understanding of the brain connectivity network. The synchronization may be attributable to the brain's anatomical, functional, or dynamical connectivity. In this study, the synchronization patterns are postulated to reflect the seizure evolution (epileptogenic process), and we shall use electrode synchronization as a similarity measure of EEG signals from different brain areas. This is fine in theory, however there are a few complexity issues in calculation of multivariate measures. First, in spite of the theoretical capability of multi-variate methods to calculate common patterns from several sources simultaneously, the calculation complexity increases exponentially with the number of sources. Therefore, we use multivariate measures for quantifying the synchronization from only 2 electrodes at a time. Specifically, a simple signal processing used to calculate the synchronization between electrode pairs is employ, -1 in this study. Then we apply a data mining technique based on network-theoretical methods to the multivariate analysis of EEG data. 7.3.2 Brain Synchronization In general, statistical similarity measures can be categorized into two groups: linear and nonlinear dependence measures. The linear measure is mainly used for measuring a linear relationship between two or more time series. For example, the most commonly used measure is cross-correlation function, which is a standard method of estimating the degree of correlation in time domain between two time series. The result of a cross correlation function can be calculated at different time lags of two time series to show the level of redundancy at different time points. Frequency coherence is another linear similarity measure, which calculates the synchrony of activities at each frequency [121]. Although the information from cross-correlation function and frequency coherence has been shown to be identical [122], the similarity between two EEG signals in different frequency bands such as delta, theta, beta, alpha and gamma, is still commonly used to investigate EEG 2.9 Statistical Distance ............... ........... .. 56 2.10 Cross-Validation .............. . . .. 57 2.11 Performance Evaluation of D-SVM .............. ...... 57 2.12 Patient Information and EEG Description ............. .. .. 58 2.13 Results ..................... ............ .... 58 2.14 Conclusions ............... .............. .. 59 3 QUANTITATIVE COMPLEXITY ANALYSIS IN MULTI-CHANNEL INTRACRANIAL EEG RECORDINGS FROM EPILEPSY BRAIN ................ .. 60 3.1 Introduction ............... . . .. 60 3.2 Patient and EEG Data Information .............. . .. 61 3.3 Proper Time Delay ............... . ..... 63 3.4 The Minimum Embedding Dimension ............. .. .. 65 3.5 Data Analysis ............... ........... .. 66 3.6 Conclusions ............... .............. .. 67 4 DISTINGUISHING INDEPENDENT BI-TEMPORAL FROM UNILATERAL ONSET IN EPILEPTIC PATIENTS BY THE ANALYSIS OF NONLINEAR CHARACTERISTICS OF EEG SIGNALS .................. ... 72 4.1 Introduction .................. ................ .. 72 4.2 Materials and Methods .................. .......... .. 75 4.2.1 EEG Description .................. ......... .. 75 4.2.2 Non-Stationarity. .................. ......... .. 76 4.2.3 Surrogate Data Technique .................. .. 77 4.2.4 Estimation of Maximum Lyapunov Exponent . . 78 4.2.5 Paired t-Test ............... ......... .. 79 4.3 Results ................... ......... .. ...... 80 4.4 Discussion ............... ............... .. 84 5 OPTIMIZATION AND DATA MINING TECHNIQUES FOR THE SCREENING OF EPILEPTIC PATIENTS. .................. ......... .. 91 5.1 Introduction ............... ................ .. 91 5.2 EEG Data Information ............. . . ... 93 5.3 Independent Component Analysis ................ .. 94 5.4 Dynamical Features Extraction ............... ... .. 95 5.4.1 Estimation of Maximum Lyapunov Exponent . . ... 95 5.4.2 Phase/Angular Frequency ..... ........... . .. 96 5.4.3 Approximate Entropy .................. ..... .. 97 5.5 Dynamical Support Vector Machine ................ . .. 98 5.6 Connectivity Support Vector Machine ............... . .. 100 5.7 Training and Testing: Cross Validation ............. . 102 5.8 Results and Discussions .................. ......... 102 the electroencephalographers can search for specific EEG configurations and link it to particular physiological states or neurological disorders. However, performing the visual inspection on long term EEG recordings is time consuming and requires continuously cautions from examiner. Inaccurate diagnosis could lead to severe consequences, especially in life-threatening conditions such as in emergency room (ER)or intensive care unit (ICU). There is currently no reliable tool for rapid EEG screening that can quickly detect and identified the abnormal configurations in EEG recordings. There is a need for developing a reliable technique which would serve as an initial medical diagnosis and prognosis tool. SVM has been successfully implemented for biomedical research on analyzing very large data sets. Moreover SVM has been recently applied for the use of epileptic seizure prediction and it has been shown to achieved 71' sensitivity and 7'- specificity for EEG recordings from 3 patients [82]. Nurettin Acir and Cuneyt Guzelis introduced a two-stage procedure SVM for the automatic epileptic spikes detection in a multi-channel EEG recordings [83]. Bruno Gonzalez-Velldnet et al., reported it is possible to detect the epileptic seizures using three features of the electroencephalogram (EEG), namely, energy, decay (damping) of the dominant frequency, and cyclostationarity of the signals [84]. Along with this directions, the abnormal EEG identification problem can be modeled as binary classification problem i.i, i or abnormal ". Embedded with neuron network and connectivity concepts we first proposed and described an application of connectivity support vector machine C-SVM, C-SVM is based on network modeling concepts and connectivity measures to compare the EEG signals recorded from different brain regions. A detail flow chart of the proposed C-SVM framework is given in Figure ??. We also uses three dynamical features of EEG 1. Angular frequency 2. Approximate entropy 3. Short-term largest lyapunov exponent to conduct the dynamical SVM in the second part of this study. [110] M.C. Mackey and L. Glass. Oscillation and chaos in physiological control systems. Science, 4300:287-289, 1977. [111] H. Petsche and M.A.B. Brazier. Synchronization of EEG A i.:;.:I; in Epilepsies. Springer, 1972. [112] L.D. Iasemidis, K.E. Pappas, R.L. Gilmore, S.N. Roper, and J.C. Sackellares. Preictal entrainment of a critical cortical mass is a necessary condition for seizure occurrence. Epilepsia, 37S(5):90, 1996. [113] L.G. Dominguez, R.A. Wennberg, W. Gaetz, D. C'!, ine, O. Snead, and J.L.P. Velazquez. Enhanced synchrony in epileptiform activity? local versus distant phase synchronization in generalized seizures. The Journal of Neuroscience, 25(35):8077-8084, 2005. [114] M.L. Anderson. Evolution of cognitive function via redeployment of brain areas. The Neuroscientist, 13(1):13-21, 2007. [115] O. Sporns and R. Ktter. Motifs in brain networks. PLoS B:.. 4. ./;, 2(11):e369, 2004. [116] H.H. Jasper. Mechanisms of propagation: Extracellular studies. In H.H. Jasper, A. A. Ward, and A. Pope, editors, Basic mechanisms of the epilepsies, pages 421-440, Boston, 1969. Little Brown. [117] G.J. Ortega, L. Menendez de la Prida, R.G. Sola, and J. Pastor. Synchronization clusters of interictal activity in the lateral temporal cortex of epileptic patients: Intraoperative electrocorticographic analysis. Epilepsia, 2007. [118] A. Brovelli, M.Z. Ding, A. Ledberg, Y.H. C'!I. i, Richard Nakamura, and Steven L. Bressler. Beta oscillations in a large-scale sensorimotor cortical network: Directional influences revealed by granger causality. Proceedings of the National A .,./. ,,i of Sciences of the United States of America, 101:9849-9854, 2004. [119] N.K. Varma, R. Kushwaha, A. Beydoun, W.J. Williams, and I. Drury. Mutual information analysis and detection of interictal morphological differences in interictal epileptiform discharges of patients with partial epilepsies. Electr ',,' *Jl,,rl. ir Clin N. ;', ,1/;, ;,.: 41 103(4):426-33, 1997. [120] S. Ken, G. Di Gennaro, G. Giulietti, F. Sebastiano, D. De Carli, G. Garreffa, C. Colonnese, R. Passariello, J. Lotterie, and B. Maraviglia. Quantitative evaluation for brain ct/mri coregistration based on maximization of mutual information in patients with focal epilepsy investigated with subdural electrodes. Magn Reson Imaging, 25(6):883-8, 2007. [121] W.H. Miltner, C. Braun, M. Arnold, H. Witte, and E. Taub. Coherence of gamma-band eeg activity as a basis for associative learning. Nature, 397:434-436, 1999. 00 oCO -H 0-H 0 *z a -H -H 00 ,' z- c .z *zo z r~- r~- z - .t00- L~ .~ .~1 .Z~ * .z *z~dd~ *z *zd~ *z* z*z* dd d d d d .z * z-H--H --H I^ o-IHO !H, O ^ OO u- O I O O O C Q .Cl l Cl tH HC H_ H .z *z z *z *z *z*z *z z *z *z * ~d~d~d~d~~HHdHH~HH~dHH~HH oo o~o C~d d $I cucu C~d "$I o i'c~3 ~id "$I ,-d d ^ d o LQ dS o cu ?o CHAPTER 6 SPATIO-TEMPORAL EEG TIME SERIES ANALYSIS 6.1 Introduction The degree of synchronization is an important indication of how information is processing in the brain. The quantification of neuronal synchronization has been investigated using different approaches, from linear cross-correlation to phase synchronization or advanced dynamical interdependence analysis. These synchronization tools have also been applied to EEG recordings and have been shown to be able to detect increased in synchronization measures prior to the seizures [89]. Synchronization can be quantified in both space and time domain. For a multi-variate system, understanding the interactions among its various variables, whose behavior can be represented along time as time-sequences, presents many challenges. One of the key aspects of highly synchronized systems with spatial extent is their ability to interact both across space and time, which complicates the ain i,~-i-; greatly. In biological systems such as the central nervous system, this difficulty is compounded by the fact that the components of interest have nonlinear complicated dynamics that can dictate overall changes in the system behavior. The exact figure of how to quantify the information exchanges in a system remains ambiguous. Studies on multi-variate time series analysis have resulted in development of a wide range of signal-processing tools for quantification of synchronization in systems. However, the general consensus on how to quantify this phenomenon is largely uncertain. In the literature, synchronization between variables can be categorized as identical synchronization, phase synchronization and generalized synchronization. In the following chapters, I undertake an in-depth analysis of preictal and interictal synchronization behavior, focusing on EEG recordings from patients with temporal lobe and generalized seizures. -1 ^ -H - ~ ~*- - L~ *X9 .z *z *z HHHH I HH 00 0 - dO dI d .z * z-Hz-H Od d d ad ^'^Mi ii o ,= 00~C0000 0d0d~d~d~d~d "-H0^0^0^0^0^ c~3 - r3*- .z *z~ u5CUO- r~*- c' .z *z z *z *z * d~~~~ddd cM 0 $-I 0 fa CUOO~n~0 z .z *z *z *z *zC~ dHHZHHdHHdHHdHH .zo* d dd 0I -H - *z s -d-d -0 -H 0z 0z ^^~3 0+1+ .z * z-Hz-H rC*- r\- .z *zd z-Hz-H .z * z-Hz-H .z *zO z-Hz- C\1 C~C~3 Od "$I c~sLn Od "$I ooo C~3d dS( C _ 0- 0 H u0 C:l S 0 - C\ 90 a 0- ^C ~'Ln C~3C\1 Lnd dSI cu o~cu ~id d $I cu c~scu d $I 0 ,- 2e a 3 us oO~ d $I us C~d d $I 00~3 C~d d $I ~300 d d $I C\1 LnC\I C~d d $I o ooc~s C~3d d $I oo oo~i ~id dS( C\1 C~C\I C~d d $I cu cuc~s C~d dS( -0 a ^ dS( C~3 ~CU "d dS( c: c: $I CHAPTER 2 EPILEPSY AND NONLINEAR DYNAMICS 2.1 Introduction The beginning and termination of epileptic seizures reflect intrinsic, but poorly understood properties of the epileptic brain. One of the most challenging tasks in the field of epilepsy research has been remained the search for basic mechanism that underlies seizures. Traditional studies into the seizure activity have focus upon neuronal apparatuses such as neurotransmitters, receptors or specific ionophores. However, a seizure involves large portions of the cerebral cortex, therefore, it is likely that investigation into the epileptic brain as a system will elucidate important greater information than traditional approaches. The development of preictal transitions can be considered as a sudden increase of synchronous neuronal firing in the cerebral cortex that may begin locally in a portion of one cerebral hemisphere or begin simultaneously in both cerebral hemisphere. By observing the occurrence of epileptic seizures, it is reasonable to believe that there are multiple states exist in a epileptic brain and the sequences of the states are not deterministic. The preictal transitions are detectable EEG dynamical changes by applying methods developed from nonlinear dynamics. Several groups have reported that seizures are not sudden transitions in and out of the abnormal ictal state; instead, seizures follow a certain dynamical transition that develops over time [32-39] see [40; 41] for review. In an study of Pijn et al. in 1991, authors were able to demonstrate decrease in the value of correlation dimension at seizure onset in the rat model. In early 1990s, Iasemidis et al., first estimated the largest Lyapunov exponent and reported seizure was initiated detectable transition period by analyzing spatiotemporal dynamics of the EEG recordings; this transition process is characterized by: (1) progressive convergence of dynamical measures among specific anatomical areas dynamicall entrainment "and (2) following the overshot brain resetting mechanism during post ictal state. Martinerie et al., (1998) Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy BRAIN DYNAMICS, SYSTEM CONTROL AND OPTIMIZATION TECHNIQUES WITH APPLICATIONS IN EPILEPSY By C('!h ig-Chia Liu August 2008 ('C! ,i: Panagote M. Pardalos Major: Biomedical Engineering The basic mechanisms of epileptogenesis remain unclear and investigators agree that no single mechanism underlies the epileptiform activity. Different forms of epilepsy are probably initiated by different mechanisms. The quantification for preictal dynamic changes among different brain cortical regions have been shown to yield important information in understanding the spatio-temporal epileptogenic phenomena in both humans and animal models. In the first part of this study, methods developed from nonlinear dynamics are used for detecting the preictal transitions. Dynamical changes of the brain, from complex to less complex spatio-temporal states, during preictal transitions were detected in intracranial electroencephalogram (EEG) recordings acquired from patients with intractable mesial temporal lobe epilepsy (\ TLE). The detection performance was further enhanced by the dynamics support vector machine (D-SVM) and a maximum clique clustering framework. These methods were developed from optimization theory and data mining techniques by utilizing dynamic features of EEG. The quantitative complexity analysis in multi-channel intracranial EEG recordings is also presented. The findings -,.;; -1 that it is possible to distinguish epilepsy patients with independent bi-temporal seizure onset zones (BTSOZ) from those with unilateral seizure onset zone (ULSOZ). Furthermore, for the ULSOZ patients, it is also possible to identify the location of the seizure onset zone in the brain. Improving clinician's certainty in identifying the Z - -~ -~ C .z *z~ z-Hz- - .z *zs z-Hz-H .z *z z-Hz-H ZHH .z *z zd d 0+1+1 CO3 z ZHH .z *z z-Hz-H ZHH o, cuo C~d d $I oocu Id "$I - 0 0-00 0 0-] 0 -0 00 0 0] 0 0 0 -- .zH0^ *z *z z z z z z z z * * ~~~~~~~00~~ ~ ~ ~ ~ HH~HHHH~~HH~0HH .z *z *z *z * ~HHZHH3HHUHH~HH So * zc ZHHd *~s r s~ i1 .z *z *z *z *z~s CHHZHH~HHcHHcHH us z1 ZHH 0 o H2 -H: H -H -H -H "I (0 o, $I s I-- oo o "d o, ?o r1 ?s 45 40 35 30 25 20 15 10 5 0 L 0 5 10 15 20 time (s) Figure 2-8. Z component of Lorenz system 25 30 35 4- Sp-value = 0.9955 0 0- i-- 5, ', 1 2 3 Patients ,, A Figure 4-10. Nonlinearities across recording areas during interictal state for BTSOZ 1 2 3 patientsPatients Figure 4-10. Nonlinearities across recording areas during interictal state for BTSOZ patients in patients with ULSOZ. Further studies on a larger sample of patients to validate these results are warranted. Success of this study will provide more much-needed information to guide electroencephalographer and clinician to improve the likelihood of successful surgery. Table 3-1. Patients and EEG data statistics for complexity analysis Patient # Gender Age Focus (RH/LH) Length of EEG (hr.) Number of seizure P1 M 19 RH 20h 37m 05s 3 P2 M 33 LH 09h 43m 57s 3 transverse and B lateral views of the brain, illustrating the depth and subdural electrode placement for EEG recordings are depicted. Subdural electrode srips are placed over the left orbitfrontal (LOF), right orbitofrontal (ROF), left subtemporal (LST), and right subtemporal (RST) cortex see Figure 3-1. The EEG recording data for epilepsy patients were obtained as part of pre-surgical clinical evaluation. They had been obtained using a Nicolet BMSI 4000 and 5000 recording system, using s 0.1 Hz high-pass and a 70 hz low pass filter. Each recording contains a total number of 28 to 32 intracranial electrodes (8 subdural and 6 hippocampal depth electrodes for each cerebral hemisphere). Prior to storage, the signals were sampled at 200Hz using an analog to digital converter with 10 bits quantization. The recordings were stored digitally onto high fidelity video type. Two epilepsy subjects (see Table 3-1) were included in this study. (A) (B) ROF 3 2 3 LOF LTD RST LST LOF LST RTD LTD Figure 3-1. Electrode placement 5. Infections: Infections of the nervous system may result in seizure activity. These include infection of the covering of the brain and the spinal fluid (meningitis), infection of the brain (encephalitis), and human immunodeficiency virus (HIV) and related infections. 6. Tumors: Cancerous (malignant) and benign brain tumors may be associated with seizures. The location of the lesion influences the risk. 7. Cerebral palsy: Epilepsy is often a symptom of cerebral palsy, which results from lack of oxygen, infection, or trauma during birth or infancy. 8. Febrile seizures: Febrile seizures occur in small children and are caused by high fever. 1.3 Classification of Epileptic Seizure (ICES 1981 revision) The ICES (Commission on Classification and Terminology of the ILAE, 1981), The ICES recognizes 18 subclasses of simple focal seizures belonging to four groups, four modalities of complex focal seizures, and three types of secondarily generalized seizures. 1.3.1 Partial Onset Seizures Partial onset seizures are those in which, in general, the first clinical and EEG changes indicate initial activation of a limited group of neurons. A partial seizure is classified primarily on the basis of whether or not consciousness is impaired during the attack. When consciousness is not impaired, the seizure is classified as a simple focal seizure. When consciousness is impaired, the seizure is classified as complex focal seizures. In patients with impaired consciousness, aberrations of behavior (automatisms) may occur. A partial onset seizure may not terminate, but instead progress to a generalized motor seizure. Impaired consciousness is defined as the inability to respond normally to exogenous stimuli by virtue of altered awareness and/or responsiveness. There is considerable evidence that partial onset seizures usually have unilateral hemispheric involvement and only rarely have bilateral hemispheric involvement; complex partial onset series acquired from electrode x is presented by electrode y and vice versa. Let X be the set of data points where its possible realizations are x, x2, x3, ..., x~ with probabilities P(xI), P(2), P(x3),...P(x,). The Shannon entropy H(X) of X is defined as Mutual information has been applied for measuring the interdependency between two time series. Many previous studies have shown its superior performance over the traditional linear measures [99-104]. Kraskov et al., 2004 introduced two classes of improved estimators for mutual information M(X, Y) from samples of random points distributed according to some joint probability density p(x, y). In contrast to conventional estimators based on histogram approach, they are based on entropy estimates from k nearest neighbour distances. Let us denote the time series of two observable variables as X = {xi} and Y ={yj}N1, where N is the length of the series and the time between consecutive observations (i.e., .',,1i].:,u. period) is fixed. Then the mutual information is given by: ( P((xi) ,yj) (66) i j where px(i) = -dx, p,(i) = -dy and p(i,j) / p(x, y)dxdy (6-17) "J "denotes the integral over bin i. If nx(i) and ny(j) are the number of data points in the ith bin of X and jth bin of Y; n(i,j) is the number of data points in the intersection bin (i,j). The probabilities are estimated as px(i) n (i)/N, px(j) n(j)/N and p(i,j) a nx(j)/N. Rather then bin approach the mutual information can be estimated from k-nearest neighbor statistics. We first estimate H(X) from X by H(X) t N P(X xi). (6-18) i= 1 Figure 5-3. Support vector machines input of C-SVM is the degree of connectivity between different brain regions. Given n time series data points, each with m time stamps, the proposed framework will decrease the number attributes by 2(n 1)/m times. Let I be the total number of data points, the dimensionality can be reduced from A E Rlxnxm to A E cRx -). The connectivity among the 18 EEG channels is calculated for each sample as shown in Figure 5-4. The connectivity are Euclidean Distance-based between channel i and channel j, i / j, for i,j = 1,..., 18. For example: let Ci and Cj denote EEG time series from channel i and channel j, respectively. Each epoch of time series has length 30 seconds, which is equal to 7,500 points. So the size of vector C, (or Cj) is 7,500. The connectivity between C, and Cj using Euclidean distance we obtain: 7500 E Uij k 1 7500 ,and 18 x 17 connectivity profiles for each sample. Thus, the C-SVM transforms each EEG time series sample into this with (18 x 17) number of attributes. C-SVM largely reduces number of attributes from m = (18 x 7500) to m = (18 x 17) and also saves memory resources and computational time. t = 0 t = oo), the structure of the trajectory (path) will shrink for a dissipative system. For a dissipative system, after a sufficient long time,the number of variables d used to describe state space reduced to a small set of A. This set of state variable is called an Ii ,1 II i .i ". An attractor can be classified into one of the following four different categories: 1. Saddle point: For any given initial conditions, after a sufficient long time, the solution may converge to the same final state (fixed point). An example for this attractor is a constant series. x(t) = x(0), t oo (2-1) 2. Limit cycle: Instead of converging to a fixed point the dynamical system may converge to a set of states, which are visited periodically. A limited cycle attractor is a closed trajectory through state space. x(t) x(t + T), (2-2) where T denotes the period of this cycle. 3. Limit tori: A limit tori attractor is a limit cycle attractor with multiple period. This attractor will no longer be closed and limited cycle becomes a limit torus. 4. Strange attractor: The existence of this type of attractor was unknown until the development of nonlinear dynamics. A strange attractor is defined as an attractor that shows sensitivity to initial conditions (exponential divergence of neighboring trajectories), it may appear to be stochastic in time domain. A strange attractor exhibits regular structure in the phase space (See Figure 2-1, 2-5 for R6ssler and Lorenz attractor). Recall an attractor is a set of state variables; geometrically an attractor can be a point, a curve, a manifold, or even a complicated set with a fractal structure known as the -li i i,.- attractor". Describing these attractors has been one of the achievements of chaos 7.4.3 Maximum Clique Algorithm In this section, we discuss the results from analyzing the structural property of the brain network using the maximum clique approach. As mentioned earlier, the idea of applying the maximum clique technique is different from the one using the spectral partitioning approach as we are only interested in the most highly synchronized group of electrodes in the brain network. We adopted the algorithm to find a maximum clique in the brain connectivity graph after deleting the insignificant arcs in the original complete graph as follows: Let G = G(V, E) be a simple, undirected graph where V = {1,..., n} is the set of vertices (nodes), and E denotes the set of arcs. Assume that there is no parallel arcs (and no self-loops joining the same vertex) in G. Denote an arc joining vertex i and j by (i, j). We define a clique of G as a subset C of vertices with the property that every pair of vertices in C is connected by an arc; that is, C is a clique if the subgraph G(C) induced by C is complete. Then, the maximum clique problem is to find a clique C with maximum cardinality (size) IC|. The maximum clique problem can be represented in many equivalent formulations (e.g., an integer programming problem, a continuous global optimization problem, and an indefinite quadratic programming). In this paper, we represent it in a simple integer programming form given by We analyzed 3 epochs of 3-hour EEG recordings, 2 hours before and 1 hour after a seizure, from Patient 2 who had the epileptogenic areas on both right and left mesial temporal lobes. Figures 7-7 and 7-8 demonstrate the electrode selection of the maximum clique group during two hours before and one hour after the seizure onset. During the period before the seizure onset, both figures manifested a pattern where all the LD electrodes were consistently selected to be in the maximum clique. During the seizure onset, the size of the maximum clique increases drastically. This is very intuitive because, in temporal lobe epilepsy, all of the brain areas are highly synchronized. We visually inspected the raw EEG recordings before and during the seizure onsets and found a similar semiological pattern of the seizure onset electrodes from the L(T)D areas z (to x(I n) end of data set z (I)" z1Q "fiduciry" trajectory Figure 2-9. Estimation of Lyapunov exponent (Lmax) and duration T. If Dt is the sampling period, then T = (N- 1)- Dt At (p- 1). -. (2-26) If the evolution time At is given in second, then the unit of L is bit/sec. The selection of p is based from Takens' embedding theorem and was estimated from epoches during ictal EEG recordings. Takens' embedding theorem is defined: f(t) A (x(t), x(t + ),...x(t + 2n T))T, (2-27) using the above defined fx, even if one only observes one variable x(t) for t oo, one can construct an embedding of the system into a p = 2m + 1 dimensional state space. The dimension of the ictal EEG attractor is found between 2 to 3 in the state space. Therefore according to Takens' the embedding dimension would be at least p = 2 3 + 1 7. The selection of '- is chosen as small as possible to capture the highest frequency component in the data. 2.6 Phase/Angular Frequency Phase/ angular frequency ,max estimates the rate of change of the stability of a dynamical system. Thus, it complements the Lyapunov exponent, which measures the local stability of the system. The difference in phase between two evolved states X(ti) and 2 -1 -2 -3 -4 -5 0 5 10 15 20 time (s) Figure 2-3. Y component of Rbssler system 25 30 35 Figure 8-1. Nonlinear interdependences for electrode FP1 before and after treatment. Although the results indicate that the mutual information and nonlinear interdependencies measures could be useful in determining the treatment effects for patients with ULD. To prove the usefulness of the proposed study, a larger patient population is needed. The approaches in this study are a bivariate measures, since a multivariate measure is not easy to model and has not been resolved. The decoupling between frontal and occipital cortical regions may be caused by decreased driving force deep inside the brain. In other words, the effect of the treatment may reduce the couple strength between thalamus and cortex in ULD subjects. Nevertheless, the limitations must be mentioned, it has been reported that it is necessary to take into account the P3 Before After Before After Before After above results -ii--.- -1 the existing the treatment effects on the coupling strength and directionality of information transport between different brain cortical regions. Table 8-2. Topographical distribution (DE)) Electrode DE for P1 (1) Fpl F3, C4, P4, F7 T4, T5, 01 (2) Fp2 C3, C4, F8, T4 T5, Pz (3) F3 Fpl, C4, P4, 02 (4) F4 C4, P4 02 (5) C3 Fp2, C4, P3, 01 (6) C4 Fpl, Fp2, F3, F4, C3 Al N/A A2 N/A (7) P3 C3, 02 (8) P4 Fpl, F3, F4 (9) 01 Fpl, C3 (10) 02 F3, F4, P3 (11) F7 Fpl, Fz, Pz (12) F8 Fp2 (13) T3 NONE (14) T4 Fpl, Fp2, Cz (15) T5 Fpl, Fp2 (16) T6 NONE (17) Fz F7 (18) Pz Fp2, F7, Cz (19) Cz T4, Pz for treatment decoupling effect (DE: Decouple Electrode DE for P2 Fp2, F3, F8, T5 Fpl, F4, T6, 02 Fpl, C3, P3, Pz Fp2, P4, 02, Fz F3, P3, 01 P4, T6, 02 N/A N/A F3, C3, T3, Pz F4, C4 C3, T5, T3, F7 Fp2, F4, C4 C3, T5, 01 Fpl, C4, P4 P3, 01 Fp2, 01 Fp2, C4 C4 Fpl, F4 F3, P3, Cz Pz DE for P3 F3, F7 F8 Fpl Cz P3, 01 P4, 02 N/A N/A 01 C4 P3, Pz C4 Fpl Fp2 NONE NONE NONE NONE Cz 02 F4. Fz DE for P4 F3, P3, Fz, T5 F8, T4, Fz C3, C4, T5, P4 Cz Fpl, F7 C4, Fz Fp2, P3, 01 Fpl, Fp2, F3 N/A N/A Fpl, C3, Cz Fp2, C3 Pz F3 Fp2, NONE Fp2, Fp2, Cz NONE Fpl, Fp2, F3 02 Fp2, T5, P3 8.5 Conclusion and Discussion The effectiveness of the new AEDs is currently accessed using myoclonus severity with the UMRS. As mentioned above, it is not easy to perform such evaluation scheme precisely especially in the later stages of the disease. Furthermore, the UMRS is a skewed measure that may not detect functional changes in a patient when these changes may be clinically important. The outcome from UMRS in this study did not evaluate the severity of the patients accurately, as the P1 was with less severity of ULD determined by the clinical experienced neurologist. The present study measure the synchronization behaviors and nonlinear interdependences, in a straightforward manner, in the cortical network during similarity patterns [123; 121]. For example, [124] used frequency coherence measures to investigate the interactions between medial limbic structures and the neocortex during ictal periods (seizure onsets). In another study by [125], the coherence pattern of cortical areas from epileptic brain was investigated to identify a cortical epileptic system during interictal (normal) and ictal (seizure) periods. Although linear measures are very useful and commonly used, they are insensitive to nonlinear coupling between signals, and non-linearities are quite common in neural contexts. To be able to investigate more of the interdependence between EEG electrodes, nonlinear measures should be applied. Nonlinear measures have been widely used to determine the interdependence among EEG signals from different brain areas. For example, [106] and [35] studied the similarity between EEG signals using nonlinear dynamical system approaches. They applied a time-delay embedding technique to reconstruct a trajectory of EEG in phase space and used the idea of generalized synchronization proposed by [126] to calculate the interdependence and causal relationships of EEG signals. We propose an approach to investigate and quantify the synchronization of the brain network, specifically tailored to study the propagation of epileptogenic processes. [127] investigated this propagation, where the average amount of mutual information during the ictal period (seizure onset) was used to identify the focal site and study the spread of epileptic seizure activity. Subsequently, [128] applied the information-theoretic approach to measure synchronization and identify causal relationships between areas in the brain to localize an epileptogenic region. Here, we apply an information-theoretic approach, called cross-mutual information, which can capture both linear and nonlinear dependence between EEG signals, to quantify the synchronization between nodes in the brain network. In order to globally model the brain network, we represent the brain synchronization network as a graph. Figure 2-11. Phase/Angular frequency of Lyapunov exponent (max) in the human neonate and in epileptic activity in electrocardiograms (Diambra, 1999) [63]. Mathematically, as part of a general theoretical framework, ApEn has been shown to be the rate of approximating a Markov chain process [62]. Most importantly, compared ApEn with Kolmogrov-Sinai (K-S) Entropy (Kolmogrov, 1958), ApEn is generally finite and has been shown to classify the complexity of a system via fewer data points via theoretical analysis of both stochastic and deterministic chaotic processes and clinical applications [62; 64-66]. Here I give brief description about ApEn calculation for a time series measured equally in time with length n. Suppose S = s, a2, ..., s is given and use the method of delay we obtain the delay vector xl, x, ..x, xn-m+l in R': Xi Si, Si+l, ., Si+m-l, (2-30) C,(r) number ofxjsuch thatd(x xj) < rf c (r) (2-31) NV- m+1 epileptogenic focus will increase the chances for better outcome of epilepsy surgery in patient with intractable MTLE. Recent advances in nonlinear dynamics performed on EEG recordings have shown the ability to characterize changes in synchronization structure and nonlinear interdependence among different brain cortical regions. Although these changes in cortical networks are rapid and often subtle, they may convey new and valuable information that are related to the state of the brain and the effect of therapeutic interventions. Traditionally, clinical observations evaluating the number of seizures during a given period of time have been gold standard for estimating the efficacy of medical treatment in epilepsy. EEG recordings are only used as a supplemental tool in clinical evaluations. In the later part of this study, a connectivity support vector machine (C-SVM) is developed for differentiating patients with epilepsy that are seizure free from those that are not. To that end, a quantitative outcome measure using EEG recordings acquired before and after anti-epileptic drug treatment is introduced. Our results indicate that connectivity and synchronization between different cortical regions at higher order EEG properties change with drug therapy. These changes could provide a new insight for developing a novel surrogate outcome measure for patients with epilepsy when clinical observations could potentially fail to detect a significant difference. False positive (FP): False positive answers denoting incorrect classifications of negative cases into the positive cases; A classification result is considered to be true positive if the D-SVM classify a interictal EEG epoch as a preictal EEG sample. False negative (FN): False negative answers denoting incorrect classifications of positive cases into the negative cases; A classification result is considered to be true positive if the D-SVM classify a preictal EEG epoch as a interictal EEG sample. The performance of the D-SVM is evaluate using sensitivity and specificity: Sensitivity (TPFN) Specificity TNP The sensitivity can be interpreted as the probability of accurately classifying EEG epochs in the positive case. Specificity can be consider as the probability of accurately classifying EEG epochs in the negative class. In general, one alv-wb wants to increase the sensitivity of classifiers by attempting to increase the correct classifications of positive cases (TP). On the other hand, false positive rate can be considered as (1 specificity) which one wants to minimize. 