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University of Southern California Dissertations and Theses
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Functional connectivity analysis and network identification in the human brain
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Functional connectivity analysis and network identification in the human brain
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FUNCTIONAL CONNECTIVITY ANAL YSIS AND NETW ORK IDENTIFICA TION IN THE HUMAN BRAIN b y Jian Li A dissertation submitted to the USC Graduate Sc ho ol in partial fulfillmen t of the requiremen ts for the degree of Do ctor of Philosoph y in Electrical Engineering Univ ersit y of Southern California Los Angeles, California Ma y 2019 © 2019 Jian Li. All righ ts res erv ed. Do ctoral Committee: Professor Ric hard M. Leah y , Chair Asso ciate Professor Justin P . Haldar Professor Gary Rosen Assistan t Professor Jessica L. Wisno wski Abstract Brain connectivit y is mo deled as a complex, segregativ e and in tegrativ e net w ork of connections b et w een differen t brain regions. Studying functional brain connectivit y can offer us an effectiv e w a y to examine ho w differen t brain net w orks relate to h uman b eha viors as w ell as ho w those net w orks ma y b e altered in neurological diseases. Ho w ev er, measuring functional connectivit y p oses a v ariet y of mathematical, signal pro cessing and neuroscience c hallenges. First, a go o d high-lev el represen tation of the data is often required in order to obtain an accurate estimation of the functional connectivit y , b ecause most of the t ypically-used linear measures are not capable of capturing the true highly non-linear brain in teractions. Second, the temp oral stationarit y of the time series assumed b y most of the studies ma y not b e realistic due to the dynamic nature of the brain. Hence, ho w to reliably estimate the spatial and temp oral dynamics of functional connectivit y sim ultaneously is a k ey c hallenge to us. Moreo v er, signals collected via almost all neuroimaging tec hniques are hea vily corrupted with noise. The inheren t lo w signal-to-noise ratio prev en ts us from obtaining a robust estimation of functional connectivit y . In this w ork, w e presen t and v alidate sev eral no v el approac hes and metho ds to address some of the c hallenges in functional connectivit y estimation and brain net w ork iden tification problems. T o address the high-lev el data represen tation issue, w e defined a bio-electrical mark er that can differen tiate the epileptogenic zone from areas of propagation in patien ts with epilepsy . W e disco v- ered a sp ecific ictal time-frequency pattern, referred as the “fingerprin t”, in the epileptogenic zone whic h con tains a com bination of sharp spik es preceding m ulti-band fast activit y concurren t with suppression of lo w er frequencies. W e dev elop ed a no v el mac hine learning system that automatically extracts eac h of these features, classifies electro de con tacts as b eing within the epileptogenic zone iii or outside the epileptogenic zone and generates individualized epileptogenic zone predictions for eac h patien t based on their anatomical magnetic resonance images. T o address the dynamic brain net w ork iden tification issue, w e dev elop ed a rank-recursiv e scal- able and robust sequen tial canonical p oly adic decomp osition framew ork that allo ws us to robustly disco v er brain net w orks whic h can o v erlap in b oth space and time in large-scale datasets. The robustness and scalabilit y w ere ac hiev ed b y using lo w er-rank solutions as the w arm start to higher- rank decomp ositions. This scalable and robust sequen tial canonical p oly adic decomp osition frame- w ork is flexible in the sense that it is not only applicable to w a v elet-transformed electro encephalog- raph y data but also to m ulti-sub ject async hronous functional magnetic resonance imaging data if the data is temp orally aligned across sub jects using the BrainSync algorithm. T o address the noise corruption issue, w e describ ed an optimization-based metho d that pro- vides a means of systematically selecting the parameter for the temp oral non-lo cal means filtering. W e further dev elop ed global PDF-based temp oral non-lo cal means, a no v el data-driv en optimized k ernel function based on Ba y es factor for the temp oral non-lo cal means filtering, whic h allo ws us to p erform global filtering with impro v ed noise reduction effects but without blurring adjacen t functional regions. Applications of these prop osed metho ds are illustrated using a v ariet y of sim ulated as w ell as in-viv o clinical data. iv Keyw ords: functional c onne ctivity, br ain networks, epilepto genic zone, tensor de c omp osition, filtering, ster e otactic ele ctr o enc ephalo gr aphy, functional magnetic r esonanc e imaging. v T o my sister vi A c kno wledgmen ts This dissertation w ould not ha v e b een p ossibly completed without the guidance, help, inspira- tion and lo v e, directly or indirectly , from a large n um b er of p eople around me. Although I can’t en umerate them one b y one, I w ould lik e to express m y deep est gratitude to all of them. I am grateful, foremost, to m y advisor and men tor, Prof. Ric hard M. Leah y , for his generous supp ort b oth financially and academically . His extremely broad kno wledge and ric h exp erience not only guided me to w ards future researc h directions, but also pro vided me with man y great opp ortunities for m y professional, scien tific as w ell as p ersonal gro wth. I am also grateful to Prof. Justin P . Haldar and Prof. Jessica L. Wisno wski for serving on m y do ctoral committee and for their insigh tful discussion ab out and significan t con tribution to the w orks in this dissertation in a v ariet y of differen t asp ects, from optimization to neuroscience. I w ould lik e to also thank other committee mem b ers of m y qualification and/or defense: Prof. Gary Rosen, Prof. Krishna S. Na y ak and Prof. John C. Mosher, for their time, effort, supp ort and advisemen t on m y researc h. I am indebted to man y p eople at Clev eland Clinic: Dr. Irene Z. W ang, Dr. Juan C. Bulacio, Dr. Imad Na jm, Dr. Kenneth T a ylor, for creating suc h a great collab oration en vironmen t and their generous help and supp ort. Sp ecial thanks to Dr. P atric k Chauv el, Dr. Jorge Gonzalez-Martinez, Dr. Dileep R. Nair and Dr. Olesy a Grinenk o for bringing me in to the researc h field of epilepsy . It is m y honor to not only co-author sev eral pap ers but also, together with all of y ou, push the cutting-edge of this researc h area forw ard, whic h could p oten tially b enefit a m yriad of p eople who suffer from epilepsy . vii I feel fortunate to b e part of the BIG lab, a friendly and jo yful researc h group with amazing colleagues and lab-mates. I w an t to thank all curren t and former group mem b ers - esp ecially Prof. Anand A. Joshi, Dr. Sy ed Ashrafulla, Dr. W en tao Zh u, Dr. Y anguang Lin, Dr. Sergül A ydöre, Dr. Chitresh Bh ushan, Dr. Divy a V aradara jan, Dr. Minqi Chong, Daeun Kim, T ae Hyung Kim, So y oung Choi, Ro drigo Lob os, Hossein Shahabi, Y unsong Liu, Dakarai McCo y , Haleh Akrami - for y our selfless discussions. I w ould lik e to ac kno wledge the National Institutes of Health, the USC Annen b erg Gradu- ate F ello wship Program, the USC Graduate Studen t G o v ernmen t, the Ming Hsieh Departmen t of Electrical and Computer Engineering, the Signal and Image Pro cessing Institute and all of their asso ciated staff for pro viding generous financial and administrativ e supp ort during m y graduate studies. Finally , I am grateful to m y wife and m y paren ts for their unlimited lo v e and endless sacrifice as w ell as to m y daugh ter and m y late grandparen ts who made me who I am. viii T a ble of Con ten ts Abstract iii Dedication vi A c kno wledgmen ts vii T a ble o f Con ten ts ix List Of Figures xiv List Of T ables xxi List of Algorithms xxii List of Abbreviations xxiii List of Notation xxvii I In t ro d uction & Bac kground 1 Chapter 1: In tro duction & Bac kground 2 1.1 Motiv ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 F unctional Brain Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Stereotactic Electro-encephalograph y . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 F unctional Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . . . . . 5 1.3 Researc h Con texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Main Con tributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 I I Lo calization of the Epileptogenic Zone 15 Chapter 2: A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 16 2.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 P atien ts and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 P atien t Selection and Data Collection . . . . . . . . . . . . . . . . . . . . . . 18 ix 2.2.2 Data Selection and Time-F requency Decomp osition . . . . . . . . . . . . . . . 19 2.3 Visual Characteristics of the Epileptogenic Zo ne . . . . . . . . . . . . . . . . . . . . 21 2.4 A utomatic Classification Pro cedure and Results . . . . . . . . . . . . . . . . . . . . . 25 2.4.1 Prepro cessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.1.1 Time-F requency Decomp osition . . . . . . . . . . . . . . . . . . . . 25 2.4.1.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.1.3 Artifact Remo v al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.2 F eature Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.2.1 F ast A ctivit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.2.2 Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4.2.3 Pre-ictal Spik es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4.3.1 F eatures and Classifier . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4.3.2 Sub ject-based Cross V alidation . . . . . . . . . . . . . . . . . . . . . 41 2.4.3.3 Lab els . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4.3.4 Classification with P artially Certain Lab e ls . . . . . . . . . . . . . . 42 2.4.3.5 V oting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4.3.6 Classification Pip eline . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5 A dditional Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.5.1 F ast A ctivit y Inside and Outside the Resection Area . . . . . . . . . . . . . . 47 2.5.2 Suppression Inside and Outside the Resection Area . . . . . . . . . . . . . . . 49 2.5.3 The Asso ciation Bet w een F ast A ctivit y and Suppression . . . . . . . . . . . . 49 2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.6.1 Defining the Fingerprin t of the Epileptogenic Zone . . . . . . . . . . . . . . . 51 2.6.2 P athoph ysiological Significance of the Time-frequency Fingerprin t . . . . . . 53 2.6.3 Limitations of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Chapter 3: Learning to Define An Biomark er of the Epileptogenic Zone 57 3.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Metho ds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.1 P atien t Selection and Data Collection . . . . . . . . . . . . . . . . . . . . . . 60 3.2.2 EZ Fingerprin t Pip eline for Individualized EZ Prediction . . . . . . . . . . . 60 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4.1 V alidation of Epileptogenic Zone Biomark ers . . . . . . . . . . . . . . . . . . 73 3.4.2 F ast A ctivities Yield Epileptogenic Zone Blurred Image . . . . . . . . . . . . 75 3.4.3 A utomated Epileptogenic Zone Fingerprin t Classification Pip eline . . . . . . 76 3.4.4 P oten tial Pitfalls in Application of the EZ Fingerprin t Metho d . . . . . . . . 77 I I I Robust Iden tification of Dynamic Brain Net w orks 79 Chapter 4: T ensor Decomp osition of Sp on taneous SEEG Data 80 4.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 Notation and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 x 4.2.1 Scalar, V ector, Matrix and T ensor . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.2 Matricization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.3 Kronec k er, Khatri-Rao and Hadamard Pro duct . . . . . . . . . . . . . . . . . 84 4.2.4 Canonical P oly adic Decomp osition . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2.5 Computation of CP Decomp osition and the ALS Algorithm . . . . . . . . . . 86 4.3 Materials and Metho ds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.1 SRSCPD F ramew ork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.2 Sim ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3.3 Application to In-viv o SEEG Dataset . . . . . . . . . . . . . . . . . . . . . . 91 4.4 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4.1 Sim ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4.2 In-viv o SEEG Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.4.2.1 Estimation of Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.4.2.2 In tra-sub ject Net w ork Comparison . . . . . . . . . . . . . . . . . . . 97 4.4.2.3 In ter-sub ject Net w ork Comparison . . . . . . . . . . . . . . . . . . . 100 4.4.2.4 Artifact Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.4.2.5 Comparison to Results Using ALS Algorithm . . . . . . . . . . . . . 101 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Chapter 5: T ensor Decomp osition of Async hronous fMRI Data 103 5.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.2.1 CP Decomp osition, the ALS Algorithm and SRSCPD . . . . . . . . . . . . . 106 5.2.2 Gradien t of the CP Mo del, A dam and Nadam . . . . . . . . . . . . . . . . . . 108 5.2.3 BrainSync . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.3 Materials and Metho ds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.3.1 NSR CPD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.3.2 Sim ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.3.3 Application to In-viv o Language T ask fMRI Data . . . . . . . . . . . . . . . . 112 5.3.4 Comparison of the Solutions and Stabilit y with Group ICA . . . . . . . . . . 113 5.3.5 The Imp ortance of BrainSync and Nadam . . . . . . . . . . . . . . . . . . . . 114 5.3.6 Application to In-viv o Resting fMRI Data . . . . . . . . . . . . . . . . . . . . 114 5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.4.1 Sim ulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.4.2 Application to In-viv o Language T ask fMRI Data . . . . . . . . . . . . . . . . 115 5.4.3 Comparison of the Solutions and Stabilit y with Group ICA . . . . . . . . . . 118 5.4.4 The Imp ortance of BrainSync and Nadam . . . . . . . . . . . . . . . . . . . . 121 5.4.5 Application to In-viv o Resting fMRI Data . . . . . . . . . . . . . . . . . . . . 121 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 IV F unctional-Boundary-Preserving Noise Reduction and Filtering 125 Chapter 6: P arameter Selection for Optimized Non-lo cal Means Filtering 126 6.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.2 Metho ds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 xi 6.2.1 T emp oral NLM Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.2.2 Optimization of tNLM P arameter h . . . . . . . . . . . . . . . . . . . . . . . 129 6.2.3 Exp erimen ts and P erformance Ev aluation . . . . . . . . . . . . . . . . . . . . 131 6.2.3.1 Sim ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.2.3.2 Application to T ask fMRI Data . . . . . . . . . . . . . . . . . . . . 133 6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Chapter 7: Global PDF-based T emp oral Non-lo cal Means Filtering 137 7.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.2 Metho d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.2.1 NLM-based Filtering and tNLM . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.2.2 Global PDF-based tNLM Filtering . . . . . . . . . . . . . . . . . . . . . . . . 139 7.2.2.1 GPDF Kernel F orm ulation . . . . . . . . . . . . . . . . . . . . . . . 139 7.2.2.2 A utomated P arameter Selection . . . . . . . . . . . . . . . . . . . . 142 7.2.2.3 Estimation of the P opulation Correlation Dis tribution . . . . . . . . 142 7.2.2.4 GPDF Filtering Algorithm . . . . . . . . . . . . . . . . . . . . . . . 143 7.3 Exp erimen ts and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.3.1 Sim ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.3.2 Application to In-viv o Resting fMRI Dataset . . . . . . . . . . . . . . . . . . 145 7.3.2.1 Dataset and Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.3.2.2 Seeded Correlation Maps . . . . . . . . . . . . . . . . . . . . . . . . 147 7.3.2.3 Correlation Matrix, Comm unit y Structure and Mo dularit y . . . . . 149 7.3.2.4 P arcellation Agreemen t with T ask fMRI A ctiv ation Maps . . . . . . 151 7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 V Concluding Remarks 161 Chapter 8: Conclusions & F uture W orks 162 8.1 Lo calization of the Epileptogenic Zone and Epileptic Net w ork Analysis . . . . . . . . 162 8.2 Resolving the Spatial and T emp oral Dynamics of Bra in Net w orks . . . . . . . . . . . 164 8.3 T o Filter or Not T o Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 App endices 167 App endix A: P atien t Selection Proto col 167 A.1 P atien t Selection in the Initial Fingerprin t Study . . . . . . . . . . . . . . . . . . . . 168 A.2 P atien t Selection in the Extended Fingerprin t Study . . . . . . . . . . . . . . . . . . 170 App endix B: Statistics of the F requency of F ast A citivit y and Suppression 171 B.1 Statistics of the F requency of F ast A ctivit y . . . . . . . . . . . . . . . . . . . . . . . 172 B.2 Statistics of the F requency of Suppression . . . . . . . . . . . . . . . . . . . . . . . . 173 App endix C: Visual Characteristics of Epileptogenic Zone 174 App endix D: Classification Result with Lesion Information 176 xii App endix E: A dditional Non-L VF A Seizures 178 App endix F: Misiden tified Fingerprin t P attern 180 App endix G: Individual Epileptogenic Zone Prediction Results 182 App endix H: Epileptogenic Zone Prediction without Seizure Clustering 184 App endix I: In terp olated Fingerprin t-based Epileptogenic Zone Prediction 186 App endix J: Con v ergence Comparison SRSCPD vs ALS 188 App endix K: A dditional Consisten t Comp onen ts F ound b y SRSCPD 190 App endix L: A dditional Non-ph ysiological Comp onen ts Using SRSCPD 192 App endix M: COR CONDIA Metric for Results Using ALS Algorithm 194 App endix N: Comp onen ts F ound Using ALS Algorithm 196 App endix O: A dditional NSR CPD Results on Language T ask fMRI Data 202 App endix P: Group ICA Results on Language T ask fMRI Data 204 App endix Q: NSR CPD Results on Resting fMRI Data 206 Reference 209 xiii List Of Figures 1.1 Scalp EEG vs SEEG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 SEEG recording and time-frequency plots of seizure onset. . . . . . . . . . . . . . . . 22 2.2 Example of pre-ictal to ictal transition s in the epileptogenic zone. The time-frequency plot sho ws the prop osed “fingerprin t”: a com bination of pre-ictal spik es, m ulti-band fast activit y and sim ultaneous suppression of slo w er bac kground frequencies. Note that fast activit y is c haracterized b y m ultiple bands that are not harmonically re- lated, c hirp at differen t frequency rat es, and whose amplitudes v ary indep enden tly across bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Time-frequency plot of seizure onset for the selected c hannel X2 -X3 from Sub ject 1 , Seizure 1 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 Time-frequency plot of the b aseline data for the selected c hannel X2-X3 from Sub ject 1, Seizure 1P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5 Time-frequency plot of the normalized data for the selected c hannel X2-X3 from Sub ject 1, Seizure 1P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.6 Original time-frequency plots of all c hannels of Sub ject 7 , Seizure 2 P with artifacts presen t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.7 Plots of indep enden t comp onen ts obtained from cICA of all c hannels of Sub ject 7 , Seizure 2 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.8 Time-frequency plots of all c hannels of Sub ject 7 , Seizure 2 P with artifact comp o- nen ts remo v ed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.9 F ast activit y extraction example. (a) Time-frequency plot of Sub ject 1 Seizure 1 P after onset. (b) F rangi filtering output. (c) The direction map. (d) First lev el initial mask. (e) The refined first lev el mask. (f ) Second lev el initial mask. (g) The refined second lev el mask. (h) The final com bined mask. . . . . . . . . . . . . . . . . . . . . 