2.12 Patient Information and EEG Description The information of the patients and the EEG recording are summarized in the table below. For each patient, we randomly selected 200 epochs from interictal state, each epoch is 10.24 seconds long in duration as input to D-SVM classification scheme. The interictal and ictal state is defined as: 1. interictal state: 1 hour away from ictal state 2. preictal state: 5 minutes data length prior to ictal state 2.13 Results The results of this study indicate that D-SVM can correctly the detect preictal state with high sensitivity and specificity. For the patients with bi-lateral seizure onset zone the performance of D-SVM is better than those with uni-lateral seizure onset zone. The activated simultaneously. Low EEG frequency indicates less responses of the brain, such as sleep, whereas higher EEG frequency implies the increased alertness. Given the above descriptions, an acquired EEG time series can be defined as a record of the fluctuating brain activity measured at different times and spaces. The high degree of synchronicity for two different brain regions implies strong connectivity among them and vice versa. We will interchangeably use the terms synchronicity and connectivity for rest of the chapter. Although the brain may have originally emerged as an organ with functionally dedicated regions, recent evidence -.-. -: --I that the brain evolved by preserving, extending, and re-combining existing network components, rather than by generating complex structures de novo [114; 115]. This is significant because it -ii:--. -I- (1) the brain network is arranged such that the functional neural complexes supporting different cognitive functions share many low-level neural components, and (2) the specific connection topology of the brain network may pl i, a significant role in seizure development. This line of thinking is also supported by [70], which demonstrates that specific connected structures are either significantly abundant or rare in cortical networks. If seizures evolve in this fashion, then we should be able to make some specific empirical hypotheses regarding the evolution of seizures, that might be borne out by investigating the synchronization between the activity in different brain areas, as revealed by quantitative analysis of EEG recordings. The goal of this study is to test the following two hypotheses. First, we should expect the brain activity in the orbitofrontal areas are highly correlated while the activity in the temporal lobe and subtemporal lobe areas are highly correlated with their own side (left only or right only) during the pre-seizure period. The high correlation can be viewed as a recruitment operation initiated by an epileptogenic area through a regular communication channel in the brain. Note that the connection of these brain areas has been a long-standing principle in normal brain functions and we believe that the same principle should hold in the case of epilepsy as well. Second, we should expect some brain regions to be consistently active, manifested the first local minimum of I(Q, S), rather than some subsequent minimum, should probably be chosen for the sampling interval T,. 3.4 The Minimum Embedding Dimension Dynamical systems processing d degree of freedom which may choose to proceed on a manifold of much lower dimension, so that only small portions of the degrees of freedom are actually active. In such case it is useful to estimate the behaviors of degrees of freedom over a period of time, and it is obvious that this information can be obtained from that dimension of attractor from the corresponding system. If one chooses the embedding dimension too low this results in points that are far apart in the original phase space being moved closer together in the reconstruction space. Takens delay embedding theorem states that a pseudo-state space can be reconstructed from infinite noiseless time series (when one choose d > 2dA) is often been used when reconstructing the delay vector [52]. There are several classical algorithms used to obtain the minimum embedding dimension [76; 74; 77]. The classical approaches usually require huge computation power and vast among of data. Another limitation of these algorithms is that they usually subjective to different types of data.We evaluated the minimum embedding dimension of the attractors from the EEG by using Caos method. The notions here followed "Practical method for determining the minimum embedding dimension of a scalar time series". Suppose that we have a time series (x, x2, X3, ..., N). Applying the method of delay we obtain the time delay vector as follows: yi(d) = (xi,Xi+ ,..., Xi+(d+l)-), i = 1,2,...,N (d 1); (3-6) where d is the embedding dimension and T is the time-delay and yi(d) means the ith reconstructed vector with embedding dimension d. Similar to the idea of the false nearest neighbor method, defining II y(d + 1) n(i,d)(d + 1) || a(, d)=) id)(d i ,2,...,N dr (3-7) II yi(d) Yn(i,d)(d) 11 1111 R.A RLu R1. E.2 - RLI RF RF3 RF2 RFI LFN LF3 LF2 LF -I RtT RT3 RT2 RTI - LT - RDI - RD - R D R D2 LDII LDW LTi1 LD3 - LDI I Minutes Figure 7-8. Electrode selection using the maximum clique algorithm for Case 2 Figure 7-9. Electrode selection using the maximum clique algorithm for Case 3 processes slowly develop themselves through a regular communication channel in the brain network, rather than abruptly disrupt, collapse, or change the way brains communicate. From this observation, we postulate that this phenomenon may be a reflection of neuronal recruitment in seizure evolution. This observation confirms our first hypothesis. In Seizure Onset- r A I Il III ..1m iiI I I I I I I I addition, we have found that nodes in the brain network are clustered during the seizure evolution. Most brain areas seem to be communicating with their physiological neighbors during the process. The key process of seizure evolution could be the step where the epileptogenic areas) govern or manipulate the other vulnerable, or easily synchronized, brain areas to communicate with their neighbors. This can be viewed as a recruitment of other brain areas done by the epileptogenic areass. In most cases, the recruitment of seizure development should start with a weaker group, which in our case is represented by a vulnerable brain area. After enough neurons have been recruited, the disorders of epileptic brains spread out abnormal functions from than localized areas of cortex or other vulnerable areas throughout the cortical networks and the entire brain network. This phenomenon was shown by the results of our maximum clique approach, which confirms our second hypothesis. In addition, a different type of maximum clique patterns may be useful in the identification of incoming seizures. This study .i i.-- -1 that, in the future, this framework may be used as a tool to provide practical seizure interventions. For example, one can locate and stimulate the brain areas that seem to be vulnerable to the seizure evolution by electrical pulses through the monitoring process of the maximum clique. This will drastically reduce the risk of seizure to epilepsy patients. 7.5 Discussion and Future Work In this study, we attempted to study seizure evolution by investigating some neuronal interactions among different brain areas. Analyzing multidimensional time series data like multichannel EEG recordings is a very complex process. The study of the brain network needs to involve the neuronal activities from not only a single source or a small group of sources, but also the entire brain network. Here we applied the cross-mutual information technique, a measure widely used in the information theory, to capture the neuronal interactions through the brain's synchronization patterns. Then we modeled the global interactions using network/graph-theoretic approaches, spectral partitioning and maximum clique. These approaches are used to generalize the brain network investigation 3-2 Average minimum embedding dimension profiles for Patient 1 (seizure 1) . 67 3-3 Average minimum embedding dimension profiles for Patient 1 (seizure 2) . 68 3-4 Average minimum embedding dimension profiles for Patient 1 (seizure 3) . 69 3-5 Average minimum embedding dimension profiles for Patient 2 (seizure 4) . 70 3-6 Average minimum embedding dimension profiles for Patient 2 (seizure 5, 6) 71 4-1 32-channel depth electrode placement ............... .... 76 4-2 Degree of nonlinearity during preictal state .................. .. 81 4-3 Degree of Nonlinearity during postictal state ................ 82 4-4 STLmax and T-index profiles during interictal state ............... ..83 4-5 STLma and T-index profiles during preictal state ................ ..84 4-6 STLmax and T-index profiles during postictal state ............... ..85 4-7 Nonlinearities across recording areas during interictal state for ULSOZ patients 86 4-8 Nonlinearities across recording areas during perictal state for ULSOZ patients 86 4-9 Nonlinearities across recording areas during postictal state for ULSOZ patients. 87 4-10 Nonlinearities across recording areas during interictal state for BTSOZ patients 88 4-11 Nonlinearities across recording areas during preictal state for BTSOZ patients 89 4-12 Nonlinearities across recording areas during postictal state for BTSOZ patients 90 5-1 Scalp electrode placement .................. ........... .. 93 5-2 EEG dynamics feature classification .................. ...... .. 99 5-3 Support vector machines .................. ............ .. 101 5-4 Connectivity support vector machine .................. ..... 102 7-1 EEG epochs for RTD2, RTD4 and RTD6 (10 seconds) . . ..... 126 7-2 Scatter plot for EEG epoch (10 seconds) of RTD2 vs. RTD4 and RTD4 vs. RTD 6 ......................... ...... .. ... . 126 7-3 Cross-mutual information for RTD4 vs. RTD6 and RTD2 vs. RTD4 ..... ..127 7-4 Complete connectivity graph (a); after removing the arcs with insignificant connectivity (b) ... .... ..... ... .................. ..... 128 7-5 Spectral partitioning .................. .............. .. 129 * Temporal depth (LTD & RTD)) * Subtemporal (LST & RST) D Orbltofrontal (LOF & ROF) p-value = 0.9975 4 3 2 1- n --- 'C.? V)''.~" Patients Figure 4-12. Nonlinearities across recording areas during postictal state for BTSOZ patients ;: "A. S *'d ',wV ,A , V.9 <.v At A* 4.~ A e/ .:I ~ -''' -~"" -' -''''' '~'~' CHAPTER 3 QUANTITATIVE COMPLEXITY ANALYSIS IN MULTI-CHANNEL INTRACRANIAL EEG RECORDINGS FROM EPILEPSY BRAIN 3.1 Introduction Epilepsy is a brain disorder characterized clinically by temporary but recurrent disturbances of brain function that may or may not be associated with destruction or loss of consciousness and abnormal behavior. Human brain is composed of more than 10 to the power 10 neurons, each of which receives electrical impulses (known as action potentials) from others neurons via synapses and sends electrical impulses via a sing output line to a similar (the axon) number of neurons (Shatz 1981). When neuronal networks are active, they produce a change in voltage potential, which can be captured by an electroencephalogram (EEG). The EEG recordings represent the time series that match up to neurological activity as a function of time. The structure of EEG recordings represent the inter activities among the groups of neurons. Many investigators have applied nonlinear dynamical methods to a broad range of medical applications. Recent developments in nonlinear dynamics have shown the abilities to explain some underlining mechanisms of brain behavior [67-71]. It is known that a dynamical system with d degree of freedom may evolve on a manifold with a lower dimension, so that only portions of the total number of degree of freedom are actually active. For a simple system with limit cycles, it is obvious that time-delay embedding produce an equivalent reconstruction of the true state. According to embedding theorem from Whitney (1936), an arbitrary D-dimension curved space can be mapped into a Cartesian (rectangular) space of 2d + 1 dimensions without having any self intersections, hence satisfying the uniqueness condition for an embedding [51]. Sauer et al. (1991) generalized Whitneys and Takens' theorem to fractural attractors with dimension Df and showed the embedding space only need to have a dimension greater than 2Df [72]. Although it is possible for a fractal to be embedded in another fractal, we only consider the integer embedding. Takens delay embedding theorem 15 10 -15 -15 I I I I 0 5 10 15 20 time (s) Figure 2-6. X component of Lorenz system 25 30 35 reported decrease in complexity quantify by the correlation density. Le Van Quyen et al. (1999) showed drop in dynamical similarity before seizures using a measure called dynamicall similarity index "; Lehnertz and Elger (1998) demonstrated seizure prediction by dynamicall complexity "time series analysis. Litt et al., (2001) showed increase in accumulated -S,1 I! energy "prior to seizure onset. See also [42]. Mormann et al., (2003) detected the preictal state based on decrease in "synchronization "measures [43]. The basic text of nonlinear dynamics and nonlinear dynamical models are presented in the following sections. Nonlinear dynamical measures namely (the largest Lyapunov exponent (Lmax), Phase/ Angular frequency (+), Approximate entropy (ApEn)) were used for detecting the preictal transitions in intracranial EEG recordings acquired from patients with intractable MTLE. Since the underlying dynamics of preictal transitions is changing from case to case, this demands sophisticate analytical tools which have the ability for identifying the changes of brain dynamics when preictal transitions occur. The preictal detection performance was further improved by proposed dynamics support vector machine (D-SVM), a classification method developed from optimization theory and data mining techniques. The detection performances were summarized in the later part of this chapter. 2.2 Dynamical Systems and State Space In this section, the basic theory about nonlinear dynamical systems will be given. A dynamical system consists of a set of d state variables, such that each state of the system map to a point p E M. Thus M is d dimensional manifold. A system is said to be a dynamical system if state of the system changes with time. Let us denote p(t) be the state of a system at time T, as time evolve (e.g., t = 0 t = 10, 000), the evolution of the state of the system through state space will form a path. This is path is call "trajectory ". If the current state p(t) uniquely determines all the future state in time, the system is said to be a deterministic dynamical system. If the mapping is not unique, the system is called a stochastic dynamical system. As p(t) evolve for a sufficient amount of time (e.g., [136] D.V. Moretti, C. Miniussi, G.B. Frisoni, C. Geroldi, O. Zanetti, G. Binetti, and P.M. Rossini. Hippocampal atrophy and eeg markers in subjects with mild cognitive impairment. Clinical N ,.,. il'l.;.: ,; 118(12):2716-2729, 2007. [137] R. Quian Quiroga, A. Kraskov, T. Kreuz, and P. Grassberger. Performance of different synchronization measures in real data: A case study on electroencephalographic signals. Phys. Rev. E, 65:041903, 2002. [138] H. Unverricht. Die MI;, I.' .-.:, Franz Deutick, L. il. i.- 1891. [139] H.B. Lundborg. Die progressive /l;,. '..ai ,-Epilepsie (Unverrichts -I;,.1. /, '.:,). Almqvist and Wiksell, Uppsala, 1903. [140] N.K. C!i. --, P. Mir, M.J. Edwards, C. Cordivari, D.M., S.A. Schneider, H.-T. Kim, N.P. Quinn, and K.P. Bhatia. The natural history of unverricht-lundborg disease: A report of eight genetically proven cases. Movement Disorders, Vol. 23, No. 1:107-113, 2007. [141] E. Ferlazzoa, A. Magauddaa, P. Strianob, N. Vi-Hongc, S. Serraa, and P. Gentonc. Long-term evolution of eeg in unverricht-lundborg disease. E1,.:I, I/"; Research, 73:219-227, 2007. [142] M.C. Salinsky, B.S. Oken, and L. Morehead. Intraindividual analysis of antiepileptic drug effects on eeg background rhythms. Electr ,' ,''., i',;,',i' 1/1,1; and Clinical N(. ;-I,,i;.: ...,I ; 90(3):186-193, 1994. [143] T.M. Cover and J.A. Thomas. Elements of Information The ..;, Wiley, New York, 1991. [144] M.R. C'!I ~i i 1: Bootstrap Methods: A Practitioner's Guide. Wiley-Interscience, 1999. [145] M. Steriade and F. Amzica. Dynamic coupling among neocortical neurons during evoked and spontaneous spike-wave seizure activity. Journal of N *,*, (i, ,'.:li. .i~; 72:2051-2069, 1994. [146] E. Sitnikova and G. van Luijtelaar. Cortical and thalamic coherence during spikewave seizures in wag/rij rats. F/'.:/* I"-; Research, 71:159-180, 2006. [147] E. Sitnikova, T. Dikanev, D. Smirnov, B. Bezruchko, and G. van Luijtelaar. Granger causality: Cortico-thalamic interdependencies during absence seizures in wag/rij rats. Journal of Neuroscience Methods, 170(2):245-254, 2008. study, prior to calculate Lma, the EEG recordings were first divided into non-overlapping window with 10.24 second in duration. The same segmentation procedure was also used by lasemidis et al., (1991) such segmentation technique is often applied especially for medical time series [55]. 4.2.3 Surrogate Data Technique The degree of nonlinearity of a signal can be examined by testing the null hypothesis: "The signal results from a Gaussian linear stochastic process ". One way to test this hypothesis is to estimate the difference in a discriminating statistic between the original EEG and its surrogate [81]. There are three different procedures for surrogate data. 1. Surrogates are realizations of independent identically distributed (iid) random variables with the same mean, variance, and probability density function as the original data. The iid surrogates were generated by randomly permuting in temporal order the samples of the original series. This shuffling process will destroy the temporal information and thus generated surrogates are mainly random observation drawn (without replacement) from the same probability distribution as original data. 2. Fourier transform (FT) surrogates are constrained realizations of linear stochastic processes with the same power spectra as the original data. FT surrogate series were constructed by computing the FT of the original series, by substituting the phase of the Fourier coefficients with random numbers in the range while keeping unchanged their modulus, and by applying the inverse FT to return to the time domain. To render completely uncoupled the surrogate pairs, two independent white noises where used to randomize the Fourier phases. 3. Auto regressive (AR) surrogates are typical realizations of linear stochastic processes with the same power spectra as the original series. By generating a Gaussian time series with the same length as the data, and reordered it to have the same rank distribution. Take the Fourier transform of this and randomize the phases (FT). Finally, the surrogate is obtained by reordering the original data to have the same Arnhold et al., (1999) introduced another nonlinear interdependence measure H(k) (X Y) as H(k)(X Y) =log R() (6-33) N 1 R}(X Y) H(k)(XIY) = 0 if X and Y are completely independent, while it is possible if closest in Y implies also closest in X for equal time indexes. H(k)(XIY) would be negative if close pairs in Y would correspond mainly to distant pairs in X. H(k)(XIY) is linear measures thus is more sensitive to weak dependencies compare to mutual information. Arnhold et al., (1999) also showed H was more robust against noise and easier to interpret than S. Since H is not normalized Quiroga et al., (2002) introduced another N(XIY): 1 R, (X) ~?R(X|Y) N(k)(XY) R ) (6-34) N R. ( X) which is normalized between 0 and 1. The opposite interdependencies S(YIX), H(YIX), and N(YIX) are defined in complete analogy and they are in general not equal to S(XIY), H(XIY), and N(XIY), respectively. Using nonlinear interdependencies on several chaotic model (Lorenz, Roessler, and Heenon models) Quiroga et al., (2000) showed the measure H is more robust than S. The .-i-vilii. I ry of above nonlinear interdependencies is the main advantage over other nonlinear measures such as the mutual information or the phase synchronization. This .,-vmmetry can give information about "driver-response "relationships but can also reflect different properties of dynamical systems when it is importance to detect causal relationships. It should be clear that the above nonlinear interdependencies measures are bivariate measures. Although it quantified the "driver-response "for given input-the whole input space under study might be driven by other unobserved systemss. 6.6 Discussions It is believed that synchronization occurs due to both local and global discharges of the neurons. From the epilepsy perspective, quantifying the changes in spatiotemporal interactions could potentially lead to the development of seizure-warning systems and For X and Y time series we define d( = I xjYd = ,I yj I as the distances for xi and yi between every other point in matrix spaces X and Y. One can rank these distances and find the knn for every xi and yi. In the space spanned by X, Y, similar distance rank method can be applied for Z = (X, Y) and for every zi = (xi, yi) one can also compute the distances d) = zj and determine the knn according to some distance measure. The maximum norm is used in this study: d() -max{ | xj, i, yj }, d) = xi xj\. (6-19) Next let '() be the distance between zi and its kh" neighbor. In order to estimate the joint probability density function (p.d.f.), we consider the probability Pk () which is the probability that for each zi the kth nearest neighbor has distance dc from zi. This probability means that k 1 points have distance less than the kth nearest neighbor and N k 1 points have distance greater than and k 1 points have distance less than -. Pk () is obtained using the multinomial distribution: Pk() k dc -(1 p)N-k-1, (6-20) where pi is the mass of the e-ball. Then the expected value of logpi will be: E(logpi) = (k)- Q(N), (6-21) where Q(-) is the 1:.it,,,,na, function: (t) F(t)-d ) (6-22) dt where F(-) is the gamma function. It holds that b(1) =C where C is the Euler - Mascheroni constant (C 0.57721). The mass of the e-ball can be approximated (if we consider the probability density function inside the ball is the same)by: (e) cdP(X= x), (6-23) from ai(yi(w xi + b) 1) = 0, for any i such that ai is not zero. D-SVM map a given EEG data set of binary labeled training data into a high dimensional feature space and separate the two classes of data linearly with a maximum margin hyperplane in the dynamical feature space. In the case of nonlinear separability, each data point x in the input space is mapped into a different dynamical feature space using some nonlinear mapping function p. Figure 2-12 and 2-13 show the 3D plot for for entropy, angular frequency, and Lmax during interictal (100 data points dynamical features 2 hours prior to seizure onset)and preictal state (100 data points dynamical features sampled 2mins prior to seizure onset). a I 5$5 4. 4" 5 L00 4D0 1 *** t$ 4 3 * * p W * 3900 S#tU 3700 360o Ap V% 3500 3 Figure 2-12. Three dimension plot for entropy, angular frequency and Lmax during interictal state [14] J.S. Lockard, W.C. Congdon, and L.L. DuC'l! ii.:,. Feasibility and safety of vagal stimulation in monkey model. Epilepsia, 31(S2):S20-S26, 1990. [15] B.M. Uthman, B.J. Wilder, J.K. Penry, C. Dean, R.E. R-i-m-i, S.A. Reid, E.J. Hammond, W.B. Tarver, and J.F. Wernicke. Treatment of epilepsy by stimulation of the vagus nerve. N ,. ,,..../,; 43(7):13338-13345, 1993. [16] G.L. MorrisIII and W.M. Mueller. Long-term treatment with vagus nerve stimulation in patients with refractory epilepsy. the vagus nerve stimulation study group e01-e05. N, ,,,. l..,. 54(8):1712, 2000. [17] D. Ko, C. Heck, S. Grafton, M.L.J. Apuzzo, W.T. Couldwell, T. C', i'. J.D. Day, V. Zelman, T. Smith, and C.M. DeGiorgio. Vagus nerve stimulation activates central nervous system structure in epileptic patients during pet blood flow imaging. Neurosurgery, 39(2):426-431, 1996. [18] T.R. Henry, R.A.E. B 1: i,-, J.R. Votaw, P.B. Pennell, C.M. Epstein, T.L. Faber, S.T. Grafton, and J.M. Hoffman. Brain blood flow alterations induced by therapeutic vagus nerve stimulation in partial epilepsy: I. acute effects at high and low levels of stimulation. Epilepsia, 39(9):983-990, 1998. [19] E. Ben-Menachem, A. Hamberger, T. Hedner, E.J. Hammond, B.M. Uthman, J. Slater, T. Treig, H. Stefan, R.E. R ,i- lic, J.F. Wernicke, and B.J. Wilder. Effects of vagus nerve stimulation on amino acids and other metabolites in the csf of patients with partial seizures. F1I.:I I/"-; Research, 20(3):221-227, 1995. [20] S.E. Krahl, K.B. Clark, D.C. Smith, and R.A. Browning. Locus coeruleus lesions suppress the seizure-attenuating effects of vagus nerve stimulation. Epilepsia, 39(7):709-714, 1998. [21] J. Malmivu and R. Plonse. Bioelectromagnetism-Principles and Applications of Bioelectric and Biomagnetic Fields. Oxford University Press, 1995. [22] E.R. Kandel, J.H. Schwartz, and T.M. Jessell. Principles of Neural Science, Fourth Edition. McGraw-Hill Medical, 2000. [23] H. Berger. Ueber das elektrenkephalogramm des menschen. Arch. P-;,.. /,.:.r. Nervenkr, 87:527-570, 1929. [24] H.H. Jasper. The ten-twenty electrode system of the international federation. Electr .. ,, lAl. i/, r Clin N, o,',i;;,-,.: 10:371-375, 1958. [25] D.A. Princea and B.W. Connors. Mechanisms of interictal epileptogenesis. 44:275-299, 1986. [26] M.E. Weinand, L.P. Carter, W.F. El-Saadany, P.J. Sioutos, D.M. Labiner, and K.J. Oommen. Cerebral blood flow and temporal lobe epileptogenicity. Journal of Neuosurgery, 86:226-232, 1997. algorithm: begin end procedure: begin 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 end Figure 7-6. Maximum clique algorithm initiated a highly organized rhythmic patterns and the patterns started to propagate throughout all the brain areas. We initially speculated that the epileptogenic areas could be the ones that are highly synchronized long before a seizure onset. In the previous case, we observed that the L(T)D electrodes are the one that started the seizure evolution. However, in a further investigation of EEG recordings from the same patient, we found some contrast results. In Figure 7-9, the electrode selection pattern of the maximum clique demonstrates a very highly synchronized group of electrodes in both left and right orbitofrontal areas during the 2-hour period preceding the seizure. After visual inspection on the raw EEG recordings, this seizure was initiated by the R(T)D area. Generally, it is maximum clique sort all nodes based on vertex ordering LIST = ordered nodes cbc = 0 current best clique size depth = 0 current depth level enter-next-depth(LIST,depth) enter-next-depth(LIST,depth) m = the number of nodes in the LIST depth=depth+l for a node in position i in the LIST if depth+(m-i)< cbc then return prune the search else mark node i if no adjacent node then cbc=depth (maximum clique found) else enter to next depth (adjacent node of i, depth) end end unmark node i if depth=1 delete node i from LIST end end (A) Focus Ara (RmJ A-B e-----4------ (B) A-C -._- - I Subtemporalarea on fcal ide (RST} dA-0 --.-- o E- - - pat uienCt 15 S BC or ...... 6 _B-O B A-F -----.----- |C-D ------------------------ v> ,C -D t 1 0 DE r -- --...- D-F -+- -m 2E-F ----. ---t I I I I I I I I I I -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 2 3 4 5 response variable: Mean Tindex vJlu Patients simultaneous 95 % confidence hmits, Tukey method Figure 4-9. Nonlinearities across recording areas during postictal state for ULSOZ patients. procedure, but also will decrease the risk of infection caused by the implanted recording electrodes. Large sample of patients with ULSOZ and BTSOZ will be required for reliable estimation of sensitivity and specificity of this method. Correct identification of the focal area in ULSOZ patients is a challenging task. An obvious question could be whether any of brain areas where the EEG signals recorded from is close enough to the actual focal area. If not, it would be very difficult to identify the focal area by an i' 1i, ~i-;' on these EEG signals. Other issues such as the number of recording areas and number of recording electrodes in each area could also affect the results of the analysis. In each of the five ULSOZ patients studied here, EEG signals were recorded from six different brain areas: left and right hippocampus, subtemporal, and orbitofrontal regions. All five patients were clinically determined to have focal area in the right hippocampus. During the interictal state, focal area (right temporal depth) consistently exhibited higher degree of nonlinearity than in the contralateral temporal depth and subtemporal areas (significant observations in 4 out of 5 patients). Similar findings were also observed during preictal and postictal states. These results -ii--.- -1 that it is possible to identify the focal area interdependencies measures among different brain regions during before and after add-on AEDs treatment for the patient with ULD. Given two time series x and y, using method of d. 1iv to obtain delay vectors (Xn, ..., Xn-(m-l1)) and y, = (x,, ..., Xn-(,r-1)), where n = 1,...N, m is the embedding dimension and r denotes the time delay [52]. Let r,,j and s,,j, j = 1, k denote the time indices of the k nearest neighbors of xT and y,/. For each xn, the mean Euclidean distance to its k neighbors is defined as k R() = ,(n r,,j)2 (6-29) j= 1 and the Y-conditioned mean squared Euclidean distance is defined by replacing the nearest neighbors by the equal time partners of the closest neighbors of y, k RXk)(X|Y) = (, xs,,j)2. (6 30) j=1 The delay 7 = 5 is estimated by auto mutual information function, the embedding dimension m = 10 is obtained using Cao's method using 10 sec EEG selected during interinctal state and a Theiler correction is set to T = 50 [73; 107]. If :, has an average squared radius R(X) = (1/N) N 1 R}N-)(X), then Rek) Rk)(X) < R(X) if the system are strongly correlated, while R}k)(XIY) R(X) > R(k)(X) if they are independent. Accordingly, it can be define and interdependence measure S(k)(X Y) as S(k)(XY) =- I )(X) (6-31) N zR (X|Y) Since Rk)(X Y) > Rk)(X) by construction, 0 < S(k)(Xly) < 1 (6-32) Low values of Sk(XIY) indicate independence between X and Y, while high values indicate synchronization. abnormal features in EEG recordings. Figure 1-7 is taken from Malmivu and Plonse (1993) [21]. 1.6.1 Scalp EEG Recording Scalp EEG recording is a most common recording method for monitoring the electrical activity of the brain. See figure 1-8. The recording electrodes are placed on the scalp of the head and record electrical potential differences between the recording electrodes. However, recordings acquired from scalp are usually contaminated by multiple sources of artifacts such as movement artifacts, chewing artifacts, eye movement, vertex waves and sleep spindles, etc. The international 10-20 electrode placement system is commonly used for routine scalp EEG recording [24]. Figure 1-8 is taken from Malmivu and Plonse (1993) [21]. A B Nasion 20% Vertex i 10% /20% 20% 020%/ P3 20% I I I IA I 10% 7 1 A\ A C3 C C T 2 ITN T"5p if1- Nasion 10% A A 10% T 20% Preaurical Inion ' g point / Inion \, V 20% Inion 10% Figure 1-8. International 10-20 electrode placement 1.6.2 Subdural EEG Recording The subdural recordings provide less unwanted information in the signals by placing the electrodes under the scalp. See figure 1-9. It requires surgical procedure to place the subdural recording electrodes and the risks of infection increase with the amount do not elicit larger the amplitude of the action potential. Therefore, the intensity of a stimulus is encoded in the frequency of action potential rather than in their amplitude. Third, the action potential "ti i,. !- along the axon without fading out because the signal is regenerated at each .,li i:ent membrane. 1.6 Recording Electric Brain Activity The first brain electrical scalp recordings of human was performed by Hans Berger in 1929 [23]. Since then the EEG recordings has been the most common diagnosis tool for epilepsy. EEG measures the electrical activity of the brain. EEG studies are particularly important when neurologic disorders are not accompanied by detectable alteration in brain structure. It is accepted that the neurons in the thalamus pl li an important role in generating the EEG signals. The synchronicity of the cortical synaptic activity reflects the degree of synchronous firing of the thalamic neurons that are generating the electrical activities. However, the purposes of these electrical activities and EEG oscillations are largely unknown. Figure 1-6. EEG recording acquired by Hans Berger in 1929 The configurations of EEG recordings pl i, an important role in determining the normal brain function from abnormal.The most obvious EEG frequencies of an awake, relaxed adult whose eyes are closed is 8-13 Hz also known as the alpha rhythm. The alpha rhythm is recorded best over the parietal and occipital lobes and is known to be associated with decreased levels of attention. When alpha rhythm are presented, subjects commonly report that they feel relaxed and happy. However, people who normally experience more alpha rhythm than usual have not been shown to be psychologically different from those with less. Another important EEG frequencies is the beta rhythm, people are attentive to Figure 1-2. Vagus nerve stimulation electrode Long term follow-up studies showed that prevention of recurrent seizures was maintained and adverse events decreased significantly over time [15; 16]. Positron emission tomography and functional MRI studies showed that VNS activates or increases blood flow in certain areas of the brain such as the thalamus [17; 18]. Cerebrospinal fluid (CSF) was analyzed in 16 subjects before, 3 months after, and 9 months after VNS treatment GABA (total and free) increased in low or high stimulation groups, aspartate marginally decreased and ethanolamine increased in the high stimulation group -ii--:. -ii.:-; an increased inhibitory effect [19]. Krahl et al., -,ir--- -I. 1 that seizure suppression induced by VNS may depend on the release of norepinephrine and they observed that acute or chronic lesions of the "Locus coeruleus" attenuated VNS-induced seizure suppression [20]. 1.5 Neuron States and Membrane Potentials 1.5.1 Neuron States The birth of Electroencephalogram (EEG) has inspired the attempts to extract the subtle alternations in brain activity. It is known the fluctuations on EEG frequency and voltage arise from spontaneous interactions between excitatory and inhibitory neurons in circuit loops. If a neuron is stimulated, the membrane potential will be altered and this alternation can be classified into two different states Figure 1-3 is taken from Malmivu and Plonse (1993)[21]. For example two planes of dimension di and d2 embedded in m dimensional space will intersect if m < di + d2, it is clear that if dl = d2 = d the embedding dimension need at least 2d + 1 to avoid the intersections in the state space. However, if only s subset of the degrees of freedom of M is represented in our measurement x(t) = ((t)), it is impossible to obtain additional information. A technique called method of delay is employ, ,1 to retreat the information from pervious times with a embedding window r and form a set of reconstructed delay vector x(t), x(t) = (x(t),x(t r), x(t 2r),...,x(t (m 1)r)), (2-19) and the duration of each embedding vector is r = (m 1) '. (2-20) A much more general situation for time-1 ,.-.- d variables constitute an adequate embedding provided the measured variable is smooth and coupled to all the other variables is proved by Takens, and the number of time lag is at least 2d + 1 [52]. 0 : M --+ R2d+l is an open and dense set in the space of pairs of smooth maps (f,h), where f is the dynamical system measure by function h. 2.5 Lyapunov Exponents The concept of Lyapunov exponents was first introduced in by A.M. Lyapunov. Lyapunov developed "Lyapunov Stability "concepts to measure the stability of a dynamical system. It quantifies the rate of separation of nearby trajectories in the state space. In this section I describe the method for estimating the Lyapunov exponents. For a dynamical system, sensitivity to initial conditions is quantified by the Lyapunov exponents. For example, consider two trajectories with nearby initial conditions on an attracting manifold. Eckmann and Ruelle (1985) pointed out that when the attractor is chaotic, the trajectories diverge, on average, at an exponential rate characterized by the largest Lyapunov exponent [53]. For a dynamical system as time evolves the sphere [81] J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. Farmer. Testing for nonlinearity in time series: the method of surrogate data. P,;,'-.:. ,. D, 58:77-94, 1992. [82] W.A. Ch'!i ..v ,lwongse, P.M. Pardalos, and O.A. Prokoyev. Electroencephalogram (EEG) Time Series Classification: Applications in Epilepsy. Annals of Operations Research, 148:227-250, 2006. [83] N. Acir and C. Guzelis. Automatic spike detection in eeg by a two-stage procedure based on support vector machines. Computers in B.:.. .,/; and Medicine, 34(7):561-575, 2004. [84] B. Gonzalez-Vellon, S. Sanei, and J.A. C'!i iihlhers. Support vector machines for seizure detection. In Proceedings of the 3rd IEEE International Symposium on S.:',iul Processing and Information T 1,-,..1..,i/; 2003. ISSPIT 2003., 2003. [85] P. Comon. Independent component analysis, a new concept. S.:j,,'rl Processing, 36:287-314, 1994. [86] J.-F. Cardoso and B. Laheld. Equivariant adaptive source separation. IEEE Transactions on S.:iil1 Processing, 44:3017-3030, 1996. [87] A.J. Bell and T.J. Sejnowski. An information-maximization approach to blind separation and blind deconvolution. Neural Computation, 7:1129-1259, 1995. [88] A. Hyvrinen, J. Karhunen, and E. Oja. Independent component ...;-. John Wiley & Sons, 2001. [89] F. Mormann, K. Lehnertz, P. David, and C.E. Elger. Mean phase coherence as a measure for phase synchronization and its application to the eeg of epilepsy patients. Pi,<; -..: D, 144:358-369, 2000. [90] C.W.J. Granger. Investigating causal relations by econometric models and cross-spectral methods. Econometrica, 37:424-438, 1969. [91] E. Rodriguez, N. George, J.P. Lachaux, J. Martinerie, B. Renault, and F.J Varela. perceptions shadow: Long-distance synchronization of human brain activity,. Nature,, 397:430-347, 1999. [92] F.H. Lopes da Silva and J.P. Pijn. The Handbook of Brain Theory and Neural Networks, chapter EEG wI .k-i- pages 348-351. MIT Press, Cambridge, MA, 1995. [93] C. Hugenii. Horoloquium oscilatorum. Paris, France, 1673. [94] M.G. Rosenblum, A.S. Pikovsky, and J. Kurths. Phase synchronization of chaotic oscillators. Phys. Rev. Lett., 76:1804-1807, 1996. [95] J.W. Freeman. C'!i o '.teristics of the synchronization of brain activity imposed by finite conduction velocities of axons. International Journal of Bifurcation and Chaos, 10:2307-2322, 2000. represented as y(w) (C- C- (6-4) The cross-coherence quantifies the degree of coupling between X and Y at given frequency w and it is also bounded between -1 and +1. 6.2.2 Partial Directed Coherence Partial Directed Coherence a frequency domain based Granger-causality technique which -,i- that an observed time series xj(n) causes another series xi(n), if knowledge of xj(n)s past significantly improves prediction of xi(n) [90]. However, the reverse case may or may not be true. To make a quantitative assessment of the amount of linear interaction and the direction of interaction among multiple time-series, the concept of Granger-causality can be used to and into the development multi autoregressive model (\!VAR). The partial directed coherence from j to i at a frequency w is given by: S(W) ( (65) where for i = j A(w) =1 Ya(r)e-j2wr; (6-6) r=i and for i / j P A-.(a;) Y- aj(r)e -2-w (6-7) r=l aij are the multivate auto-regressive (\!AR) coefficient at lag r, obtained by least-square solution of MAR model p-1 x = A'x(p r) + c, (6-8) r=l here [67] Y.-C. Lai, M.G. Frei, I. Osorio, and L. Huang. C'! i '.terization of synchrony with applications to epileptic brain signals. Phys Rev Lett, 98(10):108102, 2007. [68] P. Velazquez, J.L. Dominguez, and R.L. Wennberg. Complex phase synchronization in epileptic seizures: evidence for a devil's staircase. Phys Rev E Stat Nonlin Soft Matter Phys, 75(1):011922, 2007. [69] T.I. Netoff and S.J. Schiff. Decreased neuronal synchronization during experimental seizures. Journal of Neuroscience, 22(16):7297-307, 2007. [70] S. Sakata and T. Yamamori. Topological relationships between brain and social networks. Neural Networks, 20:12-21, 2007. [71] R. Ai,1... i .: F. Mormann, G. Widman, T. Kreuz, C. Elger, and K. Lehnertz. Improved spatial characterization of the epileptic brain by focusing on nonlinearity. Fi'.:l I,-; Research, 69:30-44, 2006. [72] T. Sauer, J.A. Yorke, and M. Casdagli. Embedology. Journal of Statistical PA, -. 65:579-616, 1991. [73] L.-Y. Cao. Practical method for determining the minimum embedding dimension of a scalar time series. P,;-,'.. D, 110(1-2):43-50, 1997. [74] M.B. Kennel, R. Brown, and H.D.I. Abarbanel. Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys Rev. A, 45:3403-34, 1992. [75] A.M. Fraser and H.L. Swinney. Independent coordinates for strange attractors from mutual information. Phys Rev A, 33:1134-1140, 1986. [76] P. Grassberger and I. Procaccia. ('C! i i:'terization of strange attractors. P,;,-.. 'l/ Review Letters, 50(5):346-349, 1983. [77] D.S. Broomhead and G.P. King. Extracting qualitative dynamics from experimental data. Ph,;,-.. D, 20:217-236, 1986. [78] M.C. Casdagli, L.D. Iasemidis, J.C. Sackellares, S.N. Roper, R.L. Gilmore, and R.S. Savit. ('! i o i.terizing nonlinearity in invasive EEG recordings from temporal lobe epilepsy. P,;,-..:',. D, 99:381-399, 1996. [79] B. Weber, K. Lehnertz, C.E. Elger, and H.G. Wieser. Neuronal complexity loss in interictal eeg recorded with foramen ovale electrodes predicts side of primary epileptogenic area in temporal lobe epilepsy: A replication study. Epilepsia, 39(9):922-927, 1998. [80] J.C. Sackellares, D.-S. Shiau, J.C. Principe, M.C.K. Yang, L.K. Dance, W. Suharitdamrong, W.A. ('C! I, i.iv ,!wongse, P.M. Pardalos, and L.D. Iasemidis. Predictibility analysis for an automated seizure prediction algorithm. Journal of Clinical N ;,', -, ... ,,/;/ 23(6):509-520, 2006. 25 20 15 10 5 0 -5 -10 -15s -20 0 5 10 15 20 time (s) Figure 2-7. Y component of Lorenz system 25 30 35 frequency bands across different regions of the brain, leading to certain clinical events such as evoked potentials [95; 96]. Similarly, it is also believed that phase synchronization across narrow frequency EEG bands, pre-seizure and at the onset of seizure may provide useful hints of the spatio-temporal interactions in epileptic brain [33; 97; 35]. Hilbert transform is used compute the instantaneous parameters ip(t) and Pb(t) of a time-signal. Consider a real-valued narrow-band signal x(t) concentrated around frequency f,. Define x(t) as 1 x = x(t) x (6-14) where x(t) can be regarded as the output of the filter with impulse respond 1 h(t) = < t < oo, (6-15) 'rt excited by an input signal x(t). This filter is call a Hillbert transformer. Hilbert transforms are accurate only when the signals have narrow-band spectrum, which is often unrealistic for most real-world signals. Pre-processing of the signal such as decomposing it into narrow frequency bands is needed before we apply Hilbert transformation to compute the instantaneous parameters. Certain conditions need to be checked to define a meaningful instantaneous frequency on a narrow-band signal. It has been reported that the distinct differences in the degree of synchronization between recordings from seizure-free intervals and those before an impending seizure, indicating an altered state of brain dynamics prior to seizure activity [89]. 6.4 Mutual Information The concept of mutual information dates back to the work of Shannon in 1948 [98]. Generally, mutual information measures the information obtained from observations of one random event for the other. It is known that mutual information has the capability to capture both linear and nonlinear relationships between two random variables since both linear and nonlinear relationships can be described through probabilistic theories. Here in our model, the mutual information measures how much information of EEG time -H -HH Sd 0 - ZHH ZHHc ZHHd ZH Sd 0 - Sd 0 - d 0 - X9 .z3 *z ZHHZH h~qz *o c *s *i*u *~ * ~dddd~~~~ ZHH us tM ^ (M sd CU~oOll~ 1'0 c~3 ~ u5 oO u5 ~ iii O, c~s c~s cudcu ood " $II" $II" $II" $I d$I ZHH d? c~sr HHI .z * Z-HZ-H ZH 0 - od 6 6 d~~ddo a 6 dmaidRO~d~ SCl ca - c^ i x^S^ r^o. x~ x~ rg.. xii ^ z rS- r^. .z *z *z z *z *z z *z *- .z *z *^ *z * HH- .z * ZHHZHH tM - ^ od 0 oo^ t o, -HHu cus i-i CM C S-00 C oo ^co _ 0-H .z *z Z-HZ-H --H- H a I ^ -H ^ - c HH 0 H cO 1 ^ ? 0 Z css oH L-I (M s M 0 c0o o a us t-I a cu c: "d le ~ 8.5 LST RST S- Seizure Onset 7.5 I(.1 E 7 6.5 0 50 100 Oe ( mi 20 250 300 350 I me (minutes) Figure 3-2. Average minimum embedding dimension profiles for Patient 1 (seizure 1) showed stable during the interictal state. In other words, the underlying degree of freedom is uniformly distributed over the interictal state in the EEG recordings. The results indicated the lowest minimum embedding dimension were found within the epileptic zone during interictal state (the RST electrodes in Figs. 2, 3, and 4; the LTD electrodes in Figs. 5 and 6). The complexity of the EEG recordings from the epileptic region is lower than that from the brain regions. The values of the minimum embedding dimension from all brain regions start decrease and converge to a lower value as the patient proceed from interictal to ictal state. The underlining dynamical changes before entering ictal period were consistently detected by the algorithm. 3.6 Conclusions In this chapter, we investigate the degree of complexity for EEG recordings by estimating the minimum embedding dimension. The algorithm we use for the minimum embedding estimation is faster and requires less data points to obtain accurate results. Right Hppocampus 44 20 l10 -- 4- 0 ...... 0 20 40 60 80 100 120 140 160 180 Figure 4-5. STL, x and T-index profiles during preictal state 4.4 Discussion In this study, we demonstrated the usefulness of nonlinear dynamics measures and investigated the degree of nonlinearity for EEG signals in different brain area for epilepsy patients with and without unilateral seizure onset zone. The degree of nonlinearity was defined as the distinction of the signal from Gaussian linear processes. The method combined the estimation of Short-Term Maximum Lyapunov Exponents (STLmax), a nonlinear discriminating statistic, and surrogate time series techniques. The hypotheses tested were that (1) there exists difference in signal nonlinearities across recording brain regions for patients with unilateral seizure onset zone, (2) EEG nonlinearities are distributed uniformly across recording brain regions for patients with bi-temporal seizure onset zone, and (3) in patients with unilateral seizure onset zone, the focal area can be identified by comparisons of EEG nonlinearities among recording brain regions. The CHAPTER 5 OPTIMIZATION AND DATA MINING TECHNIQUES FOR THE SCREENING OF EPILEPTIC PATIENTS 5.1 Introduction Detecting and identifying the important abnormal electroencephalogram (EEG) complex by visual examination is not only a time consuming task but also requires fully attentions form the electroencephalographer. In this study, we investigate the possibility for classifying EEG recordings between seizure free patients and patients still suffering from seizure attack using the support vector machine (SVM). Two multi-dimensional SVMs, connectivity SVM (C-SVM) and dynamics SVM (D-SVM), were proposed to identify the EEG recordings acquired from epileptic patients. The C-SVM uses connectivity feature that extracted from EEG recording through mutual information and D-SVM uses three dynamical measures (1. Angular frequency 2. Approximate entropy 3. Short-term largest lyapunov exponent) input for the EEG classification. One hour scalp EEG recording was acquired from each subject (5 class 1, 5 class 2) in this study. Prior to C-SVM classification, the independent component analysis (ICA) methods were applied to remove the noise in the EEG recording for improving the performance od the SVM. D-SVM achieved 94.7'. accuracy when identifying class 2 subjects compared to 69.!'. accuracy with C-SVM. Epilepsy is the most common disorders of nervous systems. Preliminary findings on the costs of epilepsy show the total cost to the nation for 2.3 million people with epilepsy was approximately $12.5 billion. The high incidence of epilepsy originates from the fact that it occurs as a result of a large number of factors, including febrile disturbance, genetic abnormal mutation, developmental deviation as well as brain insults such as central nervous system (CNS) infections, hypoxia, ischemia, and tumors. Neuron or groups of neurons generate electrical signals when interacting or transmitting information between each other. The EEG recordings capture the local field potential around electrodes that generate from neuron in the brain. Through visual inspection, The .-i-viii.ii I ry of above nonlinear interdependencies is the main advantage over other synchronization measures. This .-i-i'i::. 1 ry property can give directionality of information transport between different cortical regions. Furthermore the "driver-response "relationships but can also reflect different properties of brain functions when it is importance to detect causal relationships. It should be clear that the above nonlinear interdependencies measures are bivariate measures. Although it quantified the "driver-response "for given input-the whole input space under study might be driven by other unobserved sources. 8.4 Statistic Tests and Data Analysis In this study, the mutual information and nonlinear interdependence measures were estimated for every 10 seconds (2500 EEG data points) of continuous EEG recordings. The bootstrap re-sampling approach was adapted for deriving estimates on the measures[144]. Ten seconds of continuous EEG epoch is randomly sampled from every channels and this sampling procedure was repeated with replacement for 30 times. The reference Al and A2 channels (inactive regions) are excluded from the analysis. Two sample t-test (N 30, a = 0.05) is used to test the statistical differences on mutual information and nonlinear interdependence during before and after treatment. Low mutual information and information transport between different brain cortical regions were observed in our subjects with less severity of ULD. Furthermore, for each patient both mutual information and information transport between different brain cortical regions decrease after AEDs treatment. t-test for mutual information are summarized in Table 8-2, the topographical distribution for mutual information is also plot in heatmaps shown in Figures 8-2. The significant "driver-response "relationship is reveled by t-test. After t-test the significant information transport between Fpl and other brain cortical regions is shown in Fig. 8-1. The edges with an arrow starting from Fpl to other channel denote N(XIY) is significant larger then N(YIX), therefore Fpl is the driver, and vice versa. The 00 -H .z *z ~oOm$ od 2o c o u, rc*- i .z *z *z *z~s~ '-H oo n C^ ? L. L* 0 i0 d d c o- - * o |_ o~ 0C CH- oo s, o~oui- rU~*- * .z *z *z * - -H c o c~1 r~*- X * r~d~~*- *dd~ .z *z *z *z" *zI *zI *z *z *zI *z *zI" I"$ o~~~, r~*- o .z *z~~~0 *O *z *z*z* z L d 0 OI fa -u . *O -o'5o "dS~dS r-cu C~3d d $I ~ius Od "$I Ld n S s? p, 0 -H ~id dS( oo 0 -H SI usc LO (- d 0 - sS 'o o-c r2 ?c cs? us r 0 ? to capture synchronization patterns among different sources (brain areas). The idea of analyzing EEG recordings from several sources (multiple electrodes) is very crucial since the knowledge from local information (i.e., single electrode) is very limited. In our future study, we plan to incorporate the knowledge of general brain communication in the brain network. For example, [114] demonstrated the evolution of cognitive function through quantitative analysis of fMRI data. The proposed framework can provide a global structural patterns in the brain network and may be used in the simulation study of dynamical systems (like the brain) to predict oncoming events (like seizures). For example, an ON-OFF pattern of electrode selection in the maximum clique over one period of time can be modeled as a binary observation in a discrete state in a Markov model, which can be used to simulate the seizure evolution in the brain. In addition, the number of electrodes in the maximum clique can be used to estimate the minimum number of features and explain dynamical models or the parameters in time series regression. Note that the proposed network model represents an epileptic brain as a graph, where there exist several efficient algorithms (e.g., maximum clique, shortest path) for finding special structure of the graph. This idea has enabled us, computationally and empirically, to study the evolution of the brain as a whole. The Monte-Carlo Markov C'!i ,i (C' IC) framework may be applicable in our future study on long term EEG analysis. The MC'\ C framework has been shown very effective in data mining research [132]. It can be used to estimate the graph or clique parameters in epileptic processes from EEG recordings. Since long term EEG recordings are very massive, most simulation techniques are not scalable enough to investigate large-scale multivariate time series like EEGs. The use of MC'\ C makes it possible to approximate the brain structure parameters over time. More importantly, the MC'\ C framework can also be extended to the analysis of multi-channel EEGs by generating new EEG data points while exploring the data sequences using a Markov chain mechanism. In addition, we can integrate the MC' IC framework with a B ,i -i ,n approach. This can be Table 5-2. Results for D-SVM using 5-fold cross validation D-SVM Results D-SVM/C-SVM Dynamical features UNICA 5-fold CV 1 .'. 7' 2 9", 71,', 3 , 4 '-', 72' 6 92' 5 !', 7 II,' 7, , 8 95' 7 ,' , 9 !', 7 '", 10 95', 7.i'. Average of correctness 94.7'. 69. !' The SVM has a very long statistical foundation and assure the optimal feasible solution for a set of training data, given a set of features and the operation of the SVM. In this study, we attempted to study the separability between abnormal EEG and normal EEG using different EEG features. We tested the performance on scalp EEG recordings from normal individuals and abnormal patients. The EEG data was filtered using ICA algorithm. ICA filters the noise in EEG scalp data, keeps essential structure and makes better representable EEG data sets. The Euclidean distance based C-SVM was proposed to evaluate the connectivity among different brain regions. The dynamical features were generated as input for D-SVM, the classification results of the proposed D-SVM are very encouraging. The results indicated that D-SVM improves classification accuracy compare to C-SVM. It gives an average accuracy of 94.7'. The dynamical features provide a subset in the feature space and improve classification accuracy. One can obtain the mutual information between X and Y using the following equation [143]: I(X;Y)= H(X) + H(Y) H(X,Y), (8-2) where H(X), H(Y) are the entropies of X, Y and H(X, Y) is the joint entropy of X and Y. Entropy for X is defined by: H(X)= p(x) logp(x). (8-3) The units of the mutual information depends on the choice on the base of logarithm. The natural logarithm is used in the study therefore the unit of the mutual information is nat. We first estimate H(X) from X by N H(X) = P(X x). (8-4) i= 1 For X and Y time series we define d = ) xjY,,d = 1, yj\\ as the distances for xi and yi between every other point in matrix spaces X and Y. One can rank these distances and find the knn for every xi and yi. In the space spanned by X, Y, similar distance rank method can be applied for Z = (X, Y) and for every z = (xi, yi) one can also compute the distances d) = zjl and determine the knn according to some distance measure. The maximum norm is used in this study: d max{ -xj, y, -dyj}, d = x, xj. (8-5) Next let 6-) be the distance between zi and its kth neighbor. In order to estimate the joint probability density function (p.d.f.), we consider the probability Pk(c) which is the probability that for each zi the kth nearest neighbor has distance + dc from zi. This Pk(e) represents the probability for k 1 points have distance less than the kth nearest neighbor and N k 1 points have distance greater than C) and k 1 points have [27] C. Baumgartner, W. Series, F. Leutmezer, E. Pataraia, S. Aull, T. Czech, U. Pietrzyk, A. Relic, and I. Podreka. Preictal spect in temporal lobe epilepsy: Regional cerebral blood flow is increased prior to electroencephalogi i ii--,- i. ire onset. The Journal of Nuclear Medicin, 39(6):978-982, 1998. [28] P.D. Adelson, E. N. i,,i.,i M. Scheuer, M. Painter, J. Morgan, and H. Yonas. N. ii i ,- -ive continuous monitoring of cerebral oxygenation periictally using near-infrared spectroscopy: a preliminary report. Epilepsia, 40:1484-1489, 1999. [29] P. Federico, D.F. Abbott, R.S. Briellmann, A.S. Harvey, and G.D. Jackson. Functional mri of the pre-ictal state. Brain, 128:1811-1817, 2005. [30] R. Delamont, P. Julu, and G. Jamal. C('!i I,, in a measure of cardiac vagal activity before and after epileptic seizures. FL'.:l I"-; Res, 35:87-94, 1999. [31] D.H. Kerem and A.B. Geva. Forecasting epilepsy from the heart rate signal. Med Biol Eng Comrput, 43:230-239, 2005. [32] L.D. Iasemidis and J.C. Sackellares. Long time scale spatio-temporal patterns of entrainment in preictal ecog data in human temporal lobe epilepsy. Epilepsia, 31:621, 1990. [33] J. Martinerie, C. Van Adam, M. Le Van Quyen, M. Baulac, S. Clemenceau, B. Renault, and F.J. Varela. Epileptic seizures can be anticipated by non-linear analysis. Nature Medicine, 4:1173-1176, 1998. [34] C.E. Elger and K. Lehnertz. Seizure prediction by non-linear time series analysis of brain electrical activity. European Journal of Neuroscience, 10:786-789, 1998. [35] M. Le Van Quyen, J. Martinerie, M. Baulac, and F.J. Varela. Anticipating epileptic seizures in real time by non-linear analysis of similarity between eeg recordings. NeuroReport, 10:21492155, 1999. [36] G. Widman, K.Lehnertz, H. Urbach, and C.E. Elger. Spatial distribution of neuronal complexity loss in neocortical epilepsies. Epilepsia, 41:811-817, 2000. [37] L.D. Iasemidis, D.-S. Shiau, J.C. Sackellares, P.M. Pardalos, and A. Prasad. Dynamical resetting of the human brain at epileptic seizures: application of nonlinear dynamics and global optimization techniques. IEEE Transactions on Biomedical Engineering, 51(3):493-506, 2004. [38] F. Mormann, T. Kreuz, C. Rieke, R.G. Ail.1.. i .1 A. Kraskov, P. David, C.E. Elger, and K. Lehnertz. On the predictability of epileptic seizures. Journal of Clinical N / 'I,,. .: I ,/; 116(3):569-587, 2005. [39] K. Lehnertz, F. Mormann, 0. Hannes, A. Mller, J. Prusseit, A. C'!., i inlovskyi, M. Staniek, D. Krug, S. Bialonski, and C.E Elger. State-of-the-art of seizure prediction. Journal of clinical ';. ;,,i,,,..: l. ,i;, 24(2):147-153, 2007. included to extend this study. We test the hypothesis that, in patients with unilateral seizure onset zone, the degree of nonlinearity are different over the brain regions, and the focus areas exhibit higher degree of nonlinearity during interictal, preictal, and postictal periods. Further, for patients with independent bi-temporal seizure onset zones, the distribution of nonlinearity would be uniform over brain regions. 4.2 Materials and Methods 4.2.1 EEG Description EEG recordings were obtained from bilaterally placed depth and subdural electrodes (Roper and Gilmore, 1995), multi-electrode 28-32 common reference channels were used in this study. Figure 4-1 a inferior transverse views of the brain, illustrating approximate depth and subdural electrode placement for EEG recordings are depicted. Subdural electrode strips are placed over the left orbitofrontal (LOF), right orbitofrontal (ROF), left subtemporal (LST), and right subtemporal (RST) cortex. Depth electrodes are placed in the left temporal depth (LTD) and right temporal depth (RTD) to record hippocampal activity. EEG recordings obtained from eight patients with temporal lobe epilepsy were included in this study. See Table 4-1. Five patients were clinically determined to have unilateral seizure onset zone (ULSOZ) and the remaining three patients were determined to have independent bi-temporal seizure onset zone (BTSOZ). For each patient, three seizures were included in the EEG recordings. Segments from interictal (at least one hour before the seizure), preictal (immediately before the seizure onset) and postictal (immediately after the seizure offset) time intervals corresponding to each seizure were sampled for testing the hypothesis. Two electrodes from each brain area were included,a total of 12 electrodes were analyzed for each patient. The EEG recordings were sampled using amplifiers with input range of 0.6 mV, and a frequency range of 0.5-70Hz. The recordings were stored digitally on videotapes with a sampling rate of 200 Hz, using an different scaled spaces (marginal and joint) are not comparable To avoid this problem, instead of using a fixed k, n(i) + 1 and ny(i) + lare used in obtaining the distances (where n,(i) and ny(i) are the number of samples contained the bin [x(i) -, x(i) + (] and [y(i)- ~7, y(i) + i] respectively) in the x-y scatter diagram. The Eq.(8-12) becomes: SN H(X) = (N) (n(i) + 1) +logc c, + N log(i). (8-13) i=1 Finally the Eq.(8-2) is rewritten as: N Iknr,(X; Y) = (k) + ^(N) [ (n(i) + 1) + y(n,(i) + 1)]. (8-14) i=1 8.3.2 Nonlinear Interdependencies Arnhold et al., (1999) introduced the nonlinear interdependence measures for characterizing directional relationships (i.e. driver & response) between two time sequences [106]. Given two time series x and y, using method of delay we obtain the delay vectorsx = (x,, ..., xn-(m-i),) and y, = (x,, ..., xn-(m-_l)), where n = 1, ...N, m is the embedding dimension and r denotes the time d. 1v [52]. Let rT,j and s,j, j = 1,..., k denote the time indices of the k nearest neighbors of x, and y,. For each xn, the mean Euclidean distance to its k neighbors is defined as k R(X) = (x, xr,j)2, (8-15) j= 1 and the Y-conditioned mean squared Euclidean distance is defined by replacing the nearest neighbors by the equal time partners of the closest neighbors of y, R7k)(XlY) = (x xs.,)2. (8 16) j=1 The delay = 5 is estimated using auto mutual information function, the embedding dimension m = 10 is obtained using Cao's method and the Theiler correction is set to T 50 [73; 107]. LIST OF TABLES Table page 2-1 Patient information and EEG description ............ ... .. 59 2-2 Performance for D-SVM ............... .......... .. 59 3-1 Patients and EEG data statistics for complexity analysis . . 62 4-1 Patients and EEG data statistics .................. ....... .. 76 5-1 EEG data description .................. ............. .. 94 5-2 Results for D-SVM using 5-fold cross validation ....... . ...... 103 7-1 Patient information for clustering analysis ................ .... .. 121 8-1 ULD patient information .................. .......... 140 8-2 Topographical distribution for treatment decoupling effect (DE: Decouple Electrode (DE)) . . . . . . . . .. .. .. 146 8-3 Patient 1 before treatment nonlinear interdependencies . . ... 150 8-4 Patient 1 after treatment nonlinear interdependencies . . .... 151 8-5 Patient 2 before treatment nonlinear interdependencies . . ... 152 8-6 Patient 2 after treatment nonlinear interdependencies . . ... 153 8-7 Patient 3 before treatment nonlinear interdependencies . . ... 154 8-8 Patient 3 after treatment nonlinear interdependencies . . ..... 155 8-9 Patient 4 before treatment nonlinear interdependencies . . ... 156 8-10 Patient 4 after treatment nonlinear interdependencies . . ... 157 LIST OF FIGURES Figure page 1-1 Vagus nerve stimulation pulse generator .................. ..... 22 1-2 Vagus nerve stimulation electrode .................. ....... .. 23 1-3 Cortical nerve cell and structure of connections ................. .. 24 1-4 Membrane potentials .................. .............. .. 25 1-5 Electrical potentials .................. .............. .. 26 1-6 EEG recording acquired by Hans Berger in 1929 .. . 27 1-7 Basic EEG patterns .................. .............. .. 28 1-8 International 10-20 electrode placement .................. .. 29 1-9 Surbdural electrode placement .................. ......... .. 30 1-10 Depth electrode placement .................. ......... .. .. 31 2-1 R6ssler attractor ............... .............. .. 36 2-2 X component of R6ssler system .............. ....... .. 37 2-3 Y component of R6ssler system .............. ....... .. 38 2-4 Z component of R6essler system .............. ....... .. 