32 xiv 2.10 Suppression extraction example. (a) Time-frequency plot of Sub ject 1, Seizure 1P after onset with ideal suppression region. (b) Guided filtered time-frequency plot. (c) A phan tom of fast activit y mask. (d) Guided filtered time-frequency plot with upp er b ound set. (e) Initial mask from thresholding. (f ) The final refined mas k. . . . 35 2.11 Pre-ictal spik e extraction example. (a) Time-frequency plot of Sub ject 1, Seizure 1P b efore onset. (b) Plot of the median of upp er 25 % quan tile of the in tensit y distribution in (a) for eac h time p oin t with initial lo cal maxima. (c) Same plot as (b) with detected spik e candidates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.12 Bo xplots of maxim um frequency and timing comparison. (a) Maxim um frequency of fast activit y inside and outside resection; (b) maxim um frequency of fast activit y inside resection classified as epileptogenic-zone and non-epileptogenic-zone con tacts. (c) The start time of fast activit y inside and outside resection. (d) The start time of fast activit y inside resection classified as epileptogenic-zone and non-epileptogenic- zone con tacts. (e) The maxim um frequency of suppre ssion inside and outside re- section. (f ) The maxim um frequency of suppression inside resection classified as epileptogenic-zone and non-epileptogenic-zone con tacts. The b o xplot spans the t w o cen tral quartiles of the data around the median (red line), and the whisk ers extend to a maxim um of 1:5 times the b o x span, or to the last data p oin t, whic hev er is shorter. The remaining data p oin ts are outliers. . . . . . . . . . . . . . . . . . . . . . 48 2.13 Scatter plot of maxim um frequency of fast activit y v ersus maxim um frequency of suppression for con tacts. (a) Classified as epileptogenic zone inside the resection; (b) classified as non-epileptogenic zone inside the resection and (c) outside the resec- tion region. The presence of suppression w as determined b y thresholding 70 largest suppression areas for con tacts inside the resection region and outside separately , resulting in 40 con tacts in (a), 30 con tacts in (b) and 70 con tacts in (c). . . . . . . . 50 3.1 Epileptogenic zone fingerprin t pip eline. (a) F eature extractions for fast activit y , sup- pression and pre-ictal spik es from time-frequency maps; (b) Classification pro cedures where an SVM mo del w as trained using the original 17 patien ts in the previous study and the epileptogenic zone w as predicted for the 24 patien ts in the curren t study; (c) In terp olation of prediction scores on to patien ts’ individual MR image s. . . . . . . 62 3.2 An example of the seizure clustering pro cedure for Sub ject 112. (a) Ictal time-series illustrating v ariations in ictal patterns: Isolated pre-ictal spiking only in c hannel T’ in seizure 1 and con tin uous pre-ictal spiking sync hronous b et w een c hannel T’ and R’ in seizure 2 and 5; (b) Prediction scores obtained individually for eac h seizure; (c) Cross-correlations of the prediction scores across the electro de arra y b et w een all pairs of seizures (fiv e seizures in t otal for this sub ject); (d) Man ual clustering of seizures according to cross-correlation matrix. . . . . . . . . . . . . . . . . . . . . . . 63 3.3 Statistics of the maxim um frequency and minim um frequency of fast activit y . Ex- amples of iden tified fingerprin t pattern with gamma activit y and b eta activit y are sho wn on the righ t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 xv 3.4 T w o exemplar cases illustrating the epileptogenic zone fingerprin t prediction (b ottom- left) in terp olated on to individual patien ts MRI in comparison with fast activit y (b ottom-righ t) and p ost-op erativ e MRI (b ottom-middle) with corresp onding time- series (top-left) and time-frequency plot (top-righ t) for electro des of in terest. (a) Sub ject 106 from the seizure-free group; (b) Sub ject 220 from the non-seizure-free group. Lo cations of the electro de con tacts are illustrated in p os t-op erativ e MRI. . . 70 3.5 Bo xplot of the kurtosis of the in terp olated prediction scores for the seizure-free group on the left and the non-seizure-free group on the righ t. Example of the app earance of the epileptogenic zone fingerprin t prediction that corresp ond to lo w and high kurtosis are sho wn on the righ t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.1 An example of the sim ulated data wit h 5 comp onen ts. Eac h comp onen t is repre- sen ted b y a distinct color in all three mo des. F rom left to righ t: The c hannel (spatial) mo de sho ws the activ ated c hannels that participate in eac h net w ork; The time (tem- p oral) mo de sho ws the blo c k activ ation pattern for eac h net w ork; The sp ectrum (sp ectral) mo de sho ws the frequency sp ectrum for e ac h net w ork. . . . . . . . . . . . 90 4.2 Sim ulation results. Bo xplots of A CP o v er 100 Mon te Carlo trials are sho wn as a function of R . M denotes the n um b er of random initializations when using the original ALS algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3 Sim ulation results. Bo xplots of the F rob enius norm error o v er 100 Mon te Carlo trials are s ho wn as a function of R . M denotes the n um b er of random initializations when using the original ALS algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.4 Sim ulation results. Bo xplots of the run time in seconds o v er 100 Mon te Carlo trials are s ho wn as a function of R . M denotes the n um b er of random initializations when using original ALS algorithm. T op-left panel sho ws the zo omed-in results for a b etter comparison for lo w er rank data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5 Sim ulation results. Bo xplots of the A CP o v er 100 Mon te Carlo trials are sho wn as a function of SNR. M denotes the n um b er of random initializations when using the original ALS algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.6 COR CONDIA rank metric are sho wn as a function of rank R for t w o sessions of b oth sub jects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.7 T w o consisten t comp onen ts for Sub ject 1. F or eac h pair of consisten t comp onen ts, w e sho w mo des for session 1 in red a nd session 2 in blue in the top ro w. F rom left to righ t: The c hannel (spatial) mo de sho ws the activ ated c hannels that participate in eac h net w ork; The time (temp oral) mo de sho ws the dynamic v ariations of eac h net w ork (only the first 10 seconds is sho wn for b etter visualization); The sp ectral mo de sho ws the frequency-dep enden t comp onen t of the tensor. In the b ottom ro w the left and middle sub-figures sho w the spatial distribution of the activ ated c hannels mapp ed on to the sub ject’s smo othed cortical surface. F or visualization purp oses, a con tact or c hannel is defined as activ ated if the v alue of the (normalized) c hannel mo de at that con tact exceeds a threshold of 0:05 in b oth sessions. The righ t sub- figure sho ws the W elc h p o w er sp ectrum of the temp oral mo de. . . . . . . . . . . . . 98 xvi 4.8 T w o consisten t comp onen ts for Sub ject 2. D etails as for Figure 4.7. . . . . . . . . . . 100 5.1 Sim ulation results. Bo xplots of A CP o v er 100 Mon te Carlo trials are sho wn as a function of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2 NSR CPD results on language tfMRI data: (a) F ron to-parietal atten tional con trol net w ork (FP A CN); (b) Extended language net w ork (Lang); (c) Default mo de net- w ork (DMN); (d) Respiration activit y (Resp); (e) Reading net w ork (RN); (f ) A u- ditory net w ork (AN); (g) Visual net w ork (VN); (h) Sensorimotor activit y near the tongue area (T). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.3 Stabilit y testing results: (a) The scatter plot of the CCA pro jected spatial maps obtained using the group ICA metho d with differen t random initializations only; (b) The coun terpart to (a) when using NSR CPD; (c) Same as (a) but on the b o ot- strapp ed dataset; (b) The coun terpart to (c) when using NSR CPD. The pro jected lo cations of the iden tified brain net w orks ( 2 net w orks for ICA and 8 net w orks for NSR CPD) are sho wn with blue text aside. F or (b) and (d), for eac h comp onen t (a single dot), the color represen ts the n um b er of sub jects in v olv ed in that comp onen t. The big circle on the b ottom-righ t corner in (b) sho ws the zo omed-in v ersion of an example cluster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.4 NSR CPD results on rfMRI data: (a) Default mo de net w ork (DMN); (b) Executiv e con trol net w ork (ECN); (c) Sensorimotor net w ork (SMN); (d) Visual net w ork (VN); (e) A uditory net w ork (AN). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.1 The distribution of the elemen ts of the correlation matrixA for zero true correlation (blue) and 0:2 correlation (red), with the k ernel function in Equation (6.2) ev aluated for differen t v alues of the parameter h . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.2 Cost function in Equation (6.7) ev aluated as a function of h under the sim ulation settings. The ligh t gra y curv es are 1000 individual Mon te Corlo trials, the blue curv e is the mean and the t w o red curv es are one SD a w a y from the mean for eac h h . . . . 132 6.3 ARI metric of the clustering results ev aluated as a function ofh under the sim ulation settings. The colors of curv es ha v e the same meaning as those in Figur e 6.2 . . . . . 132 6.4 Cost function in Equation (6.7) ev aluated as a function of h for the LP task (blue) and the SC task (red), resp ectiv ely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.5 Com bined represen tativ e z-score maps ( = 0:05 ) for “Story” con trast in LP (orange) and “Random” con trast in SC (blue). R OI for analysis con taining activ ated regions in b oth con trasts is indicated in sup erior temp oral gyrus. . . . . . . . . . . . . . . . 134 6.6 Mean z score vs Dice’s co efficien t for differen t smo othing parameters and differen t lev els. The red curv e represen ts the result under tNLM smo othing and the blue curv e represen ts the result under isotropic smo othing. The parameter v alues for h and s are annotated ab o v e the curv es. . . . . . . . . . . . . . . . . . . . . . . . . . . 135 xvii 7.1 The histogram of the correlations underH 1 (blue) andH 0 (red) generated from sim- ulated data o v erlaid with tNLM k ernel functions for differen t parameter h (dotted) and GPDF k ernel function (blac k solid). . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.2 P arcellation result of sim ulated data represen ted as a VV matrix for eac h metho d and eac h hemisphere. Columns from (a) to (f ) are indicated b y their titles along upp er ro w. The ro ws represen t the t w o hemispheres. . . . . . . . . . . . . . . . . . . 144 7.3 Robustness comparison of results using Gaussian filter, global tNLM filter with op- timized parameter and GPDF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.4 Seeded correlation map for a single sub ject. Seed p oin t w as selected in the caudal pre-cuneus area sho wn as a y ello w dot in sub-figures (b) - (d). P ositiv ely correlated regions are sho wn in red, uncorrelated regions in white and negativ ely correlated regions in blue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.5 Changes of the seeded correlation v alues a fter filtering. . . . . . . . . . . . . . . . . . 155 7.6 Re-ordered unfiltered full correlat ion matrices based on the parcellation result using the unfiltered data (a), LB-filtered data (b) and GPDF-filtered data (c) for K = 7 . (d) - (f ) sho ws the corresp onding blo c k-wise median SD map o v er 160 sessions for (a) - (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.7 The net w ork mo dularit y plots as a functi on of the threshold and the filtering param- eters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.8 Maps of parcel b oundaries o v erlaid with motor task tongue vs a v erage con trast (a) - (c) and emotional task faces vs shap es (d) - (f ) for a single session of sub ject 100307 using the unfiltered data ((a) and (d)), the LB-filtered data ((b) - (e)) and the GPDF filtered data ((c) - (f )). (Num b er of parcels K = 100 ) . . . . . . . . . . . . . . . . . . 158 7.9 Bo xplots of the mean z-score v ariance of all cortical regions o v er 160 fMRI sessions for all 15 task pairs. Eac h column sho ws the b o xplot for one particular task using the unfi ltered data (left), the LB-filtered data (middle) and the GPDF filtered data (righ t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 A.1 P atien t selection w orkflo w (Chapter 2) . . . . . . . . . . . . . . . . . . . . . . . . . . 169 A.2 P atien t selection w orkflo w (Chapter 3) . . . . . . . . . . . . . . . . . . . . . . . . . . 170 B.1 Statistics of the frequency of fast activit y . Column 1: The maxim um frequency of fast activit y in epileptogenic-zone con tacts; Column 2: The minim um frequency of fast activit y in epileptogenic-zone con tacts; Column 3: The maxim um frequency of fast activit y in con tacts outside resection region; Column 4: The minim um frequency of fast activit y in con tacts outside resection region. . . . . . . . . . . . . . . . . . . . 172 xviii B.2 Statistics of the frequency o f suppression. Column 1: The maxim um frequency of suppression in epileptogenic-zone con tacts; Column 2: The minim um frequency of suppression in epileptogenic-zone con tacts; Column 3: The maxim um frequency of suppression in con tacts outside resection region; Column 4: The minim um frequency of suppression in con tacts outside resection region. . . . . . . . . . . . . . . . . . . . 173 C.1 Visual c haracteristics of th e pileptogenic zone. Con tacts in the epileptogenic zone: one exemplar time series and its corresp onding time-frequency plot is sho wn for eac h patien t. The n um b er of eac h time series plot indicates the patien t ID. Eac h plot sho ws 10 seconds prior to onset and 20 seconds after, and the frequencies are logarithmically spaced from 1 to 200 Hz. Note the c haracteristic asso ciation of pre- ictal sharp transien t, bands of fast activit y and suppression is presen t in eac h of the time frequency plot, despite its v ariations across sub jects. . . . . . . . . . . . . . . . 175 F.1 One represen tativ e con tact sho wing the epileptogenic zone fingerprin t for Sub ject 101, 103 and 140 where no epileptogenic zone w as predicted using the original EZF pip eline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 I.1 Epileptogenic zone prediction score in terp olated on to the patien ts’ pre-op erativ e MRI (left) and its comparison with the p ost-op erativ e MRI (righ t). F or Sub ject 219, t w o epileptogenic zone lo cations w ere predicted on the left and on the righ t side, the first resection via laser ablation is sho wn on the top and the resection is sho wn on the b ottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 J.1 Con v ergence comparison of CP-ALS vs SRSCPD on one sim ulated rank-7 tensor. Eac h sub-figure sho ws the mean of the a bsolute difference of the loading matrices b et w een the curren t iteration and the previous iterations o v er all mo des (see Algo- rithm V) in log scale along the y-axis as a function of n um b e r of iterations in the x-axis for ranks from 2 to 7 , where the iteration n um b er here indicates ma jor itera- tions including all s ub-problems in ALS. The stopping criterion w as set to 10 5 for all cases. The result for r = 1 w as excluded as the t w o algorithms are iden tical in that case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 K.1 A dditional consisten t comp onen ts found b y SRSCPD . . . . . . . . . . . . . . . . . . 191 L.1 A dditional non-ph ysiological mismatc hed comp onen ts found b y SRSCPD . . . . . . 193 M.1 COR CONDIA are sho wn as a function of R for t w o sessions of b oth sub jects using CP-ALS algorithm. The rank w as selected to b e R = 2;3;3;3 corresp onding to eac h of the sub-figures ab o v e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 N.1 Consisten t comp onen t for Sub ject 1 (DMN sim ilar to that in Figure 4.7 (a)) . . . . . 197 N.2 Mismatc hed comp onen t from session 1 for Sub ject 1 (Motor net w ork similar to that in Figure 4.7 (b)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 N.3 First mismatc hed comp onen t from session 2 for sub ject 1 (Unkno wn comp onen t similar to that in Figure K.1 (a)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 xix N.4 Second mismatc hed comp onen t from session 2 for Sub ject 1 (Artifact similar to that in Figure L.1 (a)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 N.5 First mismatc hed comp onen t from session 1 for Sub ject 2 (Mix of DMN and Motor net w ork) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 N.6 Second mismatc hed comp onen t from session 1 for Sub ject 2 (Artifact similar to that in Figure L.1 (b)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 N.7 Third mismatc hed comp onen t from se ssion 1 for Sub ject 2 (Another mix of DMN and Motor net w ork) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 N.8 First mismatc hed comp onen t from session 2 for Sub ject 2 (DMN similar to that in Figure 4.8 (a)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 N.9 Second mismatc hed comp onen t from session 2 for Sub ject 2 (Motor net w ork similar to that Figure 4.8 (b)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 N.10 Third mismatc hed comp onen t from session 2 for Sub ject 2 (Unkno wn comp onen t similar to that in Figure K.1 (b)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 O.1 A dditional NSR CPD results on language tfMRI data: (a) An example of plausible but un-recognized net w ork; (b) An example of sub ject-sp ecific net w ork. . . . . . . . 203 P .1 T w o recognized net w orks obtained from the language tfMRI data using the group ICA metho d: (a) Visual net w ork; (b) A uditory net w ork. . . . . . . . . . . . . . . . . 205 Q.1 Individual comp onen ts (sub-net w orks) iden tified from rfMRI data using NSR CPD, whic h constitute iden tifiable w ell-kno wn net w orks: DMN, SMN, ECN and VN. The auditory net w ork is iden tified b y itself without sub-net w orks, hence omit ted here. . . 207 xx List Of T ables 1.1 Summary of commonly used neuroimaging mo dalities . . . . . . . . . . . . . . . . . 5 2.1 Clinical c haracteristics of the included patien ts . . . . . . . . . . . . . . . . . . . . . 20 2.2 Result of automatic classification of the epilepto genic zone . . . . . . . . . . . . . . . 45 2.3 Implan tation maps with sc hematic represen tation of the resection margins (red) and bip olar SEEG c hannels iden tified b y the algorithm as TP and FP . . . . . . . . . . . 46 3.1 Clinical profiles of the patien ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Epileptogenic zone fingerprin t pre diction results and comparison with prediction results using fast activit y only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3 Comparison of the resection/laser ablation and s urgical outcomes . . . . . . . . . . . 72 4.1 Summary of the patien t data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.1 Summary of brain net w orks iden tified in language task fM RI data . . . . . . . . . . 118 7.1 Statistics (Wilco xon rank -sum) of the task v ariabilit y difference b et w een GPDF and the unfiltered case (Column 3) as w ell as b et w een GPDF and LB (Column 4 ) . . . . 