39 2-5 Lorenz system . .............. .. .......... ... ..41 2-6 X component of Lorenz system .................. ........ .. 42 2-7 Y component of Lorenz system ............... ....... .. 43 2-8 Z component of Lorenz system ............... ...... .. 44 2-9 Estimation of Lyapunov exponent (Lm,,) .................. .. .. 50 2-10 Temporal evolution of STL.ma ................ ........ .. 51 2-11 Phase/Angular frequency of Lyapunov exponent (Qma) . . .. 52 2-12 Three dimension plot for entropy, angular frequency and Lmax during interictal state ............... ................ .. 55 2-13 Three dimension plot for entropy, angular frequency and Lmax during preictal state ............... ................ .. 56 3-1 Electrode placement ............... ............ .. 62 seizures, however, frequently have bilateral hemispheric involvement. Partial onset seizure can be classified into one of the following three groups: 1. Simple partial onset seizures (consciousness not impaired) The EEG features for simple partial onset seizures are local contralateral discharge starting over the corresponding area of cortical representation and the significant feature for the background EEG is local contralateral discharge. Simple partial onset seizures may have the following clinical features: (a) With motor signs i. Focal motor without march ii. Focal motor with march (Jacksonian) iii. Versive iv. Postural v. Phonatory(Vocalization or arrest of speech) (b) With somatosensory or special sensory symptoms (simple hallucinations, e.g., tingling, light flashes, buzzing) i. Somatosensory ii. Visual iii. Auditory iv. Olfactory v. Gustatory vi. Vertiginous (c) With autonomic symptoms or signs (including epigastric sensation, pallor, sweating, flushing, piloerection and pupillary dilatation) (d) With p' ii, symptoms (disturbance of higher cerebral function). These symptoms rarely occur without impairment of consciousness and are much more commonly experienced as complex focal seizures i. Dysphasic TABLE OF CONTENTS page ACKNOW LEDGMENTS ................................. LIST O F TABLES . . . . . . . . . . LIST OF FIGURES . . . . . . . . . A B ST R A C T . . . . . . . . . . CHAPTER 1 OVERVIEW OF EPILEPSY ............................ 1.1 Introduction . . . . . . . . . 1.2 C auses of Seizure . . . . . . . . 1.3 Classification of Epileptic Seizure (ICES 1981 revision) ........... 1.3.1 Partial Onset Seizures .. .. .. .. ... .. .. .. ... .. .. . 1.3.2 Generalized Seizures .. .. .. .. ... .. .. .. ... .. .. . 1.3.3 Unclassified Seizures . .. .. ... .. .. .. ... .. .. . 1.4 Treatment for Epilepsy .. ........................ 1.4.1 Pharmacological Treatment .. ................... 1.4.2 Surgeical Section . . . . . . . 1.4.3 Neurostimulator Implant .. .................... 1.5 Neuron States and Membrane Potentials .. ............... 1.5.1 N euron States . . . . . . . . 1.5.2 M embrane Potentials .. ..................... 1.6 Recording Electric Brain Activity .. ................... 1.6.1 Scalp EEG Recording .. ..................... 1.6.2 Subdural EEG Recording .. .................... 1.6.3 Depth EEG Recording .. .................... 1.7 Conclusions and Remarks .. ...................... 2 EPILEPSY AND NONLINEAR DYNAMICS .. ............... 2.1 Introduction . . . . . . . . . 2.2 Dynamical Systems and State Space .. ................. 2.3 Fractal D im ension . . . . . . . . 2.3.1 Correlation Dimension .. .................... 2.3.2 Capacity Dim ension .. ...................... 2.3.3 Information Dimension .. .................... 2.4 State Space Reconstruction .. ..................... 2.5 Lyapunov Exponents .. ......................... 2.6 Phase/Angular Frequency .. ...................... 2.7 Approxim ate Entropy .. ......................... 2.8 Dynamical Support Vector Machine (D-SVM) .. ............. To my parents in Taiwan. Table 4-1. Patients and EEG data statistics Patient # Gender Age Focus (RH/LH) Length of EEG (hr.) P1 M 19 RH 6.1 P2 M 45 RH 5.4 P3 M 41 RH 5.8 P4 F 33 RH 5.3 P5 F 38 RH 6.3 P6 M 44 RH/LH 5.5 P7 F 37 RH/LH 4.6 P8 M 39 RH/LH 5.4 analog to digital (A/D) converter with 10 bit quantization. In this study, all the EEG recordings were viewed by two independent board certified electroencephalographers. (A) (B) ROF 2- oo oo4 LOF LTD RST LST LOF LST RTD LTD Figure 4-1. 32-channel depth electrode placement 4.2.2 Non-Stationarity Non-stationarity is an fundamental difficulty for time series analysis. The existing of non-stationarity in a measured time series will result in no invariant measures. Stationarity will cause errors for many algorithms when one is trying interpret the results of an invariant measure. In most cases, one can try to remove the stationarity by using vi I i, of filters or divided the time series into a number of shorter epochs and assume the underlying dynamics to be approximately stationary within each divided epochs. In this ( a a'12 % a' )iN AT (6-9) \ tNI tN2 NN / and xli(p -r) x2(p r) (p = x3(p r) (6-10) XN(P r) p denotes the depth of the AR model, r denotes the delay and n is the prediction error or the white noise. Note that Tr, quantifies the relative strength of the interaction of a given signal source j with regard to signal i as compared to all of js interactions to other signals. It turns out that the PDC is normalized between 0 and 1 at all frequencies. If i=j, the Partial Directed Coherence represents the casual influence from the earlier state to its current state. The MVAR approaches have been used to determine the propagation of epileptic intracranial EEG activity in temporal lobe and mesial seizures [2-3, 10, 19-20]. However, these models strictly require that the measurements be made from all the nodes, or the directional relationships could be ambiguous. In addition, there remains no clear evidence of causality relationships among the cortical regions as si,--. -1I "the nature of synchronization is mostly instantaneous or without any detectable d(. I [91]. The general, nonlinearity are commonly inherent within neuronal recordings, the above linear measures are typically restricted to measure statistical dependencies up to the second order [92]. If observations are Gaussian distributed, the 2nd order statistics are sufficient to capture all the information in the data. However, in practice, EEG data 2.3 Fractal Dimension The term "fractal" was first introduced by Mandelbrot in 1983. Roughly speaking, a fractal is a set of points that when looked at smaller scales, resembles the whole set. The concept of fractal dimension refers to a non-integer or fractional dimension originates from fractal geometry. Strange attractors often have a structure that is not simple; they are often not manifolds and actually have a highly fractured character. The dimension that is most useful takes on values that are typically not integers. These non-integer dimensions are called fractal dimensions. For any attractor, the dimension can be estimated by looking at the way in which the number of points within a sphere of radius r scales as the radius shrinks to zero. The geometric relevance of this observation is that the volume occupied by a sphere of radius r in the dimension d behaves as For regular attractors, irrespective to the origin of the sphere, the dimension would be the dimension of the attractor. But for a chaotic attractor, the dimension varies depending on the point at which the estimation is performed. If the dimension is invariant under the dynamics of the process, we will have to average the point densities of the attractor around it. For the purpose of identifying the dimension in this fashion, we find the number of points y(k) within a sphere around some phase space location x. This is defined by: N n(x,r) er- y(k)- x (2-11) k-0 where is the Heaviside step function such that (n) = 0 for n < 0, O(n) = 1 for n > 0 This counts all the points on the orbit y(k) within a radius r from the point x and normalizes this quantity by the total number of points N in the data. Also, we know that the point density, p(x), on an attractor does not need to be uniform for a strange attractor. C'!, ... -i the function as n(x; r)q-1 and defining the function C(q; r) of two variables q and r by the mean of n(x; r)q-1 over the attractor weighted with the natural distance less than (). Pk(c) is obtained using the multinomial distribution: Pk() k(N-k 1 d(c) P (1- pi)N-k-1 (86) ( k 1) dc where pi is the mass of the c-ball. Then the expected value of logpi is E(logpi) = (k) ((N), (8-7) where b(-) is the 1:li1,,i,,,a function: (t)= F(t)-d(t) (8-8) dt where F(-) is the gamma function. It holds when b(1) =C where C is the Euler - Mascheroni constant (C 0.57721). The mass of the c-ball can be approximated (if considering the p.d.f inside the c-ball is uniform) as pi(c) a cdCP(X= x), (8-9) where Cd. is the mass of the unit ball in the dx-dimensional space. From Eq.(8-9) we can find an estimator for P(X = xi) log[P(X = xi)] b (k) (N) dE(log(i)) log Cd, (8-10) finally with Eq(8-10) and Eq(8-4) we obtain the Kozachenko-Leonenko entropy estimator for X [105] N H(X) = (N) (k) + log Cd, + log (i), (8-11) i= 1 where c(i) is twice the distance from xi to its k-th neighbor in the dx dimensional space. For the joint entropy we have N H(X, Y) = (N) (k) + log(cdcd,) + da Nd log(e(i)) (8-12) i= 1 The I(X; Y) is now readily to be estimated by Eq.(8-2). The problem with this estimation is that a fixed number k is used in all estimators but the distance metric in X9~I Z -Hs . z 0 0*z 0 0 *z *z *z *z 0 0 0 0 c-^ w0 ~ t ^co c-o^ w^ ^v ^ ^^io ~ o ^d~d~d~d~d~d~d~d~d~d~d~d0d0d "-H0^0^0^0^0^0^0^0^0^0^0^0^0^ S-H -I_ -I- "$I oooo2 oOc^ 0+1+1 - cd~ cd1 rd*- d c~1 r~*- ~ - *l~~~~~~~i r~* *i *d~~iidd S' c =- l* =9 c 0 I .z *z *z *z *- *z *z d^dd~d~d~d^^d~d~ --H oo o-1 ? oo u5 0-Hd 0 a ^ 0 a ^ 0 ? u0 00 CM CS 0 - ao C 0 - Ln 0 13 0 -H o t?-c 7 Ow: 0** 0R 11U0 Lyapunov Exponent Figure 5-2. EEG dynamics feature classification infeasibilities of the constraints. With this formulation, ones wants to maximize the margin between two classes by minimizing I| w 1|2. The second term of the objective function is used to minimize the misclassification errors that are described by the slack variables ci. Introducing positive Lagrange multipliers ai to the inequality constraints in D-SVM model, we obtain the following dual formulation: i 1i j=1 n min 1 Y iyjaiajXiXj ai (5-15) n n i=1 Co r3 0 'I, C o* .. - )^* 2-"Sr0 Erropy i ' . ' where I|| || is some Euclidian distance and is given in this paper by maximum norm. Define the mean value of all a(i, d) as N-dr- E(d) N a(, d). (3-8) i= E(d) is depend only on the dimension d and the time delay r The minimum embedding dimension is founded when El(d) = E(d + 1)/E(d) saturated when d is larger than some value do if the time series comes from an attractor. The value do + 1 is the estimated minimum embedding dimension. 3.5 Data Analysis The first step in the data analysis was to divide the EEG data into non-overlapping windows of 10.24 seconds in duration for nonstationarity purposes. This procedure was to ensure that the underlining dynamical properties were approximately stationary. For each divided window, the first step of estimating the minimum embedding dimension is to construct the delay coordinates using method of delay proposed by. The time delay 7 was obtained from the first local minimum of the mutual information function. We used these time delay vectors as inputs to Cao's method for the minimum embedding dimension estimation. The minimum embedding dimension was calculated over time for EEG recordings with 29 electrodes at six brain regions (RTD, RST, ROF, LTD, LST, and LOF) from epilepsy patients. Each brain region contains 46 electrodes; the average of the minimum embedding dimension da is taken as representation to the underlining brain dynamics. We shall study the minimum embedding dimension in the following three different time periods: interictal, ictal and postictal. These three different time period are defined as follows: 1. interictal state: 1 hour away prior to ictal state 2. preictal state: 2 minutes data length prior to ictal state 3. postictal state: 1 hour after the ictal state Figures 3-2,3-3,3-4,3-5 and 3-6 show typical the minimum embedding dimension over time for six seizures. One can observe the behavior of the average minimum embedding dimension over time for six brain cortical regions. The minimum embedding dimension 1. Excited state: A neuron state with a less negative intraneural membrane potential compare to resting state of a neuron. The positive increase in voltage above the normal resting neuronal potential is called the excitatory p -vi!i itic potential (EPSP), if this potential rise high enough in the positive direction, it will elicit an action potential in the neuron. 2. Inhibited state: A neuron state with a more negative intraneural membrane potential compare to resting state of a neuron. An increase in negative beyond the normal resting member potential level is call an inhibitory ] .. 1 -vi- i)tic potential (IPSP). Afferent nerve fibers Synaptic knobs Dendrites - Cell body Figure 1-3. Cortical nerve cell and structure of connections The excitatory neurons excite the target neurons. Excitatory neurons in the central nervous system are often glutamatergic neurons. Neurons in the peripheral nervous system, such as spinal motor neurons that synapse onto muscle cells, often use ... I i*lcholine as their excitatory neurotransmitter. However, this is just a general tendency that may not ahlv- -, be true. It is not the neurotransmitter that decides excitatory or inhibitory action, but rather it is the 1 ..- I -ii ,tic receptor that is responsible for the function of c as c -i 0: Do lm g[N ] (2-15) e-o log[e] 2.3.3 Information Dimension The information dimension D1 is a generalization of the capacity that are relative probability of cubes used to cover the attractor. Let I denotes the information function: N I Pi(r) log P(r), (216) i=1 Ps(r) is the normalized probability of an element i is covered such that C,1 Pi(r) 1. Information dimension is defined as: D1 y NPi(r)logPi(r) (27) i=1 log(r) D2 < D1 < Do if elements of the fractal is equally likely to be visited in the state space. 2.4 State Space Reconstruction Most dynamical properties are contained within almost any variable and its time lags. It is not necessary to reconstruct to entire state space from the measured variable since the attractor dimension will often evolved in a much smaller dimension. The method called state-space reconstruction was proposed by Takens (1981) for reconstructing the state space for a dynamical system. For a series of observations acquired from a dynamical system, the state space reconstruction transforms the observations into stat space using an embedding coordinate map 0 : M -- S', x(t) = ((t)), (2-18) where m is the embedding dimension. The transform function 0 must be unique (i.e., has no self intersection). Whitney (1936) proved a theorem which can also be used for finding the embedding dimension [51]. 0 : M -- R2d+l; 0 embedding is an open and dense set in the space of sm 4- p-value = 0.9945 0 1 2 3 Patients Figure 4-11. Nonlinearities across recording areas during preictal state for BTSOZ patients 1-, I, 2 3 CHAPTER 4 DISTINGUISHING INDEPENDENT BI-TEMPORAL FROM UNILATERAL ONSET IN EPILEPTIC PATIENTS BY THE ANALYSIS OF NONLINEAR CHARACTERISTICS OF EEG SIGNALS 4.1 Introduction In this present study, we investigate the difference in nonlinear characteristics of electroencephalographic (EEG) recordings between epilepsy patients with independent bi-temporal seizure onset zone (BTSOZ) and those with unilateral seizure onset zone (ULSOZ). Eight adult patients with temporal lobe epilepsy were included in the study, five patients with ULSOZ and three patients with BTSOZ. The approach was based on the test of nonlinear characteristics, defined as the distinction from a Gaussian linear process, in intracranial EEG recordings. Nonlinear characteristics were tested by the statistical difference of short-term maximum Lyapunov exponent STLax, a discriminating nonlinear measure, between the original EEG recordings and its surrogates. Distributions of EEG nonlinearity over different recording brain areas were investigated and were compared between two groups of patients. The results from the five ULSOZ patients showed that the nonlinear characteristics of EEG recordings are significantly inconsistent (p < 0.01) over six different recording brain cortical regions (left and right temporal depth, sub temporal and orbitofrontal). Further, the EEG recordings acquire from focal regions of the brain exhibit higher degree of nonlinearities than the homologous contralateral regions and the nonlinear characteristics of EEG are uniformly distributed over the recording areas in all three patients with BTSOZ. These results -i-i: -1 that it is possible to efficiently and quantitatively determine whether an epileptic patient has ULSOZ based on the proposed nonlinear characteristics analysis. For the ULSOZ patients, it is also possible to identify the focal area. However, these results will have to be validated in a larger sample of patients. Success of this study can provide more essential information to patients and epileptologists and lead to successful epilepsy surgery. CHAPTER 8 TREATMENT EFFECTS ON ELECTROENCEPHALOGRAM (EEG) 8.1 Introduction Assessing the severity of myoclonus and evaluating the efficacy of antiepileptic drugs (AEDs) treatment for patients with Unverricht-Lundborg Disease (ULD) have traditionally utilized the Unified Myoclonus Rating Scale (UMRS). EEG recordings are only used as a supplemental tool for the diagnosis of epilepsy disorders. In this study, mutual information and nonlinear interdependence measures were applied on EEG recordings to identify the effect of treatment on the coupling strength and directionality of information transport between different brain cortical regions. Two 1-hour EEG recording were acquired from four ULD subjects during the time period of before and after treatment. All subjects in this study are from the same family with similar age (48 3 years) and ULD history (~37.75 years). Our results indicate that the coupling strength was low between different brain cortical regions in the patients with less severity of ULD. The effects of the treatment was associated with significant decrease of the coupling strength. The information transport between different cortex regions were reduced after treatment. These findings could provide a new insight for developing a novel surrogate outcome measure for patients with epilepsy when clinical observations could potentially fail to detect a significant difference. EEG recording system has been the most used apparatus for the diagnosis of epilepsy and other neurological disorders. It is known that changes in EEG frequency and amplitude arise from spontaneous interactions between excitatory and inhibitory neurons in the brain. Studies into the underlying mechanism of brain function have -.-.-, -1 .'1 the importance of the EEG coupling strength between different cortical regions. For example, the synchronization of EEG activity has been shown in relation to memory process [133; 134] and learning process of the brain [135]. In a pathophysiological study, different brain synchronization/desynchronization EEG patterns are shown to be induced 0' i 200 205 210 5 0 5 0 200 205 210 5 0 200 205 210 5 0 -------- 195 200 205 210 Time (mins) 10 5 0 190 195 200 205 210 215 10 5 0 190 195 200 205 210 215 10 5 0 190 195 200 205 210 215 10 5 0 190 195 200 205 210 215 10 5 0 190 195 200 205 210 215 10 5 0 190 195 200 205 210 215 Time (mins) Figure 4-2. Degree of nonlinearity during preictal state and post-ictal (Fig. 4-6) state, respectively, in a ULSOZ patient. Each figure contains two areas, one from the focal area (top two panels) and another from the homologous contralateral hippocampus area (bottom two panels). From these figures, it is clear that the EEG recorded from the focus area exhibits higher distinction from Gaussian linear processes than those recorded from the homologous contralateral hippocampus area in all three states. Further, the differences of STLax values in the focus area increased from interictal to preictal, and reached to the maximum in the postictal state, but the difference remained the stable in the homologous contralateral area. If x, has an average squared radius R(X) = (1/N) Rn -'(X), then R,) - Rk)(X) < R(X) if the system are strongly correlated, while Ref)(X Y) w R(X) > R(k)(X) if they are independent. Accordingly, it can be define and interdependence measure S(k)(XIY) as S(k)(X|Y)= 1 t (X (8-17) N ,lR (X|Y) Since Rk)(X Y) > R$ (X) by construction, 0 < s(k)(Xly) < 1 (8-18) Low values of Sk(XIY) indicate independence between X and Y, while high values indicate synchronization. Arnhold et al., (1999) introduced another nonlinear interdependence measure H(k)(X Y) as H(k)(XY) -log R ) ,(8-19) n 1 Rf(XY)' H(k)(XIY) = 0 if X and Y are completely independent, while it is possible if closest in Y implies also closest in X for equal time indexes. H(k)(XIY) would be negative if close pairs in Y would correspond mainly to distant pairs in X. H(k)(XIY) is linear measures thus is more sensitive to weak dependencies compare to mutual information. Arnhold et al., (1999) also showed H was more robust against noise and easier to interpret than S. Since H is not normalized Quiroga et al., (2002) introduced another N(XIY): 1 R(X) Rk)(XIY) N(k)(XlY) j= N -, (8-20) 1N=l R (X) which is normalized between 0 and 1. The opposite interdependencies S(YIX), H(YIX), and N(YIX) are defined in complete analogy and they are in general not equal to S(XIY), H(XIY), and N(XIY), respectively. Using nonlinear interdependencies on several chaotic model (Lorenz, Roessler, and Heenon models) Quiroga et al., (2000) showed the measure H is more robust than S. (Takens, 1981) also provided that the time 1 .-.- d variables constitute an adequate embedding provided the measured variables is smooth and couples to all the variables, and number of time lags is at least 2D + 1 [52]. For the above reasons, we employ .1 a method proposed by Cao (1997) to estimate the minimum embedding dimension of EEG time series [73]. Like some other exiting methods, Caos method is also under the concepts of false-nearest-neighbors The false-nearest-neighbors utilized on the fact that if the reconstruction space has not enough dimensions, the reconstruction will perform a projection, and hence will not be an embedding of the desired system [74]. As a of result of giving a to low embedding dimension while processing the embedding procedure, two points which is far away in the true state space will be mapped into close neighbor in the reconstruction space. These are then the false neighbors. Caos method does not require large amount of data points, is not subjective and it is not time-consuming find the proper minimum embedding dimension. The EEG recordings was divided into non-overlapping single electrode segments of 10.24 s duration, each of which was estimated for the minimum embedding dimension. Under the assumption the EEG recordings within each 10.24 s duration was approximately stationary [56], we evaluated the underlining dynamical behavior by looking at the minimum embedding dimension over time. The remaining of this chapter is organized as follows. In Sections. 2 and 3, we describe the data information and explain the algorithm for estimating the minimum embedding dimension estimation. The results from two patients with a total number of six temporal lobe epilepsy (TLE) are given in Section 4. In Section 5, we discuss the results of our findings with respect the use of this algorithm and the function of nonlinear dynamical measurements in the area of seizure control. 3.2 Patient and EEG Data Information Electrocardiogram (EEG) recordings from bilaterally placed depth and subdural electrodes (Roper and Glimore, 1995) in patients with medically refractory partial seizures of mesial temporal origin were analyzed in this study. Electrode placement. A Inferior E E LTD RTD 6 LOF ROF LST RST Sezure Onset 40 60 80 100 120 140 160 180 200 22 240 Time (minutes) Figure 3-4. Average minimum embedding dimension profiles for Patient 1 (seizure 3) large-scale multi-quadratic 0-1 programming problems. Our results also are compatible with the findings about the nature of transitions to ictal state in invasive EEG recordings from patients with seizures of mesial temporal origin. The development of multi-quadratic 0-1 programming modeling is in progress. 69 $ -H-H oO 1- 0 0 . *oz *z*o z zo d"id~dC~dC~dd dS-H0^0^0S^0S oo d? >$ "$ C 0. 0 0 00 0 I0 0 t0 0 -H- ZC 1 -H -H C -H -H -H -H -H -H -H *z *z *z cu1I X9 .zU * cu c~cu Lnd d $I r- C~3d dS( o, oO~ C~3d d $I c~s ~ic~s Lnd d $I 0 Ha .z *z CU^u2 ^^~3 0+1+ - -H z-Hz- .z *z z-Hz-Hs~ oo -c z-H-H C0 a~ -H -H d d 0 OO 0^ us oo us to, cu ooousd ddld $I Ln c~s~i d $I ~ic~s C~3d dS( o1 0 -H u (M d d o, sd 0-H --H oo-c -Id " ? LM a ? r-u t^ 1 CO C 3 2?( COd Id 0 - o r-cu d $I o ~CU C~d d $I c~s ocu C~3d d $I cu c~c~s d $I e4 O < 5? -0 Ss ?U od i$ 0 H 7 Lx 4 4 4,0 4 w, 4 U S* ak 4f : W * t%* 4. * \ } * L :l 39011 3s" , 700 360.0 Ap fEFA 3500 3 Lim. Figure 2-13. Three dimension plot for entropy, angular frequency and Lmax during preictal state 2.9 Statistical Distance In this section, we introduced Tj index, a statistical measure to estimate the difference of EEG recordings in the dynamical measures. The Tj index at time t between the dynamical profiles of EEG recording at i and j is defined as: S--ijt T(t)= D | x , N1 (2-42) where I||Dl ~ denotes the absolute value of the average of all paired differences Lt) t( w(t)), SDAj I- (L' (2-43) can be described through probabilistic theories. Here in our model, the C\ ll measures how much information of EEG time series acquired from electrode x is presented by electrode y and vice versa. Let X be the set of data points where its possible realizations are xl,, x2, 3,..., x with probabilities P(xi), P(x2), P(x3),.... The Shannon entropy H(X) of X is defined as H(X) p Inpi. (7 1) i=1 Shannon entropy measures the uncertainty content of X. It is ahv-,- positive and measured in bits, if the logarithm is taken with base 2. Now let us consider another set of data points Y, where all possible realizations of Y are yi, y2, Y3,... yn with probabilities P(y ), P(y2), P(3),.... The degree of synchronicity and connectivity between X and Y can be measured by the joint entropy of X and Y, defined as H(X, Y) 7 (7-2) i,j where pij which is the joint probability of X = Xi and Y = Yj. The cross information between X and Y, CMI(X, Y), is then given by CMI(X,Y) = H(Y)- H(XY) H(X)- H(YX) (7-3) S H(X) + H(Y) H(X,Y) (7-4) px(x)py( y) The cross mutual information is nonnegative. If these two random variables X, Y are independent, fxy(x,y) = fx(x)fy(y), then CMI(X,Y) = 0, which implies that there is no correlation between X and Y. The probabilities are estimated using the histogram based box counting method. The random variables representing the observed number of pairs of point measurements in histogram cell (i,j), row i and column j,are respectively kij, ki. and k.j. Here, we assume the probability of a pair of point measurements outside the area covered by histogram is negligible, therefore ,j Pj = 1 [75; 129]. by a higher degree of synchronization among EEG electrodes within the same region, during the pre-seizure state. We postulate that the active connection may be driven by seizure evolution, regulating abnormal communications in the epileptogenic brain areas or vulnerable areas in the brain network. To test these hypotheses, we herein propose network-theoretical methods through a multivariate statistical analysis of EEGs to study the seizure development by investigating the topological structure of the brain connectivity network. Epileptic seizures involve the synchronization of large populations of neurons [116]. Measuring the connectivity and synchronicity among different brain regions through EEG recordings has been well documented [99; 117; 69]. The structures and the behaviors of the brain connectivity have been shown to contain rich information related to the functionality of the brain [118; 67; 68]. More recently, the mathematical principles derived from information theory and nonlinear dynamical systems have allowed us to investigate the synchronization phenomena in highly non-stationary EEG recordings. For example, a number of synchronization measures were used for analyzing the epileptic EEG recordings to reach the goals of localizing the epileptogenic zones and predicting the impending epileptic seizures [99; 106; 119; 38; 120]. These studies also -i-.: -1 that epilepsy is a dynamical brain disorder in which the interactions among neuron or groups of neurons in the brain alter abruptly. Moreover, the characteristic changes in the EEG recordings have been shown to have clear associations with the synchronization phenomena among epileptogenic and other brain regions. When the conductivities between two or among multiple brain regions are simultaneously considered, the univariate analysis alone will not be able to carry out such a task. Therefore it is appropriate to utilize multivariate analysis. Multivariate analysis has been widely used in the field of neuroscience to study the relationships among sources obtained simultaneously. In this study, the cross mutual information (C' \I) approach is applied to measure the connectivity among brain regions [75]. The C' \ I approach is a bivariate measure and has been shown to have ability for ACKNOWLEDGMENTS I would like to thank Dr. Panagote M. Pardalos, Dr. Basim M. Uthman, Dr. J. Chris Sackellares, Dr. Deng-Shan Shiau, and Dr. W.A. C('!i i 1wongse for their guidance and support. During the past five years, their energy and enthusiasm have motivated me to become a better researcher. I would also like to thank Dr. Van Oostrom, Dr. Paul Carney and Dr. Steven Roper for serving on my supervisory committee and for their insightful comments and '-I'- -i in I have been very lucky to have the chance to work with many excellent people in the Brain Dynamics Laboratory (B.D.L) and Center for Applied Optimization (C.A.O). I would like to thank Linda Dance, Wichai Suharitdamrong, Dr. Sandeep Nair and Dr. Michael Bewernitz from B.D.L. and Dr. Altannar Chinchuluun, Dr. Onur Seref, Ashwin Arulselvan, Nikita Boyko, Dr. Alla Kammerdiner, Oleg Shylo, Dr. Vitally Yatsenko and Petros Xanthopoulos in C.A.O for their valuable friendship and kindness assistance. I thank the staff in the Department of Neurology at the Gainesville VA Medical Center including David Juras and Scott Bearden. I thank Joy Mitchell from North Florida Foundation for Research and Education, Inc. for providing financial support during later part of my Ph.D. study. I also like to acknowledge the financial support provided by National Institutes of Health (NIH), Optima Neuroscience Inc. and North Florida Foundation for Research and Education, Inc. (NFFRE). There are many friends I have made at the University of Florida who gave me their help and support but were not mention here, I would like to take this opportunity to thank all of them for being with me throughout this wonderful journey. Lastly, I wish to thank my lovely family in my home country Taiwan my father Ming-Chih Liu, my mother Chmni-Wei Lin, my elder sister Tsai-Lin, Liu and my younger brother C('i i"--Hao, Liu for their selfless sacrifice and constant love. |

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IwouldliketothankDr.PanagoteM.Pardalos,Dr.BasimM.Uthman,Dr.J.ChrisSackellares,Dr.Deng-ShanShiau,andDr.W.A.Chaovalitwongsefortheirguidanceandsupport.Duringthepastveyears,theirenergyandenthusiasmhavemotivatedmetobecomeabetterresearcher.IwouldalsoliketothankDr.VanOostrom,Dr.PaulCarneyandDr.StevenRoperforservingonmysupervisorycommitteeandfortheirinsightfulcommentsandsuggestions.IhavebeenveryluckytohavethechancetoworkwithmanyexcellentpeopleintheBrainDynamicsLaboratory(B.D.L)andCenterforAppliedOptimization(C.A.O).IwouldliketothankLindaDance,WichaiSuharitdamrong,Dr.SandeepNairandDr.MichaelBewernitzfromB.D.L.andDr.AltannarChinchuluun,Dr.OnurSeref,AshwinArulselvan,NikitaBoyko,Dr.AllaKammerdiner,OlegShylo,Dr.VitaliyYatsenkoandPetrosXanthopoulosinC.A.Ofortheirvaluablefriendshipandkindnessassistance.IthankthestaintheDepartmentofNeurologyattheGainesvilleVAMedicalCenterincludingDavidJurasandScottBearden.IthankJoyMitchellfromNorthFloridaFoundationforResearchandEducation,Inc.forprovidingnancialsupportduringlaterpartofmyPh.D.study.IalsoliketoacknowledgethenancialsupportprovidedbyNationalInstitutesofHealth(NIH),OptimaNeuroscienceInc.andNorthFloridaFoundationforResearchandEducation,Inc.(NFFRE).TherearemanyfriendsIhavemadeattheUniversityofFloridawhogavemetheirhelpandsupportbutwerenotmentionhere,Iwouldliketotakethisopportunitytothankallofthemforbeingwithmethroughoutthiswonderfuljourney.Lastly,IwishtothankmylovelyfamilyinmyhomecountryTaiwanmyfatherMing-ChihLiu,mymotherChing-WeiLin,myeldersisterTsai-Lin,LiuandmyyoungerbrotherChang-Hao,Liufortheirselesssacriceandconstantlove. 4 PAGE 5 page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 12 CHAPTER 1OVERVIEWOFEPILEPSY ............................ 14 1.1Introduction ................................... 14 1.2CausesofSeizure ................................ 15 1.3ClassicationofEpilepticSeizure(ICES1981revision) ........... 16 1.3.1PartialOnsetSeizures .......................... 16 1.3.2GeneralizedSeizures .......................... 19 1.3.3UnclassiedSeizures .......................... 20 1.4TreatmentforEpilepsy ............................. 20 1.4.1PharmacologicalTreatment ....................... 20 1.4.2SurgeicalSection ............................ 21 1.4.3NeurostimulatorImplant ........................ 21 1.5NeuronStatesandMembranePotentials ................... 23 1.5.1NeuronStates .............................. 23 1.5.2MembranePotentials .......................... 25 1.6RecordingElectricBrainActivity ....................... 27 1.6.1ScalpEEGRecording .......................... 29 1.6.2SubduralEEGRecording ........................ 29 1.6.3DepthEEGRecording ......................... 30 1.7ConclusionsandRemarks ........................... 31 2EPILEPSYANDNONLINEARDYNAMICS ................... 33 2.1Introduction ................................... 33 2.2DynamicalSystemsandStateSpace ..................... 34 2.3FractalDimension ................................ 45 2.3.1CorrelationDimension ......................... 46 2.3.2CapacityDimension ........................... 46 2.3.3InformationDimension ......................... 47 2.4StateSpaceReconstruction .......................... 47 2.5LyapunovExponents .............................. 48 2.6Phase/AngularFrequency ........................... 50 2.7ApproximateEntropy .............................. 51 2.8DynamicalSupportVectorMachine(D-SVM) ................ 53 5 PAGE 6 ............................... 56 2.10Cross-Validation ................................ 57 2.11PerformanceEvaluationofD-SVM ...................... 57 2.12PatientInformationandEEGDescription .................. 58 2.13Results ...................................... 58 2.14Conclusions ................................... 59 3QUANTITATIVECOMPLEXITYANALYSISINMULTI-CHANNELINTRACRANIALEEGRECORDINGSFROMEPILEPSYBRAIN ................. 60 3.1Introduction ................................... 60 3.2PatientandEEGDataInformation ...................... 61 3.3ProperTimeDelay ............................... 63 3.4TheMinimumEmbeddingDimension ..................... 65 3.5DataAnalysis .................................. 66 3.6Conclusions ................................... 67 4DISTINGUISHINGINDEPENDENTBI-TEMPORALFROMUNILATERALONSETINEPILEPTICPATIENTSBYTHEANALYSISOFNONLINEARCHARACTERISTICSOFEEGSIGNALS ..................... 72 4.1Introduction ................................... 72 4.2MaterialsandMethods ............................. 75 4.2.1EEGDescription ............................ 75 4.2.2Non-Stationarity ............................. 76 4.2.3SurrogateDataTechnique ....................... 77 4.2.4EstimationofMaximumLyapunovExponent ............. 78 4.2.5Pairedt-Test ............................... 79 4.3Results ...................................... 80 4.4Discussion .................................... 84 5OPTIMIZATIONANDDATAMININGTECHNIQUESFORTHESCREENINGOFEPILEPTICPATIENTS ............................. 91 5.1Introduction ................................... 91 5.2EEGDataInformation ............................. 93 5.3IndependentComponentAnalysis ....................... 94 5.4DynamicalFeaturesExtraction ........................ 95 5.4.1EstimationofMaximumLyapunovExponent ............. 95 5.4.2Phase/AngularFrequency ....................... 96 5.4.3ApproximateEntropy .......................... 97 5.5DynamicalSupportVectorMachine ...................... 98 5.6ConnectivitySupportVectorMachine ..................... 100 5.7TrainingandTesting:CrossValidation .................... 102 5.8ResultsandDiscussions ............................ 102 6 PAGE 7 ............. 104 6.1Introduction ................................... 104 6.2SecondOrderSynchronizationMeasures ................... 105 6.2.1CrossCorrelationFunction ....................... 105 6.2.2PartialDirectedCoherence ....................... 106 6.3PhaseSynchronization ............................. 108 6.4MutualInformation ............................... 109 6.5NonlinearInterdependencies .......................... 112 6.6Discussions ................................... 114 7CLUSTERINGELECTROENCEPHALOGRAM(EEG)SIGNALSTOSTUDYMESIALTEMPORALLOBEEPILEPSY(MTLE) ................ 117 7.1Introduction ................................... 117 7.2EpilepsyasaDynamicalBrainDisorder ................... 120 7.3DataInformation ................................ 121 7.3.1MultivariateAnalysisonEEGSignals ................. 121 7.3.2BrainSynchronization ......................... 122 7.4Graph-TheoreticModelingforBrainConnectivity .............. 124 7.4.1Cross{MutualInformation(CMI) ................... 124 7.4.2SpectralPartitioning .......................... 128 7.4.3MaximumCliqueAlgorithm ...................... 130 7.4.4ImplicationsoftheResults ....................... 132 7.5DiscussionandFutureWork .......................... 134 8TREATMENTEFFECTSONELECTROENCEPHALOGRAM(EEG) ..... 137 8.1Introduction ................................... 137 8.2DataInformation ................................ 139 8.3SynchronizationMeasures ........................... 140 8.3.1MutualInformation ........................... 140 8.3.2NonlinearInterdependencies ...................... 143 8.4StatisticTestsandDataAnalysis ....................... 145 8.5ConclusionandDiscussion ........................... 146 REFERENCES ....................................... 158 BIOGRAPHICALSKETCH ................................ 169 7 PAGE 8 Table page 2-1PatientinformationandEEGdescription ...................... 59 2-2PerformanceforD-SVM ............................... 59 3-1PatientsandEEGdatastatisticsforcomplexityanalysis ............. 62 4-1PatientsandEEGdatastatistics .......................... 76 5-1EEGdatadescription ................................ 94 5-2ResultsforD-SVMusing5-foldcrossvalidation .................. 103 7-1Patientinformationforclusteringanalysis ..................... 121 8-1ULDpatientinformation ............................... 140 8-2Topographicaldistributionfortreatmentdecouplingeect(DE:DecoupleElectrode(DE)) 146 8-3Patient1beforetreatmentnonlinearinterdependencies .............. 150 8-4Patient1aftertreatmentnonlinearinterdependencies ............... 151 8-5Patient2beforetreatmentnonlinearinterdependencies .............. 152 8-6Patient2aftertreatmentnonlinearinterdependencies ............... 153 8-7Patient3beforetreatmentnonlinearinterdependencies .............. 154 8-8Patient3aftertreatmentnonlinearinterdependencies ............... 155 8-9Patient4beforetreatmentnonlinearinterdependencies .............. 156 8-10Patient4aftertreatmentnonlinearinterdependencies ............... 157 8 PAGE 9 Figure page 1-1Vagusnervestimulationpulsegenerator ...................... 22 1-2Vagusnervestimulationelectrode .......................... 23 1-3Corticalnervecellandstructureofconnections .................. 24 1-4Membranepotentials ................................. 25 1-5Electricalpotentials ................................. 26 1-6EEGrecordingacquiredbyHansBergerin1929 .................. 27 1-7BasicEEGpatterns ................................. 28 1-8International10-20electrodeplacement ....................... 29 1-9Surbduralelectrodeplacement ............................ 30 1-10Depthelectrodeplacement .............................. 31 2-1Rosslerattractor ................................... 36 2-2XcomponentofRosslersystem ........................... 37 2-3YcomponentofRosslersystem ........................... 38 2-4ZcomponentofRoesslersystem ........................... 39 2-5Lorenzsystem ..................................... 41 2-6XcomponentofLorenzsystem ........................... 42 2-7YcomponentofLorenzsystem ........................... 43 2-8ZcomponentofLorenzsystem ........................... 44 2-9EstimationofLyapunovexponent(Lmax) ...................... 50 2-10TemporalevolutionofSTLmax 51 2-11Phase/AngularfrequencyofLyapunovexponent(max) ............. 52 2-12Threedimensionplotforentropy,angularfrequencyandLmaxduringinterictalstate .......................................... 55 2-13Threedimensionplotforentropy,angularfrequencyandLmaxduringpreictalstate .......................................... 56 3-1Electrodeplacement ................................. 62 9 PAGE 10 .... 67 3-3AverageminimumembeddingdimensionprolesforPatient1(seizure2) .... 68 3-4AverageminimumembeddingdimensionprolesforPatient1(seizure3) .... 69 3-5AverageminimumembeddingdimensionprolesforPatient2(seizure4) .... 70 3-6AverageminimumembeddingdimensionprolesforPatient2(seizure5,6) ... 71 4-132-channeldepthelectrodeplacement ........................ 76 4-2Degreeofnonlinearityduringpreictalstate ..................... 81 4-3DegreeofNonlinearityduringpostictalstate .................... 82 4-4STLmaxandT-indexprolesduringinterictalstate ................ 83 4-5STLmaxandT-indexprolesduringpreictalstate ................. 84 4-6STLmaxandT-indexprolesduringpostictalstate ................ 85 4-7NonlinearitiesacrossrecordingareasduringinterictalstateforULSOZpatients 86 4-8NonlinearitiesacrossrecordingareasduringperictalstateforULSOZpatients 86 4-9NonlinearitiesacrossrecordingareasduringpostictalstateforULSOZpatients. 87 4-10NonlinearitiesacrossrecordingareasduringinterictalstateforBTSOZpatients 88 4-11NonlinearitiesacrossrecordingareasduringpreictalstateforBTSOZpatients 89 4-12NonlinearitiesacrossrecordingareasduringpostictalstateforBTSOZpatients 90 5-1Scalpelectrodeplacement .............................. 93 5-2EEGdynamicsfeatureclassication ......................... 99 5-3Supportvectormachines ............................... 101 5-4Connectivitysupportvectormachine ........................ 102 7-1EEGepochsforRTD2,RTD4andRTD6(10seconds) .............. 126 7-2ScatterplotforEEGepoch(10seconds)ofRTD2vs.RTD4andRTD4vs.RTD6 ......................................... 126 7-3Cross-mutualinformationforRTD4vs.RTD6andRTD2vs.RTD4 ...... 127 7-4Completeconnectivitygraph(a);afterremovingthearcswithinsignicantconnectivity(b) ........................................... 128 7-5Spectralpartitioning ................................. 129 10 PAGE 11 .............................. 131 7-7ElectrodeselectionusingthemaximumcliquealgorithmforCase1 ....... 132 7-8ElectrodeselectionusingthemaximumcliquealgorithmforCase2 ....... 133 7-9ElectrodeselectionusingthemaximumcliquealgorithmforCase3 ....... 133 8-1NonlinearinterdependencesforelectrodeFP1 ................... 147 8-2Pairwisemutualinformationbetweenforallelectrodes-beforev.s.aftertreatment 149 11 PAGE 12 Thebasicmechanismsofepileptogenesisremainunclearandinvestigatorsagreethatnosinglemechanismunderliestheepileptiformactivity.Dierentformsofepilepsyareprobablyinitiatedbydierentmechanisms.Thequanticationforpreictaldynamicchangesamongdierentbraincorticalregionshavebeenshowntoyieldimportantinformationinunderstandingthespatio-temporalepileptogenicphenomenainbothhumansandanimalmodels. Intherstpartofthisstudy,methodsdevelopedfromnonlineardynamicsareusedfordetectingthepreictaltransitions.Dynamicalchangesofthebrain,fromcomplextolesscomplexspatio-temporalstates,duringpreictaltransitionsweredetectedinintracranialelectroencephalogram(EEG)recordingsacquiredfrompatientswithintractablemesialtemporallobeepilepsy(MTLE).Thedetectionperformancewasfurtherenhancedbythedynamicssupportvectormachine(D-SVM)andamaximumcliqueclusteringframework.ThesemethodsweredevelopedfromoptimizationtheoryanddataminingtechniquesbyutilizingdynamicfeaturesofEEG.Thequantitativecomplexityanalysisinmulti-channelintracranialEEGrecordingsisalsopresented.Thendingssuggestthatitispossibletodistinguishepilepsypatientswithindependentbi-temporalseizureonsetzones(BTSOZ)fromthosewithunilateralseizureonsetzone(ULSOZ).Furthermore,fortheULSOZpatients,itisalsopossibletoidentifythelocationoftheseizureonsetzoneinthebrain.Improvingclinician'scertaintyinidentifyingthe 12 PAGE 13 RecentadvancesinnonlineardynamicsperformedonEEGrecordingshaveshowntheabilitytocharacterizechangesinsynchronizationstructureandnonlinearinterdependenceamongdierentbraincorticalregions.Althoughthesechangesincorticalnetworksarerapidandoftensubtle,theymayconveynewandvaluableinformationthatarerelatedtothestateofthebrainandtheeectoftherapeuticinterventions.Traditionally,clinicalobservationsevaluatingthenumberofseizuresduringagivenperiodoftimehavebeengoldstandardforestimatingtheecacyofmedicaltreatmentinepilepsy.EEGrecordingsareonlyusedasasupplementaltoolinclinicalevaluations.Inthelaterpartofthisstudy,aconnectivitysupportvectormachine(C-SVM)isdevelopedfordierentiatingpatientswithepilepsythatareseizurefreefromthosethatarenot.Tothatend,aquantitativeoutcomemeasureusingEEGrecordingsacquiredbeforeandafteranti-epilepticdrugtreatmentisintroduced.OurresultsindicatethatconnectivityandsynchronizationbetweendierentcorticalregionsathigherorderEEGpropertieschangewithdrugtherapy.Thesechangescouldprovideanewinsightfordevelopinganovelsurrogateoutcomemeasureforpatientswithepilepsywhenclinicalobservationscouldpotentiallyfailtodetectasignicantdierence. 13 PAGE 14 1 ].Epilepsyoccursinallages,thehighestincidencesoccurininfantsandinelderly[ 2 ; 3 ].Thecauseofepilepsyresultedfromalargenumberoffactors,includingheadinjury,stroke,braintumor,centralnervoussysteminfections,developmentalanomaliesandhypoxia{ischemia. Thehallmarkofepilepsyisrecurrentseizures.Seizuresaremediatedbyabruptdevelopmentofrhythmicringoflargegroupsofneuronsinthecerebralcortex.Theserhythmicdischargesmaybeginlocallyincertainregionofonecerebralhemisphere(partialseizures)orbeginsimultaneouslyinbothcerebralhemispheres(generalizedseizure).Partialonsetseizuresmayremainconnedwithinaparticularregionofthebrainandcausenochangeinconsciousnessandrelativelymildcognitive,sensory,motororautonomicsymptoms(simplepartialseizures)ormayspreadtocauseimpairedconsciousnessduringtheseizure(complexpartialseizure)alongwithavarietyofmotorsymptoms,suchassuddenandbrieflocalizedbodyjerkstogeneralizedtonic-clonicactivity(secondarygeneralizedseizures).Primarilygeneralizedseizuresinvolvebothhemispheresofthebrain,causealteredinlossofconsciousnessduringtheoccurrence 14 PAGE 15 1. 2. 3. 4. (a) Alteredlevelsofsodium,calcium,ormagnesium(electrolyteimbalance) (b) Kidneyfailurewithincreasedureaintheblood(uremia)orchangesthatoccurwithkidneydialysis (c) Lowbloodsugar(hypoglycemia)orelevatedbloodsugar(hyperglycemia) (d) Loweredoxygenlevelinthebrain(hypoxia) (e) Severeliverdisease(hepaticfailure)andelevationofassociatedtoxins 15 PAGE 16 6. 7. 8. 16 PAGE 17 1. (a) Withmotorsigns i. Focalmotorwithoutmarch ii. Focalmotorwithmarch(Jacksonian) iii. Versive iv. Postural v. Phonatory(Vocalizationorarrestofspeech) (b) Withsomatosensoryorspecialsensorysymptoms(simplehallucinations,e.g.,tingling,lightashes,buzzing) i. Somatosensory ii. Visual iii. Auditory iv. Olfactory v. Gustatory vi. Vertiginous (c) Withautonomicsymptomsorsigns(includingepigastricsensation,pallor,sweating,ushing,piloerectionandpupillarydilatation) (d) Withpsychicsymptoms(disturbanceofhighercerebralfunction).Thesesymptomsrarelyoccurwithoutimpairmentofconsciousnessandaremuchmorecommonlyexperiencedascomplexfocalseizures i. Dysphasic 17 PAGE 18 Dysmnesia iii. Cognitive(e.g.,dreamystates,distortionsoftimesense) iv. Aective(fear,anger,etc.) v. Illusions(e.g.,macropsia) vi. Structuredhallucinations(e.g.,music,scenes) 2. (a) Withimpairmentofconsciousnessatonset i. Withimpairmentofconsciousnessonly ii. Withautomatisms (b) Simplepartialonsetfollowedbyimpairmentconsciousness i. Withsimplefocalfeaturesfollowedbyimpairedconsciousness ii. Withautomatisms 3. (a) Simplepartialseizureevolvingtogeneralizedseizures (b) Complexpartialseizureevolvingtogeneralizedseizures (c) Simplepartialseizureevolvingtocomplexfocalseizuresevolvingtogeneralizedseizures 18 PAGE 19 1. (a) (b) 2. 3. PAGE 20 4. 5. 6. 1.4.1PharmacologicalTreatment 20 PAGE 21 4 ].TheevidencefrombothexperimentalandclinicalstudieswhichsuggeststhatlossofecacyofAEDsmaydevelopduringtheirlong-termuseinaminorityofpatients[ 5 ].Themostcommonmechanismisanincreaseintherateofmetabolismofthedrug.Forinstance,many"rstgeneration"AEDs,includingphenobarbital(PB),phenytoin(PHT),andcarbamazepine(CBZ),stimulatetheproductionofhigherlevelsofhepaticmicrosomalenzymes,causingmorerapidremovalandbreakdownoftheseAEDsfromthecirculation[ 6 ]. 7 ].However,studieshavealsoshownthatonlyindividualwithunilateralonsetzoneproducessignicantreductionsinseizurefrequencywithcurrentsurgicalroutines[ 8 ]. 9 ]basedonpreviousobservationsthatVNSblockedinterictalEEGspiking[ 10 ]ordesynchronizedEEGinthethalamusorcortex[ 11 ; 12 ]incats.ZabaraextendedthisideatodemonstrateanticonvulsantactivityofVNSindogs[ 9 ].Lateron,McLachlandemonstratedsuppressionofinterictalspikesandseizuresby 21 PAGE 22 13 ].Furthermore,chronicintermittentVNShasbeenshowntopreventrecurrenceofepilepticseizuresinmonkeys[ 14 ].Theseobservationsleadtoclinicaltrialstoinvestigatethefeasibility,safety,andecacyofVNSinhumanpatients.Thersthumanimplantwasperformedin1988andtherstrandomizedactivecontrolstudywasperformedin1992.EncouragingresultsfromthersttwopilotstudiesleadtorandomizeddoubleblindclinicaltrialsthatresultedinFDAapprovalofVNSasadjunctivetherapyforintractableepilepsyinJuly,1997.ImplantationoftheNCPsystem(VNSpulsegeneratorandelectricalleads)isnormallyperformedundergeneralanesthesiaasanoutpatientbasis.Theprocedureusuallytakestwohoursandrequirestwoskinincisions;oneintheleftupperchestforhousingthepulsegeneratorinapocketundertheskinandtheotherisinleftneckareaabovethecollarbonetogainaccesstothevagusnerveandplacetwosemi-circularelectrodesandaneutraltetheraroundtheleftvagusnerve(Figure 1-1 ). Figure1-1. Vagusnervestimulationpulsegenerator 22 PAGE 23 Vagusnervestimulationelectrode Longtermfollow-upstudiesshowedthatpreventionofrecurrentseizureswasmaintainedandadverseeventsdecreasedsignicantlyovertime[ 15 ; 16 ].PositronemissiontomographyandfunctionalMRIstudiesshowedthatVNSactivatesorincreasesbloodowincertainareasofthebrainsuchasthethalamus[ 17 ; 18 ].Cerebrospinaluid(CSF)wasanalyzedin16subjectsbefore,3monthsafter,and9monthsafterVNStreatmentGABA(totalandfree)increasedinloworhighstimulationgroups,aspartatemarginallydecreasedandethanolamineincreasedinthehighstimulationgroupsuggestinganincreasedinhibitoryeect[ 19 ].Krahletal.,suggestedthatseizuresuppressioninducedbyVNSmaydependonthereleaseofnorepinephrineandtheyobservedthatacuteorchroniclesionsofthe"Locuscoeruleus"attenuatedVNS-inducedseizuresuppression[ 20 ]. 1.5.1NeuronStates 1-3 istakenfromMalmivuandPlonse(1993)[ 21 ]. 23 PAGE 24 2. Figure1-3. Corticalnervecellandstructureofconnections Theexcitatoryneuronsexcitethetargetneurons.Excitatoryneuronsinthecentralnervoussystemareoftenglutamatergicneurons.Neuronsintheperipheralnervoussystem,suchasspinalmotorneuronsthatsynapseontomusclecells,oftenuseacetylcholineastheirexcitatoryneurotransmitter.However,thisisjustageneraltendencythatmaynotalwaysbetrue.Itisnottheneurotransmitterthatdecidesexcitatoryorinhibitoryaction,butratheritisthepostsynapticreceptorthatisresponsibleforthe 24 PAGE 25 Membranepotentials actionoftheneurotransmitter.Inhibitoryneuronsinhibittheirtargetneurons.Inhibitoryneuronsareofteninterneurons.Theoutputofsomebrainstructures(neostriatum,globuspallidus,cerebellum)areinhibitory.TheprimaryinhibitoryneurotransmittersareGABAandglycine.Modulatoryneuronsevokemorecomplexeectstermedneuronmodulation.Theseneuronsuseneurotransmitterssuchasdopamine,acetylcholine,serotoninandothers[ 22 ].Theeectofsummingsimultaneouspostsynapticpotentialsbyactivatingmultipleterminalsonwidelyspacedareasofthemembraneiscalledspatialsummationandsuccessivepresynapticdischargesfromasinglepresynapticterminal,thistypeofsummationiscalltemporalsummation. 1-4 andgure 1-5 istakenfromMalmivuandPlonse(1993)[ 21 ].Thesourcesofelectricalpotentialcanbecategorizedintothefollowingfourdierentcategorizes: 25 PAGE 26 Electricalpotentials 1. 2. 3. 4. 26 PAGE 27 23 ].SincethentheEEGrecordingshasbeenthemostcommondiagnosistoolforepilepsy.EEGmeasurestheelectricalactivityofthebrain.EEGstudiesareparticularlyimportantwhenneurologicdisordersarenotaccompaniedbydetectablealterationinbrainstructure.ItisacceptedthattheneuronsinthethalamusplayanimportantroleingeneratingtheEEGsignals.Thesynchronicityofthecorticalsynapticactivityreectsthedegreeofsynchronousringofthethalamicneuronsthataregeneratingtheelectricalactivities.However,thepurposesoftheseelectricalactivitiesandEEGoscillationsarelargelyunknown. Figure1-6. EEGrecordingacquiredbyHansBergerin1929 ThecongurationsofEEGrecordingsplayanimportantroleindeterminingthenormalbrainfunctionfromabnormal.ThemostobviousEEGfrequenciesofanawake,relaxedadultwhoseeyesareclosedis8-13Hzalsoknownasthealpharhythm.Thealpharhythmisrecordedbestovertheparietalandoccipitallobesandisknowntobeassociatedwithdecreasedlevelsofattention.Whenalpharhythmarepresented,subjectscommonlyreportthattheyfeelrelaxedandhappy.However,peoplewhonormallyexperiencemorealpharhythmthanusualhavenotbeenshowntobepsychologicallydierentfromthosewithless.AnotherimportantEEGfrequenciesisthebetarhythm,peopleareattentiveto 27 PAGE 28 BasicEEGpatterns anexternalstimulusorarethinkinghardaboutsomething,thealpharhythmisreplacedbylower-amplitude,high-frequency(>13Hz).Thisisbetarhythmoscillations.ThistransformationisalsoknownasEEGarousalandisassociatedwiththeactofpayingattentiontostimuluseveninadarkroomwithnovisualinputs.Atransientisaneventwhichclearlystandsoutagainstthebackground.Asharpwaveisatransientwith70ms{200msinduration.Aspikewaveisatransitionwithlessthan70msinduration.Aspikethatfollowbyaslowwaveiscalledaspike-and-wavecomplex,whichcanbeseeninpatientswithtypicalabsenceseizure.Inthecaseshavingtwoormorespikeoccurinsequenceformingmultiplespikecomplexcallpolyspikecomplex,iftheyarefollowedbyaslowwave,theyarecalledpolyspike-and-wavecomplex.Spikeandsharpwavesare 28 PAGE 29 1-7 istakenfromMalmivuandPlonse(1993)[ 21 ]. 1-8 .Therecordingelectrodesareplacedonthescalpoftheheadandrecordelectricalpotentialdierencesbetweentherecordingelectrodes.However,recordingsacquiredfromscalpareusuallycontaminatedbymultiplesourcesofartifactssuchasmovementartifacts,chewingartifacts,eyemovement,vertexwavesandsleepspindles,etc.Theinternational10-20electrodeplacementsystemiscommonlyusedforroutinescalpEEGrecording[ 24 ].Figure 1-8 istakenfromMalmivuandPlonse(1993)[ 21 ]. Figure1-8. International10-20electrodeplacement 1-9 .Itrequiressurgicalproceduretoplacethesubduralrecordingelectrodesandtherisksofinfectionincreasewiththeamount 29 PAGE 30 Figure1-9. Surbduralelectrodeplacement 1-10 .Thedepthrecording 30 PAGE 31 Figure1-10. Depthelectrodeplacement 25 ].Likemanyphenomenaoccurinnature,thereexistcertainbuildupperiodpriortosomemajorevents. 31 PAGE 32 26 { 31 ].Fromabovendings,itisreasonabletohypothesizethataseizureisstartingfromsmallabnormallydischargingfromneuronsthatrecruitandentraintheneighboringneuronsintoalargerorfullseizure.Thishypothesisisparticularlyclearforthefocalonsetseizures.Theserecruitment,entrainmentandtransitionphenomenatakeplaceinabrainstate,thepreictalstate.Recently,forpatientswithMTLEseveralauthorshaveshownitispossibletodetectthepreictaltransitionsusingquantitativeEEGanalysis[ 32 { 39 ]. Inchapter2,weinvestigatedtheexistenceofpreictalstates.WequantiedanddetectedthechangesinEEGdynamicsthatareassociatedwithpreictalstateinEEGrecordings.Threedierentdynamicalmeasureswereused:1.LargestLyapunovexponent,ameasureisknownformeasuringthechaoticityofthesteadystateofadynamicalsystem,2.PhaseinformationofthelargestLyapunovspectrum,basedontheoryfromtopologyandinformationtheoryand3.Approximateentropy,amethodformeasuringtheregularityorpredictabilityoftimeseries.Thedynamicssupportvectormachine(D-SVM)wassubsequentlyintroducedforimprovingtheperformanceofthepreictaldetection.Inchapter3,weanalyzedthecomplexityofEEGrecordingsusingmethodsdevelopedfromnonlineardynamicsandshowedtheEEGcomplexitychangespriortotheseizureonsets.Inthechapter4,weinvestigatedthedierencesinnonlinearcharacteristicsbetweenpatientswithindependentbi-temporalseizureonsetzone(BTSOZ)andpatientswithunilateralseizureonsetzone(ULSOZ).Inchapter5,weintroducedthesupportvectormachinestoclassifyEEGrecordingsacquiredfromseizurefreeandnoneseizurefreepatients.Inchapter6,wediscusseddierentmethodsforspatiotemporalEEGtimeseriesanalysis.Inchapter7,weproposedamaximumcliqueframeworktostudythebraincorticalnetworks.Inthechapter8,westudiedthemedicaltreatmenteectsonthestructureofbraincorticalnetworks. 32 PAGE 33 Thedevelopmentofpreictaltransitionscanbeconsideredasasuddenincreaseofsynchronousneuronalringinthecerebralcortexthatmaybeginlocallyinaportionofonecerebralhemisphereorbeginsimultaneouslyinbothcerebralhemisphere.Byobservingtheoccurrenceofepilepticseizures,itisreasonabletobelievethattherearemultiplestatesexistinaepilepticbrainandthesequencesofthestatesarenotdeterministic.ThepreictaltransitionsaredetectableEEGdynamicalchangesbyapplyingmethodsdevelopedfromnonlineardynamics.Severalgroupshavereportedthatseizuresarenotsuddentransitionsinandoutoftheabnormalictalstate;instead,seizuresfollowacertaindynamicaltransitionthatdevelopsovertime[ 32 { 39 ]see[ 40 ; 41 ]forreview.InanstudyofPijnetal.in1991,authorswereabletodemonstratedecreaseinthevalueofcorrelationdimensionatseizureonsetintheratmodel.Inearly1990s,Iasemidisetal.,rstestimatedthelargestLyapunovexponentandreportedseizurewasinitiateddetectabletransitionperiodbyanalyzingspatiotemporaldynamicsoftheEEGrecordings;thistransitionprocessischaracterizedby:(1)progressiveconvergenceofdynamicalmeasuresamongspecicanatomicalareas\dynamicalentrainment"and(2)followingtheovershotbrainresettingmechanismduringpostictalstate.Martinerieetal.,(1998) 33 PAGE 34 42 ].Mormannetal.,(2003)detectedthepreictalstatebasedondecreasein\synchronization"measures[ 43 ]. Thebasictextofnonlineardynamicsandnonlineardynamicalmodelsarepresentedinthefollowingsections.Nonlineardynamicalmeasuresnamely(thelargestLyapunovexponent(Lmax),Phase/Angularfrequency(),Approximateentropy(ApEn))wereusedfordetectingthepreictaltransitionsinintracranialEEGrecordingsacquiredfrompatientswithintractableMTLE.Sincetheunderlyingdynamicsofpreictaltransitionsischangingfromcasetocase,thisdemandssophisticateanalyticaltoolswhichhavetheabilityforidentifyingthechangesofbraindynamicswhenpreictaltransitionsoccur.Thepreictaldetectionperformancewasfurtherimprovedbyproposeddynamicssupportvectormachine(D-SVM),aclassicationmethoddevelopedfromoptimizationtheoryanddataminingtechniques.Thedetectionperformancesweresummarizedinthelaterpartofthischapter. 34 PAGE 35 1. 2. whereTdenotestheperiodofthiscycle. 3. 4. 2-1 2-5 forRosslerandLorenzattractor). Recallanattractorisasetofstatevariables;geometricallyanattractorcanbeapoint,acurve,amanifold,orevenacomplicatedsetwithafractalstructureknownasthe\strangeattractor".Describingtheseattractorshasbeenoneoftheachievementsofchaos 35 PAGE 36 Rosslerattractor theory.Inthefollowingsections,severalmethodsforqualifyingthedynamicalattractorsaredescribed;resultsonapplicationforrealworldEEGrecordingarealsoincluded. 44 ] _x=(y+z); _y=x+ay; _z=b+xzcz: (2{6) (x;y;z)2<3 Initialconditions:[00.01-0.01]. 36 PAGE 37 XcomponentofRosslersystem 37 PAGE 38 YcomponentofRosslersystem 38 PAGE 39 ZcomponentofRoesslersystem 39 PAGE 40 45 ].ThestrangeattractorinthiscaseisafractalofHausdordimensionbetween2and3.Grassberger(1983)hasestimatedtheHausdordimensiontobe2.060.01andthecorrelationdimensiontobe2.050.01[ 46 ]. (2{7) _y=x(z)y _z=xyz (2{10) (x;y;z)2<3 Initialconditions:[00.01-0.01]; Length:40seconds; Samplerate:50Hz. 40 PAGE 41 Lorenzsystem 41 PAGE 42 XcomponentofLorenzsystem 42 PAGE 43 YcomponentofLorenzsystem 43 PAGE 44 ZcomponentofLorenzsystem 44 PAGE 45 Forregularattractors,irrespectivetotheoriginofthesphere,thedimensionwouldbethedimensionoftheattractor.Butforachaoticattractor,thedimensionvariesdependingonthepointatwhichtheestimationisperformed.Ifthedimensionisinvariantunderthedynamicsoftheprocess,wewillhavetoaveragethepointdensitiesoftheattractoraroundit.Forthepurposeofidentifyingthedimensioninthisfashion,wendthenumberofpointsy(k)withinaspherearoundsomephasespacelocationx.Thisisdenedby: whereistheHeavisidestepfunctionsuchthat(n)=0forn<0,(n)=1forn0 Thiscountsallthepointsontheorbity(k)withinaradiusrfromthepointxandnormalizesthisquantitybythetotalnumberofpointsNinthedata.Also,weknowthatthepointdensity,(x),onanattractordoesnotneedtobeuniformforastrangeattractor.Choosingthefunctionasn(x;r)q1anddeningthefunctionC(q;r)oftwovariablesqandrbythemeanofn(x;r)q1overtheattractorweightedwiththenatural 45 PAGE 46 ThisC(q;r)iswellknowncorrelationintegralandthefractaldimensionofthesystemisestimated: forsmallarthatthefunctionlog[C(q,r)]behavelinearlywithlog[r]fortruedimension. CorrelationDimensionhasbeenshowntohavetheabilityincapturingthepreictaltransitioninmanystudies.Forexample,A.BabloyantzandA.Destexhe,(1986)showntheexistenceofchaoticattractorinphasespacefromEEGacquiredfromanapatientwithabsenceseizure[ 47 ].Pijnetal.,(1997)showedintemporallobeepilepsypatientsthatepilepticseizureactivityoften,butnotalways,emergesasalow-dimensionaloscillation[ 48 ].Itwasalsofoundthatcorrelationdimensiondecreasesindeepsleepstages,thusreectingasynchronizationoftheEEG.AdecreaseincorrelationdimensionhasbeenrelatedtotheabnormalsynchronizationbehaviorsonEEGrecordingsinepilepsyandotherpathologiessuchasAlzheimers,dementia,Parkinson[ 48 { 50 ]. 46 PAGE 47 log[]: Informationdimensionisdenedas: log(r);(2{17)D2D1D0ifelementsofthefractalisequallylikelytobevisitedinthestatespace. wheremistheembeddingdimension.Thetransformfunction#mustbeunique(i.e.,hasnoselfintersection).Whitney(1936)provedatheoremwhichcanalsobeusedforndingtheembeddingdimension[ 51 ].#:M!<2d+1;#embeddingisanopenanddensesetinthespaceofsmoothmaps. 47 PAGE 48 andthedurationofeachembeddingvectoris Amuchmoregeneralsituationfortime-laggedvariablesconstituteanadequateembeddingprovidedthemeasuredvariableissmoothandcoupledtoalltheothervariablesisprovedbyTakens,andthenumberoftimelagisatleast2d+1[ 52 ].#:M!<2d+1isanopenanddensesetinthespaceofpairsofsmoothmaps(f,h),wherefisthedynamicalsystemmeasurebyfunctionh. 53 ].Foradynamicalsystemastimeevolvesthesphere 48 PAGE 49 54 ].Forandimensionalsystem,astimeevolves,theorderLyapunovexponentiscorrespondingtothemostexpandedtothemostcontractedprincipalaxes. Iasemidisetal.rstusedthemaximumLyapunovexponenttoshowtheEEGrecordingsexhibitabrupttransientdropsinchaoticitybeforeseizureonset[ 55 { 61 ].ThemaximumLyapunovexponentisdenedby: jXi;j(0)j;(2{22) whereti=t0+(j1)t,withandi2[1;Na]andt2[0;t],tismaximumevolutiontimeforXi;j. Xi;j(0)=X(ti)X(tj);(2{24) isperturbationoftheducialorbitatti,and Xi;j(t)=X(ti+t)X(tj+t);(2{25) istheevolutionofXi;j(0)aftert. TheX(ti)isvectoroftheducialtrajectoryt(X(t0)),wheret=t0+(i1)t,X(t0)=x(t0;:::;X(t0+(p1))T,andX(tj)isaproperlychosenvectoradjacenttoX(ti)inthestatespace.Naisnecessarynumberofiterationsforthesearchthroughreconstructedstatespace(withembeddingdimensionpanddelay),fromNdatapoints 49 PAGE 50 EstimationofLyapunovexponent(Lmax) anddurationT.IfDtisthesamplingperiod,then Iftheevolutiontimetisgiveninsecond,thentheunitofLisbit=sec.TheselectionofpisbasedfromTakens'embeddingtheoremandwasestimatedfromepochesduringictalEEGrecordings.Takens'embeddingtheoremisdened: usingtheabovedenedfx,evenifoneonlyobservesonevariablex(t)fort!1,onecanconstructanembeddingofthesystemintoap=2m+1dimensionalstatespace. ThedimensionoftheictalEEGattractorisfoundbetween2to3inthestatespace.ThereforeaccordingtoTakens'theembeddingdimensionwouldbeatleastp=23+1=7.Theselectionofischosenassmallaspossibletocapturethehighestfrequencycomponentinthedata. 50 PAGE 51 TemporalevolutionofSTLmax =1 whereNisthetotalnumberofphasedierencesestimatedfromtheevolutionofX(ti)toX(ti+(t))inthestatespace,and i=jarccos(X(ti)X(ti+t) 62 ].Itcandierentiatebetweenregularandirregulardataininstanceswheremomentstatistics(e.g.meanandvariance)approachesfailtoshowasignicantdierence.Applicationsincludeheartrateanalysis 51 PAGE 52 Phase/AngularfrequencyofLyapunovexponent(max) inthehumanneonateandinepilepticactivityinelectrocardiograms(Diambra,1999)[ 63 ].Mathematically,aspartofageneraltheoreticalframework,ApEnhasbeenshowntobetherateofapproximatingaMarkovchainprocess[ 62 ].Mostimportantly,comparedApEnwithKolmogrov-Sinai(K-S)Entropy(Kolmogrov,1958),ApEnisgenerallyniteandhasbeenshowntoclassifythecomplexityofasystemviafewerdatapointsviatheoreticalanalysisofbothstochasticanddeterministicchaoticprocessesandclinicalapplications[ 62 ; 64 { 66 ].HereIgivebriefdescriptionaboutApEncalculationforatimeseriesmeasuredequallyintimewithlengthn.SupposeS=s1;s2;:::;snisgivenandusethemethodofdelayweobtainthedelayvectorx1;x2;:::;xnm+1inRm: 52 PAGE 53 m(rf)=nm+1Xi=1lnCmi(rf) Finallytheapproximateentropyisgivenby: Theparameterrfcorrespondstoanapriorixeddistancebetweenneighboringtrajectoryandrfischosenaccordingtothestandarddeviationestimatedfromdata.Hence,rfcanbeviewedasalteringlevelandtheparametermistheembeddingdimensiondeterminingthedimensionofthephasespace.Heuristically,ApEnquantiesthelikelihoodthatsubsequencesinSofpatternsthatarecloseandwillremaincloseonthenextincrement.ThelowerApEnvalueindicatesthatthegiventimeseriesismoreregularandcorrelated,andlargerApEnvaluemeansthatitismorecomplexandindependent. D-SVMperformsclassicationbyconstructinganN-dimensionalhyperplanethatseparatesthedataintotwodierentclasses.ThemaximalmarginclassierruleisusedtoconstructtheD-SVM.TheobjectiveofmaximalmarginD-SVMistominimizethebond 53 PAGE 54 Ahyperplane(w;b)iscalledacanonicalhyperplanesuchthat 2kwk2+C subjectto whereCisaparametertobechosenbytheuser,wisthevectorperpendiculartotheseparatinghyperplane,bistheosetandarereferringtotheslackvariablesforpossibleinfeasibilityoftheconstraints.Withthisformulation,oneswantstomaximizethemarginbetweentwoclassesbyminimizingkwk2.Thesecondtermoftheobjectivefunctionisusedtominimizethemisclassicationerrorsthataredescribedbytheslackvariablesi.IntroducingpositiveLagrangemultipliersitotheinequalityconstraintsinDSVMmodel,weobtainthefollowingdualformulation: min1 2i=1Xnj=1XnyiyjijxixjnXi=1i(2{39) s.t. 0iC;i=1;:::n:(2{41) Thesolutionoftheprimalproblemisgivenbyw=Piiyixi,wherewisthevectorthatisperpendiculartotheseparatinghyperplane.Thefreecoecientbcanbefound 54 PAGE 55 2-12 and 2-13 showthe3Dplotforforentropy,angularfrequency,andLmaxduringinterictal(100datapointsdynamicalfeatures2hourspriortoseizureonset)andpreictalstate(100datapointsdynamicalfeaturessampled2minspriortoseizureonset). Figure2-12. Threedimensionplotforentropy,angularfrequencyandLmaxduringinterictalstate 55 PAGE 56 Threedimensionplotforentropy,angularfrequencyandLmaxduringpreictalstate wherek 56 PAGE 57 TN+1;t T];(2{44) whereNdenotesnumberofLmaxinthemovingwindowand^ijtdenotesthestandarddeviationofthesampleDijwithinw(t).Asymptotically,Tij(t)followsthet-distributionwithN1degreeoffreedom.WeusedN=30(i.e.averagesof30paireddierencesofvaluesfromdynamicalprolespermovingwindow). AclassicationresultisconsideredtobetruepositiveiftheD-SVMclassifyapreictalEEGepochasapreictalEEGsample. AclassicationresultisconsideredtobetruenegativeiftheD-SVMclassifyainterictalEEGepochasainterictalEEGsample. 