153 D.1 Implan tation maps with sc hematic represen tation of the resection margins (shaded in red) for patien ts with confirmed or susp ected lesions. Bip olar SEEG c hannels inside epileptogenic lesion iden tified (epileptogenic-zone) and not Iden tified (Non- epileptogenic-zone) b y the algorithm as epile ptogenic are named for eac h individual patien t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 E.1 A dditional non-L VF A seizures excluded from the study . . . . . . . . . . . . . . . . . 179 G.1 Individual epileptogenic-zone predictions with conside ring differen t seizure clusters . 183 H.1 Epileptogenic zone prediction results without conside ring differen t seizure clusters . 185 xxi List of Algorithms I F ast activit y detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 I I Suppression detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 I I I Pre-ictal spik e detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 IV Classification of the epileptogenic zone . . . . . . . . . . . . . . . . . . . . . . . . . . 43 V Alternating least square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 VI Scalable and robust sequen tial canonical p oly adic decomp osition . . . . . . . . . . . 89 VI I Nadam-accelerated scalable and robust canonical p oly adic decomp osition . . . . . . 110 VI I I Optimal selection of parameter h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 IX GPDF filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 xxii List of Abbreviations 2D 2 - D imensional 3D 3 - D imensional A CP Av eraged C ongruence P ro duct A dam A da ptiv e M omen t Estimation ALS A lternating L east S quare AN A uditory N et w ork ARI A djusted R and I ndex BOLD B lo o d- O xygen L ev el D ep enden t CC C onnected C omp onen t CCA C urvilinear C omp onen t A nalysis cICA C omplex I ndep enden t C omp onen t Analysis COR CONDIA Cor e Con sistency Dia gnostic CP C anonical P oly adic CT C omputed T omograph y DC D irect C urren t DMN D efault M o de N et w ork ECoG E lectro co rtico g raph y EEG E lectro e ncephalo g raph y EZF E pileptogenic Z one F ingerprin t xxiii F CD F o cal C ortical D ysplasia fMRI F unctional M agnetic R esonance I maging FN F alse N egativ e FP F alse P ositiv e FP A CN F ron to- p arietal Atten tional C on trol N et w ork FPR F alse P ositiv e R ate FSIN F ast S omatic I nhibitory In ter- N eurons FWHM F ull- W idth- H alf- M axim u m GLM G eneralized L inear M o del GMM G aussian M ixture M o del GPDF G lobal PDF -based tNLM HCP H uman C onnectome P ro ject ICA I ndep enden t C omp o nen t A nalysis IQR I n ter q uartile R ange IRB I nstitutional R eview B oard IRPF I ncremen ted- R ank P o w er F actorization Lang Extended Lang uage net w ork LB L aplace- B eltrami LP L anguage P ro cessing L VF A L o w V oltage F ast A ctivit y MEG M agneto e ncephalo g raph y MRI M agnetic R esonance I maging Nadam N estero v-accelerated A da ptiv e M omen t estimation NCuts N ormalized Cuts NLM N on- l o cal M eans NSR CPD N adam-accelerated S calable and R obust CP D ecomp osition xxiv PCA P rincipal C omp onen t A nalysis PDF P robabilit y D ensit y F unction PET P ositron E mission T omograph y PICA P robabilistic I ndep enden t C omp o nen t Analysis PMG P oly m icro g yria. PNH P eriv en tricular N o d ular H eterotopia PPV P ositiv e P redictiv e V alue rfMRI R esting F unctional M agnetic R esonance I maging Resp Resp iration RN R eading N et w ork R OI R egion o f I n terest SC S o cial R ecognition SD S tandard D eviation SEEG S tereotactic E lectro e ncephalo g raph y SF G S up erior F ron tal G yrus SMA S upplemen tal M otor A rea SMN S ensori m otor N et w ork SNR S ignal-to- N oise R atio SPECT S ingle- p hoton E mission C omputed T omograph y SRSCPD S calable and R obust S equen tial C anonical P oly adic D ecomp osition SVD S ingular V alue D ecomp osition SVM S upp ort V ector M ac hine T T ongue tfMRI T ask F unctional M agnetic R esonance I maging TN T rue N egativ e tNLM T emp oral N on- l o cal M eans xxv TP T rue P ositiv e VN V isual N et w ork WM W orking M emory xxvi List of Notation Notation x2R A real scalar x2R N A real v ector with length N X2R MN A real matrix with dimension M b y N x i;j The (i;j) th elemen t inX X2R IJK A real third-order tensor with dimension I b y J b y K x i;j;k The (i;j;k) th elemen t i nX Unary Op erations ∥x∥ The l 2 norm of v ectorx X T The transp ose of matrixX X y The Mo ore-P enrose pseudo-in v erse of matrixX X 1 The in v erse of matrixX ∥X∥ F The F rob enius norm of matrixX ∥X∥ The norm of tensorX X (n) The matricized tensorX along the n th dimension xxvii Binary Op erations a◦b The outer pro duct of v ectora andb X Y The Kronec k er pro duct of matrixX andY X⊙Y The Khatri-Rao pro duct of matrixX andY XY The Hadamard pro duct of matrixX andY M i X The matrix-tensor m ultiplication in mo de i xxviii P art I In tro duction & Bac kground 1 Chapter 1 In tro duction & Bac kground , , , — 1.1 Motiv ation The brain, the most complex organ in the h uman b o dy , is kno wn to ha v e coun tless m ysteries, the ma jorit y of whic h still ha v e not b een disco v ered. The brain is a highly complex but v ery efficien t net w ork, consisting of billions of w ell-organized neurons that p o w er the brain to hierarc hically pro cess a v ariet y of information at all lev els, suc h as sensing, con trolling, thinking, memorizing [ 39 ]. Differen t brain regions ha v e their o wn functionalities but also con tin uously comm unicate with eac h other, forming a complex, segregativ e and in tegrativ e net w ork. In the past few decades, a ric h history of b oth structural and functional neuroscience researc h has pro vided a large amoun t of kno wledge and deep er and deep er understanding of the brain, esp ecially ab out the roles and functions of eac h brain region. Thanks to the recen t adv ances in neuroimaging tec hniques, w e are no w able to measure brain activities, examine functional in teractions b et w een differen t brain regions and analyze brain connectivit y . Brain connectivit y is defined as a pattern of either structural connections or links formed b y ax- onal path w a ys (structural connectivit y [ 83 ]) or temp oral statistical de p endencies in measuremen ts 2 Chapter 1. In tro duction & Bac kground 3 of functional brain activit y usually computed as correlation or coherence (functional connectiv- it y [ 191 ]) b et w een anatomically segregated brain regions. In this w ork, w e fo cus on functional c onne ctivity as it pro vides new insigh ts ab out large-scale neuronal comm unications in the brain and facilitates us to b etter understand the brain functioning mec hanism. It also offers us an effec- tiv e w a y to examine ho w differen t brain net w orks relate to h uman b eha viors as w ell as ho w those net w orks with differen t functional sp ecializations ma y b e altered in neurological diseases [ 39 , 86 ]. A t ypical generic form ulation of functional connectivit y b et w een t w o differen t regions of the brain can b e written as: r ij (t) =g(f(x i (t));f(y j (t))) (1.1) wherex i (t) andy i (t) are t w o time series from lo cation i and j , resp ectiv ely; f is a generalization (feature extraction) function that maps ra w time series to some more represen tativ e and meaningful features, e.g., f can b e an unit mapping, f(x) = x , when no feature extraction is applied; g is a measure of functional connectivit y strength. e.g., g(x;y) = x T y/∥x∥ 2 ∥y∥ 2 represen ts the most commonly used P earson’s correlation co efficien t measure; r ij (t) is the resulting functional connectivit y measuremen t as a function of t . Although measuring functional connectivit y can b e fo rm ulated as simply as computing the P earson’s correlation co efficien ts b et w een time series, it p oses a v ariet y of mathematical, signal pro cessing and neuroscience c hallenges. Signal represen tations. What is a go o d and concise high-lev el represen tation of ra w time series, i.e., the functionf in Equation ( 1.1 ), that allo ws us to map ra w data to some meaningful feature space suc h that a n accurate estimate of functional connectivit y can b e obtained? A go o d feature extraction is particularly useful when a linear connectivit y estimation function (g in Equation ( 1.1 )) is used. This is b ecause a linear function itself is not capable of capturing the true highly non-linear brain in teractions without using some meaningful (p ossibly non- linear) feature extraction f together. In fact, most estimation functions used in practice are linear, e.g., correlation, coherence. 4 1.2. F unctional Brain Imaging Spatio-temp oral dynamics. Un til recen tly , most of the studies assume stationarit y of functional connectivit y , i.e., Equation ( 1.1 ) is not a function of t an ymore and reduces to r ij =g(f(x i );f(y j )) , a pure function of spatial lo cations. Ho w ev er, the stationarit y assump- tion ma y not b e realistic due to the dynamic nature of the brain [ 148 ]. Moreo v er, differen t functional regions ma y spatially o v erlap while b eing correlated in time [ 104 ]. Therefore, reliable estimation of the spatial and temp oral dynamics of functional connectivit y sim ulta- neously is a k ey c hallenge to us. Robustness. Signals collected via almost all neuroimaging tec hniques, esp ecially for those functional neuroimaging mo dalities including but not limited to electro-encephalograph y (EEG), magneto-encephalograph y (MEG), functional magnetic resonance imaging (fMRI), p ositron emission tomograph y (PET), are often hea vily corrupted with noise. The lo w signal- to-noise ratio (SNR) prev en ts us from obtaining a r obust estimation of functional connectivit y . Giv en a pair of observ ations (x i ;y j ), ho w far can the estimater ij v ary from the true functional connectivit y? Can w e reduce this v ariance b y some means? 1.2 F unctional Brain Imaging F unctional brain imaging comprises a v ariet y of neuroimaging tec hniques dev oted to a b et- ter understanding of the h uman brain via measuring the electroph ysiological, hemo dynamic, and metab olic pro cesses that underlie either normal or pathological brain functions. Commonly used brain imaging mo dalities include: EEG, MEG, Electro corticograph y (ECoG), stereotactic EEG (SEEG), fMRI and PET. Eac h of these mo dalities has their o wn pros and cons in terms of the scale of the signal sensitivit y , in v asiv eness, spatial resolution, and temp oral resolution. T able 1.1 summaries the main prop erties of eac h mo dalit y . Belo w the less w ell kno wn SEEG tec hnique and relativ ely w ell kno wn fMRI imaging mo dalit y will b e briefly in tro duced and review ed, as those are the t w o mo dalities primarily used in this w ork. Chapter 1. In tro duction & Bac kground 5 T able 1.1: Summary of commonl y used neuroimaging mo dalities Imaging Mo dalit y Measuring (Se nsitiv e to) In v asiv e Spatial Resolution * T emp oral Resolution * EEG Scalp electr ic p oten tial No P o or Excellen t MEG Scalp magne tic field No P o or Excellen t ECoG Cortical electric p oten tial Y es F air † Excellen t SEEG Cortical lo cal field p oten tial Y es Go o d † Excellen t fMRI BOLD ‡ c hanges No Go o d P o or PET Radioactivit y of tracers No F air P o or * A relativ e description comparing mo dal ities listed in this table † Spatial sampling/co v erage is limited ‡ BOLD - Blo o d-Oxygen Lev el Dep enden t. 1.2.1 Stereotactic Electro-encephalograph y Stereotactic electro-encephalograph y is an in v asiv e pre-surgical mapping mo dalit y for patien ts who ha v e pharmaco-resistan t fo cal epilepsy and a negativ e magnetic resonance imaging (MRI) with the absence of a clear epileptogenic lesion. In con trast to the regular scalp EEG where electric p oten tial differences b et w een pairs of scalp electro des are measured f rom sensors glued to the skin (Figure 1.1 ( a )), in SEEG, depth electro des are implan ted in to the patien t’s brain (Figure 1.1 ( b ) and Figure 1.1 ( c )) directly measuring the lo cal field p oten tials from cortical regions of in terest. SEEG w as first used in 1965 [ 15 ] sp ecially for the purp ose of iden tification of the epileptogenic zone for patien ts with epilepsy . Direct measuremen t in the brain with SEEG a v oids the highly ill-p osed problem of source lo calization encoun tered with scalp EEG. SEEG has the b enefit of high temp oral resolution (on the scale of milliseconds), the same as EEG, and b ecause electro des are implan ted through small b ore holes rather than via craniotom y , it can b e view ed as less in v asiv e than ECoG. 1.2.2 F unctional Magnetic Resonance Imaging F unctional magnetic resonance imaging is a non-in v asiv e w a y to measure brain activities. The most widely used metho d is based on Blo o d-Oxygen Lev el Dep enden t (BOLD) signal c hanges that 6 1.3. Researc h Con texts (a) EEG cap 1 (b) SEEG implan tation (c) X-ra y of implan ted electro des Figure 1.1: Scalp EEG vs SEEG are due to the hemo dynamic resp onse to neuronal activit y . Although the temp oral resolution is limited b y the relativ ely slo w hemo dynamic resp onse (on the scale of seconds), when compared to electrical neural activit y that can b e captured using SEEG, fMRI studies are capable of pro ducing spatial resolutions as high as 13 mm. Th us, fMRI has b ecome the most widely used metho d in the past few decades for its facilitation of whole-brain mapping and functional connectivit y researc h. 1.3 Researc h Con texts This researc h addresses the functional connectivit y estimation and net w ork iden tification prob- lems discussed ab o v e in the follo wing three con texts: Lo calization of the epileptogenic zone. T reatmen t of in tractable fo cal epilepsy b y resec- tion of the epileptogenic zone is often effectiv e pro vided that the epileptogenic zone can b e reliably iden tified. F o cal epilepsy , ho w ev er, is fundamen tally a net w ork-based disease. The epileptogenic zone is connected to an ictal net w ork whose other no des ma y also exhibit abnor- mal neural activit y either concurren tly or subsequen tly . In patien ts without MRI detectable lesions, differen tiation of the onset zone from these other no des in the net w ork can b e diffi- cult ev en with the use of in v asiv e recordings [ 67 , 140 , 141 ]. T o ha v e a b etter understanding of brain net w orks and connectivit y in patien ts with epilepsy , w e consider defining a represen- tativ e ictal electro-biomark er (in a signal pro cessing p ersp ectiv e) of the epileptogenic zone, whic h is referred as the epileptogenic zone “fingerprin t”, so that it can uniquely iden tify the 1 Image cred it: Wikip edia:Electro encephalograph y Chapter 1. In tro duction & Bac kground 7 epileptogenic zone from areas of seizure propagation as w ell as regions that are not in v olv ed in epilepsy . Robust iden tification of dynamic brain net w orks. Exploring functional connectiv- it y is a ric h approac h to s tudying brain net w orks [ 74 ]. Of particular recen t in terest is the dynamic nature of functional connectivit y [ 148 ]. The most commonly used strategy for de- co ding dynamic functional connectivit y is to compute correlation or coherence using a sliding windo w [ 46 , 91 ]. Ho w ev er, using a long temp oral windo w to obtain robust functional connec- tivit y estimates inevitably leads to o v er-smo othing of dynamic c hanges [ 99 , 146 ]. T o o v ercome this difficult y , principal comp onen t analysis (PCA)-based and indep enden t comp onen t anal- ysis (ICA)-based approac hes ha v e b een prop osed. Although they do not in tro duce temp oral smo othing, a limitation of those metho ds is that either the time series of eac h net w ork is required to b e indep enden t (temp oral ICA) [ 168 ] or the spatial mo des of the net w orks are disjoin t (spatial ICA) [ 23 , 42 ] or orthogonal (PCA) [ 73 ], whereas real net w orks can o v erlap and b e correlated in b oth space and time [ 104 ]. Here w e dev elop no v el algorithms to robustly iden tify the spatio-temp oral dynamics of brain net w orks, i.e., obtain b oth spatial maps and asso ciated time series sim ultaneously , from either EEG or fMRI recordings, but without im- p osing an y unrealistic constrain t. F unctional-b oundary-preserving noise reduction and filtering. With the in v en tion and adv ances of fMRI, no w ada ys it b ecomes p ossible to non-in v asiv ely infer information ab out the neuronal activities of the brain in health y sub jects b y measuring the BOLD signal fluctu- ations. In the past t w o decades, man y studies fo cused on measuring and exploring functional connectivit y using in-viv o resting fMRI (rfMRI) dataset. Ho w ev er, functional connectivit y measuremen ts are usually v ery unreliable due to the inheren t lo w SNR in BOLD signals. Prepro cessing of fMRI data often includes a spatial smo othing step using Gaussian filter- ing to reduce noise but at the exp ense of blurring adjacen t but distinct functional regions. Non-lo cal means (NLM) filtering is an edge-preserving metho d originally designed for natural image denoising [ 36 ] and a v arian t of NLM, called temp oral NLM (tNLM) [ 26 ], w as recen tly 8 1.4. M ain Con tributions prop osed for filtering rfMRI data but without blurring across functional b oundaries. Ho w- ev er, the exp onen tial k ernel function as w ell as the parameters used in computing the filtering w eigh ts is c hosen heuristically . Hence, in this w ork, w e consider designing and optimizing the filter k ernel as w ell as its asso ciated parameters to ac hiev e an optimal trade-off b et w een noise reduction and functional b oundary preserv ation in functional connectivit y . 1.4 Main Con tributions W e defined a bio-electrical mark er for the epileptogenic zone based on an ob jectiv e description of in ter-ictal to ictal transition, with the aim of iden tifying a time-frequency pattern that can differen tiate the epileptogenic zone from areas of propagation. W e h yp othesized that a sp ecific ictal time-frequency pattern in the epileptogenic zone should con tain a com bination of (i) sharp transien ts or spik es; preceding (ii) m ulti-band fast activit y; concurren t with (iii) suppression of lo w er frequencies. W e dev elop ed a soft w are pip eline that automatically extracted eac h of these three features from the time-frequency data and used a supp ort v ector mac hine (SVM) to classify eac h con tact-pair as b eing within the epileptogenic zone or not. Our mac hine learning system successfully iden tified this pattern in 15 of 17 patien ts who w ere v erified to b e seizure-free for at least 12 mon ths after surgical resection. The total n um b er of iden tified con tacts across all patien ts w as 64 , with 58 lo calized inside the resected area. Subsequen t quan titativ e analysis sho w ed only using the maxim um frequency or the timing of fast activit y can not differen tiate con tacts either b et w een resected and non-resected regions or b et w een con tacts iden tified as epileptogenic v ersus non-epileptogenic. The classification and analysis results indicated that it is the time-frequency pattern consisting of the com bination of the three features that is able to differen tiate the epileptogenic zone from areas of propagation and, as suc h, represen ts an epileptogenic zon e “fingerprin t” . Chapter 1. In tro duction & Bac kground 9 W e examined the genera lizabilit y of the “fingerprin t” pattern b y applying the epileptogenic zone fingerprin t (EZF) pip eline to an indep enden t consecutiv e series of patien ts (11 seizure- free and 13 non-seizure-free after surgery) who underw en t stereotactic EEG (SEEG) ev alu- ation and had seizure onset c haracterized b y lo w v oltage fast activit y (L VF A). W e sho w ed that the fingerprin t could also b e iden tified in seizures with lo w er frequency (suc h as b eta or alpha) oscillatory activities at the onset. W e further extended the EZF metho d to a fully automated end-to-end classification pip eline, whic h allo w ed us to estimate an individualized anatomical estimate of the epile ptogenic zone exten t for eac h patien t, b y in terp olating the pre- diction scores on to individual patien t’s MRI images. When the fingerprin t w as mapp ed on to the p ost-op erativ e MRI using the extended EZF pip eline, w e demonstrated that fingerprin t- based epileptogenic zone estimate in seizure-free patien ts presen ted as relativ ely restricted areas and w as almost alw a ys inside the resected region. On the other hand, of a large fraction of the non-seizure-free patien ts, the estimated epileptogenic zone w as not w ell-lo calized and lo cated partially or completely outside the resection, whic h ma y explain surgical failure in suc h cases. In con trast, when using the same extended EZF pip eline but with fast activ- it y only , the epileptogenic zone size w as o v er-estimated, indicating a reduced discriminativ e abilit y relativ e to the use of the full fingerprin t f or lo calization of the epileptogenic zone. W e approac hed the dynamic brain net w ork iden tification problem using a tensor-based mo del. W e dev elop ed a rank-recursiv e scalable and robust sequen tial canonical p oly adic decomp o- sition (SRSCPD) framew ork to decomp ose a tensor in to sev eral rank-1 comp onen ts. The robustness and scalabilit y are ac hiev ed using a w arm start for eac h rank based on the results from the previous rank. In sim ulations, w e sho w that SRSCPD consisten tly outp erforms the m ulti-start alternating least square (ALS) algorithm o v er a range of ranks and SNRs, with lo w er computation cost. When applying SRSCPD to resting in-viv o SEEG data from t w o sub jects with epilepsy , consisten t brain net w orks corresp onding to the default mo de net w ork (DMN) and the motor net w ork and their dynamic b eha viors w ere iden tified in b oth sub jects. These net w orks w ere also highly consisten t within sub ject b et w een t w o sessions recorded sev eral hours apart. 