57 PAGE 58 AclassicationresultisconsideredtobetruepositiveiftheD-SVMclassifyainterictalEEGepochasapreictalEEGsample. AclassicationresultisconsideredtobetruepositiveiftheD-SVMclassifyapreictalEEGepochasainterictalEEGsample. TheperformanceoftheD-SVMisevaluateusingsensitivityandspecicity: Sensitivity=TP 58 PAGE 59 PatientinformationandEEGdescription Patientno.GenderAgeFocusregion(s)NumberofseizureLengthofrecording 1F45R.H128days2F42R.H/L.H1812days3M30R.H65days4M39R.H/L/H43days5M52R.H95days6F65R.H76days Table2-2. PerformanceforD-SVM Patientno.SensitivitySpecicity 191.3%91.4%293.2%95.2%390.9%91.7%494.1%96.4%584.7%92.0%689.3%94.5% resultsalsoconrmthatthepreictalandinterictalbrainstatearedierentiableusingapproachesdevelopedfromnonlineardynamics. 59 PAGE 60 67 { 71 ].Itisknownthatadynamicalsystemwithddegreeoffreedommayevolveonamanifoldwithalowerdimension,sothatonlyportionsofthetotalnumberofdegreeoffreedomareactuallyactive.Forasimplesystemwithlimitcycles,itisobviousthattime-delayembeddingproduceanequivalentreconstructionofthetruestate.AccordingtoembeddingtheoremfromWhitney(1936),anarbitraryD-dimensioncurvedspacecanbemappedintoaCartesian(rectangular)spaceof2d+1dimensionswithouthavinganyselfintersections,hencesatisfyingtheuniquenessconditionforanembedding[ 51 ].Saueretal.(1991)generalizedWhitneysandTakens'theoremtofracturalattractorswithdimensionDfandshowedtheembeddingspaceonlyneedtohaveadimensiongreaterthan2Df[ 72 ].Althoughitispossibleforafractaltobeembeddedinanotherfractal,weonlyconsidertheintegerembedding.Takensdelayembeddingtheorem 60 PAGE 61 52 ].Fortheabovereasons,weemployedamethodproposedbyCao(1997)toestimatetheminimumembeddingdimensionofEEGtimeseries[ 73 ].Likesomeotherexitingmethods,Caosmethodisalsoundertheconceptsoffalse-nearest-neighborsThefalse-nearest-neighborsutilizedonthefactthatifthereconstructionspacehasnotenoughdimensions,thereconstructionwillperformaprojection,andhencewillnotbeanembeddingofthedesiredsystem[ 74 ].Asaofresultofgivingatolowembeddingdimensionwhileprocessingtheembeddingprocedure,twopointswhichisfarawayinthetruestatespacewillbemappedintocloseneighborinthereconstructionspace.Thesearethenthefalseneighbors.Caosmethoddoesnotrequirelargeamountofdatapoints,isnotsubjectiveanditisnottime-consumingndtheproperminimumembeddingdimension.TheEEGrecordingswasdividedintonon-overlappingsingleelectrodesegmentsof10.24sduration,eachofwhichwasestimatedfortheminimumembeddingdimension.UndertheassumptiontheEEGrecordingswithineach10.24sdurationwasapproximatelystationary[ 56 ],weevaluatedtheunderliningdynamicalbehaviorbylookingattheminimumembeddingdimensionovertime. Theremainingofthischapterisorganizedasfollows.InSections.2and3,wedescribethedatainformationandexplainthealgorithmforestimatingtheminimumembeddingdimensionestimation.Theresultsfromtwopatientswithatotalnumberofsixtemporallobeepilepsy(TLE)aregiveninSection4.InSection5,wediscusstheresultsofourndingswithrespecttheuseofthisalgorithmandthefunctionofnonlineardynamicalmeasurementsintheareaofseizurecontrol. 61 PAGE 62 PatientsandEEGdatastatisticsforcomplexityanalysis Patient#GenderAgeFocus(RH/LH)LengthofEEG(hr.)Numberofseizure P1M19RH20h37m05s3P2M33LH09h43m57s3 transverseandBlateralviewsofthebrain,illustratingthedepthandsubduralelectrodeplacementforEEGrecordingsaredepicted.Subduralelectrodesripsareplacedovertheleftorbitfrontal(LOF),rightorbitofrontal(ROF),leftsubtemporal(LST),andrightsubtemporal(RST)cortexseeFigure 3-1 .TheEEGrecordingdataforepilepsypatientswereobtainedaspartofpre-surgicalclinicalevaluation.TheyhadbeenobtainedusingaNicoletBMSI4000and5000recordingsystem,usings0.1Hzhigh-passanda70hzlowpasslter.Eachrecordingcontainsatotalnumberof28to32intracranialelectrodes(8subduraland6hippocampaldepthelectrodesforeachcerebralhemisphere).Priortostorage,thesignalsweresampledat200Hzusingananalogtodigitalconverterwith10bitsquantization.Therecordingswerestoreddigitallyontohighdelityvideotype.Twoepilepsysubjects(seeTable 3-1 )wereincludedinthisstudy. Figure3-1. Electrodeplacement 62 PAGE 63 75 ].Theconceptofmutualinformationisgivenasbelow Mutualinformationisoriginatedfrominformationtheoryandithasbeenusedformeasuringinterdependencebetweentwoseriesofvariables.LetusdenotethetimeseriesoftwoobservablevariablesasX=fxigNi=1andY=fyjgNj=1,whereNisthelengthoftheseriesandthetimebetweenconsecutiveobservations(i.e.samplingrate)isxed.Themutualinformationbetweenobservationsxiandyiisdenedas: wherePx;y(xi;yj)isthejointprobabilitydensityofxandyevaluatedat(xi;yj)andPx(xi),Py(yj)arethemarginalprobabilitydensitiesofxandyevaluatedatxiandyjrespectively.Theunitofmutualinformationisinbit,whenbased2logarithmistaken. Ifxandyarecompletelyindependent,thejointprobabilitydensityPx;y(xi;yj)equalstotheproductofitstwomarginalprobabilitiesandthemutualinformationbetween 63 PAGE 64 FraserandSwinney(1986)showedonecouldobtaintheaveragemutualinformationbetweentwotimeseriesSandQwithlengthns1;s2;:::;snandq1;q2;:::qnusingtheentropiesH(S),H(Q)andH(S;Q).Supposewehaveobservedtherstseriesofinterestwithasetofnoutcomess1;s2;:::;sn;andeachoutcomeisassociatedwithprobabilitiesPs(s1);Ps(s2);:::Ps(sn)andsameasQ. Theaverageamountofuncertaintythatameasurementofsreducestheuncertaintyofqisgiven Inotherwords,\Byknowingameasurementofs,howmanybitsonaveragecanbepredictedaboutq?" SupposeavariablevisinvestigatedbybeingsampledwithsamplingintervalTs.LetsuchprocessbethecontextofsystemSandsystemQ,letsbethemeasurementofvattimet,andletqbethemeasurementattimet+Ts.UsingthesemeasurementtodenesystemsSandQ,mutualinformationI(Q;S)canbecalculated.Thus,mutualinformationbecomesafunctionofTs.Forthisproblem,mutualinformationwillbethenumberofbitsofv(t+Ts)thatcanbepredicated,onaverage,whenv(t)isknown.OnewantstopickTsshouldbechosensothatv(t+Ts)isasunpredictableaspossible.Maximumunpredictabilityoccursatminimumofpredictability;thatis,attheminimuminthemutualinformation.Becauseoftheexponentialdivergenceofchaotictrajectories, 64 PAGE 65 52 ].Thereareseveralclassicalalgorithmsusedtoobtaintheminimumembeddingdimension[ 76 ; 74 ; 77 ].Theclassicalapproachesusuallyrequirehugecomputationpowerandvastamongofdata.Anotherlimitationofthesealgorithmsisthattheyusuallysubjectivetodierenttypesofdata.WeevaluatedtheminimumembeddingdimensionoftheattractorsfromtheEEGbyusingCaosmethod.Thenotionsherefollowed\Practicalmethodfordeterminingtheminimumembeddingdimensionofascalartimeseries".Supposethatwehaveatimeseries(x1;x2;x3;:::;xN).Applyingthemethodofdelayweobtainthetimedelayvectorasfollows: wheredistheembeddingdimensionandisthetime-delayandyi(d)meanstheithreconstructedvectorwithembeddingdimensiond.Similartotheideaofthefalsenearestneighbormethod,dening kyi(d)yn(i;d)(d)k;i=1;2;:::;Nd(3{7) 65 PAGE 66 3-2 3-3 3-4 3-5 and 3-6 showtypicaltheminimumembeddingdimensionovertimeforsixseizures.Onecanobservethebehavioroftheaverageminimumembeddingdimensionovertimeforsixbraincorticalregions.Theminimumembeddingdimension 66 PAGE 67 AverageminimumembeddingdimensionprolesforPatient1(seizure1) showedstableduringtheinterictalstate.Inotherwords,theunderlyingdegreeoffreedomisuniformlydistributedovertheinterictalstateintheEEGrecordings.Theresultsindicatedthelowestminimumembeddingdimensionwerefoundwithintheepilepticzoneduringinterictalstate(theRSTelectrodesinFigs.2,3,and4;theLTDelectrodesinFigs.5and6).ThecomplexityoftheEEGrecordingsfromtheepilepticregionislowerthanthatfromthebrainregions.Thevaluesoftheminimumembeddingdimensionfromallbrainregionsstartdecreaseandconvergetoalowervalueasthepatientproceedfrominterictaltoictalstate.Theunderliningdynamicalchangesbeforeenteringictalperiodwereconsistentlydetectedbythealgorithm. 67 PAGE 68 AverageminimumembeddingdimensionprolesforPatient1(seizure2) Itiscomputationallyecientandcertainlylesstimeconsumingcomparedtosomeclassicalproceduresforestimatingembeddingdimensionestimation.TheresultsofthisstudyconrmthatitispossibletopredictanseizurebasedonnonlineardynamicsofmultichannelintracranialEEGrecordings.Formajorityofseizuresthespatiotemporaldynamicalfeaturesofthepreictaltransitionaresimilartothatoftheprecedingseizure.Thissimilaritymakesitpossibletoapplyoptimizationtechniquestoidentifyelectrodesitesthatwillparticipateinthenextpreictaltransition,basedontheirbehaviorduringthepreviouspreictaltransition.Atpresenttheelectrodeselectionproblemsweresolvedecientlyandthesolutionswereoptimallyattained.However,futuretechnologymayallowphysicianstoimplantthousandsofelectrodesitesinthebrain.Thisprocedurewillhelpustoobtainmoreinformationandallowtohaveabetterunderstandingabouttheepilepticbrain.Therefore,inordertosolveproblemswithlargernumberofrecordingelectrodes,thereisaneedtodevelopcomputationallyfastapproachesforsolving 68 PAGE 69 AverageminimumembeddingdimensionprolesforPatient1(seizure3) large-scalemulti-quadratic0{1programmingproblems.OurresultsalsoarecompatiblewiththendingsaboutthenatureoftransitionstoictalstateininvasiveEEGrecordingsfrompatientswithseizuresofmesialtemporalorigin.Thedevelopmentofmulti-quadratic0{1programmingmodelingisinprogress. 69 PAGE 70 AverageminimumembeddingdimensionprolesforPatient2(seizure4) 70 PAGE 71 AverageminimumembeddingdimensionprolesforPatient2(seizure5,6) 71 PAGE 72 72 PAGE 73 1 ].Epilepsymaybetreatedwithdrugs,surgery,aspecialdiet,oranimplanteddeviceprogrammedtostimulatethevagusnerve(VNStherapy).ForthepatientwhodoesnotreacttoAnti{epilepticdrugs(AEDs)orothertypesofthetremens,epilepsysurgeryisoftenconsideredbecauseitoersthepotentialforcureofseizuresandsuccessfulpsychosocialrehabilitation.Theusefulnessofresectivesurgeryforthetreatmentofcarefullyselectedpatientswithmedicallyintractable,localization-relatedepilepsyisclear.Seizure-freeratefollowingtemporallobectomyareconsistently65%to70%inadults.Epilepsysurgeryliesoncarefullyevaluationofthecandidatesforsurgery,surgicalinterventionmaybecarriedoutwithahighprobabilityofsuccessiftheareaofseizureonsetisconsistentlyandrepeatedlyfromthesameportionofthebrain.Thefocuslocalizationprocedurebecomesoneofthemostimportanttasksinthepre-surgicalexamination.Someseizuresareresultedfromcorticaldamage.Neuroimagingcanhelpinidentifyingandlocalizingthedamageregionsinthebrainandtherefore,thefocus.Currently,brainmagneticresonanceimaging(MRI)providesthebeststructuralimagingstudy.However,themostcommontoolinepilepsydiagnosisiselectroencephalogram(EEG).TheEEGrecordingsreectinteractionsbetweenneuronsinthebrain.Forroutineper-surgicalevaluation,thepatienthastostayintheepilepsymonitoringunit(EMU)for4to5daysandisexpectedtoobtain4seizuresintherecording.However,insomecasesthehospitalstaymaybelongerinordertohaveenoughseizurestoidentifythefocusarea.Therefore,aecientlymethodfordeterminingthesuitabilityofapatientbasedontheanalysisoftheshorterEEGrecordingswouldnotonlyimprovetheoutcomeoftheseizurecontrolbutalsoreducethepatients'nancialburdenforthepre-surgicalevaluationprocedures.Ingeneral,EEGcapturesthespatiotemporalinformationoftheunderlyingneuronsactivitiesnearbytherecordingelectrodes.ItisacceptedthatEEGrecordingcontainsnon-linearmechanismsatmicroscopiclevel.ManystudieshaveshownthepresencesofnonlinearityintheEEGrecordingsfrombothhumans 73 PAGE 74 78 ].Andrzejaketal.(2006)showedbyfocusingonnonlinearityandacombinationofnonlinearmeasureswithsurrogatesappearsasthekeytoasuccessfulcharacterizationofspatiotemporaldistributionofepilepticprocess[ 71 ].B.Weberetal.(1998)alsoshownevidencesfortheusefulnessofnonlineartimeseriesanalysisforthecharacterizationofthespatio-temporaldynamicsoftheprimaryepileptogenicareainpatientswithTLE[ 79 ].UsingcorrelationdimensionK.Lehnertzetal.(1995)reportedthevarianceoftheEEGdimensionduringinterictalallowedtheprimaryepileptogenicareatobecharacterizedinexactagreementwiththeresultsofthepresurgicalwork-up[ 49 ].ManyresearchersalsohaveshownthattheintracranialEEGrecordingsexhibitcertaincharacteristicsthataresimilartochaoticsystems.Forexample,Sackellares,Iasemidisetal.rstusedthemaximumLyapunovexponent,ameasureofchaoticity,toshowtheEEGrecordingsexhibitabrupttransientdropsinchaoticitybeforeseizureonset[ 32 ; 60 ; 80 ].Byfollowingtheconceptofspatiotemporaldynamicalentrainment(i.e.,similardegreeofchaoticitybetweentwoEEGsignals),thisgroupfurtherreportedthat,duringtheinterictalstate,thenumberofrecordingsitesentrainedtotheepileptogenicmesialtemporalfocuswassignicantlylessthanthatofthehomologouscontralateralelectrodesites[ 57 ].TheseresultsuggestedthatitispossibletoidentifytheepileptogenicfocusbyexaminingthedynamicalcharacteristicsoftheinterictalEEGsignals.Therstpartofthisstudy,aphaserandomizationsurrogatedatatechniquewasusedtogeneratesurrogateEEGsignals.ByrandomizingthephaseoftheFourieramplitudes,allinformationwhichisnotcontainedinthepowerspectrumislost.ThesurrogateEEGwillhavethesamelinearpropertiesthusthesamepowerspectrumandequalcoecientsofalinearautoregressive(AR)model.IftheoriginaldistributionofLmaxissignicantlydierentfromitssurrogate,itwillbetheevidenceforthenonlinearity.Inthesecondpartofthisstudy,eightadultpatientswithtemporallobeepilepsywere 74 PAGE 75 4.2.1EEGDescription 4-1 ainferiortransverseviewsofthebrain,illustratingapproximatedepthandsubduralelectrodeplacementforEEGrecordingsaredepicted.Subduralelectrodestripsareplacedovertheleftorbitofrontal(LOF),rightorbitofrontal(ROF),leftsubtemporal(LST),andrightsubtemporal(RST)cortex.Depthelectrodesareplacedinthelefttemporaldepth(LTD)andrighttemporaldepth(RTD)torecordhippocampalactivity. EEGrecordingsobtainedfromeightpatientswithtemporallobeepilepsywereincludedinthisstudy.SeeTable 4-1 .Fivepatientswereclinicallydeterminedtohaveunilateralseizureonsetzone(ULSOZ)andtheremainingthreepatientsweredeterminedtohaveindependentbi-temporalseizureonsetzone(BTSOZ).Foreachpatient,threeseizureswereincludedintheEEGrecordings.Segmentsfrominterictal(atleastonehourbeforetheseizure),preictal(immediatelybeforetheseizureonset)andpostictal(immediatelyaftertheseizureoset)timeintervalscorrespondingtoeachseizureweresampledfortestingthehypothesis.Twoelectrodesfromeachbrainareawereincluded,atotalof12electrodeswereanalyzedforeachpatient.TheEEGrecordingsweresampledusingamplierswithinputrangeof0.6mV,andafrequencyrangeof0.5{70Hz.Therecordingswerestoreddigitallyonvideotapeswithasamplingrateof200Hz,usingan 75 PAGE 76 PatientsandEEGdatastatistics Patient#GenderAgeFocus(RH/LH)LengthofEEG(hr.) P1M19RH6.1P2M45RH5.4P3M41RH5.8P4F33RH5.3P5F38RH6.3P6M44RH/LH5.5P7F37RH/LH4.6P8M39RH/LH5.4 analogtodigital(A/D)converterwith10bitquantization.Inthisstudy,alltheEEGrecordingswereviewedbytwoindependentboardcertiedelectroencephalographers. Figure4-1. 32-channeldepthelectrodeplacement 76 PAGE 77 55 ]. 81 ].Therearethreedierentproceduresforsurrogatedata. 1. Surrogatesarerealizationsofindependentidenticallydistributed(iid)randomvariableswiththesamemean,variance,andprobabilitydensityfunctionastheoriginaldata.Theiidsurrogatesweregeneratedbyrandomlypermutingintemporalorderthesamplesoftheoriginalseries.Thisshuingprocesswilldestroythetemporalinformationandthusgeneratedsurrogatesaremainlyrandomobservationdrawn(withoutreplacement)fromthesameprobabilitydistributionasoriginaldata. 2. Fouriertransform(FT)surrogatesareconstrainedrealizationsoflinearstochasticprocesseswiththesamepowerspectraastheoriginaldata.FTsurrogateserieswereconstructedbycomputingtheFToftheoriginalseries,bysubstitutingthephaseoftheFouriercoecientswithrandomnumbersintherangewhilekeepingunchangedtheirmodulus,andbyapplyingtheinverseFTtoreturntothetimedomain.Torendercompletelyuncoupledthesurrogatepairs,twoindependentwhitenoiseswhereusedtorandomizetheFourierphases. 3. Autoregressive(AR)surrogatesaretypicalrealizationsoflinearstochasticprocesseswiththesamepowerspectraastheoriginalseries.BygeneratingaGaussiantimeserieswiththesamelengthasthedata,andreorderedittohavethesamerankdistribution.TaketheFouriertransformofthisandrandomizethephases(FT).Finally,thesurrogateisobtainedbyreorderingtheoriginaldatatohavethesame 77 PAGE 78 Inthispresentedstudy,weemployedthesecondalgorithmtogeneratethesurrogatedata.Byshuingthephasesbutkeepingtheamplitudeofthecomplexconjugatepairsatthesametimethesurrogateswillhavethesamepowerspectrum(autocorrelation)asthedata,butwillhavenononlineardeterminism.ForeachEEGepoch,tensurrogateswillbeproducedtoinsuredFourierphasesarecompletelyrandomized.Thissurrogatealgorithmhasbeenappliedandcombiningwithcorrelationintegral,ameasuresensitivetoawidevarietyofnon-linearities,wasusedfordetectionfornonlinearitybyCasdigalietal.(1996)[ 78 ]. 55 ; 54 ].Inthissection,wewillonlygiveashortdescriptionandbasicnotationofourmathematicalmodelsusedtoestimateSTLmax.First,letusdenethefollowingnotation. 78 PAGE 79 LetLbeanestimateoftheshorttermmaximumLyapunovexponent,denedastheaverageoflocalLyapunovexponentsinthestatespace.Lcanbecalculatedbythefollowingequation: jXi;j(0)j:(4{1) 79 PAGE 80 wherek overamovingwindoww(t)denedas TN+1;t T](4{4) whereNdenotesnumberofLmaxinthemovingwindowand^ijtdenotesthestandarddeviationofthesampleDijwithinw(t).Asymptotically,Tij(t)followsthet-distributionwithN1degreeoffreedom.WeusedN=30(i.e.averagesof30paireddierencesofLmaxvaluespermovingwindow).SinceeachLmaxvaluewasderivedfroma10.24secondofEEGepoch,thelengthofourmovingwindowisapproximatelyabout5minutes.AcriticalvalueT 4-2 4-3 showtheLmaxprolefrombothoriginalanditssurrogatesrespectively,foranalysispurposes,herewedeneinterictalstateisthetimedurationatleast60minspriortoseizureonset,preictalisthetimeduration10minspriortoseizureonset,nallypostictalis5minsafterseizureonset.Theresultsclearshowedthenonlinearitywasdetectedinthreedierentstates,thisfurtherclariedLmaxwascapabletocaptureinformationwhichisnotcontainedinlinearARmodel.ThedierencesbetweenoriginalLmaxanditsurrogateswasfoundlargestintheEEGrecordedfromepilepticfocusregions. TheSTLmaxvalues(thediscriminatingstatistic)estimatedfromoriginalEEGanditssurrogates,andT-indexprolesduringinterictal(Fig. 4-4 ),pre-ictal(Fig. 4-5 ) 80 PAGE 81 Degreeofnonlinearityduringpreictalstate andpost-ictal(Fig. 4-6 )state,respectively,inaULSOZpatient.Eachgurecontainstwoareas,onefromthefocalarea(toptwopanels)andanotherfromthehomologouscontralateralhippocampusarea(bottomtwopanels).Fromthesegures,itisclearthattheEEGrecordedfromthefocusareaexhibitshigherdistinctionfromGaussianlinearprocessesthanthoserecordedfromthehomologouscontralateralhippocampusareainallthreestates.Further,thedierencesofSTLmaxvaluesinthefocusareaincreasedfrominterictaltopreictal,andreachedtothemaximuminthepostictalstate,butthedierenceremainedthestableinthehomologouscontralateralarea. 81 PAGE 82 DegreeofNonlinearityduringpostictalstate Figures 4-7 (A), 4-8 (A)and 4-9 (A)showthedegreeofnonlinearity(quantiedmeanT-indexvalues)inEEGforULSOZpatients,duringinterictal,preictalandpostictalstates,respectively.Figures 4-7 (B), 4-8 (B)and 4-9 (B)showMultiplecomparisonsofnonlinearitiesineachpairofrecordingareas(A=LTD,B=RTD,C=LST,D=RST,E=LOF,F=ROF). Theresultsdemonstratedthatthenonlinearitieswereinconsistentacrossrecordingareasforallvepatients.TheresultsfromANOVAshowedthatthereexistsignicantlyrecordingareaeectsonthedegreeofnonlinearityinallthreestates(p-values=0.0019,0.0012,0.0015forinterictal,preictalandpostictal,respectively).Further,multiplecomparisons(showninFigures 4-7 b, 4-8 band 4-9 b)revealedthatsignicantlydierences 82 PAGE 83 (p-value<0.05)inthedegreeofnonlinearityexistsbetweenthefocusarea(RightHippocampus,RTD)anditshomologouscontralateralbrainarea(LeftHippocampus,LTD)inthreestates,withhigherdegreeofnonlinearityinfocalarea.ThedegreeofnonlinearityacrossrecordedareasforthreeBTSOZpatientswasshowninFigures 4-10 4-11 and 4-12 forinterictal,preictalandpostictalstates,respectively.Itisobservedthatthedegreeofnonlinearitywasuniformlydistributedoverrecordedareas.TheresultsfromANOVArevealedthattherecordingareaeectsonthedegreeofnonlinearityinstateswerenotsignicant(p-values=0.9955,0.9945,0.9975forinterictal,preictalandpostictal,respectively). 83 PAGE 84 84 PAGE 85 resultsofthisstudysuggestthatthedistributionofEEGsignalnonlinearitiesacrossrecordingbrainareasinULSOZpatientsisdierentfromBTSOZpatients.IneachofthevetestpatientswithULSOZ,theEEGnonlinearitiesweresignicantlyinconsistentamongrecordingareas.Ontheotherhand,theEEGnonlinearitieswereuniformlydistributedacrossbrainareasineachofthethreetestpatientswithBTSOZ.Theseresultswereconsistentduringtheinterictal,preictalandpostictalperiods.Thusitmaybepossibletoecientlyandquantitatively,withashortdurationofEEGrecording,determinewhetheranepilepticpatienthasunilateralfocalareathathe/shecouldbeacandidateforepilepsysurgerytreatment.Iftheseresultscanbevalidatedinalargesamplepatient,thedurationofEEGmonitoringprocedureforaBTSOZepilepticpatientcouldbegreatlyshortened.ThiswillnotonlyreducethecostoftheEEGmonitoring 85 PAGE 86 NonlinearitiesacrossrecordingareasduringinterictalstateforULSOZpatients Figure4-8. NonlinearitiesacrossrecordingareasduringperictalstateforULSOZpatients 86 PAGE 87 NonlinearitiesacrossrecordingareasduringpostictalstateforULSOZpatients. procedure,butalsowilldecreasetheriskofinfectioncausedbytheimplantedrecordingelectrodes.LargesampleofpatientswithULSOZandBTSOZwillberequiredforreliableestimationofsensitivityandspecicityofthismethod.CorrectidenticationofthefocalareainULSOZpatientsisachallengingtask.AnobviousquestioncouldbewhetheranyofbrainareaswheretheEEGsignalsrecordedfromiscloseenoughtotheactualfocalarea.Ifnot,itwouldbeverydiculttoidentifythefocalareabyananalysisontheseEEGsignals.Otherissuessuchasthenumberofrecordingareasandnumberofrecordingelectrodesineachareacouldalsoaecttheresultsoftheanalysis.IneachoftheveULSOZpatientsstudiedhere,EEGsignalswererecordedfromsixdierentbrainareas:leftandrighthippocampus,subtemporal,andorbitofrontalregions.Allvepatientswereclinicallydeterminedtohavefocalareaintherighthippocampus.Duringtheinterictalstate,focalarea(righttemporaldepth)consistentlyexhibitedhigherdegreeofnonlinearitythaninthecontralateraltemporaldepthandsubtemporalareas(signicantobservationsin4outof5patients).Similarndingswerealsoobservedduringpreictalandpostictalstates.Theseresultssuggestthatitispossibletoidentifythefocalarea 87 PAGE 88 NonlinearitiesacrossrecordingareasduringinterictalstateforBTSOZpatients inpatientswithULSOZ.Furtherstudiesonalargersampleofpatientstovalidatetheseresultsarewarranted.Successofthisstudywillprovidemoremuch-neededinformationtoguideelectroencephalographerandcliniciantoimprovethelikelihoodofsuccessfulsurgery. 88 PAGE 89 NonlinearitiesacrossrecordingareasduringpreictalstateforBTSOZpatients 89 PAGE 90 NonlinearitiesacrossrecordingareasduringpostictalstateforBTSOZpatients 90 PAGE 91 Epilepsyisthemostcommondisordersofnervoussystems.Preliminaryndingsonthecostsofepilepsyshowthetotalcosttothenationfor2.3millionpeoplewithepilepsywasapproximately$12.5billion.Thehighincidenceofepilepsyoriginatesfromthefactthatitoccursasaresultofalargenumberoffactors,including,febriledisturbance,geneticabnormalmutation,developmentaldeviationaswellasbraininsultssuchascentralnervoussystem(CNS)infections,hypoxia,ischemia,andtumors. Neuronorgroupsofneuronsgenerateelectricalsignalswheninteractingortransmittinginformationbetweeneachother.TheEEGrecordingscapturethelocaleldpotentialaroundelectrodesthatgeneratefromneuroninthebrain.Throughvisualinspection, 91 PAGE 92 SVMhasbeensuccessfullyimplementedforbiomedicalresearchonanalyzingverylargedatasets.MoreoverSVMhasbeenrecentlyappliedfortheuseofepilepticseizurepredictionandithasbeenshowntoachieved76%sensitivityand78%specicityforEEGrecordingsfrom3patients[ 82 ].NurettinAcirandCuneytGuzelisintroducedatwo-stageprocedureSVMfortheautomaticepilepticspikesdetectioninamulti-channelEEGrecordings[ 83 ].BrunoGonzalez-Velldnetetal.,reporteditispossibletodetecttheepilepticseizuresusingthreefeaturesoftheelectroencephalogram(EEG),namely,energy,decay(damping)ofthedominantfrequency,andcyclostationarityofthesignals[ 84 ].Alongwiththisdirections,theabnormalEEGidenticationproblemcanbemodeledasbinaryclassicationproblem{\normalorabnormal".EmbeddedwithneuronnetworkandconnectivityconceptswerstproposedanddescribedanapplicationofconnectivitysupportvectormachineC-SVM,C-SVMisbasedonnetworkmodelingconceptsandconnectivitymeasurestocomparetheEEGsignalsrecordedfromdierentbrainregions.AdetailowchartoftheproposedC-SVMframeworkisgiveninFigure??.WealsousesthreedynamicalfeaturesofEEG1.Angularfrequency2.Approximateentropy3.Short-termlargestlyapunovexponenttoconductthedynamicalSVMinthesecondpartofthisstudy. 92 PAGE 93 Scalpelectrodeplacement 5-1 showsthelocationoftheelectrodesonthescalp. Table 8-1 showstheEEGdescriptionfrom10subjects,EEGsignalswererecordedatsamplingrate250Hz.Forconsistency,weanalyzeandinvestigateEEGtimeseriesusingbipolarelectrodesonlyfrom18standardchannelsforeverypatient.EEGrecordingsfromeachsubjectswereinspectedbycerticatedelectroencephalographers.Werandomlyanduniformlysampletwo30-secondEEGepochsfromeachsubject.SinceEEGrecordingsweredigitizedatthesamplingrateof250Hz,thelengthofeachEEGepoch7,500points. 93 PAGE 94 EEGdatadescription EEGdata Patient Duration(minutes) Length(points) A1 28.71 430,650 A2 29.87 448,025 A3 20.89 313,375 A4 30.19 452,875 A5 29.94 449,150 N1 31.50 472,464 N2 32.90 493,464 N3 28.06 420,964 N4 21.90 328,464 N5 33.33 499,464 Total 287.29 4,309,395 85 ].TheICAalgorithmsconsiderthehigher-orderstatisticsoftheseparatedatamapsrecordedatdierenttimepoints,withnoregardforthetimeorderinwhichthemapsoccur.[ 86 { 88 ].Assumingalinearstatisticalmodel,wehave wherexandyarerandomvectorswithzeromeanandnitecovariance;Aisarectangularmatrixwithatmostasmanycolumnsasrows. TheElementsofvectorxaretheindependentcomponents,x1;:::;xn,whicharenlinearmixturesobserved.Elementsofvectoryaretheindependentcomponents,y1;:::;ym.Withoutlossofgenerality,zeromeanassumptioncanalwaysbemade.Iftheobservablevariablesxidonothavezeromean,itcanalwaysbecenteredbysubtractingthesamplemeanthatproducesthezero-meanmodel.Ahaselementsaijfori=1;:::;nandj=1;:::;m.Valuesofnandmmaybedierent. 94 PAGE 95 Inthisstudy,WeusedtheGaussianKerneltoimprovetheperformanceofC-SVM.TheGaussianisdenedasK(xi;xj)=exp(jjxixjjj2 5.4.1EstimationofMaximumLyapunovExponent 55 ].Inthissection,wewillonlygiveashortdescriptionandbasicnotationofourmathematicalmodelsusedtoestimateSTLmax.First,letusdenethefollowingnotation. 95 PAGE 96 LetLbeanestimateoftheshorttermmaximumLyapunovexponent,denedastheaverageoflocalLyapunovexponentsinthestatespace.Lcanbecalculatedbythefollowingequation: jXi;j(0)j:(5{3) =1 whereNisthetotalnumberofphasedierencesestimatedfromtheevolutionofX(ti)toX(ti+(t))inthestatespace,and i=jarccos(X(ti)X(ti+t) 96 PAGE 97 62 ].Itcandierentiatebetweenregularandirregulardataininstanceswheremomentstatistics(e.g.meanandvariance)approachesfailtoshowasignicantdierence.Applicationsincludeheartrateanalysisinthehumanneonateandinepilepticactivityinelectrocardiograms(Diambra,1999)[ 63 ].Mathematically,aspartofageneraltheoreticalframework,ApEnhasbeenshowntobetherateofapproximatingaMarkovchainprocess[ 62 ].Mostimportantly,comparedApEnwithKolmogrov-Sinai(K-S)Entropy(Kolmogrov,1958),ApEnisgenerallyniteandhasbeenshowntoclassifythecomplexityofasystemviafewerdatapointsviatheoreticalanalysisofbothstochasticanddeterministicchaoticprocessesandclinicalapplications[ 62 ; 64 { 66 ].HereIgivebriefdescriptionofApEncalculationforatimeseriesmeasuredequallyintimewithlengthn,S=s1;s2;:::;snisgivenbyrstformasequenceofvectorx1;x2;:::;xnm+1inRmusing: wheremisgivenasanintegerandrfisapositiverealnumber.ThevalueoflisthelengthofcomparedsubsequencesinS,andrfspeciesatolerancelevel. m(rf)=nm+1Xi=1lnCmi(rf) Finallytheapproximateentropyisgivenby: 97 PAGE 98 Ahyperplane(w;b)iscalledacanonicalhyperplanesuchthat 2kwk2+C subjectto 98 PAGE 99 EEGdynamicsfeatureclassication infeasibilitiesoftheconstraints.Withthisformulation,oneswantstomaximizethemarginbetweentwoclassesbyminimizingkwk2.Thesecondtermoftheobjectivefunctionisusedtominimizethemisclassicationerrorsthataredescribedbytheslackvariablesi.IntroducingpositiveLagrangemultipliersitotheinequalityconstraintsinD-SVMmodel,weobtainthefollowingdualformulation: min1 2i=1Xnj=1XnyiyjijxixjnXi=1i(5{15) 99 PAGE 100 0iC;i=1;:::n(5{17) Thesolutionoftheprimalproblemisgivenbyw=Piiyixi,wherewisthevectorthatisperpendiculartotheseparatinghyperplane.Thefreecoecientbcanbefoundfromi(yi(wxi+b)1)=0,foranyisuchthatiisnotzero.D-SVMmapagivenEEGdatasetofbinarylabeledtrainingdataintoahighdimensionalfeaturespaceandseparatethetwoclassesofdatalinearlywithamaximummarginhyperplaneinthedynamicalfeaturespace.Inthecaseofnonlinearseparability,eachdatapointxintheinputspaceismappedintoadierentspaceusingsomenonlinearmappingfunction'.Anonlinearkernelfunction,k(x;x),canbeusedtosubstitutethedotproduct<'(x);'(x)>.ThiskernelfunctionallowstheD-SVMtooperateecientlyinanonlinearhigh-dimensionalfeaturespacewithoutbeingadverselyaectedbydimensionalityofthatspace. LetGbeanundirectedgraphwithverticesV1;:::;Vn,whereVirepresentselectrodei.Thereisanedge(link)withtheweightwijforeverypairofnodesViandVjcorrespondingtotheconnectivityofthebraindynamicsbetweenthesetwoelectrodes.Theconnectivityorsynchronizationcanbeviewedasbeenactivatedbyinteractionsbetweenneuronsinthelocalcircuitryunderlyingtherecordingelectrodes.Figure 5-4 representsahypotheticalbraingraphinwhicheachconnectedpathdenotestheunderlyingconnectivity.Withthisgraphmodel,theattributesofC-SVMinputsarethepair-wisedrelationbetweentwotimeseriesprolesratherthantimestampsofatimeseriesprole.Inthiscontext,the 100 PAGE 101 Supportvectormachines inputofC-SVMisthedegreeofconnectivitybetweendierentbrainregions.Givenntimeseriesdatapoints,eachwithmtimestamps,theproposedframeworkwilldecreasethenumberattributesby2(n1)=mtimes.Letlbethetotalnumberofdatapoints,thedimensionalitycanbereducedfromA2 PAGE 102 Connectivitysupportvectormachine 5-2 .TheproposedC-SVMwithoutGaussiankernelproducedaverageaccuracyof69.4%andD-SVMusingdynamicalfeaturesobtainedfromEEGrecordingsproduced94.7%inclassication. 102 PAGE 103 ResultsforD-SVMusing5-foldcrossvalidation D-SVM Results D-SVM/C-SVM DynamicalfeaturesUNICA 5-foldCV1 46%72% 2 92%76% 3 94%68% 4 98%72% 5 96%66% 6 92%54% 7 96%70% 8 95%76% 9 94%70% 10 95%70% Average%ofcorrectness 94.7%69.4% TheSVMhasaverylongstatisticalfoundationandassuretheoptimalfeasiblesolutionforasetoftrainingdata,givenasetoffeaturesandtheoperationoftheSVM.Inthisstudy,weattemptedtostudytheseparabilitybetweenabnormalEEGandnormalEEGusingdierentEEGfeatures.WetestedtheperformanceonscalpEEGrecordingsfromnormalindividualsandabnormalpatients.TheEEGdatawaslteredusingICAalgorithm.ICAltersthenoiseinEEGscalpdata,keepsessentialstructureandmakesbetterrepresentableEEGdatasets.TheEuclideandistancebasedC-SVMwasproposedtoevaluatetheconnectivityamongdierentbrainregions.ThedynamicalfeaturesweregeneratedasinputforD-SVM,theclassicationresultsoftheproposedD-SVMareveryencouraging.TheresultsindicatedthatD-SVMimprovesclassicationaccuracycomparetoC-SVM.Itgivesanaverageaccuracyof94:7%.Thedynamicalfeaturesprovideasubsetinthefeaturespaceandimproveclassicationaccuracy. 103 PAGE 104 89 ].Synchronizationcanbequantiedinbothspaceandtimedomain.Foramulti-variatesystem,understandingtheinteractionsamongitsvariousvariables,whosebehaviorcanberepresentedalongtimeastime-sequences,presentsmanychallenges.Oneofthekeyaspectsofhighlysynchronizedsystemswithspatialextentistheirabilitytointeractbothacrossspaceandtime,whichcomplicatestheanalysisgreatly.Inbiologicalsystemssuchasthecentralnervoussystem,thisdicultyiscompoundedbythefactthatthecomponentsofinteresthavenonlinearcomplicateddynamicsthatcandictateoverallchangesinthesystembehavior.Theexactgureofhowtoquantifytheinformationexchangesinasystemremainsambiguous.Studiesonmulti-variatetimeseriesanalysishaveresultedindevelopmentofawiderangeofsignal-processingtoolsforquanticationofsynchronizationinsystems.However,thegeneralconsensusonhowtoquantifythisphenomenonislargelyuncertain.Intheliterature,synchronizationbetweenvariablescanbecategorizedasidenticalsynchronization,phasesynchronizationandgeneralizedsynchronization.Inthefollowingchapters,Iundertakeanin-depthanalysisofpreictalandinterictalsynchronizationbehavior,focusingonEEGrecordingsfrompatientswithtemporallobeandgeneralizedseizures. 104 PAGE 105 6.2.1CrossCorrelationFunction where=N1;N;:::;0;:::;N1.ThiscrosscorrelationissymmetricRxy()=Rxy().Itcanalsobeshownthat wherex; yaretheestimatedmeanofx(n)andy(n). Cross-correlationcoecientbetweenx(n)andy(n)isdenedascrosscovariancenormalizedbytheproductofthesquarerootofthevariancesoftwoobservedseries,asfollows: wherexyarethestandarddeivationofx(n)andy(n).Cross-correlationcoecientboundedbetween-1and+1. Theasthefrequencydomain,denex(!)=F(x(n))andY(!)=F(y(n))astheFouriertransformequivalentsofx(n)andy(n).Ifthecross-spectrumCxy(!)andauto-spectrumsCxx(!)andCyy(!)aredenedbynormalizedcross-coherenceandcanbe 105 PAGE 106 (Cxx(!)Cyy(!))1 2:(6{4) Thecross-coherencequantiesthedegreeofcouplingbetweenXandYatgivenfrequency!anditisalsoboundedbetween-1and+1. 90 ].However,thereversecasemayormaynotbetrue.Tomakeaquantitativeassessmentoftheamountoflinearinteractionandthedirectionofinteractionamongmultipletime-series,theconceptofGranger-causalitycanbeusedtoandintothedevelopmentmultiautoregressivemodel(MVAR).Thepartialdirectedcoherencefromjtoiatafrequency!isgivenby: wherefori=j Aij(!)=1pXr=1aij(r)ej2!r;(6{6) andfori6=j here 106 PAGE 107 and pdenotesthedepthoftheARmodel,rdenotesthedelayandnisthepredictionerrororthewhitenoise. Notethat!quantiestherelativestrengthoftheinteractionofagivensignalsourcejwithregardtosignaliascomparedtoallofjsinteractionstoothersignals.ItturnsoutthatthePDCisnormalizedbetween0and1atallfrequencies.Ifi=j,thePartialDirectedCoherencerepresentsthecasualinuencefromtheearlierstatetoitscurrentstate. TheMVARapproacheshavebeenusedtodeterminethepropagationofepilepticintracranialEEGactivityintemporallobeandmesialseizures[2-3,10,19-20].However,thesemodelsstrictlyrequirethatthemeasurementsbemadefromallthenodes,orthedirectionalrelationshipscouldbeambiguous.Inaddition,thereremainsnoclearevidenceofcausalityrelationshipsamongthecorticalregionsassuggested\thenatureofsynchronizationismostlyinstantaneousorwithoutanydetectabledelay"[ 91 ]. Thegeneral,nonlinearityarecommonlyinherentwithinneuronalrecordings,theabovelinearmeasuresaretypicallyrestrictedtomeasurestatisticaldependenciesuptothesecondorder[ 92 ].IfobservationsareGaussiandistributed,the2ndorderstatisticsaresucienttocapturealltheinformationinthedata.However,inpractice,EEGdata 107 PAGE 108 78 ]. 93 ].Conceptually,iftherhythmsofonesignalareinharmonywiththatoftheother,thetwosignalsareknowntobephaselocked.Phasesynchronizationcanthereforebedenedasthedegreetowhichtwosignalsarephaselocked.ForthisHuygens'classicalcase,phasesynchronizationisusuallydenedaslockingofthephasesoftwooscillators. wherenandmareintegers,'a(t)and'b(t)denotethephasesoftheoscillators,andn;misdenedastheirrelativephase. Rosenblumetal,(1996)generalizedtheabovephaselockingformulabytheweakerconditionofphaseentrainment[ 94 ]: byevenweakerconditionoffrequencylocking: dtmgdbt dt=0;(6{13)edenotesaveragingovertime,and'n;mtherelativefrequencyofthesystem. Mostofrealworldsignalshavebroadspectra.Forexample,EEGsignalrecordingsareusuallyintherangeof0.1to1000Hzeventhoughtheyareusuallybandpasslteredbetween0.1and70Hzsinceamajorportionoftheenergyiscontainedinthatspectrum.TheEEGcanbeclassiedroughlyintove(5)dierentfrequencybands,namelythedelta(0-4Hz),theta(4-8Hz),alpha(8-12Hz),Beta(12-16Hz)andtheGamma(16-80Hz)frequencybands.FreemandemonstratedevidenceofphaselockingbetweenEEG 108 PAGE 109 95 ; 96 ].Similarly,itisalsobelievedthatphasesynchronizationacrossnarrowfrequencyEEGbands,pre-seizureandattheonsetofseizuremayprovideusefulhintsofthespatio-temporalinteractionsinepilepticbrain[ 33 ; 97 ; 35 ].Hilberttransformisusedcomputetheinstantaneousparameters'a(t)and'b(t)ofatime-signal.Considerareal-valuednarrow-bandsignalx(t)concentratedaroundfrequencyfc.Denex(t)as where excitedbyaninputsignalx(t).ThislteriscallaHillberttransformer.Hilberttransformsareaccurateonlywhenthesignalshavenarrow-bandspectrum,whichisoftenunrealisticformostreal-worldsignals.Pre-processingofthesignalsuchasdecomposingitintonarrowfrequencybandsisneededbeforeweapplyHilberttransformationtocomputetheinstantaneousparameters.Certainconditionsneedtobecheckedtodeneameaningfulinstantaneousfrequencyonanarrow-bandsignal.Ithasbeenreportedthatthedistinctdierencesinthedegreeofsynchronizationbetweenrecordingsfromseizure-freeintervalsandthosebeforeanimpendingseizure,indicatinganalteredstateofbraindynamicspriortoseizureactivity[ 89 ]. 98 ].Generally,mutualinformationmeasurestheinformationobtainedfromobservationsofonerandomeventfortheother.Itisknownthatmutualinformationhasthecapabilitytocapturebothlinearandnonlinearrelationshipsbetweentworandomvariablessincebothlinearandnonlinearrelationshipscanbedescribedthroughprobabilistictheories.Hereinourmodel,themutualinformationmeasureshowmuchinformationofEEGtime 109 PAGE 110 99 { 104 ]. Kraskovetal.,2004introducedtwoclassesofimprovedestimatorsformutualinformationM(X;Y)fromsamplesofrandompointsdistributedaccordingtosomejointprobabilitydensity(x;y).Incontrasttoconventionalestimatorsbasedonhistogramapproach,theyarebasedonentropyestimatesfromknearestneighbourdistances.LetusdenotethetimeseriesoftwoobservablevariablesasX=fxigNi=1andY=fyjgNj=1,whereNisthelengthoftheseriesandthetimebetweenconsecutiveobservations(i.e.,samplingperiod)isxed.Thenthemutualinformationisgivenby: wherepx(i)=Ridx,py(i)=Ridyand \Ri"denotestheintegraloverbini.Ifnx(i)andny(j)arethenumberofdatapointsintheithbinofXandjthbinofY;n(i;j)isthenumberofdatapointsintheintersectionbin(i;j).Theprobabilitiesareestimatedaspx(i)nx(i)=N,px(j)nx(j)=Nandp(i;j)nx(j)=N.Ratherthenbinapproachthemutualinformationcanbeestimatedfromk-nearestneighborstatistics. WerstestimatebH(X)fromXby 110 PAGE 111 Nextlet(i) 2bethedistancebetweenzianditskthneighbor.Inordertoestimatethejointprobabilitydensityfunction(p:d:f:),weconsidertheprobabilityPk()whichistheprobabilitythatforeachzithekthnearestneighborhasdistance(i) 2dfromzi.Thisprobabilitymeansthatk1pointshavedistancelessthanthekthnearestneighborandNk1pointshavedistancegreaterthan(i) 2andk1pointshavedistancelessthan(i) 2.Pk()isobtainedusingthemultinomialdistribution: wherepiisthemassofthe-ball.Thentheexpectedvalueoflogpiwillbe: where()isthedigammafunction: where()isthegammafunction.Itholdsthat(1)=CwhereCistheEuler-Mascheroniconstant(C0:57721).Themassofthe-ballcanbeapproximated(ifweconsidertheprobabilitydensityfunctioninsidetheballisthesame)by: 111 PAGE 112 6{23 )wecanndanestimatorforP(X=xi): log[P(X=xi)](k)(N)dE(log(i))logcdx;(6{24) nallywithEq( 6{24 )andEq( 6{25 )weobtaintheKozachenko-LeonenkoentropyestimatorforX[ 105 ]: ^H(X)=(N)(k)+logcdx+dx where(i)istwicethedistancefromxitoitsk-thneighborinthedxdimensionalspace.Forthejointentropywehave: ^H(X;Y)=(N)(k)+log(cdxcdy)+dx+dy TheI(X;Y)isnowcanbeobtainedbyEq.( 6{16 ).Theproblemwiththismethodisthataxedkisusedinallestimatorsbutthedistancemetricindierentscaledspaces(marginalandjoint)arenotcomparable.Toavoidsuchproblem,insteadofusingaxedk,nx(i)+1andny(i)+1areusedinobtainingthedistances(wherenx(i)andny(i)arethenumberofsamplescontainedthebin[x(i)(i) 2;x(i)+(i) 2]and[y(i)(i) 2;y(i)+(i) 2]respectively)inthex{yscatterdiagram.TheEq.( 6{26 )becomes: ^H(X)=(N)(nx(i)+1)+logcdx+dx FinallytheEq.( 6{16 )isrewrittenas: 106 ].Inthissection,weinvestigatedirectionalrelationshipsusingthenonlinear 112 PAGE 113 Giventwotimeseriesxandy,usingmethodofdelaytoobtaindelayvectorsxn=(xn;:::;xn(m1))andyn=(xn;:::;xn(m1)),wheren=1;:::N,mistheembeddingdimensionanddenotesthetimedelay[ 52 ].Letrn;jandsn;j,j=1;:::;kdenotethetimeindicesoftheknearestneighborsofxnandyn.Foreachxn,themeanEuclideandistancetoitskneighborsisdenedas andtheY-conditionedmeansquaredEuclideandistanceisdenedbyreplacingthenearestneighborsbytheequaltimepartnersoftheclosestneighborsofyn Thedelay=5isestimatedbyautomutualinformationfunction,theembeddingdimensionm=10isobtainedusingCao'smethodusing10secEEGselectedduringinterinctalstateandaTheilercorrectionissettoT=50[ 73 ; 107 ]. IfxnhasanaveragesquaredradiusR(X)=(1=N)PNn=1R(N1)n(X),thenR(k)nR(k)n(X) PAGE 114 whichisnormalizedbetween0and1.TheoppositeinterdependenciesS(YjX),H(YjX),andN(YjX)aredenedincompleteanalogyandtheyareingeneralnotequaltoS(XjY),H(XjY),andN(XjY),respectively.Usingnonlinearinterdependenciesonseveralchaoticmodel(Lorenz,Roessler,andHeenonmodels)Quirogaetal.,(2000)showedthemeasureHismorerobustthanS. Theasymmetryofabovenonlinearinterdependenciesisthemainadvantageoverothernonlinearmeasuressuchasthemutualinformationorthephasesynchronization.Thisasymmetrycangiveinformationabout\driver-response"relationshipsbutcanalsoreectdierentpropertiesofdynamicalsystemswhenitisimportancetodetectcausalrelationships.Itshouldbeclearthattheabovenonlinearinterdependenciesmeasuresarebivariatemeasures.Althoughitquantiedthe\driver-response"forgiveninput-thewholeinputspaceunderstudymightbedrivenbyotherunobservedsystem(s). 114 PAGE 115 108 ].Incognitivetaskstudies,In1980,Freemanfound\moreregular"spatiotemporalactivitiesinEEGforabriefperiodoftimewhentheanimalinhaledafamiliarodoruntiltheanimalexhaled[ 109 ; 96 ].Someearlierstudiesalsoindicatedthesignicantroleofsynchronizationforphysiologicalsystemsinhumans;thedetectablealtersinsynchronizationphenomenahavebeenassociatedtoanumberofchronic,acutediseasesorthenormalityofbrain.[ 110 ; 111 ].Intheeldofepilepsyresearch,severalauthorshavesuggesteddirectrelationshipbetweenaltersinsynchronizationphenomenaandonsetoftheepilepticseizuresusingEEGrecordings.Forexample,Iasemidisetal.,(1996)reportedfromintracranialEEGthattheentrainmentinthelargestLyapunovexponentsfromcriticalcorticalregionsisanecessaryconditionforonsetofseizuresforpatientswithtemporallobeepilepsy[ 112 ; 61 ];LeVanQuyenetal.,showedepilepticseizurecanbeanticipatedbynonlinearanalysisofdynamicalsimilaritybetweenrecordings[ 35 ].Mormannetal.,showedthepreictalstatecanbedetectedbasedonadecreaseinsynchronizationonintracranialEEGrecordings[ 89 ; 43 ].ThehighlycomplexbehaviorontheEEGrecordingsisconsideredtonormalityofbrainstate,whiletransitionsintoalowercomplexitybrainstateareregardedasapathological 115 PAGE 116 113 ].Innextchapters,wewillusetheconceptsandmathematicalframeworksintroducedinthischaptertoclustertheEEGbetweennormalandabnormalepochesandtodetecttheeectinEEGresultedfromAEDstreatments. 116 PAGE 117 Neuralactivityismanifestedbyelectricalsignalsknownasgradedandactionpotentials.Berger'sdemonstrationin1929hasshownthatitispossibletorecordtheelectricalactivityfromthehumanbrain,particularlytheneuronslocatednearthesurfaceofthebrain[ 23 ].Whileweoftenthinkofelectricalactivityinneuronsintermsofactionpotentials,theactionpotentialsdonotusuallycontributedirectlytotheelectroencephalogram(EEG)recordings.Infact,forscalpEEGrecordings,theEEGpatternsaremainlythegradedpotentialsaccumulatedfromhundredsofthousandsofneurons.TheEEGpatternsvarygreatlyinbothamplitudeandfrequency.TheamplitudeoftheEEGreectsthedegreeofsynchronousringoftheneuronslocatedaroundtherecordingelectrodes.Ingeneral,thehighEEGamplitudeindicatesthatneuronsare 117 PAGE 118 Althoughthebrainmayhaveoriginallyemergedasanorganwithfunctionallydedicatedregions,recentevidencesuggeststhatthebrainevolvedbypreserving,extending,andre-combiningexistingnetworkcomponents,ratherthanbygeneratingcomplexstructuresdenovo[ 114 ; 115 ].Thisissignicantbecauseitsuggests:(1)thebrainnetworkisarrangedsuchthatthefunctionalneuralcomplexessupportingdierentcognitivefunctionssharemanylow-levelneuralcomponents,and(2)thespecicconnectiontopologyofthebrainnetworkmayplayasignicantroleinseizuredevelopment.Thislineofthinkingisalsosupportedby[ 70 ],whichdemonstratesthatspecicconnectedstructuresareeithersignicantlyabundantorrareincorticalnetworks.Ifseizuresevolveinthisfashion,thenweshouldbeabletomakesomespecicempiricalhypothesesregardingtheevolutionofseizures,thatmightbeborneoutbyinvestigatingthesynchronizationbetweentheactivityindierentbrainareas,asrevealedbyquantitativeanalysisofEEGrecordings.Thegoalofthisstudyistotestthefollowingtwohypotheses.First,weshouldexpectthebrainactivityintheorbitofrontalareasarehighlycorrelatedwhiletheactivityinthetemporallobeandsubtemporallobeareasarehighlycorrelatedwiththeirownside(leftonlyorrightonly)duringthepre-seizureperiod.Thehighcorrelationcanbeviewedasarecruitmentoperationinitiatedbyanepileptogenicareathrougharegularcommunicationchannelinthebrain.Notethattheconnectionofthesebrainareashasbeenalong-standingprincipleinnormalbrainfunctionsandwebelievethatthesameprincipleshouldholdinthecaseofepilepsyaswell.Second,weshouldexpectsomebrainregionstobeconsistentlyactive,manifested 118 PAGE 119 116 ].MeasuringtheconnectivityandsynchronicityamongdierentbrainregionsthroughEEGrecordingshasbeenwelldocumented[ 99 ; 117 ; 69 ].Thestructuresandthebehaviorsofthebrainconnectivityhavebeenshowntocontainrichinformationrelatedtothefunctionalityofthebrain[ 118 ; 67 ; 68 ].Morerecently,themathematicalprinciplesderivedfrominformationtheoryandnonlineardynamicalsystemshaveallowedustoinvestigatethesynchronizationphenomenainhighlynon-stationaryEEGrecordings.Forexample,anumberofsynchronizationmeasureswereusedforanalyzingtheepilepticEEGrecordingstoreachthegoalsoflocalizingtheepileptogeniczonesandpredictingtheimpendingepilepticseizures[ 99 ; 106 ; 119 ; 38 ; 120 ].Thesestudiesalsosuggestthatepilepsyisadynamicalbraindisorderinwhichtheinteractionsamongneuronorgroupsofneuronsinthebrainalterabruptly.Moreover,thecharacteristicchangesintheEEGrecordingshavebeenshowntohaveclearassociationswiththesynchronizationphenomenaamongepileptogenicandotherbrainregions.Whentheconductivitiesbetweentwooramongmultiplebrainregionsaresimultaneouslyconsidered,theunivariateanalysisalonewillnotbeabletocarryoutsuchatask.Thereforeitisappropriatetoutilizemultivariateanalysis.Multivariateanalysishasbeenwidelyusedintheeldofneurosciencetostudytherelationshipsamongsourcesobtainedsimultaneously.Inthisstudy,thecrossmutualinformation(CMI)approachisappliedtomeasuretheconnectivityamongbrainregions[ 75 ].TheCMIapproachisabivariatemeasureandhasbeenshowntohaveabilityfor 119 PAGE 120 101 ; 97 ].ThebrainconnectivitygraphisthenconstructedwhereverticesinthegraphrepresenttheEEGelectrodes. EverydistinctpairofverticesisconnectedbyanarcwiththelengthequaltotheconnectivityquantiedbyCMI.Afterconstructingabrainconnectivitygraph,whichisacompletegraph,wethenremovearcsofconnectivitybelowaspeciedthresholdvaluetopreserveonlystrongcouplingsofelectrodepairs.Finally,weemployamaximumcliquealgorithmtondamaximumcliqueinwhichthebrainregionsarestronglyconnected.Themaximumcliquesizecanbe,inturn,usedtorepresenttheamountoflargestconnectedregionsinthebrain.Themaximumcliquealgorithmreducesthecomputationaleortforsearchingintheconstructedbrainconnectivitygraph.Theproposedgraph-theoreticapproachoersaneasyprotocolforinspectingthestructuresofthebrainconnectivityovertimeandpossiblyidentifyingthebrainregionswhereseizuresareinitiated. 120 PAGE 121 Table7-1. Patientinformationforclusteringanalysis PatientGenderAgeNumberofSeizureDurationofEEGNumberofelectrodesonsetzonerecordings(days)seizures 1Male2926R.Hippocampus6.07192Male3730L./R.Hippocampus9.8811 121 PAGE 122 Therefore,weusemultivariatemeasuresforquantifyingthesynchronizationfromonly2electrodesatatime.Specically,asimplesignalprocessingusedtocalculatethesynchronizationbetweenelectrodepairsisemployedinthisstudy.Thenweapplyadataminingtechniquebasedonnetwork-theoreticalmethodstothemultivariateanalysisofEEGdata. 121 ].Althoughtheinformationfromcross-correlationfunctionandfrequencycoherencehasbeenshowntobeidentical[ 122 ],thesimilaritybetweentwoEEGsignalsindierentfrequencybandssuchasdelta,theta,beta,alphaandgamma,isstillcommonlyusedtoinvestigateEEG 122 PAGE 123 123 ; 121 ].Forexample,[ 124 ]usedfrequencycoherencemeasurestoinvestigatetheinteractionsbetweenmediallimbicstructuresandtheneocortexduringictalperiods(seizureonsets).Inanotherstudyby[ 125 ],thecoherencepatternofcorticalareasfromepilepticbrainwasinvestigatedtoidentifyacorticalepilepticsystemduringinterictal(normal)andictal(seizure)periods. Althoughlinearmeasuresareveryusefulandcommonlyused,theyareinsensitivetononlinearcouplingbetweensignals,andnon-linearitiesarequitecommoninneuralcontexts.TobeabletoinvestigatemoreoftheinterdependencebetweenEEGelectrodes,nonlinearmeasuresshouldbeapplied.NonlinearmeasureshavebeenwidelyusedtodeterminetheinterdependenceamongEEGsignalsfromdierentbrainareas.Forexample,[ 106 ]and[ 35 ]studiedthesimilaritybetweenEEGsignalsusingnonlineardynamicalsystemapproaches.Theyappliedatime-delayembeddingtechniquetoreconstructatrajectoryofEEGinphasespaceandusedtheideaofgeneralizedsynchronizationproposedby[ 126 ]tocalculatetheinterdependenceandcausalrelationshipsofEEGsignals. Weproposeanapproachtoinvestigateandquantifythesynchronizationofthebrainnetwork,specicallytailoredtostudythepropagationofepileptogenicprocesses.[ 127 ]investigatedthispropagation,wheretheaverageamountofmutualinformationduringtheictalperiod(seizureonset)wasusedtoidentifythefocalsiteandstudythespreadofepilepticseizureactivity.Subsequently,[ 128 ]appliedtheinformation-theoreticapproachtomeasuresynchronizationandidentifycausalrelationshipsbetweenareasinthebraintolocalizeanepileptogenicregion.Here,weapplyaninformation-theoreticapproach,calledcross-mutualinformation,whichcancapturebothlinearandnonlineardependencebetweenEEGsignals,toquantifythesynchronizationbetweennodesinthebrainnetwork.Inordertogloballymodelthebrainnetwork,werepresentthebrainsynchronizationnetworkasagraph. 123 PAGE 124 70 ].Inanearlierstudy,[ 115 ]demonstratedthatthebrainevolvedahighlyecientnetworkarchitecturewhosestructuralconnectivity(ormotif)iscapableofgeneratingalargerepertoireoffunctionalstates.Inanotherrecentstudy,thebrainnetworkgraphwasinvestigatedtoverifythatthere-useofexistingneuralcomponentsplayedasignicantroleintheevolutionarydevelopmentofcognition[ 114 ].Applyingnetwork/graph-theoreticmethodstoEEGsignals,wecanmodelthebrainconnectivity/synchronizationnetworkasacompletegraphG(V;E),whereVisasetofverticesandEisasetofedges.Vertices(alsocallednodes)arerepresentedbyEEGelectrodes(alsoreferredaschannels).Edges(alsocalledarcs)arerepresentedbythesynchronization/similaritybetween2EEGelectrodeswhosedegreescorrespondtotheedgeweights.Inshort,abrainconnectivitynetworkcanthenbeconstructedasagraphwhoseverticesareEEGelectrodesandtheweightededgesarethecouplingstrengthofelectrodepairs.Everypairofverticesisconnectedbyaweightededge.Inthisstudy,wefocusonthestructuralchangesinthebrainconnectivitynetworkthatmayberelatedtotheseizureevolution.Thestructuralchangescouldberepresentedbyconnectivityfractions/partitionsthroughaggregationandsegregationofthebrainnetwork.Ininthisstudy,weproposetwonetwork-theoreticapproaches,spectralpartitioningandmaximumclique,toidentifyindependent/segregatedandclusteredbrainareas. 98 ].Generally,CMImeasurestheinformationobtainedfromobservationsofonerandomeventfortheother.ItisknownthatCMIhasthecapabilitytocapturebothlinearandnonlinearrelationshipsbetweentworandomvariablessincebothlinearandnonlinearrelationships 124 PAGE 125 ShannonentropymeasurestheuncertaintycontentofX.Itisalwayspositiveandmeasuredinbits,ifthelogarithmistakenwithbase2.NowletusconsideranothersetofdatapointsY,whereallpossiblerealizationsofYarey1;y2;y3;:::ynwithprobabilitiesP(y1);P(y2);P(y3);::::.ThedegreeofsynchronicityandconnectivitybetweenXandYcanbemeasuredbythejointentropyofXandY,denedas wherepXYijwhichisthejointprobabilityofX=XiandY=Yj:ThecrossinformationbetweenXandY,CMI(X;Y),isthengivenby (7{3) =H(X)+H(Y)H(X;Y) (7{4) =ZZpXY(x;y)log2pXY(x;y) Thecrossmutualinformationisnonnegative.IfthesetworandomvariablesX;Yareindependent,fXY(x;y)=fX(x)fY(y),thenCMI(X;Y)=0,whichimpliesthatthereisnocorrelationbetweenXandY.Theprobabilitiesareestimatedusingthehistogrambasedboxcountingmethod.Therandomvariablesrepresentingtheobservednumberofpairsofpointmeasurementsinhistogramcell(i;j),rowiandcolumnj,arerespectivelykij;ki:andk:j.Here,weassumetheprobabilityofapairofpointmeasurementsoutsidetheareacoveredbyhistogramisnegligible,thereforePi;jPij=1[ 75 ; 129 ]. 125 PAGE 126 EEGepochsforRTD2,RTD4andRTD6(10seconds) ScatterplotforEEGepoch(10seconds)ofRTD2vs.RTD4andRTD4vs.RTD6 Figure 7-3 showstheCMIvaluesmeasuredfromrightmesialtemporaldepth(R(T)D)regions.Figure 7-2 displaysthescatterplotsfortheEEGrecordedfromthesame(R(T)D)brainregion.Fromthescatterplot,itisclearthatEEGrecordingsbetweenR(T)D2andR(T)D4haveweaklinearcorrelationwhichhavealsoyieldedlowerCMIvaluesinFigure 7-3 .ThestrongerlinearrelationshipisdiscoveredbetweenR(T)D4andR(T)D6andthislinearcorrelationpatternhasresultedinhigherCMIvalues.PriortomeasuringtheCMI,werstdividedEEGrecordingsintosmallernon-overlappingEEGepochs.Thesegmentationprocedureiswidelyutilizedtosubduethenon-stationarynatureoftheEEGrecordings.ThechangesofEEGpatterntendtoappearverybriey, 126 PAGE 127 Cross-mutualinformationforRTD4vs.RTD6andRTD2vs.RTD4 examplesincludesharpwavetransients,spikes,spike-wavecomplexes,andspindles.WorkingonshorterEEGepochswillinsurethestationarityfortheunderlyingprocessesandthusanychangeintheconnectivitycanbedetected.Therefore,aproperlengthofEEGepochshastobedeterminedformeasuringtheconnectivityamongEEGrecordings.WechosethelengthoftheEEGepochsequalto10.24seconds(2048points),whichhasalsobeenutilizedinmanyperviousEEGresearchstudies[ 58 ; 130 ].ThebrainconnectivitymeasuredusingCMIformthecompletegraph,inwhicheachnodehasanarctoeveryotheradjacentvertex.Intheprocedureforremovingtheinsignicantarcs(weakconnectionbetweenbrainregions),werstestimatedanappropriatethresholdvaluebyutilizingthestatisticaltests.Wedeterminedthisthresholdbyobservingthestaticalsignicanceoverthecompleteconnectivitygraph,thisthresholdvaluewassettobeavaluewherethesmallnoiseiseliminated,butyettherealsignalisnotdeleted[ 131 ]. 127 PAGE 128 Completeconnectivitygraph(a);afterremovingthearcswithinsignicantconnectivity(b) 7-5 (a)and 7-5 (b). Notethatthismatrixissymmetricbecausethemutualinformationmeasurehasnocouplingdirection.Ineachrowandcolumnofthisbitmap,thecolorrepresentsthesynchronizationlevel.Notethatwewillignorethediagonalofthematrixbecausewecanalwaysndaveryhighlevelofself-synchronization. Afterapplyingthenormalizedcut,wecalculatedaneigenvectorcorrespondingtothesecondsmallesteigenvalue.Subsequently,weseparatedelectrodesintotwogroupswiththeminimumcutorseparationwithminimumcostbyapplyingthethresholdvalueat0.UsingtheeigenvectorinFigure 7-5 c,electrodeswereseparatedintotwoclusters.Itiseasytoobserveaclearseparationofthesetwoclustersthroughthevalueofeigenvector,inwhichasharptransientfromR(S)T4toL(O)F1isusedasaseparatingpoint.TherstgroupofsynchronizedelectrodesisfromL(S)T,L(T)D,R(S)TandR(T)Dareas.ThesecondgroupisfromL(O)FandR(O)Fareas.Afterthe 128 PAGE 129 Spectralpartitioning rstiteration,itisclearthatthesynchronizationintheLD-LT-RD-RTclusterisnotuniformthroughoutallelectrodesinthecluster.Therefore,weconsequentlyperformedanotheriterationofspectralpartitioningontheLD-LT-RD-RTclustertondhighlysynchronizedgroupsofelectrodeswithinthecluster.Thisprocedurecanbeviewedasahierarchicalclustering.Afterrearrangingtheelectrodesbasedonthesynchronizationlevel,wefoundtwosub-clustersofelectrodesinthebitmapshowninFigure 7-5 b.AsshowninFigure 7-5 d,thevalueofeigenvectorindicatesthattherearetwosub-clusterswithintheLD-LT-RD-RTclusterbyapplyingthethresholdof0.TheLD-LT-RD-RTclusterwasseparatedintotwosub-clusters:LD-LTandRD-RT.Thisobservationsuggeststhatthereexistsahighlysynchronizedpatterninthesamesideoftemporallobeaswellasintheentireorbitofrontalarea.Thisndingcanbeconsideredasaproofofconceptthattheseizureevolutionalsofollowsaregularcommunicationpatterninthebrainnetwork. 129 PAGE 130 Weanalyzed3epochsof3-hourEEGrecordings,2hoursbeforeand1hourafteraseizure,fromPatient2whohadtheepileptogenicareasonbothrightandleftmesialtemporallobes.Figures 7-7 and 7-8 demonstratetheelectrodeselectionofthemaximumcliquegroupduringtwohoursbeforeandonehouraftertheseizureonset.Duringtheperiodbeforetheseizureonset,bothguresmanifestedapatternwherealltheLDelectrodeswereconsistentlyselectedtobeinthemaximumclique.Duringtheseizureonset,thesizeofthemaximumcliqueincreasesdrastically.Thisisveryintuitivebecause,intemporallobeepilepsy,allofthebrainareasarehighlysynchronized.WevisuallyinspectedtherawEEGrecordingsbeforeandduringtheseizureonsetsandfoundasimilarsemiologicalpatternoftheseizureonset-electrodesfromtheL(T)Dareas 130 PAGE 131 begin sortallnodesbasedonvertexordering LIST=orderednodes cbc=0currentbestcliquesize depth=0currentdepthlevel enter-next-depth(LIST,depth) end begin 1m=thenumberofnodesintheLIST 2depth=depth+1 3foranodeinpositioniintheLIST 4ifdepth+(m-i)cbcthen 6else 8ifnoadjacentnodethen 10else 12end 15ifdepth=1 16deletenodeifromLIST 17end Maximumcliquealgorithm initiatedahighlyorganizedrhythmicpatternsandthepatternsstartedtopropagatethroughoutallthebrainareas.Weinitiallyspeculatedthattheepileptogenicareascouldbetheonesthatarehighlysynchronizedlongbeforeaseizureonset.Inthepreviouscase,weobservedthattheL(T)Delectrodesaretheonethatstartedtheseizureevolution.However,inafurtherinvestigationofEEGrecordingsfromthesamepatient,wefoundsomecontrastresults.InFigure 7-9 ,theelectrodeselectionpatternofthemaximumcliquedemonstratesaveryhighlysynchronizedgroupofelectrodesinbothleftandrightorbitofrontalareasduringthe2-hourperiodprecedingtheseizure.AftervisualinspectionontherawEEGrecordings,thisseizurewasinitiatedbytheR(T)Darea.Generally,itis 131 PAGE 132 ElectrodeselectionusingthemaximumcliquealgorithmforCase1 132 PAGE 133 ElectrodeselectionusingthemaximumcliquealgorithmforCase2 ElectrodeselectionusingthemaximumcliquealgorithmforCase3 processesslowlydevelopthemselvesthrougharegularcommunicationchannelinthebrainnetwork,ratherthanabruptlydisrupt,collapse,orchangethewaybrainscommunicate.Fromthisobservation,wepostulatethatthisphenomenonmaybeareectionofneuronalrecruitmentinseizureevolution.Thisobservationconrmsourrsthypothesis.In 133 PAGE 134 134 PAGE 135 114 ]demonstratedtheevolutionofcognitivefunctionthroughquantitativeanalysisoffMRIdata. Theproposedframeworkcanprovideaglobalstructuralpatternsinthebrainnetworkandmaybeusedinthesimulationstudyofdynamicalsystems(likethebrain)topredictoncomingevents(likeseizures).Forexample,anON-OFFpatternofelectrodeselectioninthemaximumcliqueoveroneperiodoftimecanbemodeledasabinaryobservationinadiscretestateinaMarkovmodel,whichcanbeusedtosimulatetheseizureevolutioninthebrain.Inaddition,thenumberofelectrodesinthemaximumcliquecanbeusedtoestimatetheminimumnumberoffeaturesandexplaindynamicalmodelsortheparametersintimeseriesregression.Notethattheproposednetworkmodelrepresentsanepilepticbrainasagraph,wherethereexistseveralecientalgorithms(e.g.,maximumclique,shortestpath)forndingspecialstructureofthegraph.Thisideahasenabledus,computationallyandempirically,tostudytheevolutionofthebrainasawhole.TheMonte-CarloMarkovChain(MCMC)frameworkmaybeapplicableinourfuturestudyonlongtermEEGanalysis.TheMCMCframeworkhasbeenshownveryeectiveindataminingresearch[ 132 ].ItcanbeusedtoestimatethegraphorcliqueparametersinepilepticprocessesfromEEGrecordings.SincelongtermEEGrecordingsareverymassive,mostsimulationtechniquesarenotscalableenoughtoinvestigatelarge-scalemultivariatetimeserieslikeEEGs.TheuseofMCMCmakesitpossibletoapproximatethebrainstructureparametersovertime.Moreimportantly,theMCMCframeworkcanalsobeextendedtotheanalysisofmulti-channelEEGsbygeneratingnewEEGdatapointswhileexploringthedatasequencesusingaMarkovchainmechanism.Inaddition,wecanintegratetheMCMCframeworkwithaBaysianapproach.Thiscanbe 135 PAGE 136 136 PAGE 137 EEGrecordingsystemhasbeenthemostusedapparatusforthediagnosisofepilepsyandotherneurologicaldisorders.ItisknownthatchangesinEEGfrequencyandamplitudearisefromspontaneousinteractionsbetweenexcitatoryandinhibitoryneuronsinthebrain.StudiesintotheunderlyingmechanismofbrainfunctionhavesuggestedtheimportanceoftheEEGcouplingstrengthbetweendierentcorticalregions.Forexample,thesynchronizationofEEGactivityhasbeenshowninrelationtomemoryprocess[ 133 ; 134 ]andlearningprocessofthebrain[ 135 ].Inapathophysiologicalstudy,dierentbrainsynchronization/desynchronizationEEGpatternsareshowntobeinduced 137 PAGE 138 136 ]. Inepilepsyresearch,severalauthorshavesuggesteddirectrelationshipbetweenchangeinsynchronizationphenomenaandonsetofepilepticseizuresusingEEGrecordings.Forexample,Iasemidisetal.,reportedfromintracranialEEGthatthenonlineardynamicalentrainmentfromcriticalcorticalregionsisanecessaryconditionforonsetofseizuresforpatientswithtemporallobeepilepsy[ 112 ; 37 ; 60 ];LeVanQuyenetal.,showedepilepticseizurecanbeanticipatedbynonlinearanalysisofdynamicalsimilaritybetweenrecordings[ 35 ];Mormannetal.,showedthepreictalstatecanbedetectedbasedonadecreaseinsynchronizationonintracranialEEGrecordings[ 89 ; 43 ].ThehighlycomplexbehavioronEEGrecordingsisconsideredasnormalityofbrainstate,whiletransitionsintoalowercomplexitybrainstateareregardedasapathologicalnormalitylosses.Synchronizationpatternswerealsofoundtodiersomewhatdependingonepilepticsyndromes,withprimarygeneralizedabsenceseizuresdisplayingmorelong-rangesynchronyinfrequencybands(3-55Hz)thangeneralizedtonicmotorseizuresofsecondary(symptomatic)generalizedepilepsyorfrontallobeepilepsy[ 113 ]. Inthisstudy,mutualinformationandnonlinearinterdependencemeasureswereappliedontheEEGrecordingstoidentifytheeectoftreatmentonthecouplingstrengthanddirectionalityofinformationtransportbetweendierentbraincorticalregions[ 137 ; 100 { 102 ; 104 ].TheEEGrecordingswereobtainedfrompatientswithULD. ULDisonetypeofProgressiveMyoclonicEpilepsy(PME);arareepilepsydisorderwithcomplexinheritance.ULDwasrstdescribedbyUnverrichtin1891andLundborgin1903[ 138 ; 139 ].AEDsismainstayforthetreatmentofULDwithoverallunsatisfactoryecacy.Duetotheprogressionoftheseverityofmyoclonus,theecacyofAEDstreatmentisdiculttobedeterminedclinicallyespeciallyinthelaterstagesofthedisease.TheEEGrecordingsforULDsubjectsusuallydemonstrateabnormalslowbackground,generalizedhigh-amplitude3{5Hzspikewavesorpolyspikeandwave 138 PAGE 139 140 ; 141 ].Furthermore,AEDsassociatedgeneralizedslowingofEEGbackgroundrhythmshasbeenreportedwithhighlyvariablefrompatienttopatient[ 142 ].However,itwasdiculttodeterminehowstrongthecorrelationbetweenEEGslowinganddiseaseprogressionwassincetheintensicationofdrugtreatmentduringthelaterstagesofillnessmighthavecontributedtotheEEGslowing[ 140 ]. TheclinicalobservationshavebeenthemostcommonmethodforevaluatingtheinuenceandeectivenessofAEDsinterventionsinpatientswithepilepsyandotherneurologicaldisorders.Morespecically,ecacyoftreatmentisusuallymeasuredbycomparingtheseizurefrequencyduringtreatmenttoanitebaselineperiod.EEGrecordingsaremainlyusedassupplementaldiagnostictoolsinmedicaltreatmentevaluations.Otherthancountingthenumberofseizuresasameasurefortreatmenteectthereiscurrentlynoreliabletoolforevaluatingtreatmenteectsinpatientswithseizuredisorders.AquantitativesurrogateoutcomemeasureusingEEGrecordingsforpatientswithepilepsyisdesired. Thisrestofthischapterisorganizedasfollows.ThebackgroundofthepatientsandtheparametersoftheEEGrecordingsaregiveninsectionII.InsectionIII.Themethodsforidentifythecouplingstrengthanddirectionalityofinformationtransportbetweendierentbraincorticalregionsaredescribed.Thequantitativeanalysis,statisticaltestsandresultsarepresentedinSectionIV.TheconclusionanddiscussionaregiveninSectionV. 139 PAGE 140 Table8-1. ULDpatientinformation 1Female47998482Male451080653Male501250664Male51116854 8.3.1MutualInformation 103 ].LetusdenotethetimeseriesoftwoobservablevariablesasX=fxigNi=1andY=fyjgNj=1,whereNisthelengthoftheseriesandthetimebetweenconsecutiveobservations(i.e.,samplingperiod)isxed.Thenthemutualinformationisgivenby; 140 PAGE 141 143 ]: whereH(X);H(Y)aretheentropiesofX;YandH(X;Y)isthejointentropyofXandY.EntropyforXisdenedby: Theunitsofthemutualinformationdependsonthechoiceonthebaseoflogarithm.Thenaturallogarithmisusedinthestudythereforetheunitofthemutualinformationisnat.WerstestimatebH(X)fromXby ForXandYtimeserieswedened(x)ij=kxixjk;d(y)ij=kyiyjkasthedistancesforxiandyibetweeneveryotherpointinmatrixspacesXandY.Onecanrankthesedistancesandndtheknnforeveryxiandyi.InthespacespannedbyX;Y,similardistancerankmethodcanbeappliedforZ=(X;Y)andforeveryzi=(xi;yi)onecanalsocomputethedistancesd(z)ij=kzizjkanddeterminetheknnaccordingtosomedistancemeasure.Themaximumnormisusedinthisstudy: Nextlet(i) 2bethedistancebetweenzianditskthneighbor.Inordertoestimatethejointprobabilitydensityfunction(p:d:f:),weconsidertheprobabilityPk()whichistheprobabilitythatforeachzithekthnearestneighborhasdistance(i) 2dfromzi.ThisPk()representstheprobabilityfork1pointshavedistancelessthanthekthnearestneighborandNk1pointshavedistancegreaterthan(i) 2andk1pointshave 141 PAGE 142 2.Pk()isobtainedusingthemultinomialdistribution: wherepiisthemassofthe-ball.Thentheexpectedvalueoflogpiis where()isthedigammafunction: where()isthegammafunction.Itholdswhen(1)=CwhereCistheEuler-Mascheroniconstant(C0:57721).Themassofthe-ballcanbeapproximated(ifconsideringthep:d:finsidethe-ballisuniform)as wherecdxisthemassoftheunitballinthedx-dimensionalspace.FromEq.( 8{9 )wecanndanestimatorforP(X=xi) log[P(X=xi)](k)(N)dE(log(i))logcdx;(8{10) nallywithEq( 8{10 )andEq( 8{4 )weobtaintheKozachenko-LeonenkoentropyestimatorforX[ 105 ] ^H(X)=(N)(k)+logcdx+dx where(i)istwicethedistancefromxitoitsk-thneighborinthedxdimensionalspace.Forthejointentropywehave ^H(X;Y)=(N)(k)+log(cdxcdy)+dx+dy TheI(X;Y)isnowreadilytobeestimatedbyEq.( 8{2 ).Theproblemwiththisestimationisthataxednumberkisusedinallestimatorsbutthedistancemetricin 142 PAGE 143 2;x(i)+(i) 2]and[y(i)(i) 2;y(i)+(i) 2]respectively)inthex{yscatterdiagram.TheEq.( 8{12 )becomes: ^H(X)=(N)(nx(i)+1)+logcdx+dx FinallytheEq.( 8{2 )isrewrittenas: 106 ].Giventwotimeseriesxandy,usingmethodofdelayweobtainthedelayvectorsxn=(xn;:::;xn(m1))andyn=(xn;:::;xn(m1)),wheren=1;:::N,mistheembeddingdimensionanddenotesthetimedelay[ 52 ].Letrn;jandsn;j,j=1;:::;kdenotethetimeindicesoftheknearestneighborsofxnandyn.Foreachxn,themeanEuclideandistancetoitskneighborsisdenedas andtheY-conditionedmeansquaredEuclideandistanceisdenedbyreplacingthenearestneighborsbytheequaltimepartnersoftheclosestneighborsofyn Thedelay=5isestimatedusingautomutualinformationfunction,theembeddingdimensionm=10isobtainedusingCao'smethodandtheTheilercorrectionissettoT=50[ 73 ; 107 ]. 143 PAGE 144 SinceR(k)n(XjY)R(k)n(X)byconstruction, 0 PAGE 145 144 ].TensecondsofcontinuousEEGepochisrandomlysampledfromeverychannelsandthissamplingprocedurewasrepeatedwithreplacementfor30times.ThereferenceA1andA2channels(inactiveregions)areexcludedfromtheanalysis.Twosamplet-test(N=30,=0:05)isusedtotestthestatisticaldierencesonmutualinformationandnonlinearinterdependenceduringbeforeandaftertreatment.LowmutualinformationandinformationtransportbetweendierentbraincorticalregionswereobservedinoursubjectswithlessseverityofULD.Furthermore,foreachpatientbothmutualinformationandinformationtransportbetweendierentbraincorticalregionsdecreaseafterAEDstreatment.t-testformutualinformationaresummarizedinTable 8-2 ,thetopographicaldistributionformutualinformationisalsoplotinheatmapsshowninFigures 8-2 Thesignicant\driver-response"relationshipisreveledbyt-test.Aftert-testthesignicantinformationtransportbetweenFp1andotherbraincorticalregionsisshowninFig. 