10 1.4. M ain Con tributions In application of SRSCPD to fMRI data, our goal is to robustly iden tify common brain net- w orks and their corres p onding temp oral dynamics across sub jects in resting or async hronous fMRI signals. W e first temp orally align async hronous fMRI data using the orthogonal Brain- Sync transform, allo wing us to study common brain net w orks across sessions and sub jects. W e then map the sync hronized fMRI data in to a 3 -dimensional (3D) tensor (v ertices time session/sub ject) and p erform a greedy canonical p oly adic (CP) decomp osition, reducing the rank to 20 to impro v e the SNR. Finally , w e apply Nestero v-accelerated adaptiv e momen t estimation (Nadam) within our SRSCPD framew ork to the rank-reduced tensor. W e suc- cessfully iden tified eigh t brain net w orks with their corresp onding temp oral dynamics from 40 sub jects of the Human Connectome Pro ject’s (HCP) language task fMRI (tfMRI) data with- out using an y prior information regarding the task designs. F our of these sho w the sub jects’ resp onses to cues at the b eginning of eac h task blo c k; one corresp onds to the resp onse to the language task; one sho ws the DMN that exhibits deactiv ation during the tasks; the remaining t w o net w orks reflect non-task-related activities. These net w orks w ere not all found using the group ICA metho d or when either BrainSync or Nadam w as not used. F urthermore, b o ot- strap analysis demonstrates sup erior robustness of those net w orks that are found using our tensor approac h relativ e to group ICA. Finally , when our metho d w as applied to the rfMRI data from the same set of sub jects, fiv e large-scale net w orks w ere successfully iden tified. Commonly used surface based Laplace-Beltrami (LB) filtering or v olumetric Gaussian filter- ing tends to undermine the individual differences detected b y analysis of BOLD signal b y smo othing signals across b oundaries of differen t functional areas. T o o v ercome this issue, a tNLM filtering metho d w as dev elop ed to denoise fMRI data while preserving spatial struc- tures but the k ernel and parameters for tNLM filter need to b e c hosen carefully in order to ac hiev e optimal results. W e first prop osed an optimization-based metho d that pro vided a means of systematically selecting the parameter for the original tNLM filtering. W e sho w ed that the optimal v alue coincided with the p oin t at whic h w e ac hiev ed the maxim um enhance- men t in SNR without blurring b et w een distinct functional regions in b oth sim ulation and tfMRI exp erimen t. W e further presen ted a global PDF-based tNLM filtering (GPDF), a Chapter 1. In tro duction & Bac kground 11 no v el, data-driv en optimized k ernel function based on Ba y es factor for tNLM filtering. W e demonstrated its sup erior p erformance and robustness o v er the LB and the original tNLM filtering using b oth sim ulated data and in-viv o rfMRI datasets. Results sho w ed that GPDF filtering enabled us to p erform global filtering with impro v ed noise reduction effects without blurring adjacen t functional regions. 1.5 Organization of the Dissertation P art I I presen ts the w ork on lo calization of the epileptogenic zone. In Chapter 2 , a fingerprin t pattern is disco v ered and defined on the time-frequency represen tation of ictal SEEG data and a mac hine-learning-based algorithm is dev elop ed and v alidated on a set of 17 seizure-free patien ts. Then, in Chapter 3 , this fingerprin t metho d is further v alidated on a completely indep enden t set of patien ts with broader selection criteria, including b oth seizure-free patien ts and non-seizure- free patien ts. Also the classification pip eline is extended to patien ts’ anatomical space and the iden tification p erformance using the fingerprin t metho d w as compared with that using fast activit y features only . P art I I I presen ts the w ork on robust iden tification of dynamic brain net w orks. A SRSCPD framew ork is dev elop ed and applied to resting SEEG data in Chapter 4 and is further extended to fMRI data while com bining the BrainSync tec hnique in Chapter 5 . P art IV presen ts the w ork on functional-b oundary-preserving fMRI filtering. W e will describ e ho w to optimally select the parameter for the original tNLM filtering (Chapter 6 ) and ho w to design an optimal tNLM k ernel in order to p erform a global filtering (Chapter 7 ). Finally , P art V pro vides some concluding remarks and a short discussion of p oten tial future directions regarding eac h of the topics. The materials in the follo wing c hapters w as tak en from the follo wing publications. Eac h c hapter is a self-con tained story with some necessary bac kground in tro duced at the b eginning, whic h ma y o v erlap, to some exten t, across c hapters and can b e skipp ed at the readers’ discretion. 12 1.5. Organization o f the Dissertation Chapter 2 : O. Grinenk o † , J. Li † , J. C. Mosher, Z. W ang, J. Bulacio, J. Gonzalez-Martinez, D. Nair, I. Na jm, R. M. Leah y , P . Chauv el, “A fingerprin t of the epileptogenic zone in h uman epilepsies”, Br ain , v ol. 141, no. 1, pp. 117-131, 2018. https://doi.org/10.1093/brain/awx306. Chapter 3 : O. Grinenk o, J. Li , J. C. Mosher, J. Bulacio, J. Gonzalez-Martinez, I. Na jm, P . Chauv el, R. M. Leah y , “In searc h of biomark ers for the epileptogenic zone: a mac hine learn- ing approac h”, 72nd A nnual Me eting of the A meric an Epilepsy So ciety , New Orleans, Dec. 2018. Chapter 4 : J. Li , J. P . Haldar, J. C. Mosher, D. R. Nair, J. Gonzalez-Martinez, R. M. Leah y , “Scalable and robust tensor decomp osition of sp on taneous stereotactic EEG data”, IEEE T r ans- actions on B iome dic al Engine ering , 2018. (Preprin t). https://doi.org/10.1109/TBME.2018.2875467 . J. Li , J. C. Mosher, D. Nair, J. Gonzalez-Martinez, R. M. Leah y , “Robust tensor decomp o- sition of resting brain net w orks in stereotactic EEG”, IEEE 51st A silomar Confer enc e on Signal, S ystem and Computers , P acific Gro v e, Oct. 2017. https://doi.org/10.1109/ACSSC.2017.8335616 . Chapter 5 J. Li , J. L. Wisno wski, A. A. Joshi, R. M. Leah y , “Brain net w ork iden tification in asyn- c hronous task fMRI data using robust and scalable tensor decomp osition”, Pr o c. SPIE Me dic al Imagin g 2019: Image Pr o c essing , San Diego, Mar. 2019. https://doi.org/10.1117/12.2512684 † Co-first authors with equal c on tribution. Chapter 1. In tro duction & Bac kground 13 J. Li , J. L. Wisno wski, A. A. Joshi, R. M. Leah y , “Iden tifying brain net w orks using tensor decomp osit ion of m ultiple sub ject async hronous task fMRI”, 24th A nnual Me eting of the Or ganization for Human Br ain Mapping , Singap ore, Jun. 2018. A. A. Joshi, M. Chong, J. Li , S. Y. Choi, R. M. Leah y , “Are y ou thinking what I’m thinking? Sync hronization of resting fMRI time-series across sub jects”, Neur oimage , v ol. 172, pp. 740-752, 2018. https://doi.org/10.1016/j.neuroimage.2018.01.058 . Chapter 6 J. Li , R. M. Leah y , “P arameter selection for optimized non-lo cal means filtering of task fMRI”, IEEE 14th International Symp osium on Biome dic al Imaging , pp. 476-480, Mel- b ourne, Apr. 201 7. https://doi.org/10.1109/ISBI.2017.7950564 . Chapter 7 J. Li , S. Y. Choi, A. A. Joshi, J. L. Wisno wski, R. M. Leah y , “Global PDF-based temp oral non-lo cal means filtering rev eals individual differences of brain connectivit y”, IEEE 15th Intern ational Symp osium on Biome dic al Imaging , W ashington, D.C., Apr. 2018. https://doi.org/10.1109/ISBI.2018.8363513 . J. Li , S. Y. Choi, R. M. Leah y , “Global PDF-based non-lo cal means filtering of resting fMRI data”, 23r d A nnual Me eting of the Or ganization for Human Br ain Mapping , V ancouv er, Jun. 2017. This page in ten tionally left blank P art I I Lo calization of the Epileptogenic Zone 15 Chapter 2 A Fingerprin t of the Epileptogenic Zone in Human Epilepsies , , , — 2.1 In tro duction The epilepto genic zone , defined as the site of primary organization of ictal disc harge, refers to the areas b ound together through an excessiv e sync hronization at seizure onset [ 14 , 204 ]. “F ast” or “rapid” activit y at ictal onset has b een recognized as the main feature of the epileptogenic zone since the infancy of SEEG electro des [ 15 ]. Since the dev elopmen t of sub dural ECoG recordings, m uc h atten tion has also b een paid to the time precedence of phasic transien ts, esp ecially spiking activities [ 31 , 144 ]. In the past 15 y ears, iden tification of high frequency oscillations during in ter-ictal and ictal p erio ds in exp erimen tal mo dels reorien ted researc h in terest to w ards high-gamma activities in h uman epilepsies as a p oten tial epileptogenic zone mark er [ 32 , 132 , 200 , 213 ]. In parallel, direct curren t (DC) recordings exemplified the concomitance of ultra-slo w and fast frequencies [ 79 , 100 , 181 , 207 ]. F ast activit y frequen tly o ccurs quasi-sim ultaneously in m ultiple areas so that visual discrimination can b e cum b ersome and lead to sub jectiv e in terpretations. Sev eral metho ds of ob jectiv e lo calization of 16 Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 17 the epileptogenic zone based on ictal SEEG signal pro cessing ha v e b een dev elop ed during the past10 y ears. Bartolomei et al. [ 20 ] measured the relativ e onset times of c hanges in energy ratio b et w een fast and slo w activities in differen t areas in v olv ed at seizure onset to define an epileptogenicit y index. Da vid et al. [ 56 ] mapp ed the anatomical lo cation of the fastest activities and their slo w propagation during p eri-onset time on to the patien ts’ MRIs and compared the lo cation with the surgical resection area. A differen t approac h, frequency lo calization, w as used b y Gnatk o vsky et al. [ 80 ]. After defining frequencies of in terest and plotting their p o w er c hange o v er time, they lo calized the distribution of frequencies of in terest in differen t con tacts of the depth electro des. The epileptogenic zone, defined as the area exhibiting frequency c hanges at seizure onset, could then b e delineated. In a retrosp ectiv e and prosp ectiv e study of patien ts in v estigated using SEEG, the same metho d w as applied to test three p oten tial biomark ers of the epileptogenic zone, namely fast activit y , signal flattening and slo w p oten tial shift. These biomark ers co-lo calized with the epileptogenic zone as defined b y standard SEEG criteria and p ost-resection seizure outcome [ 79 ]. Differen tiating the primary epileptogenic zone from regions of propagation is difficult using fast activit y signal analysis. In practice, fast frequencies and their time of o ccurrence can b e easily iden tified in temp oral lob e seizures [ 20 ]; ho w ev er, a degree of epileptogenicit y cannot b e similarly defined in neo cortical epilepsies, where these c hanges o ccur abruptly and sim ultaneously o v er widespread cortical areas. F urthermore, if the highest frequency is an indicator of the most epileptogenic areas, it is not clear whether lo w er frequencies w ould mark less or non-epileptogenic ones. F ailures in epilepsy surgery for fo cal cortical dysplasia are often due to difficult y in estimating the exten t of fast-activit y-generating cortex to b e remo v ed for a seizure-free outcome [ 9 ]. Dep ending on the region in v olv ed and/or the underlying pathology , fast activit y ma y b e pre- ceded b y or emerge from a sharp w a v e or rh ythmical high-amplitude spiking [ 117 , 144 , 171 ]. Seizure onset is not a monomorphic but a complex phenomenon. Characterization of the significance of fast activit y cannot b e appraised in isolation. A holistic approac h is absen t from the literature, motiv ating us to ob jectiv ely describ e in ter- ictal/pre-ictal/ictal transition and ictal ev olution using time-frequency analysis applied to all elec- tro des in patien ts undergoing SEEG. After qualitativ ely iden tifying a time-frequency pattern only 18 2.2. P atien ts and Materials presen t in the epileptogenic zone, w e dev elop ed a SVM-based learning system to distinguish time- frequency patterns inside the epileptogenic zone from those outside the epileptogenic zone and demonstrated its effectiv eness. This pattern lasts for a p erio d of time and is b ey ond strict seizure on- set and it can differen tiate the epileptogenic zone from areas of propagation. Characterized b y com- bined v ariations in high and lo w frequencies, p ossibly reflecting c hanges in neuronal unit activities lik e those rep orted using sim ultaneous ECoG and micro-electro de unit recordings [ 133 , 187 , 198 , 199 ], the time-frequency pattern w e describ e represen ts an epileptogenic zone “fingerprin t” . 2.2 P atien ts and Materials 2.2.1 P atien t Selection and Data Collection W orking under an Institutional Review Board (IRB) appro v ed proto col at the Clev eland Clinic, w e included 17 patien ts who underw en t SEEG ev aluation in our Epilepsy Cen ter. Our inclusion criteria w ere: (i) tailored resection or laser ablation guided b y SEEG; (ii) three or more seizures recorded during SEEG that w ere c haracterized b y sustained (3 -s duration or longer) gamma activit y at the onset; (iii) no seizures, including auras, after surgery . Details of the patien t selection proto col are presen ted in App endix A . The SEEG ev aluation w as a clinically determined in v asiv e pre-surgical mo dalit y for patien ts with pharmaco-resistan t fo cal epilepsy , incongruen t non-in v asiv e data, or a negativ e MRI with the absence of a clear epileptogenic lesion. Anatomo-electro clinical h yp otheses w ere form ulated individually for eac h patien t during a m ulti-disciplinary patien t managemen t conference based on a v ailable non-in v asiv e data: clinical history , video EEG, MRI, PET, ictal single-photon emission computed tomograph y (SPECT) and MEG. SEEG i mplan tation w as p erformed using m ulti-lead depth electro des (A dT ec h, In tegra, or PMT). The electro des w ere implan ted according to the T alairac h stereotactic metho d using orthogonal or oblique tra jectories [ 84 ]. Anatomical lo cations of the electro de leads w ere c hec k ed b y the digital fusion of a p ost- implan tation thin-sliced computed tomograph y (CT) 3D image with a pre-op erativ e T1-w eigh ted v olumetric MRI. Images w ere aligned and v erified using CURR Y 7 (Compumedics NeuroScan). Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 19 The SEEG signals w ere recorded on Nihon K ohden EEG system with a sampling rate of 500 Hz (un til 2012, 7 patien ts in our study) or 1000 Hz (2012 and later, 10 patien ts). After the SEEG ev aluation, patien ts underw en t a tailored resection, or a laser ablation of the iden tified epileptogenic zone. A p ost-op erativ e MRI w as acquired 16 mon ths after surgery , aligned to the CT, and used to i den tify electro de con tacts that w ere p ositioned in the resected/ablated area. F ollo w-up information w as collected from a review of the medical records. W e also collected retrosp ectiv e data ab out age of epilepsy onset and epilepsy duration, lo calization and t yp e of lesion based on MRI, surgical neuropathology , and resection details. Se v en teen patien ts matc hed inclusion criteria. All patien ts w ere seizure-free with a mean follo w up duration of 42:9 mon ths [standard deviation (SD) 20:4 mon ths]. T able 2.1 sho ws the clinical profil es of these patien ts. 2.2.2 Data Selection and Time-F requency Decomp osition Three t ypical clinical seizures c haracterized b y sustained gamma band activit y at the onset w ere selected for eac h patien t. Sixteen patien ts had a sing le stereot ypical ictal pattern. One patien t had t w o distinct ictal patterns; b oth patterns w ere included in the analysis. F or eac h iden tified seizure onset, w e extracted a windo w of 40 s of SEEG data: 20 s b efore and 20 s after the seizure onset. W e also collected 40 s of baseline SEEG data 2 min b efore the seizure onset to use as a statistical baseline for the seizure. All analyses w ere based on bip olar signals formed as the difference b et w een pairs of adjacen t con tacts on eac h electro de. Channels con taining visually ob vious artifacts w ere remo v ed. T w o v ariations of complex Morlet w a v elet transform w ere applied to eac h c hannel of data. F or the visual assessmen t, w e used Brainstorm [ 178 ] to generate Morlet w a v elet transforms ev aluated at 50 frequencies spaced on a logarithmic scale from 1 to 200 Hz. The Morlet time-frequency index w as set to 5 (i.e., 5 w a v elengths full-width-half-maxim um (FWHM) at a giv en frequency). W e used Brainstorm’s “flattening” option, whic h amplifies the higher frequencies to comp ensate for the “1/f ” c haracteristic of the p o w er sp ectrum and giv es a more visually app ealing image. An example of this pro cessing is sho wn in Figure 2.1 . F or the automated pro cessing discussed in the “F eature extraction” section b elo w, the second Morlet w a v elet transform v ariation used linear spacing in the 20 2.2. P atien ts and Materials T a ble 2 .1: Clinical c haracteristics of the included pati en ts Sub ject ID Age (y ears) Epilepsy duration (y ears) MRI lesion Resection (or abla- tion) details Surgical pathol- ogy F ollo w-up duration (mon ths) * Outcome Anatomical lo- cation of the epileptogenic zone 1 43 37 F CD, insu- lar/fron tal op- erculum An terior insula/fron tal op erculum F CD, t yp e 2B 13 Seizure-free Insular/fron tal op- erculum 2 29 22 Negativ e T emp oral-parietal- o ccipital F CD, t yp e 1C 49 Seizure-free Occipito-temp oral 3 33 17 Hipp o campal scle- rosis An terior temp oral lob e Hipp o campal scle- rosis t yp e 1 48 Seizure-free T emp oral 4 17 8 Negativ e Laser ablation, sup e- rior fron tal gyrus No pathology 19 Seizure-free F ron tal 5 16 1 Benign neoplasm, p osterior parahip- p o campal gyrus P osterior parahip- p o campus gyrus and neoplasm Lo w grade glial/glioneuronal neoplasm 39 Seizure-free Basal p osterior temp oral 6 46 41 F CD, mesial- fron tal Prefron tal lob e Non-sp ecific c hanges 38 Seizure-free F ron tal 7 5 1 Negativ e Sup erior fron tal gyrus, sup erior fron tal sulcus, fron tal p ole F CD, t yp e 2B 21 Seizure-free Sup erior fron tal gyrus/ sup erior fron tal sulcus 8 63 14 Negativ e Orbitofron tal F CD, t yp e 1 44 Seizure-free Orbitofron tal/pars orbitalis 9 33 19 Gliotic p ostop c hanges An terior temp oral lob e F CD, t yp e 1B 40 Seizure-free T emp oral 10 21 11 Negativ e Occipital lob e Grey matter het- erotopia, F CD t yp e 1B 12 Seizure-free Cuneus 11 32 27 F CD, precen tral gyrus Precen tral gyrus Non conclusiv e 77 Seizure-free Precen tral gyrus 12 22 3 F CD, sup erior fron tal sulcus Sup erior and middle fron tal gyri, an terior cingulate F CD t yp e 2B 78 Seizure-free F ron tal 13 19 18 Negativ e Middle fron tal gyrus F CD t yp e 1 48 Seizure-free Inferior fron tal sul- cus/middle fron tal gyrus 14 30 18 Negativ e F ron tal op erculum F CD t yp e 2B 47 Seizure-free F ron tal op ercu- lum/sub cen tral region 15 20 11 Negativ e F ron tal lob e F CD, t yp e 1 82 Seizure-free Sup erior fron tal gyrus/sup erior fron tal sulcus 16 65 25 Negativ e An terior temp oral lob e F CD 1C 39 Seizure-free T emp oral 17 65 9 Negativ e An terior temp oral lob e F CD 1C 36 Seizure-free T emp oral * F ollo w up information curren t as of July 2017. Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 21 frequency range of 1 to 200 Hz with in terv al 1 Hz, and a Morlet time-frequency index of 8 . The baseline SEEG data w ere pro cessed iden tically , and the baseline Morlet w a v elet co efficien ts w ere used to normalize the seizure Morlet w a v elet co efficien ts b y dividing the p o w er in eac h frequency line b y its baseline p o w er (see Section 2.4.1.2 ). 2.3 Visual Characteristics of the Epileptogenic Zone F or eac h patien t in T able 2.1 , w e computed time-frequency maps of eac h SEEG con tact, b oth inside and outside the epileptogenic zone. W e sho w one suc h exemplar map in Figure 2.1 ( b ) for all con tacts in a single patien t and, in Figure 2.1 ( a ), the corresp onding SEEG time series for this patien t. Despite the v ariation in anatomical lo cations of the epileptogenic zone and the t yp e of epilep- togenic lesion from patien t to patien t, a c haracteristic time-frequency pattern emerged for con tacts lo cated inside the epileptogenic zone. Figure 2.2 sho ws a t ypical example of this pattern. A t seizure onset, w e observ ed three predominan t features from the time-frequency map as illustrated in Figure 2.2 ( b ): (i) single or m ultiple pre-ictal sharp transien t(s) or spik e(s) (dep ending on their slo w comp onen t duration); (ii) narro w frequency bands of fast activit y; with (iii) sim ultaneous sup- pression of slo w pre-ictal frequencies. The comparison b et w een c hannels that w ere lo calized inside v ersus outside resection regions rev ealed the predominance of this pattern inside resected regions (Figure 2.1 ( b )). Pre-ictal spik es and sha rp transien ts app eared as either a burst of pre-ictal spik es, o r as a single spik e. These spik es w ere c haracterized b y t w o comp onen ts, a slo w ( delta to theta) activit y , and a sim ultaneous fast activit y (high frequency oscillations) in a frequency range from 60 to 200 Hz. These pre-ictal spik es or sharp transien ts w ere part of the pattern in 49 of 51 seizures. In the other t w o seizures, the transien t w as missing at ictal onset, only a delta burst w as observ ed. The duration of pre-ictal spiking preceding the seizure onset v aried in length from a single sharp transien t to con tin uous spiking without clear in ter-ictal/pre-ictal transition. The second notable feature w as the narro w-band fast activit y , c haracterized b y t w o or three (four in one case) high-in tensit y bands (Figure 2.2 ( b )). The median of maxim um frequency of fast 22 2.3. Visual Characteristics of the Epileptogenic Zone (a) The SEEG time series. Bip olar mon tage w as applied b y taking the difference b et w een signals from t w o adjacen t con tacts. Ictal onset is mark ed as “0”, full duration of the segmen t is 40 s (from20 s to 20 s). (b) The Morlet time-frequency maps corresp onding to eac h bip olar con tact pair in (a). Eac h graph sho ws 20 s b efore the seizure onset to 20 s after the seizure onset, along the horizon tal axis. The v ertical axis represen ts frequencies from 1 Hz to 200 Hz, logarithmically spaced and sp ectrally flattened to emphasize the higher frequencies. Con tacts in the resected area are outlined in blue. The pattern c haracterized b y the com bination of a spik e, m ulti-band fast activit y and suppression is lo calized inside the resected area (y ello w arro w), while broadband fast activit y (blac k arro w) and suppression without sustained fast activit y (white arro w) are lo calized outside the resected area. Figure 2.1: SEEG recording and time-frequency plots of seizure onset. Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 23 (a) Channel R5-R6 from Figure 2.1 ( a ) from 5 s b efore to 20 s after the ictal onset (b) The corresp onding time-frequency plot (logarithmic scale) of Figure 2.2 ( a ) Figure 2.2: Example of pre-ictal to ictal transitions in the epileptogenic zone. The time-frequency plot sho ws the prop osed “fingerprin t”: a com bination of pre-ictal spik es, m ulti-band fast activit y and sim ultaneous suppression of slo w er bac kground frequencies. Note that fast activit y is c harac- terized b y m ultiple bands that are not harmonically related, c hirp at differen t frequency rates, and whose amplitudes v ary indep enden tly across bands. 24 2.3. Visual Characteristics of the Epileptogenic Zone activit y w as 97 Hz [in terquartile range (IQR) 33 Hz] across all con tacts expressing the c haracteristic time-frequency pattern and the median of minim um frequency of fast activit y w as 43 Hz (IQR 21 Hz) (for statistics, see App endix B Figure B.1 ). The transition from spik es to fast activit y w as immediate: either narro w band fast activit y app eared righ t after the last spik e (7 patien ts) or fast activit y initially app eared across a broad band of frequencies and then in 1 to 5 s it is organized in to sev eral narro w bands (10 patien ts). F or examples of the v ariation of the transition, see App endix C Figure C.1 . The initial fast activit y w as often asso ciated with a brief increase of activities in some lo w frequencies, e.g., the delta/theta range. T w o other k ey features of the fast activit y w ere a decreasing frequency shift (also kno wn as “do wn-c hirping”) with the progression of the seizure, and a pulsing amplitude c hange. The m ultiple bands w ere generally not harmonically related; rather, they c hirp ed at differen t frequencies, pulsed in their o wn amplitudes indep enden tly . In fiv e patien ts narro w bands of fast activit y p ersisted through the full seizure and ended with a brief burst of spik es. In the rest of the patien ts, narro w band fast activit y transitioned in to rh ythmical ictal spik es. The maxim um duration of the fast activit y , as observ ed o v er all con tacts for a patien t, v aried from 3 to 91 s (median 15 s). Of 17 patien ts, fast activit y app eared sim ultaneously b oth inside and outside the resected area in 11 patien ts; four patien ts also had fast activit y b oth inside and outside the resected area but there w as a dela y from inside to outside that ranged from 100 ms to 3 s; in the remaining 2 patien ts, fast activit y w as observ ed only inside the resected area. F ast activit y outside the resection area frequen tly app eared only as a single and broad band of frequencies (Figure 2.1 ( b ), blac k arro w). The median of the maxim um frequency of fast activit y across all con tacts outside the resection area w as 74 Hz (IQR 28 Hz) and median of the minim um frequency w as 44 Hz (IQR 22 Hz) (see App endix B Figure B.1 ). The third notable feature w as the suppression of lo w-frequency pre-ictal activit y at seizure onset (a decrease of signal p o w er in the lo w er frequencies in the time-frequency plots), whic h app eared sim ultaneously with the fast activit y (Figure 2.2 ( b )). The suppression w as alw a ys most pronounced in the delta-theta frequencies but commonly extended up to gamma frequency range. The median Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 25 of the maxim um frequencies that suppressed across all con tacts expressing the pattern w as 45 Hz (IQR 20 Hz) (see App endix B Figure B.2 ). The duration of the suppression w as related to the duration of the fast activit y , with the suppre ssion often terminated b y a burst of in tense slo w activit y , as the fast activit y decreased or arrested. Suppression w as also observ ed in the con tacts outside the resection, but it w as also less in tense, more diffuse, and uncorrelated with the fast activit y (Figure 2.1 ( b ), white arro w). The median of the maxim um frequencies that suppressed across all con tacts outside the resection w as 25 Hz (IQR 22 Hz) (se e App endix B Figure B.2 ). Summarizing the visual iden tification pro cess, w e iden tified a consisten t pattern within the epileptogenic zone that w as c haracterized b y the com bination of initial sharp transien ts/spik es, follo w ed b y m ulti-band fast activit y and a concurren t suppression of lo w er frequency activities. W e displa y a represen tativ e con tact from the resected zone from eac h of 15 of the patien ts in App endix C Figure C.1 , highligh ting this pattern in b oth the time and time-frequency domains. Con v ersely , con tacts outside the resected zone rev ealed a p ossible subset of these features, but not a full com bination. 2.4 A utomatic Classification Pro cedure and Results With the predominan t features of the epileptogenic zone visually iden tified, w e dev elop ed an algorithm to automatically extract these features from the time-frequency data. W e then used mac hine learning metho ds (SVM) [ 51 ] to automatically classify a con tact as b eing inside or outside the epileptogenic zone. 2.4.1 Prepro cessing 2.4.1.1 Time-F r equency Decomp osition The con tin uous Morlet w a v elet transform with linear frequency scale from 1 to 200 Hz w as applied to the ra w SEEG data for eac h c hannel. The bip olar c hannel pair X2 -X3 from Sub ject 1 , Seizure 1 P w as c hosen as an example for illustration. The FWHM of the Gaussian k ernel in the 26 2.4. A utomatic Classification Pro cedure and Results time domain for the mother w a v elet w as c hosen to b e 8 cycles as a trade-off b et w een temp oral and sp ectral resolution. The resulting time-frequency plot is sho wn in Figure 2.3 . Figure 2.3: Time-frequency plot of seizure onset for the selected c hannel X2 -X3 from Sub ject 1 , Seizure 1 P . Figure 2.4: Time-frequency plot of the baseline data for the selected c hannel X2-X3 from Sub ject 1, Seizure 1P 2.4.1.2 Normalization T w o min utes b efore seizure onset, a baseline data set w as also recorded for eac h seizure and transformed in the same w a y to a time-frequency plot as sho wn in Figure 2.4 . The seizure data w ere then normalized with resp ect to the baseline segmen t for eac h frequency comp onen t. Let Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 27 Figure 2.5: Time-frequency plot of the normalized data for the selected c hannel X2-X3 from Sub ject 1, Seizure 1P . x(t;f) b e the seizure onset time-frequency data, let y(t;f) b e the baseline time-frequency data and let x n (t;f) b e the normalized time-frequency data. Then the normalization is p erformed as x n (t;f) = x(t;f) mean t fy(t;f)g std t fy(t;f)g (2.1) where mean t fg and std t fg denotes the mean and SD with resp ect to time, resp ectiv ely . Figure 2.5 sho ws the result of normalization. 2.4.1.3 Artifact Remo v al Indep enden t comp onen t analysis has pro v e n to b e an effectiv e algorithm in the analysis of EEG/MEG data [ 130 , 197 ], and is widely used to iden tify artifacts, suc h as ey e mo v emen t, ey e blink and electrical artifact [ 60 , 189 ]. Here w e applied complex ICA (cICA) [ 27 ] to the time- frequency data to iden tify and remo v e artifacts. This pro cedure w as applied separately to the seizure and baseline data b efore normalization. Figure 2.6 sho ws an example of the time-frequency plots of all c hannels for Sub ject 7 , Seizure 2 P . W e observ ed that one horizon tally-structured artifact are presen t across almost all c hannels. A 3D tensor w as formed from the time-frequency data as c h annel time frequency b y concatenating all c hannels along the first dimension, denoted as a tensorY = Y(c;t;f) , where c2 S C = [1;2;:::;N] , t2 S T = [20;:::;20] , f2 S F = [1;:::;200] 28 2.4. A utomatic Classification Pro cedure and Results and N is the total n um b er of c hannels for that sub ject. Let tensor X = X(c;t;f) b e the source signals in the same format and letM2R NN b e the un-mixing matrix. The cICA decomp osition can b e expressed as X =M 1 Y (2.2) where i is the standard matrix-tensor m ultiplication in mo de i (see [ 112 ]). Figure 2.7 sho ws the indep enden t comp onen ts matricized fromX . Let S A b e the set of indices of the iden tified artifact comp onen ts and let s k , k2 S A , b e the k th artifact comp onen t. F or example, in this case the last comp onen t, s 42 , captured the artifact w e observ ed in Figure 2.6 . Then this artifact comp onen t can b e remo v ed b y zeroing out the k th column ofM for k2S A and the “artifact-free” time-frequency plots, denoted asY ′ =Y ′ (c;t;f) , can b e reconstructed as Y ′ =M ′ 1 X (2.3) whereM ′ (i;j) =M 1 (i;j) for all i and j / 2 S A andM ′ (i;j) = 0 for all i and j2 S A . Figure 2.8 sho ws the time-frequency plots when this artifact comp onen t is remo v ed from the original time- frequency plots. It can b e seen that the artifacts are suppressed w ell for most c hannels. 2.4.2 F eature Extraction Iden tification of eac h of the three features that c haracterize the epileptogenic zone fingerprin t is describ ed in the follo wing subsections with the follo wing t w o steps: (i) candidate extraction, whic h c hec ks for the existence of the feature and finds its lo cation, and (ii) descriptor extraction, whic h generates a n umerical description of the candidates. W e b egin with the time-frequency plots after artifact remo v al, as describ ed in Section 2.4.1 . W e use c hannel X2 -X3 from Sub ject 1 , Seizure 1 P , as sho wn in Figure 2.5 , for illustrativ e purp oses. Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 29 Figure 2.6: Original time-frequency plots of all c hannels of Sub ject 7 , Seizure 2 P with artifacts presen t Figure 2.7: Plots of indep enden t comp onen ts obtained from cICA of all c hannels of Sub ject 7 , Seizure 2 P . 30 2.4. A utomatic Classification Pro cedure and Results Figure 2.8: Time-frequency plots of all c hannels of Sub ject 7 , Seizure 2 P with artifact comp onen ts remo v ed. 2.4.2.1 F ast A ctivit y Figure 2.9 ( a ) sho ws the time-frequency plot from Figure 2.5 from seizure onset. F ast activit y is highly v ariable across sub jects and c hannels. In some instances, only one band ma y b e presen t. In others, m ultiple bands can o ccur sim ultaneously as in this example. Moreo v er, the fast activit y often exhibits “do wn-c hirping” in whic h the frequency bands shift lo w er as a function of time. F ast activit y detection uses Algorithm I , for whic h w e include pseudo-co de b elo w. In order to capture the ridge-shap ed structures, a no v el m ulti-scale F rangi filter algorithm w as dev elop ed. F rangi et al. [ 72 ] describ ed a m ulti-scale Hessian-based filter mainly for blo o d v essel enhancemen t. W e use the MA TLAB (The Math W orks, Inc., Natic k, Massac h usetts, United States) implemen tation b y Kro on [ 114 ]. F rangi’s metho d uses differen t scales of Gaussian smo othing to allo w detection of v essels of differen t sizes. Our mo dification of this approac h applies differen t scales of thresholding to iden tify fast activities of differing strengths. A t eac h scale, false detections are eliminated using information ab out the orien tation, area, p osition and eccen tricit y of fast activit y . The final fast Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 31 activit y mask is generated b y com bining all candidates from differen t scales of detection. Finally , n u merical descriptors, as a represen tation of fast activit y , are extracted from the mask. F a st activit y Candidate Extraction The pro cedure F astA ctivityDetection in Algorithm I tak es a normalized artifact-cleaned p ost-onset time-frequency plot, denoted as TF righ t , as input (Figure 2.9 ( a )) and the F rangi filter is applied. This yields a filter output, denoted b y TF F , and a direction map, denoted b y D. Eac h pixel in TF F reflects the degree of “banding” and eac h pixel in D represen ts the direction or orien tation of these bands. Figure 2.9 ( b ) sho ws an example of TF F and Figure 2.9 ( c ) sho ws an example of D where the angle ranges from90 degree (negativ e y-axis) to +90 degree (p ositiv e y-axis), rotating an ti-clo c kwise as indicated b y the color bar. F or eac h lev el of thresholding, the follo wing op erations are p erformed: 1. Binarize TF F to generate the initial mask, denoted as M new , b y thresholding using the curren t lev el k . 2. F or an y pixel in M new , if the corresp onding pixel in D exceeds some limit on the maxim um angle, it is flagged to b e false in M new . 3. Find all the connected comp onen ts (CCs) in M new . F or eac h connected comp onen t, if an y of the follo wing criteria is satisfied, then t hat connected comp onen t is remo v ed from M new . (a) If the area of the connected comp onen t is sm aller than some threshold areaTh. (b) If the cen ter of the connected comp onen t is outside the region restricted b y some thresh- old b drTh. (c) If the eccen tricit y of the conne cted comp onen t is smaller than some threshold eccenTh. 4. Remo v e an y banding area that has already b een detected from the previous iteration and only k eep the newly detected band(s). 5. Merge the mask in to the mask from the previous iteration. The initial mask from the first lev el thresholding is sho wn in Figure 2.9 ( d ), with refineme n t based on step 2 and 3 resulting in Figure 2.9 ( e ). Step 4 and 5 are not in v olv ed in this first iteration. 32 2.4. A utomatic Classification Pro cedure and Results (a) (b) (c) (d) (e) (f ) (g) (h) Figure 2.9: F ast activit y extraction example. (a) Time-frequency plot of Sub ject 1 Seizure 1 P after onset. (b) F rangi filtering output. (c) The direction map. (d) First lev el initial mask. (e) The refined first lev el mask. (f ) Second lev el initial mask. (g) The refined second lev el mask. (h) The final c om bined mask. Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 33 Algorithm I F ast activit y detection pro cedure F astA ctivityDetection (TF righ t ) Initialize M old Initialize lev els N n um b er of lev els Initialize angleTh, areaTh, b drTh, eccenTh (TF F , D) F rangiFilter(TF righ t ) for k = 1;2;:::;N do M new threshold(TF F , lev el k ) for all pixels with index (i;j ) in M new do if D(i;j)> angleTh then M new (i;j) 0 end if end for for all CC with index i in M new do if the are a of CC i < areaTh then remo v e CC i from M new end if if the ce n ter of CC i outside b drTh then remo v e CC i from M new end if if the ec cen tricit y of CC i < eccenTh then remo v e CC i from M new end if end for for all CC with index i in M new do if CC i ∩ M old ̸=∅ then remo v e CC i from M new end if end for M old M new ∪ M old end for return M old end pro cedure During the second iteration, the initial mask obtained from the second lev el thresholding is sho wn in Figure 2.9 ( f ). After step 3 and 4 , an additional w eak er banding region is found successfully as sho wn in Figure 2.9 ( g ). Finally , the t w o masks are merged as sho wn in Figure 2.9 ( h ). F a st A ctivit y Descriptor Extraction There are 17 n umerical descriptors extracted from the mask. The first descriptor is simply the total n um b er of “banding” features found in the candidate 34 2.4. A utomatic Classification Pro cedure and Results extraction step ab o v e, denoted as N B . In our c hosen example, this n um b er is 3 . Among all detected bandings, only the t w o largest are retained and sorted based in order of ascending cen ter frequency . F o r e ac h banding feature the follo wing 8 descriptors are extracted: Area B - the area of the banding region. Orien tation B - the orien tation of the banding region. T BS - the starting time of t he banding with resp ect to onset time. T BE - the ending time of the ba nding with resp ect to onset time. F BS - the corresp onding a v erage frequency at time T BS . F BE - the corresp onding a v erage frequency at time T BE . F BMax - t he maxim um frequency that the banding reac hes. F BMin - the minim um frequency that the banding reac hes. 2.4.2.2 Suppression Lo w frequency suppression is apparen t immediately after seizure onset, sho wn as a dark region in the lo w er frequencies of the time-frequency plot in Figure 2.5 . Although the lev el of suppression pla ys an imp ortan t role in c haracterizing the suppression region, determining the existence of suppression and describing its app earance n umerically remains c hallenging. Figure 2.10 ( a ) sho ws the ideal suppression region for this example, where the b oundary is sho wn as the blue curv e. A simple thresholding will not w ork w ell in most cases for the follo wing reasons. First, for those c hannels where fast activit y exists, it is not uncommon that the fast activit y only partially spans the suppression region. In these cases, thresholding will incorp orate some non-suppression dark regions, where the frequency is higher than the frequency of fast activit y bands, in to the suppression region. This phenomenon could o ccur, for example, ab o v e the suppression region in the left side of Figure 2.10 ( a ). Second, for those c hannels where fast activit y do es not exist, there is no upp er b oundary to define the edge of the suppression region. Third, the in tensit y inside the Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 35 suppression region v aries considerably . As a result, the thresholded suppression region w ould b e brok en in to sev eral sub-regions. This phenomenon could o ccur, for example, in the lo w er righ t corner of the suppression region in Figure 2.10 ( a ). Hence, w e prop osed a suppression detection pip eline to remedy the difficulties discussed ab o v e as sho wn in Algorithm I I . The time-frequency plot w as smo othed b y a guided filter [ 94 ] and a b oundary w as set based on the banding information to restrict the detection region. Then a thresholding w as p erformed follo w ed b y morphological op erations in order to retain the top ological in tegrit y of the detected mask. Finally , similar to the banding detection pro cedure, n umerical descriptors w ere extracted from the mask. (a) (b) (c) (d) (e) (f ) Figure 2.10: Suppression extraction example. (a) Time-frequency plot of Sub ject 1, Seizure 1P after onset with ideal suppression region. (b) Guided filtered time-frequency plot. (c) A phan tom of fast activit y mask. (d) Guided filtered time-frequency plot with upp er b ound set. (e) Initial mask from thresholding. (f ) The final refined mask. 36 2.4. A utomatic Classification Pro cedure and Results Suppression Candidate Extraction First, a guided filter [ 94 ] is applied to the original time- frequency plot for the sak e of edge-preserving smo othing. The resulting time-frequency plot, de- noted as TF GF in the algorithm, can b e seen in Figure 2.10 ( b ), where the region inside the sup- pression is w ell smo othed but the b oundary b et w een the suppression region and the fast activit y is k ept clear. Second, w e define an upp er b ound for the suppression due to the partial co v erage of the band- ing discussed ab o v e. This is sho wn in the sub-pro cedure Bound ar yDetection in Algorithm I I and illustrated in Figure 2.10 ( c ). W e n um b er eac h three bandings (1 , 2 and 3 resp ectiv ely in Figure 2.10 ( c ) in blac k). Then w e define four sen tinel p oin ts: P 1 is the left most p oin t of banding 1 . P 2 is the righ t most p oin t of banding 2 . P 3 is the left most p oin t of banding 3 and P 4 is the righ t most p oin t of banding 3 . Finally , w e also define four exemplar p oin ts Q 1 to Q 4 (sho wn in y ello w in Figure 2.10 ( c ). The co ordinates of p oin t P i will b e denoted as x P i and y P i . If there is at least one banding found in the fast activit y detection step, there are four differen t situations to consider when determining the b oundary for eac h time p oin t k . They will b e discussed as follo ws: Case 1: When there is no banding at timek and all the bandings are on the righ t side of time k , i.e., when k <x P 1 , e.g., p oin t Q 1 . Then the b oundary Bdr(k ) is defined as y P 1 Case 2: When there is a banding or m ultiple bandings at time k , i.e., when x P 1 <k <x P 2 or when x P 3 < k < x P 4 . e.g., p oin t Q 2 . Then the b oundary Bdr(k ) is defined as the minim um frequency that the bandings reac h at t ime k . Case 3: When there is no banding at time k and time k is in a gap b et w een bandings, i.e., when x P 2 < k < x P 3 . e.g., p oin t Q 3 . Then the b oundary Bdr(k ) is defined as the linear in terp olation of y P 2 and y P 3 . Case 4: When there is no banding at time k and all the bandings are on the left side of time k , i.e., when k >x P 4 . e.g., p oin t Q 4 . Then the b oundary Bdr(k ) is defined a s y P 4 . If there is no banding detected at all, a default b oundary is used. The default b oundary is defined as the minim um of all b oundaries across those c hannels where at least one banding is found and h ence the b oundary can b e w ell defined. Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 37 Third, w e mask off the region outside the b oundary b y setting its in tensit y to the maxim um in tensit y of the TF GF as sho wn in Figure 2.10 ( d ) to ensure it is excluded from the suppression region in the subsequen t thresholding. F ourth, a thresholding is applied to obtain a binary initial mask for the suppression region, as sho wn in Figure 2.10 ( e ). Next, some morphological op erations (dilation, hole-filling and erosion, denoted as M dilate , M fill_holes and M ero de resp ectiv ely) are p erformed sequen tially to ensure the top ological in tegrit y of the detected region. Finally , only the largest connected comp onen t is returned as the detected suppression mask as sho wn in Figure 2.10 ( f ). Suppression Descriptor Extraction There are 9 n umerical descriptors extracted f rom the mask: A S - the area of the suppression area . V S - the v ariance of the in tensit y of t he suppression area. IMed S - the median in tensit y of t he suppression area. IMax S - the maxim um in tensit y of the suppression area. T SS - the starting time of the suppression with resp ect to onset time. T SE - the ending time of the suppression with resp ec t to onset time. F SMax - the maxim um freque ncy that the suppression reac hes. F SMin - the minim um frequency that the suppression reac hes. R tP S - the ratio b et w een the IMed S and the median in tensit y of the pre-ictal region. The pre-ictal region is a rectangle area where the20 s and 0 s define the b oundary along the temp oral axis and the F SMin and F SMax define the b oundary along the sp ectral axis. 38 2.4. A utomatic Classification Pro cedure and Results Algorithm I I Suppression detection pro cedure SuppressionDetection (TF righ t , Mask B ) Initialize Th TF GF GuildedFilter(TF righ t ) maxV al max (TF GF ) Bdr BoundaryDetection(Mask B ) T n um b e r of time samples along x-axis N maxim um frequency along y-axis for k = 1;2;:::T do TF GF (k , Bdr(k ), Bdr(k )+1, …, N ) maxV al end for TF GFT threshold(TF GF , Th) TF GFTM M ero de (M fill_holes (M dilate (TF GFT ))) return the largest connected comp onen t in T F GFTM end pro cedure pro cedure Bound ar yDetection (Mask B ) if Mask B ̸=∅ then for k = 1;2;:::;T do if k <x P 1 then Bdr(k) y P 1 else if x P 1 k <x P 2 or x P 3 k <x P 4 then Bdr(k) min(Mask B (k;:) ) else if x P 2 k <x P 3 then Bdr(k) linear in terp olation o f y P 2 and y P 3 else if kx P 4 then Bdr(k) y P 4 end if end for return Bdr else return default b oundary end if end pro cedure 2.4.2.3 Pre-ictal Spik es The third feature is spiking during the pre-ictal p erio d, from20 s to onset. The time-frequency plot for the pre-ictal p erio d for our exemplar c hannel, i.e., the left side of Figure 2.5 and denoted TF Left , is re-dra wn here in Figure 2.11 ( a ). W e define a spik e as activit y of v ery short duration that spans almost all frequencies. This can b e w ell c haracterized b y t w o prop erties: the strength of the spik e and the sharpness of the spik e. Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 39 (a) (b) (c) Figure 2.11: Pre-ictal spik e extraction example. (a) Time-frequency plot of Sub ject 1, Seizure 1P b efore onset. (b) Plot of the median of upp er 25 % quan tile of the in tensit y distribution in (a) for eac h time p oin t with initial lo cal maxima. (c) Same plot as (b) with detected spik e candidates. Pre-ictal Spik e Candidate Extraction The extraction pro cedur e is sho wn in Algorithm I I I . First, for eac h time p oin t, the upp er 25 % quan tile of the in tensit y distribution of TF Left is found and its median determined. This results in a one-dimensional signal, denoted as P , as sho wn in Figure 2.11 ( b ). Second, the lo cations of the lo cal maxima, denoted maxP os, and their v alues maxV al, are found (sho wn as red circles in Figure 2.11 ( b )). Third, w e compute the ratio of p eaks to their corresp onding neigh b ors as a measure of the sharpness of the spik es as follo ws: F or eac h p eak i at maxP os(i ), the k neigh b ors o f that p eak are found, denoted as n bP os. The median of the resp onse P at n bP os is c omputed. The ratio b et w een the maxP os(i ) and the median v alue w e obtained from step t w o is calcu- lated. If the ratio is less than some desired threshold, the candidacy of the p eak i is remo v ed. The final spik e candidates are sho wn as red circles in Figure 2.11 ( c ). It can b e seen that only those large and sharp spik es are left after the extraction pro cedure. 40 2.4. A utomatic Classification Pro cedure and Results Algorithm I I I Pre-ictal spik e detection pro cedure Preict alSpikeDetection (TF left ) Initialize n um b er of neigh b ors k Initialize ratio threshold Th T n um b e r of time samples along x-axis for k = 1;2;:::T do P(i ) median of upp er 25 % quan tile of TF left (k;: ) end for maxV al lo cal maxim um of P maxP os time p oin ts where maxV al o ccur N n um b er of maxV al found for k = 1;2;:::N do n bP os neigh b ors of maxP os(k ) ratio maxV al(k ) / median(P(n bP os)) if ratio < Th then remo v e m axP os(k ) and maxV al(k ) end if end for return maxV al and maxP os end pro cedure Pre-ictal Spik e Descriptor Extraction There are total 3 n umerical descriptors extracted from the spik e candidates. N S - the total n um b er of spik e candidates fo und. M S - the mean of the ratios w e obtained from the second stage ab o v e. T PE - the time the last spik e o ccurs with resp ect to the onset. 2.4.3 Classification 2.4.3.1 F eatures and Classifier In con trast to the usage of “feature” in Section 2.4.2 , here w e use “feature”, as is standard in mac hine learning, to refer to the (concatenated) n umerical descriptors of fast activit y , suppression and pre-ictal spik es. Let N F b e the total n um b er of features a v ailable for eac h sample. Here, N F = 17 + 9 + 3 = 29 . Let N S b e the total n um b er of samples w e ha v e for all c hannels and all 3 seizures from eac h of 17 sub jects. Let C i b e the n um b er of c hannels for sub ject i . Therefore, Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 41 N S = ( ∑ i C i )317 . Then the feature matrix, denoted as F2R N S N F , along with the lab els describ ed in the follo wing section, w as used as the input to a SVM classifier. An RBF (Gaussian) k ernel with empirical k ernel scale 7:5 w as applied and the regularization parameter w as c hosen to b e 3:5 for p enalizing those samples that violate the margin. Therefore, a soft-margin k ernel SVM w as formed, computed using standard con v ex quadratic programming. 2.4.3.2 Sub ject-based Cross V alidation In order to prev en t o v erfitting the data and to test accuracy , w e use cross-v alidation. T w o ma jor cross-v alidation metho ds are commonly used: random split k -fold cross-v alidation and lea v e-one- sample-out cross-v alidation. K -fold cross-v alidation divides the dataset in to k folds with equal size via random sampling without replacemen t. F or eac h fold i , the other k 1 folds are com bined together for training the classifier. Then the prediction for the i th fold is p erformed based on the trained classifier. Lea v e-one-sample-out cross-v alidation can b e treated as a sp ecial case of k -fold cross-v alidation where k equals the total n um b er of samples N S . Therefore, k -fold cross-v alidation is m ore computationally efficien t than lea v e-one-sample-out for k <N S . Unfortunately , standard k -fold cross-v alidation is not applicable to our dataset b ecause of the strong dep endency across con tact pairs and seizures for eac h sub ject. i.e., an assumption of inde- p en dence b et w een samples do es not hold. Also, it has b een sho wn that lea v e-one-sample-out cross- v alidation tends to in tro duce (optimistic) bias due to the insufficien t testing data [ 195 ]. Therefore, w e use lea v e-one-sub ject-out cross-v alidation metho d: letM b e the total n um b er of sub jects, where in our case M = 17 . F or eac h sub ject i , w e first gather the data from all other M 1 sub jects and train the SVM. W e use the resulting SVM to iden tify putativ e epileptogenic zone and this pro cedure w as carried out for sub ject i . W e then rep eat for eac h of M = 17 sub jects. 2.4.3.3 Lab els Ground truth lab els are crucial for sup ervised classification problem. F or the epileptogenic zone iden tification purp ose, the ideal binary class lab el should b e p ositiv e for those con tacts inside the epileptogenic zone and negativ e outside the epileptogenic zone. Ho w ev er, this ideal ground truth is 42 2.4. A utomatic Classification Pro cedure and Results not kno wn precisely with our data. Rather w e kno w whic h con tacts lie within the resected zone and, since these patien ts w ere all seizure free, w e can assume that these are a sup erset of the electro des in the epileptogenic zone. Therefore, for those con tacts that w ere inside the resected region, the resection lab el w as defined as p ositiv e and for those outside the resected region, the resection lab el w as defined as negativ e. 2.4.3.4 Classification with P artially Certain Lab els There is an asymmetry in the lab els w e ha v e: w e kno w that outside the resection zone there are non-epileptogenic-zone con tacts, but within it there are a mix of epileptogenic-zone and non- epileptogenic-zone con tacts. This app ears to b e a no v el problem in classification from partially uncertain lab elled data. W e use a clustering-guided algorithm to reduce uncertain t y in the partially certain lab els. Let S b e the en tire dataset. Before w e train the SVM w e first p erform the follo wing clustering pro cedure. This is rep eated separately for eac h of the 17 cross-v alidation runs. First, an unsup ervised k -means clustering w as p erformed on all samples from S , clustering in to t w o sets, P U and N U . The subscript U stands for “unsup ervised” . W e kno w that if none of the desired features (fast activit y , suppression and pre-ictal spik e) is detected, then the concatenated descriptors formed is a zero v ector. So let x2 S b e the sample that is c losest to a zero v ector in its Euclidean norm. Th us, x is the most unlik ely sample to lie in the epileptogenic zone and therefore least lik ely to b e misclassified in the k -means clustering. Then P U is c hosen as the set that do es not con tain x and N U is the negativ e set con taining x . Note that P U and N U are m utually exclusiv e, i.e., P U ∩ N U = ∅ and P U ∪ N U = S . Also, the resection lab els, denoted as L R , naturally con tained t w o groups as w e discussed b efore. Let P R and N R b e the p ositiv e and negativ e resection group resp ectiv ely , where the subscript R stands for “resection” . P R and N R are also m utually exclusiv e. Then, P S =P R ∩ P U is defined as the epileptogenic-zone-p ositiv e set, and N S = N R as the epileptogenic-zone-negativ e set for the sup ervised training of the SVM. In short, this pre-lab elling step can b e view ed as a filtering pro cedure b ecause the set P R ∩ N U is eliminated from S as most lik ely to ha v e incorrect lab els b efore the sup ervised SVM training. Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 43 2.4.3.5 V o ting The prediction lab el, denoted asL P w as obtained from the classification. Although L P con tains the lab els for all c hannels from all sub jects, it still do es not directly indicate the estimated lo cation of the epileptogenic zone. This is b ecause for eac h sub ject, w e ha v e 3 seizures recorded and the prediction lab els across 3 seizures for a particular c hannel of a sub ject ma y not b e consisten t. Therefore, w e use a ma jorit y v oting mec hanism across the three seizures to dra w the final conclusion. Algorithm IV Classification of the epileptogenic zone pro cedure EpileptogenicZoneClassifica tion (F , L R ) for i = 1;2;:::M do P U ;N U k -means(F i;:;: ) P R ;N R L R i;:;: P S P R ∩ P U N S N R Mo del SVMT raining(F i;:;: ;P S ;N S ) L P i;:;: ; Score i;:;: Predict( Mo del;F i;:;: ) for k = 1;2;:::;C i do if ∑ j Score i;j;k > 0 or ∑ j L P i;j;k 2 then L V i;k T rue else L V i;k F alse end if end for end for return L V end pro cedure 2.4.3.6 Classification Pip eline In summary , the en tire pip eline of the classification is sho wn in Algorithm IV. Note that the sub- scripts i;j;k follo wing v ariableF;L R ;L P and Score represen t the indices for sub jecti2 [1;2;:::;M] , seizure j2 [1;2;3] and c hannel k2 [1;2;:::;C i ] . Colon is used when all indices are queried. A mi- n us sign b efore an index indicates all indices but that one. F or example, F i;3;: is the feature matrix for all c hannels in the 3 rd seizure of the i th sub ject. L R j;:;: is the resection lab el for all c hannels and all seizures of all sub jects excluding the j th one. One exception is that the subscript i;j for 44 2.4. A utomatic Classification Pro cedure and Results v ariableL V represen ts the indices for sub jecti2 [1;2;:::;M] and c hannelj2 [1;2;:::;C i ] without the indices for seizures as w e sum o v er results across all 3 seizures. Moreo v er, when coun ting the n um b e r of p ositiv e prediction lab els for a c hannel, the prediction lab el L P is treated n umerically as 1 when true and 0 when false. Finally , w e summarize all v ariables and notations used in this Section 2.4.3 b elo w for easy reference and understanding of Algorit hm IV . F2R N S N F - The data matr ix with N S n um b er of samples and N F n um b er of features . L R - Resection lab els. M - T otal n um b er of sub jects. P U ;N U - P ositiv e and negat iv e lab el set obtained from k-means unsup ervised clustering. P R ;N R - P ositiv e and negativ e lab el se t obtained directly from L R . P S ;N S - P ositiv e and negativ e lab el set used for SVM sup ervised trainin g. L P - Prediction lab els. Score - Scores indicating the distance fro m the decision b oundary for eac h sample. L V - V oting lab els. 2.4.4 Results Our goal in dev eloping the automatic classifier w as to determine whether the time-frequency patterns seen inside and outside the resection zone could indeed b e used to statistically separate the t w o classes. The algorithm w as able to ob jectiv ely iden tify the epileptogenic zone in 15 of 17 patien ts. In the other t w o patien ts (P atien ts 6 and 16 ), none of the con tacts w ere iden tified as epileptogenic. The total n um b er of epileptogenic-zone-iden tified con tacts across all patien ts using the SVM w as 64 , with 58 (p ositiv e predictiv e v alue (PPV) 58/64 = 90:6 %) lo calized inside the resected area (T able 2.2 ). The program also iden tified 1149 con tacts as not in the epileptogenic zone, of whic h 827 w ere outside the resection area and therefore assumed to b e correct (false p ositiv e rate (FPR) 6/827 = 0:7 %). The remaining 322 con tacts not iden tified as epileptogenic zone w ere Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 45 T able 2.2: Result of automatic classification of the epileptogenic zone Prediction T rue Prediction F alse Within Resect 58 (TP) 322 (FN) Outside Resect 6 (FP) 827 (TN) 0.7% (FPR) 90.6% (PPV) found in the resection zone, but this is not surprising, since w e assumed that the region of resection migh t b e larger than the actual epileptogenic zone. W e note that in one patien t who underw en t laser ablation surgery (Sub ject 4 , T able 2.1 ), only 2 con tacts w ere in the ablated region and that b o th of them w ere correctly classified as epileptogenic (T able 2.3 ). W e also found that our classifier iden tified as epileptogenic zone most of the con tacts lo calized inside p oten tially epileptogenic lesions [fo cal cortical dysplasia (F CD) t yp e 2B, hipp o campal sclerosis, b enign tumour] (see App endix D T able D.1 ). The only 6 con tacts in all patien ts that w ere iden tified as epileptogenic zone b y the program but found outside the region of resection are sho wn in T able 2.3 , along with all of the other correctly iden tified con tacts. As sho wn in App endix C Figure C.1 , most of the succ essfully iden tified con tacts had a unique com bination of suppression and t w o or more narro w frequency bands of fast activit y , mostly with a preceding sharp transien t (2 of 15 sub jects did not ha v e the clear pre-ictal spiking). The global similarit y of the pattern across sub jects could b e observ ed v ery clearly , y et with some v ariance of the pattern. 