8-1 .TheedgeswithanarrowstartingfromFp1tootherchanneldenoteN(XjY)issignicantlargerthenN(YjX),thereforeFp1isthedriver,andviceversa.The 145 PAGE 146 Table8-2. Topographicaldistributionfortreatmentdecouplingeect(DE:DecoupleElectrode(DE)) (1)Fp1F3,C4,P4,F7Fp2,F3,F8,T5F3,F7F3,P3,Fz,T5T4,T5,O1F8,T4,Fz(2)Fp2C3,C4,F8,T4Fp1,F4,T6,O2F8C3,C4,T5,P4T5,PzCz(3)F3Fp1,C4,P4,O2Fp1,C3,P3,PzFp1Fp1,F7(4)F4C4,P4O2Fp2,P4,O2,FzCzC4,Fz(5)C3Fp2,C4,P3,O1F3,P3,O1P3,O1Fp2,P3,O1(6)C4Fp1,Fp2,F3,F4,C3P4,T6,O2P4,O2Fp1,Fp2,F3A1N/AN/AN/AN/AA2N/AN/AN/AN/A(7)P3C3,O2F3,C3,T3,PzO1Fp1,C3,Cz(8)P4Fp1,F3,F4F4,C4C4Fp2,(9)O1Fp1,C3C3,T5,T3,F7P3,PzC3(10)O2F3,F4,P3Fp2,F4,C4C4Pz(11)F7Fp1,Fz,PzC3,T5,O1Fp1F3(12)F8Fp2Fp1,C4,P4Fp2Fp2,(13)T3NONEP3,O1NONENONE(14)T4Fp1,Fp2,CzFp2,O1NONEFp2,(15)T5Fp1,Fp2Fp2,C4NONEFp2,Cz(16)T6NONEC4NONENONE(17)FzF7Fp1,F4CzFp1,Fp2,F3(18)PzFp2,F7,CzF3,P3,CzO2O2(19)CzT4,PzPzF4,FzFp2,T5,P3 146 PAGE 147 NonlinearinterdependencesforelectrodeFP1 beforeandaftertreatment.AlthoughtheresultsindicatethatthemutualinformationandnonlinearinterdependenciesmeasurescouldbeusefulindeterminingthetreatmenteectsforpatientswithULD.Toprovetheusefulnessoftheproposedstudy,alargerpatientpopulationisneeded.Theapproachesinthisstudyareabivariatemeasures,sinceamultivariatemeasureisnoteasytomodelandhasnotbeenresolved.Thedecouplingbetweenfrontalandoccipitalcorticalregionsmaybecausedbydecreaseddrivingforcedeepinsidethebrain.Inotherwords,theeectofthetreatmentmayreducethecouplestrengthbetweenthalamusandcortexinULDsubjects.Nevertheless,thelimitationsmustbementioned,ithasbeenreportedthatitisnecessarytotakeintoaccountthe 147 PAGE 148 145 { 147 ]. 148 PAGE 149 Pairwisemutualinformationbetweenforallelectrodes-beforev.s.aftertreatment 149 PAGE 150 Patient1beforetreatmentnonlinearinterdependencies Fp1-F30.370.441.401.380.690.700.880.100.080.390.320.100.090.05 Fp1-F40.200.340.770.770.410.440.590.070.270.630.360.200.180.25 Fp1-C30.230.310.880.750.490.480.710.060.040.400.240.140.110.09 Fp1-C40.130.220.460.400.280.270.420.090.130.400.280.230.180.28 Fp1-P30.170.260.820.560.400.350.580.040.060.480.280.170.110.12 Fp1-P40.150.250.620.410.320.280.500.080.060.600.250.190.140.15 Fp1-O10.140.360.400.210.170.100.150.040.990.510.200.200.100.26 Fp1-O20.130.230.390.210.220.140.180.030.020.280.140.150.090.21 Fp1-F70.390.371.461.420.680.680.840.110.110.400.480.170.170.16 Fp1-F80.200.240.650.600.370.360.400.060.060.330.230.180.140.33 Fp1-T30.260.311.191.080.590.570.760.080.050.520.470.170.170.20 Fp1-T40.150.240.590.460.300.260.330.040.130.510.320.230.170.29 Fp1-T50.180.280.760.540.380.340.560.050.120.490.330.190.170.22 Fp1-T60.120.210.320.160.180.100.180.020.020.240.110.140.090.17 Fp1-Fz0.270.671.131.180.570.640.800.101.510.650.340.170.110.09 Fp1-Cz0.190.360.740.650.420.420.630.050.380.380.230.170.140.11 Fp1-Pz0.160.270.630.410.350.290.490.040.060.490.220.190.140.17 PAGE 151 Patient1aftertreatmentnonlinearinterdependencies Fp1-F30.390.481.561.370.720.700.870.110.070.490.250.100.080.05 Fp1-F40.240.320.870.830.460.500.710.080.070.580.340.200.150.14 Fp1-C30.220.300.880.810.500.500.720.070.040.290.180.120.090.08 Fp1-C40.190.250.620.660.360.420.630.060.040.320.220.180.120.16 Fp1-P30.160.270.540.490.330.330.580.050.030.250.140.130.100.09 Fp1-P40.140.240.470.420.270.290.530.040.040.320.170.170.100.14 Fp1-O10.130.230.340.270.180.180.440.030.030.300.150.150.100.12 Fp1-O20.120.190.230.210.100.110.310.030.040.210.130.130.090.16 Fp1-F70.200.170.600.620.280.290.360.260.230.770.840.360.370.44 Fp1-F80.210.200.870.430.490.5340.640.050.090.630.220.220.250.47 Fp1-T30.260.310.930.950.510.540.720.120.080.440.380.240.190.23 Fp1-T40.150.210.480.380.250.220.320.040.060.480.300.220.180.35 Fp1-T50.150.250.520.460.310.300.580.040.030.250.150.130.100.08 Fp1-T60.140.230.330.260.180.180.380.040.040.320.180.180.120.21 Fp1-Fz0.270.381.031.060.550.600.790.100.060.510.320.160.110.09 Fp1-Cz0.190.280.690.680.400.440.650.070.040.370.180.170.090.10 Fp1-Pz0.160.260.500.460.300.310.570.060.030.270.140.130.090.08 PAGE 152 Patient2beforetreatmentnonlinearinterdependencies Fp1-F30.530.581.901.750.790.780.860.140.130.350.350.070.080.09 Fp1-F40.400.511.581.400.710.680.790.090.080.400.450.110.130.14 Fp1-C30.340.471.371.400.540.630.740.090.070.430.460.230.120.11 Fp1-C40.250.411.031.040.430.490.640.080.130.440.560.210.190.19 Fp1-P30.280.391.161.380.460.580.690.070.050.460.540.260.160.13 Fp1-P40.231.000.951.090.430.490.620.063.440.420.550.200.180.20 Fp1-O10.220.340.871.150.330.470.570.070.070.430.570.290.200.19 Fp1-O20.220.340.921.040.420.460.540.050.060.370.490.170.170.22 Fp1-F70.430.471.681.760.710.740.850.150.150.440.390.130.110.11 Fp1-F80.300.401.301.240.610.600.680.060.100.390.410.140.150.29 Fp1-T30.370.411.451.890.590.700.750.110.120.500.540.250.150.13 Fp1-T40.250.370.961.050.410.470.580.070.080.430.580.230.200.25 Fp1-T50.250.311.091.410.440.550.630.060.050.420.620.230.210.17 Fp1-T60.210.360.951.150.390.490.550.060.140.470.570.270.190.24 Fp1-Fz0.4310.631.781.550.760.720.860.1155.360.280.360.070.100.07 Fp1-Cz0.320.441.391.330.610.630.750.070.040.370.470.150.150.13 Fp1-Pz0.240.350.921.040.370.490.610.060.060.400.510.240.170.17 PAGE 153 Patient2aftertreatmentnonlinearinterdependencies Fp1-F30.330.461.261.300.660.690.830.050.030.210.160.060.050.03 Fp1-F40.210.310.780.720.440.440.620.050.050.230.250.110.130.11 Fp1-C30.160.280.550.490.350.340.550.030.020.120.120.070.070.06 Fp1-C40.140.240.390.390.220.250.340.030.010.080.170.060.110.10 Fp1-P30.130.240.290.230.170.160.140.030.030.090.080.070.060.13 Fp1-P40.130.230.290.270.160.180.120.030.020.110.120.080.080.11 Fp1-O10.120.220.240.260.130.16-0.030.030.010.120.110.090.080.20 Fp1-O20.130.230.270.300.140.20-0.080.030.030.100.120.080.080.14 Fp1-F70.290.331.121.090.540.530.660.180.200.650.630.310.300.37 Fp1-F80.220.230.780.510.450.300.510.050.060.280.240.120.150.25 Fp1-T30.200.280.750.720.430.430.640.030.040.130.140.070.080.07 Fp1-T40.140.190.420.290.240.150.190.030.040.120.150.080.140.25 Fp1-T50.140.240.330.180.190.110.180.020.030.110.080.060.060.14 Fp1-T60.130.120.280.280.160.180.030.020.020.100.080.080.060.09 Fp1-Fz0.310.431.211.180.620.640.790.070.050.300.240.090.070.04 Fp1-Cz0.150.230.400.430.240.290.450.040.040.090.150.060.080.07 Fp1-Pz0.130.230.260.260.140.180.120.020.020.070.100.060.070.11 PAGE 154 Patient3beforetreatmentnonlinearinterdependencies Fp1-F30.370.492.001.940.750.750.840.110.090.941.030.160.160.13 Fp1-F40.320.891.551.560.640.670.680.0932.560.840.860.180.160.21 Fp1-C30.241.031.521.430.600.630.700.083.590.730.710.200.190.23 Fp1-C40.150.260.780.810.360.400.40.110.170.720.700.280.290.34 Fp1-P30.165.370.880.820.400.400.480.0424.10.670.620.240.220.25 Fp1-P40.150.710.880.790.370.380.370.062.640.700.690.220.200.28 Fp1-O10.159.500.780.690.360.350.330.0528.220.620.550.220.210.32 Fp1-O20.040.050.660.800.220.210.330.040.050.660.800.220.210.33 Fp1-F70.391.682.082.040.750.720.830.136.701.031.210.180.240.21 Fp1-F80.210.281.191.310.490.5340.640.090.090.921.020.280.250.26 Fp1-T30.220.401.271.330.520.530.640.100.511.001.170.250.280.23 Fp1-T40.198.141.171.250.510.540.610.0830.120.760.880.220.220.29 Fp1-T50.1714.481.231.250.500.480.550.0743.450.961.110.240.250.26 Fp1-T60.0614.220.740.810.240.200.280.164.320.900.990.380.440.38 Fp1-Fz0.372.661.811.730.730.720.790.126.680.740.840.120.18 Fp1-Cz0.1727.751.101.050.450.490.530.0688.790.680.730.190.190.18 Fp1-Pz0.643.620.950.920.400.410.442.7218.610.710.790.240.240.27 PAGE 155 Patient3aftertreatmentnonlinearinterdependencies Fp1-F30.370.492.001.940.750.750.840.110.090.941.030.160.160.13 Fp1-F40.320.891.551.560.640.670.680.0932.560.840.860.180.160.21 Fp1-C30.241.031.521.430.600.630.700.083.590.730.710.200.190.23 Fp1-C40.150.260.780.810.360.400.40.110.170.720.700.280.290.34 Fp1-P30.165.370.880.820.400.400.480.0424.10.670.620.240.220.25 Fp1-P40.150.710.880.790.370.380.370.062.640.700.690.220.200.28 Fp1-O10.159.500.780.690.360.350.330.0528.220.620.550.220.210.32 Fp1-O20.040.050.660.800.220.210.330.040.050.660.800.220.210.33 Fp1-F70.391.682.082.040.750.720.830.136.701.031.210.180.240.21 Fp1-F80.210.281.191.310.490.5340.640.090.090.921.020.280.250.26 Fp1-T30.220.401.271.330.520.530.640.100.511.001.170.250.280.23 Fp1-T40.198.141.171.250.510.540.610.0830.120.760.880.220.220.29 Fp1-T50.1714.481.231.250.500.480.550.0743.450.961.110.240.250.26 Fp1-T60.0614.220.740.810.240.200.280.164.320.900.990.380.440.38 Fp1-Fz0.372.661.811.730.730.720.790.126.680.740.840.120.18 Fp1-Cz0.1727.751.101.050.450.490.530.0688.790.680.730.190.190.18 Fp1-Pz0.643.620.950.920.400.410.442.7218.610.710.790.240.240.27 PAGE 156 Patient4beforetreatmentnonlinearinterdependencies Fp1-F30.450.521.631.280.730.680.780.070.050.280.230.080.060.07 Fp1-F40.250.350.950.790.490.470.590.080.060.310.220.140.110.09 Fp1-C30.210.280.660.530.360.340.470.040.020.280.280.160.130.13 Fp1-C40.140.210.410.350.230.220.280.100.120.310.290.180.170.21 Fp1-P30.170.240.470.360.240.230.210.060.030.220.180.140.110.15 Fp1-P40.160.470.430.290.210.190.180.051.290.210.170.130.100.18 Fp1-O10.170.230.450.270.210.170.030.050.030.260.200.160.120.14 Fp1-O20.160.230.460.300.220.180.060.050.040.240.200.140.130.20 Fp1-F70.450.501.761.510.770.720.870.100.110.280.260.060.050.05 Fp1-F80.360.391.270.910.650.540.740.050.070.380.290.110.130.11 Fp1-T30.210.800.810.610.440.380.550.051.980.250.260.120.130.10 Fp1-T40.180.280.510.330.270.210.350.070.060.310.250.190.160.18 Fp1-T50.170.250.450.240.200.160.150.050.030.260.190.180.120.14 Fp1-T60.160.240.370.230.170.140.100.060.040.260.150.180.100.14 Fp1-Fz0.410.481.471.190.700.650.770.060.040.220.220.060.060.06 Fp1-Cz0.180.630.610.490.280.300.400.062.100.280.250.190.130.13 Fp1-Pz0.170.240.480.290.230.200.160.060.020.230.160.160.100.15 PAGE 157 Patient4aftertreatmentnonlinearinterdependencies Fp1-F30.320.461.241.250.640.670.820.040.030.270.150.080.050.03 Fp1-F40.210.310.760.680.430.420.600.060.050.240.260.110.130.11 Fp1-C30.180.290.580.520.350.360.550.030.020.120.120.070.060.07 Fp1-C40.150.240.380.400.220.260.320.030.020.080.150.060.100.10 Fp1-P30.140.250.330.220.200.160.150.030.030.090.080.070.050.15 Fp1-P40.120.250.270.270.150.170.090.020.020.100.090.070.060.10 Fp1-O10.140.220.250.220.150.14-0.170.030.010.140.110.100.080.18 Fp1-O20.130.240.260.280.140.17-0.090.030.030.080.140.070.090.15 Fp1-F70.380.431.461.440.700.690.860.100.120.210.190.070.070.05 Fp1-F80.230.240.910.590.500.320.450.050.080.250.280.100.180.31 Fp1-T30.200.290.720.690.410.410.630.040.030.150.140.110.080.08 Fp1-T40.150.200.440.280.260.150.190.020.040.170.150.100.130.27 Fp1-T50.130.250.300.170.180.110.190.020.030.070.060.050.050.13 Fp1-T60.130.230.280.250.160.150.010.020.020.100.090.080.060.10 Fp1-Fz0.300.421.191.170.610.630.780.050.030.240.200.080.060.04 Fp1-Cz0.150.250.410.420.240.280.430.040.040.100.140.080.070.10 Fp1-Pz0.130.250.280.250.150.170.100.020.030.080.090.060.060.10 PAGE 158 [1] WorldHealthOrganizationWHO.Epilepsy:aetiogy,epidemiologyandprognosis.WorldHealthOrganization2008.Allrightsreserved,pagehttp://www.who.int/mediacentre/factsheets/fs165/en/,2008. [2] J.Loiseau,P.Loiseau,B.Duch,M.Guyot,J.-F.Dartigues,andB.Aublet.Asurveyofepilepticdisordersinsouthwestfrance:Seizuresinelderlypatients.AnnalsofNeurology,27(3):232{237,1990. [3] I.Kotsopoulos,M.deKrom,F.Kessels,J.Lodder,J.Troost,M.Twellaar,T.vanMerode,andA.Knottnerus.Incidenceofepilepsyandpredictivefactorsofepilepticandnon-epilepticseizures.Seizure,14(3):175{182,2005. [4] CollaborativeGroupfortheStudyofEpilepsy.Prognosisofepilepsyinnewlyreferredpatients:Amulticenterprospectivestudyoftheeectsofmonotherapyonthelong-termcourseofepilepsy.Epilepsia,33(1):45{51,1992. [5] N.Pocock.Reviewevidencefortolerancetoantiepilepticdrugs.Epilepsia,47:1253{1284,2006. [6] W.LscherandD.Schmidt.Criticalreview:Experimentalandclinicalevidenceforlossofeect(tolerance)duringprolongedtreatmentwithantiepilepticdrugs.Epilepsia,47(8):1253{1284,2006. [7] M.R.Sperling,A.J.Saykin,F.D.Roberts,J.A.French,andM.J.O'Connor.Occupationaloutcomeaftertemporallobectomyforrefractoryepilepsy.Neurol-ogy,45(5):970{977,1995. [8] G.F.Rossi.Evaluationofsurgicaltreatmentoutcomeinepilepsy.CriticalReviewsinNeurosurgery,8(5):282{289,1998. [9] J.Zabara.Inhibitionofexperimentalseizuresincaninesbyrepetitivevagalstimulation.Epilepsia,33(6):1005{1012,1992. [10] A.Zanchetti,S.C.Wang,andG.Moruzzi.Theeectofvagalaerentstimulationontheeegpatternofthecat.ElectroencephalographyandClinicalNeurophysiology,4(3):357{361,August1952. [11] J.Magnes,G.Moruzzi,andO.Pompeiano.Synchronizationofthee.e.g.producedbylow-frequencyelectricalstimulationoftheregionofthesolitarytract.Arch.Ital.Biol.,99:33{67,1961. [12] M.H.Chase,Y.Nakamura,C.D.Clemente,andM.B.Sterman.Aerentvagalstimulation:neurographiccorrelatesofinducedeegsynchronizationanddesynchronization.BrainResearch,5(2):236{49,1967. [13] R.S.McLachlan.Suppressionofinterictalspikesandseizuresbystimulationofthevagusnerve.Epilepsia,34(5):918{923,1993. 158 PAGE 159 J.S.Lockard,W.C.Congdon,andL.L.DuCharme.Feasibilityandsafetyofvagalstimulationinmonkeymodel.Epilepsia,31(S2):S20{S26,1990. [15] B.M.Uthman,B.J.Wilder,J.K.Penry,C.Dean,R.E.Ramsay,S.A.Reid,E.J.Hammond,W.B.Tarver,andJ.F.Wernicke.Treatmentofepilepsybystimulationofthevagusnerve.Neurology,43(7):13338{13345,1993. [16] G.L.MorrisIIIandW.M.Mueller.Long-termtreatmentwithvagusnervestimulationinpatientswithrefractoryepilepsy.thevagusnervestimulationstudygroupe01-e05.Neurology,54(8):1712,2000. [17] D.Ko,C.Heck,S.Grafton,M.L.J.Apuzzo,W.T.Couldwell,T.Chen,J.D.Day,V.Zelman,T.Smith,andC.M.DeGiorgio.Vagusnervestimulationactivatescentralnervoussystemstructureinepilepticpatientsduringpetbloodowimaging.Neurosurgery,39(2):426{431,1996. [18] T.R.Henry,R.A.E.Bakay,J.R.Votaw,P.B.Pennell,C.M.Epstein,T.L.Faber,S.T.Grafton,andJ.M.Homan.Brainbloodowalterationsinducedbytherapeuticvagusnervestimulationinpartialepilepsy:I.acuteeectsathighandlowlevelsofstimulation.Epilepsia,39(9):983{990,1998. [19] E.Ben-Menachem,A.Hamberger,T.Hedner,E.J.Hammond,B.M.Uthman,J.Slater,T.Treig,H.Stefan,R.E.Ramsay,J.F.Wernicke,andB.J.Wilder.Eectsofvagusnervestimulationonaminoacidsandothermetabolitesinthecsfofpatientswithpartialseizures.EpilepsyResearch,20(3):221{227,1995. [20] S.E.Krahl,K.B.Clark,D.C.Smith,andR.A.Browning.Locuscoeruleuslesionssuppresstheseizure-attenuatingeectsofvagusnervestimulation.Epilepsia,39(7):709{714,1998. [21] J.MalmivuandR.Plonse.Bioelectromagnetism{PrinciplesandApplicationsofBioelectricandBiomagneticFields.OxfordUniversityPress,1995. [22] E.R.Kandel,J.H.Schwartz,andT.M.Jessell.PrinciplesofNeuralScience,FourthEdition.McGraw-HillMedical,2000. [23] H.Berger.Ueberdaselektrenkephalogrammdesmenschen.Arch.Psychiatr.Nervenkr,87:527{570,1929. [24] H.H.Jasper.Theten-twentyelectrodesystemoftheinternationalfederation.ElectroencephalogrClinNeurophysiol,10:371{375,1958. [25] D.A.PrinceaandB.W.Connors.Mechanismsofinterictalepileptogenesis.44:275{299,1986. [26] M.E.Weinand,L.P.Carter,W.F.El-Saadany,P.J.Sioutos,D.M.Labiner,andK.J.Oommen.Cerebralbloodowandtemporallobeepileptogenicity.JournalofNeuosurgery,86:226{232,1997. 159 PAGE 160 C.Baumgartner,W.Serles,F.Leutmezer,E.Pataraia,S.Aull,T.Czech,U.Pietrzyk,A.Relic,andI.Podreka.Preictalspectintemporallobeepilepsy:Regionalcerebralbloodowisincreasedpriortoelectroencephalography-seizureonset.TheJournalofNuclearMedicin,39(6):978{982,1998. [28] P.D.Adelson,E.Nemoto,M.Scheuer,M.Painter,J.Morgan,andH.Yonas.Noninvasivecontinuousmonitoringofcerebraloxygenationperiictallyusingnear-infraredspectroscopy:apreliminaryreport.Epilepsia,40:1484{1489,1999. [29] P.Federico,D.F.Abbott,R.S.Briellmann,A.S.Harvey,andG.D.Jackson.Functionalmriofthepre-ictalstate.Brain,128:1811{1817,2005. [30] R.Delamont,P.Julu,andG.Jamal.Changesinameasureofcardiacvagalactivitybeforeandafterepilepticseizures.EpilepsyRes,35:87{94,1999. [31] D.H.KeremandA.B.Geva.Forecastingepilepsyfromtheheartratesignal.MedBiolEngComput,43:230{239,2005. [32] L.D.IasemidisandJ.C.Sackellares.Longtimescalespatio-temporalpatternsofentrainmentinpreictalecogdatainhumantemporallobeepilepsy.Epilepsia,31:621,1990. [33] J.Martinerie,C.VanAdam,M.LeVanQuyen,M.Baulac,S.Clemenceau,B.Renault,andF.J.Varela.Epilepticseizurescanbeanticipatedbynon-linearanalysis.NatureMedicine,4:1173{1176,1998. [34] C.E.ElgerandK.Lehnertz.Seizurepredictionbynon-lineartimeseriesanalysisofbrainelectricalactivity.EuropeanJournalofNeuroscience,10:786{789,1998. [35] M.LeVanQuyen,J.Martinerie,M.Baulac,andF.J.Varela.Anticipatingepilepticseizuresinrealtimebynon-linearanalysisofsimilaritybetweeneegrecordings.NeuroReport,10:2149{2155,1999. [36] G.Widman,K.Lehnertz,H.Urbach,andC.E.Elger.Spatialdistributionofneuronalcomplexitylossinneocorticalepilepsies.Epilepsia,41:811{817,2000. [37] L.D.Iasemidis,D.-S.Shiau,J.C.Sackellares,P.M.Pardalos,andA.Prasad.Dynamicalresettingofthehumanbrainatepilepticseizures:applicationofnonlineardynamicsandglobaloptimizationtecniques.IEEETransactionsonBiomedicalEngineering,51(3):493{506,2004. [38] F.Mormann,T.Kreuz,C.Rieke,R.G.Andrzejak,A.Kraskov,P.David,C.E.Elger,andK.Lehnertz.Onthepredictabilityofepilepticseizures.JournalofClinicalNeurophysiology,116(3):569{587,2005. [39] K.Lehnertz,F.Mormann,O.Hannes,A.Mller,J.Prusseit,A.Chernihovskyi,M.Staniek,D.Krug,S.Bialonski,andC.EElger.State-of-the-artofseizureprediction.Journalofclinicalneurophysiology,24(2):147{153,2007. 160 PAGE 161 C.E.Elger.Futuretrendsinepileptology.CurrentOpinioninNeurology,14(2):185{186,2001. [41] F.Mormann,R.G.Andrzejak,C.E.Elger,andK.Lehnertz.Seizureprediction:thelongandwindingroad.Brain,130(2):314{333,2006. [42] S.Gigola,F.Ortiz,C.E.DAttellis,W.Silva,andS.Kochen.Predictionofepilepticseizuresusingaccumulatedenergyinamultiresolutionframework.JournalofNeuroscienceMethods,138:107{111,2004. [43] F.Mormann,R.G.Andrzejak,T.Kreuz,C.Rieke,P.David,C.E.Elger,andK.Lehnertz.Automateddetectionofapreseizurestatebasedonadecreaseinsynchronizationinintracranialelectroencephalogramrecordingsfromepilepsypatients.Phys.Rev.E,67(2):021912,2003. [44] O.Rossler.Anequationforcontinuouschaos.Phys.Lett.,35A:397{398,1976. [45] E.N.Lorenz.Deterministicnonperiodicow.JournalofAtmosphericSciences,20:130{141,1963. [46] P.GrassbergerandI.Procaccia.Measuringthestrangenessofstrangeattractors.PhysicaD:NonlinearPhenomena,9:189{208,1983. [47] A.BabloyantzandA.Destexhe.Lowdimensionalchaosinaninstanceofepilepsy.Proc.Natl.Acad.Sci.USA,83:3513{3517,1986. [48] J.P.M.Pijn,D.N.Velis,M.J.vanderHeyden,J.DeGoede,C.W.M.vanVeelen,andF.H.LopesdaSilva.Nonlineardynamicsofepilepticseizuresonbasisofintracranialeegrecordings.BrainTopography,9:249{270,1997. [49] K.LehnertzandC.E.Elger.Spatio-temporaldynamicsoftheprimaryepileptogenicareaintemporallobeepilepsycharacterizedbyneuronalcomplexityloss.Electroen-cephalogr.Clin.Neurophysiol.,95:108{117,1995. [50] U.R.Acharya,O.Fausta,N.Kannathala,T.Chuaa,andS.Laxminarayan.Non-linearanalysisofeegsignalsatvarioussleepstages.ComputerMethodsandProgramsinBiomedicine,80(1):37{45,2005. [51] H.Whitney.Dierentiablemanifolds.TheAnnalsofMathematics,SecondSeries,37(3):645{680,1936. [52] F.Takens.Detectingstrangeattractorsinturbulence.InD.A.RandandL.S.Young,editors,DynamicalSystemsandTurbulence,LectureNotesinMathematics,volume898,pages366{381.Springer-Verlag,1981. [53] J.-P.EckmannandD.Ruelle.Ergodictheoryofchaosandstrangeattractors.Rev.Mod.Phys.,57:617{656,1985. 161 PAGE 162 A.Wolf,J.B.Swift,H.L.Swinney,andJ.A.Vastano.Determininglyapunovexponentsfromatimeseries.PhysicaD,16:285{317,1985. [55] L.D.Iasemidis.Onthedynamicsofthehumanbrainintemporallobeepilepsy.PhDthesis,UniversityofMichigan,AnnArbor,1990. [56] L.D.Iasemidis,A.Barreto,R.L.Gilmore,B.M.Uthman,S.N.Roper,andJ.C.Sackellares.Spatio-temporalevolutionofdynamicalmeasuresprecedesonsetofmesialtemporallobeseizures.Epilepsia,35S:133,1993. [57] L.D.Iasemidis,J.C.Principe,J.M.Czaplewski,R.L.Gilmore,S.N.Roper,andJ.C.Sackellares.Spatiotemporaltransitiontoepilepticseizures:anonlineardynamicalanalysisofscalpandintracranialeegrecordings.InF.L.Silva,J.C.Principe,andL.B.Almeida,editors,SpatiotemporalModelsinBiologicalandArticialSystems,pages81{88.IOSPress,1997. [58] L.D.Iasemidis,J.C.Principe,andJ.C.Sackellares.Measurementandquanticationofspatiotemporaldynamicsofhumanepilepticseizures.InM.Akay,editor,Nonlinearbiomedicalsignalprocessing,pages294{318.Wiley{IEEEPress,vol.II,2000. [59] L.D.Iasemidis,D.-S.Shiau,P.M.Pardalos,andJ.C.Sackellares.Phaseentrainmentandpredictabilityofepilepticseizures.InP.M.PardalosandJ.C.Principe,editors,Biocomputing,pages59{84.KluwerAcademicPublishers,2001. [60] P.M.Pardalos,J.C.Sackellares,P.R.Carney,andL.D.Iasemidis.QuantitativeNeuroscience.KluwerAcademicPublisher,2004. [61] L.D.Iasemidis,P.M.Pardalos,D.-S.Shiau,W.A.Chaovalitwongse,K.Narayanan,A.Prasad,K.Tsakalis,P.R.Carney,andJ.C.Sackellares.Longtermprospectiveon-linereal-timeseizureprediction.JournalofClinicalNeurophysiology,116(3):532{544,2005. [62] S.M.Pincus.Approximateentropyasameasureofsystemcomplexity.InProceed-ingsoftheNationalAcademyofSciencesoftheUnitedStatesofAmerica,pages2297{2301,1991. [63] L.DiambraandC.P.Malta.Nonlinearmodelsfordetectingepilepticspikes.Phys.Rev.E,59:929{937,1999. [64] S.M.Pincus.Approximateentropy(apen)asacomplexitymeasure.CHAOS,5(1):110{117,1995. [65] D.T.Kaplan,M.I.Furman,S.M.Pincus,S.M.Ryan,L.A.Lipsitz,andA.L.Goldberger.Agingandthecomplexityofcardiovasculardynamics.BiophysicalJournal,59:945{949,1991. [66] A.N.Kolmogorov.Anewmetricinvariantoftransientdynamicalsystemsandautomorphismsinlebesguespace.DokladyAkademiiNaukSSSR,119:861{864,1958. 162 PAGE 163 Y.-C.Lai,M.G.Frei,I.Osorio,andL.Huang.Characterizationofsynchronywithapplicationstoepilepticbrainsignals.PhysRevLett,98(10):108102,2007. [68] P.Velazquez,J.L.Dominguez,andR.L.Wennberg.Complexphasesynchronizationinepilepticseizures:evidenceforadevil'sstaircase.PhysRevEStatNonlinSoftMatterPhys,75(1):011922,2007. [69] T.I.NetoandS.J.Schi.Decreasedneuronalsynchronizationduringexperimentalseizures.JournalofNeuroscience,22(16):7297{307,2007. [70] S.SakataandT.Yamamori.Topologicalrelationshipsbetweenbrainandsocialnetworks.NeuralNetworks,20:12{21,2007. [71] R.Andrzejak,F.Mormann,G.Widman,T.Kreuz,C.Elger,andK.Lehnertz.Improvedspatialcharacterizationoftheepilepticbrainbyfocusingonnonlinearity.EpilepsyResearch,69:30{44,2006. [72] T.Sauer,J.A.Yorke,andM.Casdagli.Embedology.JournalofStatisticalPhysics,65:579{616,1991. [73] L.-Y.Cao.Practicalmethodfordeterminingtheminimumembeddingdimensionofascalartimeseries.PhysicaD,110(1{2):43{50,1997. [74] M.B.Kennel,R.Brown,andH.D.I.Abarbanel.Determiningembeddingdimensionforphase-spacereconstructionusingageometricalconstruction.PhysRev.A,45:3403{34,1992. [75] A.M.FraserandH.L.Swinney.Independentcoordinatesforstrangeattractorsfrommutualinformation.PhysRevA,33:1134{1140,1986. [76] P.GrassbergerandI.Procaccia.Characterizationofstrangeattractors.PhysicalReviewLetters,50(5):346{349,1983. [77] D.S.BroomheadandG.P.King.Extractingqualitativedynamicsfromexperimentaldata.PhysicaD,20:217{236,1986. [78] M.C.Casdagli,L.D.Iasemidis,J.C.Sackellares,S.N.Roper,R.L.Gilmore,andR.S.Savit.CharacterizingnonlinearityininvasiveEEGrecordingsfromtemporallobeepilepsy.PhysicaD,99:381{399,1996. [79] B.Weber,K.Lehnertz,C.E.Elger,andH.G.Wieser.Neuronalcomplexitylossininterictaleegrecordedwithforamenovaleelectrodespredictssideofprimaryepileptogenicareaintemporallobeepilepsy:Areplicationstudy.Epilepsia,39(9):922{927,1998. [80] J.C.Sackellares,D.-S.Shiau,J.C.Principe,M.C.K.Yang,L.K.Dance,W.Suharitdamrong,W.A.Chaovalitwongse,P.M.Pardalos,andL.D.Iasemidis.Predictibilityanalysisforanautomatedseizurepredictionalgorithm.JournalofClinicalNeurophysiology,23(6):509{520,2006. 163 PAGE 164 J.Theiler,S.Eubank,A.Longtin,B.Galdrikian,andJ.Farmer.Testingfornonlinearityintimeseries:themethodofsurrogatedata.PhysicaD,58:77{94,1992. [82] W.A.Chaovalitwongse,P.M.Pardalos,andO.A.Prokoyev.Electroencephalogram(EEG)TimeSeriesClassication:ApplicationsinEpilepsy.AnnalsofOperationsResearch,148:227{250,2006. [83] N.AcirandC.Guzelis.Automaticspikedetectionineegbyatwo-stageprocedurebasedonsupportvectormachines.ComputersinBiologyandMedicine,34(7):561{575,2004. [84] B.Gonzalez-Vellon,S.Sanei,andJ.A.Chambers.Supportvectormachinesforseizuredetection.InProceedingsofthe3rdIEEEInternationalSymposiumonSignalProcessingandInformationTechnology,2003.ISSPIT2003.,2003. [85] P.Comon.Independentcomponentanalysis,anewconcept.SignalProcessing,36:287{314,1994. [86] J.-F.CardosoandB.Laheld.Equivariantadaptivesourceseparation.IEEETransactionsonSignalProcessing,44:3017{3030,1996. [87] A.J.BellandT.J.Sejnowski.Aninformation-maximizationapproachtoblindseparationandblinddeconvolution.NeuralComputation,7:1129{1259,1995. [88] A.Hyvrinen,J.Karhunen,andE.Oja.Independentcomponentanalysis.JohnWiley&Sons,2001. [89] F.Mormann,K.Lehnertz,P.David,andC.E.Elger.Meanphasecoherenceasameasureforphasesynchronizationanditsapplicationtotheeegofepilepsypatients.PhysicaD,144:358{369,2000. [90] C.W.J.Granger.Investigatingcausalrelationsbyeconometricmodelsandcross-spectralmethods.Econometrica,37:424{438,1969. [91] E.Rodriguez,N.George,J.P.Lachaux,J.Martinerie,B.Renault,andF.JVarela.perceptionsshadow:Long-distancesynchronizationofhumanbrainactivity,.Nature,,397:430{347,1999. [92] F.H.LopesdaSilvaandJ.P.Pijn.TheHandbookofBrainTheoryandNeuralNetworks,chapterEEGanalysis,pages348{351.MITPress,Cambridge,MA,1995. [93] C.Hugenii.Horoloquiumoscilatorum.Paris,France,1673. [94] M.G.Rosenblum,A.S.Pikovsky,andJ.Kurths.Phasesynchronizationofchaoticoscillators.Phys.Rev.Lett.,76:1804{1807,1996. [95] J.W.Freeman.Characteristicsofthesynchronizationofbrainactivityimposedbyniteconductionvelocitiesofaxons.InternationalJournalofBifurcationandChaos,10:2307{2322,2000. 164 PAGE 165 J.W.FreemanandJ.M.Barrie.Analysisofspatialpatternsofphaseinneocorticalgammaeegsinrabbit.JournalofNeurophysiology,84:1266{1278,2000. [97] M.LeVanQuyen,J.Foucher,J.P.Lachaux,E.Rodriguez,A.Lutz,J.Martinerie,andF.J.Varela.Comparisonofhilberttransformandwaveletmethodsfortheanalysisofneuronalsynchrony.JoutnalofNeuroscienceMethods,111(2):83{98,2001. [98] C.E.Shannon.Amathematicaltheoryofcommunication.BellSystemTechnicalJournal,27:379{423,623{656,1948. [99] R.QuianQuiroga,A.Kraskov,T.Kreuz,andP.Grassberger.Performanceofdierentsynchronizationmeasuresinrealdata:Acasestudyonelectroencephalographicsignals.Phys.Rev.E,65:041903,2002. [100] R.QuianQuiroga,A.Kraskov,T.Kreuz,andP.Grassberger.Replyto"commenton`performanceofdierentsynchronizationmeasuresinrealdata:Acasestudyonelectroencephalographicsignals'".Phys.Rev.E,67:063902,2003. [101] R.B.DuckrowandA.M.Albano.Commentonperformanceofdierentsynchronizationmeasuresinrealdata:Acasestudyonelectroencephalographicsignals.Phys.Rev.E,67(6):063901,Jun2003. [102] N.NicolaouandS.J.Nasuto.Commenton"performanceofdierentsynchronizationmeasuresinrealdata:Acasestudyonelectroencephalographicsignals".Phys.Rev.E,72:063901,2005. [103] A.Kraskov,H.Stogbauer,andP.Grassberger.Estimatingmutualinformation.Phys.Rev.E,69:066138,2004. [104] R.QuianQuiroga,A.Kraskov,andP.Grassberger.Replyto"commenton`performanceofdierentsynchronizationmeasuresinrealdata:Acasestudyonelectroencephalographicsignals'".Phys.Rev.E,72:063902,2005. [105] L.F.KozachenkoandN.N.Leonenko.Sampleestimateofentropyofarandomvector.ProblemsofInformationTransmission,23:95{101,1987. [106] J.Arnhold,P.Grassberger,K.Lehnertz,andC.E.Elger.Arobustmethodfordetectinginterdependences:applicationtointracraniallyrecordedeeg.PhysicaD,134:419{4{30,1999. [107] J.Theiler.Spuriousdimensionfromcorrelationalgorithmsappliedtolimitedtime-seriesdata.Phys.Rev.A,34:2427{2432,1986. [108] G.L.HolmesandY.Ben-Ari.Theneurobiologyandconsequencesofepilepsyinthedevelopingbrain.PediatricResearch,49(3):320{325,2001. [109] J.W.Freeman.Spatialpropertiesofaneegeventintheolfactorybulbandcortex.ElectroencephalogrClinicalNeurophysiology,44:586{605,1978. 165 PAGE 166 M.C.MackeyandL.Glass.Oscillationandchaosinphysiologicalcontrolsystems.Science,4300:287{289,1977. [111] H.PetscheandM.A.B.Brazier.SynchronizationofEEGActivityinEpilepsies.Springer,1972. [112] L.D.Iasemidis,K.E.Pappas,R.L.Gilmore,S.N.Roper,andJ.C.Sackellares.Preictalentrainmentofacriticalcorticalmassisanecessaryconditionforseizureoccurrence.Epilepsia,37S(5):90,1996. [113] L.G.Dominguez,R.A.Wennberg,W.Gaetz,D.Cheyne,O.Snead,andJ.L.P.Velazquez.Enhancedsynchronyinepileptiformactivity?localversusdistantphasesynchronizationingeneralizedseizures.TheJournalofNeuroscience,25(35):8077{8084,2005. [114] M.L.Anderson.Evolutionofcognitivefunctionviaredeploymentofbrainareas.TheNeuroscientist,13(1):13{21,2007. [115] O.SpornsandR.Ktter.Motifsinbrainnetworks.PLoSBiology,2(11):e369,2004. [116] H.H.Jasper.Mechanismsofpropagation:Extracellularstudies.InH.H.Jasper,A.A.Ward,andA.Pope,editors,Basicmechanismsoftheepilepsies,pages421{440,Boston,1969.LittleBrown. [117] G.J.Ortega,L.MenendezdelaPrida,R.G.Sola,andJ.Pastor.Synchronizationclustersofinterictalactivityinthelateraltemporalcortexofepilepticpatients:Intraoperativeelectrocorticographicanalysis.Epilepsia,2007. [118] A.Brovelli,M.Z.Ding,A.Ledberg,Y.H.Chen,RichardNakamura,andStevenL.Bressler.Betaoscillationsinalarge-scalesensorimotorcorticalnetwork:Directionalinuencesrevealedbygrangercausality.ProceedingsoftheNationalAcademyofSciencesoftheUnitedStatesofAmerica,101:9849{9854,2004. [119] N.K.Varma,R.Kushwaha,A.Beydoun,W.J.Williams,andI.Drury.Mutualinformationanalysisanddetectionofinterictalmorphologicaldierencesininterictalepileptiformdischargesofpatientswithpartialepilepsies.ElectroencephalogrClinNeurophysiol,103(4):426{33,1997. [120] S.Ken,G.DiGennaro,G.Giulietti,F.Sebastiano,D.DeCarli,G.Garrea,C.Colonnese,R.Passariello,J.Lotterie,andB.Maraviglia.Quantitativeevaluationforbrainct/mricoregistrationbasedonmaximizationofmutualinformationinpatientswithfocalepilepsyinvestigatedwithsubduralelectrodes.MagnResonImaging,25(6):883{8,2007. [121] W.H.Miltner,C.Braun,M.Arnold,H.Witte,andE.Taub.Coherenceofgamma-bandeegactivityasabasisforassociativelearning.Nature,397:434{436,1999. 166 PAGE 167 D.Olivier,D.Cosmelli,andK.J.Friston.Evaluationofdierentmeasuresoffunctionalconnectivityusinganeuralmassmodel.NeuroImage,21:659{673,2004. [123] J.-P.Lachaux,E.Rodriguez,J.Martinerie,andF.J.Varela.Coherenceofgamma-bandeegactivityasabasisforassociativelearning.Nature,8:194{208,1999. [124] F.Bartolomei,F.Wendling,J.Vignal,S.Kochen,J.Bellanger,BadierJ.,R.L.Bouquin-Jeannes,andP.Chauvel.Seizuresoftemporallobeepilepsy:Identicationofsubtypesbycoherenceanalysisusingstereoelectro-encephalography.ClinicalNeurophysiology,110:1741{1754,1999. [125] V.L.Towle,R.K.Carder,L.Khorasani,andD.Lindberg.Electrocorticographiccoherencepatterns.JournalofClinicalNeurophysiology,16:528{547,1999. [126] H.D.I.Abarbanel,R.Brown,andM.Kennel.Variationoflyapunovexponentsinchaoticsystems:Theirimportanceandtheirevalutionusingobserveddata.JournalofNonliearScience,2:343{365,1992. [127] N.Mars,P.Thompson,andR.Wilkus.Spreadofepilepticactivityinhumans.Epilepsia,26:85{94,1985. [128] M.Palus,V.Albrecht,andI.Dvorak.Informationtheoretictestofnonlinearlityintimeseries.Phys.Rev.A.,34:4971{4972,1993. [129] R.Moddemeijer.Onestimationofentropyandmutualinformationofcontinuousdistributions.SignalProcessing,16(3):233{246,1989. [130] L.D.Iasemidis,D.-S.Shiau,W.A.Chaovalitwongse,J.C.Sackellares,P.M.Pardalos,P.R.Carney,J.C.Principe,A.Prasad,B.Veeramani,andK.Tsakalis.Adaptiveepilepticseizurepredictionsystem.IEEETransactionsonBiomedicalEngineering,5(5):616{627,2003. [131] A.J.ButteandI.S.Kohane.Mutualinformationrelevancenetworks:functionalgenomicclusteringusingpairwiseentropymeasurements.PacSympBiocomput,pages418{29,2000. [132] C.Andrieu,N.D.Freitas,A.Doucet,andMJordan.Anintroductiontomcmcformachinelearning.MachineLearning,50:5{43,2003. [133] W.Klimesch.Memoryprocesses,brainoscillationsandeegsynchronization.InternationalJournalofPsychophysiology,24(1-2):61{100,1996. [134] W.Klimesch.Eegalphaandthetaoscillationsreectcognitiveandmemoryperformance:areviewandanalysis.BrainResearchReviews,29(2-3):169{195,1999. [135] W.Singer.Synchronizationofcorticalactivityanditsputativeroleininformationprocessingandlearning.AnnualReviewofPhysiology,55:349{374,1993. 167 PAGE 168 D.V.Moretti,C.Miniussi,G.B.Frisoni,C.Geroldi,O.Zanetti,G.Binetti,andP.M.Rossini.Hippocampalatrophyandeegmarkersinsubjectswithmildcognitiveimpairment.ClinicalNeurophysiology,118(12):2716{2729,2007. [137] R.QuianQuiroga,A.Kraskov,T.Kreuz,andP.Grassberger.Performanceofdierentsynchronizationmeasuresinrealdata:Acasestudyonelectroencephalographicsignals.Phys.Rev.E,65:041903,2002. [138] H.Unverricht.DieMyoclonie.FranzDeutick,Leipzig,1891. [139] H.B.Lundborg.DieprogressiveMyoclonus-Epilepsie(UnverrichtsMyoclonie).AlmqvistandWiksell,Uppsala,1903. [140] N.K.Chew,P.Mir,M.J.Edwards,C.Cordivari,D.M.,S.A.Schneider,H.-T.Kim,N.P.Quinn,andK.P.Bhatia.Thenaturalhistoryofunverricht-lundborgdisease:Areportofeightgeneticallyprovencases.MovementDisorders,Vol.23,No.1:107{113,2007. [141] E.Ferlazzoa,A.Magauddaa,P.Strianob,N.Vi-Hongc,S.Serraa,andP.Gentonc.Long-termevolutionofeeginunverricht-lundborgdisease.EpilepsyResearch,73:219{227,2007. [142] M.C.Salinsky,B.S.Oken,andL.Morehead.Intraindividualanalysisofantiepilepticdrugeectsoneegbackgroundrhythms.ElectroencephalographyandClinicalNeurophysiology,90(3):186{193,1994. [143] T.M.CoverandJ.A.Thomas.ElementsofInformationTheory.Wiley,NewYork,1991. [144] M.R.Chernick.BootstrapMethods:APractitioner'sGuide.Wiley-Interscience,1999. [145] M.SteriadeandF.Amzica.Dynamiccouplingamongneocorticalneuronsduringevokedandspontaneousspike-waveseizureactivity.JournalofNeurophysiology,72:2051{2069,1994. [146] E.SitnikovaandG.vanLuijtelaar.Corticalandthalamiccoherenceduringspikewaveseizuresinwag/rijrats.EpilepsyResearch,71:159{180,2006. [147] E.Sitnikova,T.Dikanev,D.Smirnov,B.Bezruchko,andG.vanLuijtelaar.Grangercausality:Cortico-thalamicinterdependenciesduringabsenceseizuresinwag/rijrats.JournalofNeuroscienceMethods,170(2):245{254,2008. 168 PAGE 169 Chang-ChiaLiuearnedhisB.S.degreeinspring2000fromtheDepartmentofIndustrialEngineering,Da-YehUniversityinTaiwan.HewenttotheUnitedStatesinspring2002andjoinedtheUniversityofFloridainfall2002.HereceiveddualM.S.degreesfromDepartmentsofIndustrialandSystemsEngineeringandJ.CraytonPruittFamilyBiomedicalEngineeringinspring2004andfall2007,respectively.Hisresearchinterestsincludeglobaloptimization,timeseriesanalysis,chaostheory,andnonlineardynamicswithapplicationsinbiomedicine. 169 |