2.5 A dditional Studies T o determine the uniqueness of the fingerprin t mo del, w e underto ok additional studies of the fast activit y and suppression and p ossible in teractions b et w een them, to find out if either feature b y itself could differen tiate epileptogenic from propagation areas. 46 2.5. A dditional Studies T able 2.3: Implan tation maps with sc hematic represen tation of the resection margins (red) and bip olar SEEG c hannels iden tified b y the algorithm as TP and FP ID Map * TP FP ID Map * TP FP ID Map * TP FP 1 R1-R2 R2-R3 R3-R4 X1-X2 X2-X3 X3-X4 X5-X6 W2-W3 W3-W4 7 M3-M4 M5-M6 M7-M8 Q1-Q2 13 W’6-W’7 2 O1-O2 O2-O 3 O3-O4 V1-V2 V3-V4 V4-V5 V6-V7 8 O7-O8 T1-T2 14 R4-R5 R5-R6 M4-M5 3 B1-B2 B2-B3 B3-B4 C2-C 3 B4-B5 B7-B8 B8-B9 E1-E2 E2-E3 E6-E7 9 A3-A4 15 L1-L2 L2-L3 L3-L4 L4-L5 L5-L6 O3-O4 O5-O6 Q5-Q6 4 L’1-L’2 L’3-L’4 L’5-L’6 10 V7-V8 16 5 L’1-L’2 11 L5-L6 L6-L7 L7-L8 17 B’1-B’2 B’3-B’4 T’1-T’2 T’4-T’5 T’5-T’6 T’7-T’8 T’8-T’9 C’3-C’4 6 12 K4-K5 K5-K6 K7-K8 * Electro des on the maps are mark ed: (i) as red if they con tain true p ositiv e (TP) c hannels (p oten tially epileptogenic ins ide the resection); (ii) as green if they con tain false p ositiv e (FP) c hannels (p oten tially epileptogenic outside the resection); (iii) as blac k if they con tain only true negativ e (TN) c hannels (not p oten tially epileptogenic outside the resection); (iv) as blac k in the red-shaded area if they con tain false negativ e (FN) c hannels (not p oten tially epileptogenic inside the resection). Boundaries of prior resections are sc hematically shaded in y ello w (only P atien ts 3 and 9 had a previous resection). Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 47 2.5.1 F ast A ctivit y Inside and Outside the Resection Area Some previous approac hes to lo calize epileptogenic zone w ere based on frequency c haracteristics of the ictal fast activit y as w ell as the timing of the frequency c hanges [ 20 , 56 , 79 ]. F ast activit y frequency , whether or not asso ciated with EEG flattening, w as used to delineate the epileptogenic zone prop osing that the regions with the earliest c hange and highest frequency represen ted “the most epileptogenic” zone. Ho w ev er, the significance of fast frequencies as an absolute mark er of epileptogenicit y w as questioned in a n um b er of clinical studies [ 29 , 132 ]. T o explore the significance of frequency c haracteristics and the timing of frequency c hange in the iden tified epileptogenic zone fingerprin t, w e p erformed the follo wing additional analyses. First, w e extracted all con tact pairs exhibiting fast activit y across all seizures and all sub jects. W e computed the maxim um frequency and the timing of fast activit y for eac h of the time-frequency plots. W e then compared maxim um frequency and timing of fast activit y across the con tacts inside the resection region with that outside the resection region, using the clinical resection lab els. W e further sub divided the con tacts inside the resection region in to an epileptogenic-zone group and a non-epileptogenic-zone group as iden tified b y the classification pro cedure describ ed ab o v e. W e then compared the maxim um frequency and timing of fast activit y b et w een the con tacts within the epileptogenic zone and those outside the epileptogenic zone. In total (across all seizures and patien ts), 662 con tacts (175 epileptogenic-zone con tacts + 487 non-epileptogenic-zone con tacts) with iden tified fast activit y w ere lo calized inside the resection region and 764 con tacts outside. There is a statistically significan t difference ( P = 2:2610 6 in Figure 2.12 ( a )) in the p opulation mean of the maxim um frequency of fast activit y across con tacts within the resected region compared to those outside and also across epileptogenic-zone con tacts compared to non-epileptogenic-zone con tacts (P = 1:0210 28 in Figure 2.12 ( b )). Ho w ev er, the v ariance of the measure is sufficien tly large for b oth cases, so that they cannot b e used to differen tiate b et w een these regions using data from a single sub ject. In addition to the group analysis, w e underto ok additional analysis for eac h patien t individually . W e found no statistically significan t difference b et w een the maxim um frequency of fast activit y 48 2.5. A dditional Studies (a) (b) (c) (d) (e) (f ) Figure 2.12: Bo xplots of maxim um frequency and timing comparison. (a) Maxim um frequency of fast activit y inside and outside resection; (b) maxim um frequency of fast activit y inside resection classified as epileptogenic-zone and non-epileptogenic-zone con tacts. (c) The start time of fast activit y inside and outside resection. (d) The start time of fast activit y inside resection classified as epileptogenic-zone and non-epileptogenic-zone con tacts. (e) The maxim um frequency of suppression inside and outside resection. (f ) The maxim um frequency of suppression inside resection classified as epileptogenic-zone and non-epileptogenic-zone con tacts. The b o xplot spans the t w o cen tral quartiles of the data around the median (red line), and the whisk ers extend to a maxim um of 1:5 times the b o x span, or to the last data p oin t, whic hev er is shorter. The remaining data p oin ts are outliers. inside the resection and that outside in 15 of 17 patien ts. F or the other t w o patien ts, the maxim um frequency of fast activit y inside the resection w as significan tly higher than that outside for one and significan tly lo w er for the other. Regarding the timing of fast activit y , there is no statistically significan t difference ( P = 0:08 in Figure 2.12 ( c )) in the p opulation mean of the timing of fast activit y across con tacts within the resected region compared to those outside. There is a statistically significan t difference ( P = 2:1010 28 in Figure 2.12 ( d )) across epileptogenic-zone con tacts compared to non-epileptogenic- zone con tacts. Ho w ev er, similar to the result of the maxim um frequency of fast activit y , the v ariance of the measure is sufficien tly large that they cannot b e used to differen tiate b et w een these regions using data from a single sub ject. Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 49 2.5.2 Suppression Inside and Outside the Resection Area W e used the same pro cedure as with the ab o v e fast activit y study to also test for differences in the maxim um frequency of suppression. W e selected only those con tacts where suppression o ccurred. The presence of suppression w as determined b y thresholding the area ratio in the time- frequency plot b et w een the suppression region and the en tire region after seizure onset. The threshold ratio used to define suppression w as set to 3:1 %, whic h is the a v erage ratio for the con tacts inside the resection zone. Suppression w as iden tified in 16 of 17 patien ts b oth inside and outside the resection. In total (across all seizures and patien ts), suppression w as found in 467 con tacts (139 epileptogenic-zone con tacts + 328 non-epileptogenic-zone con tacts) inside the resection region and 474 con tacts outside the resection region. W e observ ed that highest frequency of suppression could extend in to the lo w gamma range (Figure 2.12 ( e )). W e found that there is no statistically significan t difference ( P = 0:105 in Figure 2.12 ( e )) in the p opulation mean of the maxim um frequency of suppression across con tacts within the resected region compared to those outside as w ell as across epileptogenic-zone con tacts compared to non- epileptogenic-zone con tacts (P = 0:205 in Figure 2.12 ( f )). 2.5.3 The Asso ciation Bet w een F ast A ctivit y and Suppression T o explore the relationship b et w een fast activit y and suppression, the maxim um frequency of fast activit y (in case where there are m ultiple bands, w e use the maxim um frequency of the lo w est band as it is used as the upp er b ound for suppression) (see Section 2.4.2.2 ) and the maxim um frequency of suppression w ere extracted as describ ed ab o v e on those con tacts where b oth fast activit y and suppression w ere presen t. W e then p erformed a regression to study the relationship b et w een the t w o maxim um frequencies for con tacts (i) classified as epileptogenic zone inside the resection; (ii) classified as non-epileptogenic zone inside the resection; and (iii) outside the resection region. 50 2.5. A dditional Studies (a) (b) (c) Figure 2.13: Scatter plot of maxim um frequency of fast activit y v ersus maxim um frequency of suppression for con tacts. (a) Classified as epileptogenic zone inside the resection; (b) classified as non-epileptogenic zone inside the resection and (c) outside the resection region. The presence of suppression w as determined b y thresholding 70 largest suppression areas for con tacts inside the resection region and outside separately , resulting in 40 con tacts in (a), 30 con tacts in (b) and 70 con tacts in (c). In con tacts where b oth fast activit y and suppression w ere found, w e observ ed stronger correla- tion (r = 0:642 in Figure 2.13 ( a )) b et w een the maxim um frequency of fast activit y and suppres- sion for epileptogenic-zone con tacts than the correlation (r = 0:552 in Figure 2.13 ( b )) for non- epileptogenic-zone con tacts inside the resection. Moreo v er, the correlation for b oth epileptogenic- zone and non-epileptogenic-zone con tacts, i.e., for con tacts inside the resection, are m uc h higher than that (r = 0:336 in Figure 2.13 ( c )) for con tacts outside the resection region. The presence of suppression w as determined b y thresholding the k largest suppression areas for con tacts inside the resection region and outside separately . Figure 2.13 sho ws resu lts for k = 70 . Based on b o otstrap confidence in terv als, w e observ ed significan t separation b et w een the correlation of con tacts inside the resection region and those outside for v alues of k in the range from 50 to 250 , indicating the robustness of this phenomenon. Moreo v er, the Dice co efficien t [ 63 ] b et w een the set of con tacts that con tained fast activit y and the set of con tacts that con tained suppression w as m uc h larger inside the resection region (0:54 ) than that outside the resection region (0:27 ) when the same threshold ratio 3:1 % is applied for determining the presence of suppression. This indicates that there is a stronger asso ciation b et w een fast activit y and suppression inside the resection region than outside. Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 51 These data suggest that suppression and fast activit y migh t represen t a com bined ph ysiological phenomenon inside the epileptogenic zone. Outside the epileptogenic zone, they are less correlated and ma y also app ear indep enden tly due to a p ossible disso ciated propagation of these t w o features. 2.6 Discussion 2.6.1 Defining the Fingerprin t of the Epileptogenic Zone The time-frequency fingerprin t of the epileptogenic zone at seizure onset is c haracterized b y the asso ciation of three elemen ts: pre-ictal spik es, fast activit y and suppression. A visual analysis of the time-frequency plots in a giv en seizure rev eals this c haracterization. This fingerprin t is reinforced b y the homogeneit y of this pattern across patien ts whatev er the aetiologies and areas in v olv ed. T o automatically iden tify the epileptogenic zone pattern, differen tiate it from the areas of prop- agation, and quan titativ ely c haracterize its features, w e dev elop ed a SVM-based mac hine learning system. A no v el pip eline (see Section 2.4.3 ) w as prop osed to c haracterize eac h feature in the pat- tern and classify eac h con tact as b eing inside the epileptogenic zone or not accurately . T o the b est of our kno wledge, this pip eline, for the first time, enables us to directly lo calize the epileptogenic zone, rather than merely detecting ictal/in ter-ictal spik es in SEEG recordings. Ov erall, the pro- p osed algorithm could successfully iden tify the epileptogenic zone in a p opulation of patien ts with fo cal epilepsy that w ere seizure-free after surgery , whic h in turn supp orts our h yp ot hesis that the time-frequency fingerprin t c haracterizes the epileptogenic zone. The efficacy of the detector in our selected patien t p opulation suggests that the three elemen ts are not indep enden t but define a pattern. The fast activit y comp onen t is structured: it is not broadband, but rather consists o f m ultiple narro w bands. Its duration is not limited to ictal onset but ma y extend un til the end of the seizure. F ast activit y is time-related to a wide band (delta to lo w gamma) suppression of lo w er frequency activit y , and the suppression cut-off frequency is correlated to the highest frequency of fast activit y . The onset of fast activit y/suppression is con tiguous from a sharp transien t (a short ev en t c haracterized b y a sim ultaneous b o ost of slo w and fast frequencies): it can b e a single spik e 52 2.6. Discussion or the last in a burst of rh ythmical spik es (“pre-ictal” spik es). This transien t marks the onset of fast activit y (F ast activit y is dev elop ed from the fast comp onen t of transien ts) and suppression (suppression starts b y in terruption of the slo w comp onen t of transien ts). The frequency con tiguit y b et w een the fast comp onen t of the sharp transien t and the fast activit y onset is apparen t in the transition from pre-ictal to ictal. None of the three individual fingerprin t elemen ts w e ha v e iden tified is sufficien t to c haracterize the epileptogenic zone, as eac h of these features can b e iden tified separately , with equal or reduced in tensit y , at v ariable distances from the epileptogenic zone (F ast activit y without suppression or suppression without fast activit y , and either with or without pre-ictal spik es). This signature lik ely indicates disso ciated propagation of the epileptogenic zone elemen ts. A ccordingly , a strong correlation b et w een maxim um frequency of fast activit y and maxim um frequency of suppression w as observ ed within the epileptogenic zone but not outside. On the other hand, the narro w banding of fast activit y in high-gamma frequencies sim ultaneous together with suppression in lo w er frequencies is indeed a p eculiar pattern. Although it is apparen t in most of the ictal time-frequency plots published so far, suppression (and its electrographic expression “flattening”) has to date b een disregarded as a k ey phenomenon of ictogenicit y . Often, visual analysis of SEEG can barely distinguish b et w een v ery fast activit y and suppression. The t w o distinct features can b e clearly visualized using the time-frequency plots only in recen t studies. In terestingly , suppression is not only limited to the lo w er delta frequency range but also sometimes reac hes theta, b eta and ev en the lo w gamma range. A t the end of suppression, there is a reb ound of activities in those frequencies that w ere previously suppressed. The fast activit y/suppression pattern is alw a ys preceded b y a sharp transien t. The transien t w a v eform can b e a sharp high-amplitude w a v e or a series of spik es with v arying amplitude (often with increasing amplitude b efore fast activit y onset). This initial short duration elemen t is quite distinct from the slo w p olarizing p oten tial underlying dev elopmen t of fast activit y [ 80 ] but p ossibly could initiate it. Ov ert pre-ictal spik es w ere mainly describ ed in asso ciation with epileptogenic lesions suc h as F CD t yp e 2 and hipp o campal sclerosis [ 144 , 171 ], but their imp ortance has b een underestimated as a general feature of fo cal epilepsies. Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 53 Epileptogenicit y indices dev elop ed so far ha v e tak en a differen t approac h, emphasizing analysis of fast activit y , for example Bartolomei et al. ’s [ 20 ] energy ratio, dividing fast (b eta + gamma) b y slo w (theta + alpha) frequency energy . Our time-frequency analysis sho ws that b eta and lo w gamma are frequen tly suppressed, while p o w er in fast activit y is highly v ariable. The energy ratio mixes these t w o phenomena and do es not accoun t for our observ ation that frequency range and duration can v ary b oth across sub jects and b et w een the epileptogenic zones and non-epileptogenic zones. Da vid et al. [ 56 ] lo calized fast activit y and its early spread represen ted in the patien t’s MRI. This can b e effectiv e in some cases when the epileptogenic zone is v ery fo cal. Ho w ev er, w e ha v e demonstrated that there ma y b e no significan t difference b et w een fast activit y inside and outside the epileptogenic zone. Inadequacy of fast activit y as an absolute mark er has b een do cumen ted in a n um b er of clinical studies [ 29 , 173 ]. Gnatk o vsky et al. [ 79 , 80 ] measured the p o w er of fast activit y and found correlations with a slo w p olarizing shift to lo calize the epileptogenic zone. Similarly , a slo w or DC comp onen t has b een sho wn to co-lo calize with fast activit y [ 100 , 181 ]. 2.6.2 P athoph ysiological Significance of the Time-frequency Fingerprin t The epileptogenic zone pattern can b e in terpreted in the ligh t of previous w ork on pre-ictal/ictal transition in seizures c haracterized b y fast activit y at the onset. Correlation or functional coupling studies ha v e sho wn that sync hronization b et w een neuronal p opulations lo cated in distinct areas o ccurs b efore fast activit y onset, during the pre-ictal spiking p erio d [ 19 , 202 ], then de-correlation o ccurs throughout the en tire fast activit y p erio d and sync hronization increases later b efore seizure termination. Pre-ictal sync hronicit y is maximal as the slo w comp onen t of the sharp transien t in- creases in size (duration and frequency range) then disapp ears, whereas its fast comp onen t prolongs without in terruption. During the pre-ictal p erio d, recen t micro-electro de studies in h uman epilep- sies ha v e consisten tly rep orted a progressiv e increase in a rate of fast-disc harging in ter-neuronal activit y while slo w p yramidal activit y is still presen t [ 57 , 136 , 155 , 187 , 200 ]. Then, during fast activ- it y , acceleration of in ter-neuronal activit y sim ultaneous with slo wing or arrest of p yramidal activit y is observ ed with no evidence of sync hronization at the neuronal lev el [ 187 ]. 54 2.6. Discussion The narro w banding of fast activit y is lik ely to reflect a lo cal high frequency oscillatory activit y in a homogeneous neuronal p opulation [ 152 , 158 ], and as suc h migh t corresp ond to increased activit y of fast inhibitory in ter-neurons mo dulating mem brane p oten tial of p yramidal cells [ 176 ]. The arrest of slo w paced p yramidal disc harge as a consequence of fast inhibitory in ter-neuronal activit y can also explain the suppression of lo w er frequencies during the fast activit y p erio d [ 155 , 187 ], as w ell as reco v ering of lo w er frequencies as fast activit y ends. The increase in fast inhibitory in ter-neuronal activit y can b e caused b y the decreased inhibition from slo w inhibitory neurons. Sev eral studies ha v e sho wn an effect of slo w inhibitory (dendritic) activit y on fast inhibitory (p eri-somatic) in ter-neurons [ 16 , 145 ]. A deficien t slo w inhibition disin- hibits fast in ter-neurons in exp erimen tal hipp o campal epilepsy [ 52 , 107 ]. In the neo cortex, enhanced GABA ergic inhibition w as iden tified in F CD t yp e 2 [ 45 ]. A mo del of fast activit y generation based on suc h selectiv e deficiency of slo w inhibitory pro cesses [ 175 ] has b een elab orated b y W endling et al. [ 203 , 204 ] and Bartolomei et al. [ 19 ]. Giv en a sufficien t lev el of global excitabilit y , the transition from pre-ictal sync hronization to desync hronized fast activit y can result from a brisk decrease of slo w inhibition, coinciding with a concomitan t increase or steadiness of fast inhibition. Comparison b et w een sim ulated and real SEEG activities clearly sho ws that this condition can pro duce fast activit y at seizure onset [ 203 ]. F urther correlations with neurobiology can b e put forw ard based on recen t pap ers. There is strong evidence that the parv albumin-con taining, fast-spiking bask et cells are critical for gamma rh ythmogenesis b oth in vitro and in viv o. There is also strong evidence that cortical fast-spiking in ter-neurons are resonators, meaning that they ha v e an abrupt onset of firing at a threshold frequency . Those features as demonstrated b y Tikidji-Ham bury an et al. [ 182 ] allo w more robust sync hronization in the presence of noise and heterogeneit y . So, the spreading gamma activities inside and outside the epileptogenic zone w ould corresp ond to mec hanisms differen t from a sto c hastic p opulation oscillator. These resonator mec hanisms (if op erational in long-range connections) [ 186 ] w ould lead to the notion of sustained fast activit y pro jecting in to a parallel propa gation net w ork th us critically c hallenging a classical serial view of fo cal seizure propagation. Chapter 2. A Fingerprin t of the Epileptogenic Zone in Human Epilepsies 55 In our study , w e observ ed a m ultiband fast activit y pattern. Ho w could this pattern b e ex- plained? If neurons fire out of phase, their summed activit y can b e at a higher frequency than the disc harge of an y one neuron. So higher gamma activities are compatible with desync hronization b et w een sync hronized p opulations at the neuronal lev el [ 110 ]. Another explanation is giv en from the same authors: with the inhomogeneous spread, phase differences b et w een groups of neurons, where neurons fire sync hronously within eac h group, can result in frequency m ultiplication at the lev el of the field p oten tial. This could also b e the explanation for the existence of fast activit y at similar frequencies outside the epileptogenic zone: groups of neurons firing sync hronous bursts out of phase with other groups due to differences in conduction dela ys as activit y spreads remotely from the epileptogenic zone to m ultiple spread rela y areas in the seizure net w ork [ 147 ]. Another p ossibilit y w ould b e the co existence of t w o indep enden t generators of gamma activit y . Keeley et al. [ 106 ] prop osed a mo del with m ultiple gamma oscillators in a single net w ork made of t w o inhibitory comp eting subp opulations. The curren t results are consisten t with a new h yp othesis dev elop ed b y de Curtis et al. [ 57 ] on fo cal seizure onset mec hanisms. In seizures c haracterized b y lo w v oltage fast activit y similar to those studied in our w ork, they ha v e gathered exp erimen tal evidence from animal exp erimen ts [ 81 , 190 ] and pre-surgical micro-electro de recordings in fa v our of a piv otal inhibitory mec hanism (sync hronous activ ation of GABA-A receptors). They prop ose that the same mec hanism leads to an extracellular p otassium increase whic h w ould facilitate seizure propagation. The imp ortance of pre- ictal spik es [ 77 , 117 , 170 , 171 , 196 ] and their underlying sync hronizing mec hanism is also emphasized in the deCurtis-A v oli mo del. Their o ccurrence had b een underestimated, p ossibly due to depth electro de sampling bias. Therefore, the sequence of sim ultaneous ev en ts comp osing the time-frequency pattern that w e ha v e iden tified could b e understo o d as follo ws. The pre-ictal spik e(s) w ould corresp ond to a progressiv e sync hronization of p yramidal cells (slo w comp onen t) activ ating disinhibited fast somatic inhibitory in ter-neurons (fast comp onen t). Successiv e bursts of fast in ter-neuron activities w ould then merge in to a sustained disc harge (m ulti-band fast activit y) leading to p yramidal silencing 56 2.6. Discussion (suppression). The last part of the seizure (end of suppression coinciding with fast activit y decrease) could b e due to lo cal and remote p ost-inhibitory reb ound bursting p yramidal neurons activit y [ 161 ]. The time-frequency pattern evidenced in our study reflects a summation at the lo cal field p oten tial lev el of the complex in tricate neuronal mec hanisms op erating at seizure onset. This is presumably wh y its elemen ts are in ter-related. The unique com bination of the three features (fast activit y , suppression, and spiking) can b e view ed as a fingerprin t of the epileptogenic zone. 2.6.3 Limitations of the Study One limitation of the presen t study aris es from the appro ximation in estimating the real epilep- togenic zone b oundaries. W e ha v e opted for the con tours of the cortical resection in seizure-free patien ts to delineate the area within whic h the epileptogenic zone lies. Ho w ev er, w e are a w are that for large resections, the epileptogenic zone is lik ely to b e more restricted in size, and that only for v ery limited surgeries, suc h as laser ablation, will the difference b et w een the epileptogenic zone and resected/ablated b oundaries b e negligible. Resection v olume is an op en question in epilepsy surgery where no standard based on morphological lesion is scien tifically defendable. Hence the need for a biomark er is crucial. Spatial sampling is an in trinsic restriction of an y in tracranial study . This limitation can also accoun t for the v ariable SNR of the fingerprin t pattern in our patien t group. A non-optimal p osition of the electro des can w ell explain those v ariations. The algorithm failed to iden tify the epileptogenic zone in t w o cases, and in fiv e there w ere misiden tified con tacts. Source reconstruction metho ds suc h as minim um-norm estimation [ 12 ] could b e applied to the SEEG data to p oten tially impro v e lo calization when the con tacts do not lie directly within the epileptogenic zone. Could this fingerprin t b e considered a generic trait of fo cal epilepsies? Its main features and putativ e pathoph ysiology are tigh tly related to the existence of fast activit y at seizure onset. This fast activit y w as a patien t selection criterion of the curren t study . Ev en though this is the most common electroph ysiological signature [ 165 ], other slo w er rh ythmical activit y patterns made of spik es or spik e-and-w a v es ha v e also b een observ ed. Chapter 3 Learning to Define An Electrical Biomark er of the Epileptogenic Zone , , — 3.1 In tro duction New vistas in the pathoph ysiology of fo cal epilepsies and attempts to impro v e the lo calization of the epileptogenic zone in epilepsy surgery con v erge on a common question: what is the role of high frequency cortical activit y in seizure initiation? After the iden tification of in terictal fast ripples as a p oten tial mark er for epileptogenicit y , sev eral clinical studies ha v e in v estigated their p ossible co-lo calization with the seizure onset using statistical approac hes. Ho w ev er, a consisten tly iden tifiable transition from in ter-ictal high frequency activit y and/or spik es to seizure dev elopmen t is far from eviden t in practice [ 85 ]. Therefore, a clearer definition of the “seizure onset” is of high imp ortance as it is b oth a prerequisite to v alidate in ter-ictal features and a rational basis to delineate the epileptogenic zone. Ev en though v arious mo des of onset can b e observ ed [ 118 , 165 ], a feature generally ac kno wledged b y all authors is the sharp or progressiv e frequency pattern c hange to a higher frequency range in some areas, often called the “seizure onset zone” . 57 58 3.1. In tro duction Defining a bio-electrical mark er for the epileptogenic zone is b oth useful for epilepsy surgery and meaningful for understanding the pathoph ysiology of epilepsy . An accurate mark er w ould help to fill the gap b et w een “in tra-cranial EEG” phenomenological classifications and neuroph ysiologi- cal/mo deling studies. Sev eral attempts to measure the epileptogenic zone exten t based on SEEG signal pro cessing ha v e b een made o v er the past decade. Bartolomei et al. [ 20 ] prop osed to quan tify the relativ e onset times of the fast activit y recorded in differen t areas. Detecting sp ectral c hanges during the pre-ictal/ictal transition, the signal energy ratio b et w een high and lo w frequency bands w as calcu- lated and its detection time plotted in eac h recording c hannel. Application to mesial temp oral lob e epilepsies as opp osed to “lateral” temp oral lob e epilepsies differen tiated epileptogenicit y among the t w o lo calizations. Da vid et al. [ 56 ] underto ok a neuroimaging study to represen t an epilepto- genicit y index based on quan tification of fast activities using the Statistical P arametric Mapping. This study allo w ed mapping the lo cation of the fastest activities and their “slo w propagation” dur- ing p eri-onset time, and comparison with the surgically resected area. This w ork added to the v alidation of high frequency activities as a mark er of seizure onset zone. A differen t approac h of “frequency lo calization” w as used b y Gnatk o vsky et al. [ 80 ], considering that frequency c hanges in differen t bands ma y o ccur successiv ely or sim ultaneously during seizure onset. Therefore, the epileptogenic zone, defined as the area of frequency c hanges at seizure onset, could b e delineated whatev er the p eculiarit y of frequency patterns recorded in differen t seizures in a giv en patien t. In a prosp ectiv e study of patien ts in v estigated with SEEG, the same metho d w as applied to test three biomark ers of the epileptogenic zone, namely fast activit y , signal flattening, and slo w p oten tial shift. These biomark ers co-lo calized with the lo cation of the epileptogenic zone as defined b y stan- dard neuroph ysiological means and p ost-surgical seizure outcome [ 79 ]. Ho w ev er, the three mark ers w ere analyzed as discrete phenomena, although the fast activit y and the slo w p oten tial shift are probably related, and the signal flattening w as pro v ed not to b e discriminativ e. A “fingerprin t” of the epileptogenic zone w as iden tified in Chapter 2 via a time-frequency approac h in a series of patien ts who w ere ev aluated with SEEG and seizure-free after surgery [ 88 ]. A time-frequency pattern of the in ter-ictal to ictal transition w as common across all patien ts inside the Chapter 3. Learning to Define An Biomark er of the Epileptogenic Zone 59 resected areas. Although this pattern consisted of narro w-band fast activities, only the com bination of fast activities with pre-ictal spik e(s) and sim ultaneous suppression of lo w er frequencies w as the discriminating factor. Based on the epileptogenic zone fingerprin t (EZF) pattern, a SVM-based classification algorithm w as dev elop ed to automatically iden tify features of the fingerprin t pattern and distinguish epileptogenic-zone con tacts from non-epileptogenic-zone con tacts. Since the EZF w as demonstrated in seizure-free patien ts with ictal fast activities in the gamma band, some questions still remain unansw ered: Is the EZF a common feature o f all fo cal seizures, ev en for seizure onset patterns with oscillatory activit y in lo w er frequency range? W ould the EZF b e recorded outside the resected region in non-seizure-free patien ts, meaning that surgical failures can b e explained b y the fact that EZF-p ositiv e areas w ere not resected? Previous studies had treated fast activities as a single category whereas Grinenk o et al. [ 88 ] sho w ed a distinction b et w een broad- band and narro w-band gamma activities. So, a comparison b et w een the accuracy of narro w-band fast activit y iden tified b y our metho d and the EZF in lo calizing the epileptogenic zone is needed. In this c hapter, w e ev aluate p erformance of the original EZF metho d on a broader p opulation of patien ts, including b oth seizure-free and non-seizure-free patien ts. W e sho w that the EZF can b e iden tified in patien ts with oscillatory activities as lo w as b eta or alpha frequency . Classification results sho w substan tial difference b et w een the seizure-free and non-seizure-free group of patien ts, where a large n um b er of epileptogenic-zone con tacts are predicted outside the resection in the non- seizure-free group. W e further extend the EZF metho d to a fully automated end-to-end classification pip eline b y in terp olating the prediction scores on to individual patien t’s MR images, whic h allo ws us to predict and visualize the exten t of the epileptogenic zone with resp ect to eac h individual anatom y . By applying this extended EZF pip eline, w e demonstrated that fingerprin t-based epileptogenic zone anatomical estimation presen ted as w ell-circumscrib ed areas globally lo cated inside the resected region in seizure-free patien ts. In con trast, of a large fraction of the non-seizure-free patien ts, the estimated epileptogenic zone w as not w ell-lo calized, and lo cated partially or completely outside the resection, whic h ma y explain the surgical failure. F urthermore, if using only the fast activit y , follo wing the same extended EZF pip eline, and mapping only the fast activit y on to the MRI, w e sho w ed that the epileptogenic zone could b e o v er-estimated. 60 3.2. Metho ds 3.2 Metho ds 3.2.1 P atien t Selection and Data Collection W orking under a proto col appro v ed b y the IRB, w e included a consecutiv e series of patien ts in y ear 2015 who had seizures that b egan with L VF A in the b eta or gamma bands and w ere op erated after SEEG ev aluation. The details of the patien t selection proto col are presen ted in App endix A.2 Figure A.2 . SEEG is an in v asiv e pre-surgical pro cedure for patien ts with pharmaco-resistan t fo cal epilepsy . Anatomo-electro-clinical h yp otheses w ere form ulated individually for eac h patien t during a m ulti- disciplinary patien t managemen t conference based on a v ailable nonin v asiv e data: clinical history , video EEG, MRI, PET, ictal SPECT and MEG. Multi-lead depth electro des (A dT ec h, Racine, Wisconsin; In tegra, Plainsb oro, New Jersey; or PMT, Chanhassen, Minnesota) w ere implan ted according to the T alairac h stereotactic metho d. SEEG signals w ere re corded on a Nihon K ohden (Irvine, California) EEG mac hine with a sampling rate of 1000 Hz. Anatomical lo cations of the electro de leads w ere iden tified b y a digital fusion of the p ost- implan tation thin-sliced CT image with the pre-op erativ e T1-w eigh ted MR image using CURR Y 7 (Compumedics NeuroScan, Ham burg, German y) and visualized on the pre-op erativ e MR image using Brainstorm [ 178 ]. A p ost-op erativ e MR image w as also acquired after the surgery (except for Sub ject 231 where a p ost-op erativ e CT image w as acquired) and rigidly co-registered to the p ost-implan tation CT image to iden tify the p ositions of the electro de con tacts with resp ect to the lo cation of the resected/ablated region. 3.2.2 Epileptogenic Zone Fingerprin t Pip eline for Individualized Epileptogenic Zone Prediction In Chapter 2 , w e describ ed an EZF metho d that distinguished electro de con tacts within the epileptogenic zone from non-epileptogenic-zone con tacts 1 . W e approac hed the problem b y first 1 An op en-source soft w are w as dev elop ed and released for researc h purp ose only , publicly a v ailable at: https: //silencer1127.github.io/software/EZ_Fingerprint/ezf_main Chapter 3. Learning to Define An Biomark er of the Epileptogenic Zone 61 transforming the ra w SEEG time-series in to time-frequency represen tations using the Morlet w a v elet transform, follo w ed b y an artifact remo v al pro cedure using cICA. W e then normalized the onset time-frequency maps against the baseline time-frequency maps for eac h frequency . The normalized time-frequency maps w ere then used to extract three distinct features: fast activit y , suppression, and pre-ictal spik es. Finally , an SVM classifier w as trained on the com bination of the three features and a sub ject-based cross-v alidation w as p erformed on a set of 17 patien ts who had seizures with sustained gamma activities and w ere seizure-free after the surgery . In this c hapter, w e v alidate the effectiv eness of the EZF on a completely indep enden t consecutiv e series of patien ts that underw en t SEEG ev aluation in 2015 , including b oth seizure-free and non- seizure-free groups of patien ts (see Section 3.2.1 for details). The en tire pro cessing pip eline is sho wn in Figure 3.1 . First, an SVM-based mo del w as trained using the patien ts’ data from our previous study [ 88 ]. Then w e pro cessed all a v ailable seizures for patien ts in the curren t study using exactly the same pip eline and applied the trained mo del to b oth the seizure-free and non-seizure-free group of patien ts to predict epileptogenic-zone con tacts. Also, as b efore, w e man ually lab eled con tacts as b eing within or outside the resected area based on the patien ts’ co-regi stered p ost-op erativ e MR image. Finally , w e compared the predicted lab els with the resection lab els. In tra-sub ject v ariation of time series c haracteristics of ictal patterns, as exemplified in Fig- ure 3.2 (a), can significan tly affect prediction results and lead to underestimation of the epilep- togenic zone. Suc h a v ariation do es not affect SEEG visual analysis as the exp ert classifies them as the same seizure t yp e but with subtle w a v eform alterations or differen t latencies in the prop- agation net w ork. Ho w ev er, the SVM classifier is sensitiv e to suc h formal v ariation. T o tak e this v ariation in to consideration, a clustering of seizures w as p erformed for eac h sub ject as follo ws: for eac h seizure, w e first applied the EZF pip eline to the ictal time-series Figure 3.2 (a) to obtain a prediction score (hereafter “score”) (Figure 3.2 (b)) for eac h con tact from the trained SVM mo del. In con trast to binary prediction results (a con tact is predicted to b e either an epileptogenic-zone con tact or non-epileptogenic-zone con tact) from the v oting algorithm sho wn in Section 2.4.3.5 , the “ra w” scores offer a con tin uous predicted v alue for eac h con tact, roughly reflecting ho w lik ely eac h con tact is to b e in the epileptogenic zone. A p ositiv e score indicates a higher probabilit y of the 62 3.2. Metho ds Figure 3.1: Epileptogenic zone fingerprin t pip eline. (a) F eature extractions for fast activit y , suppression and pre-ictal spik es from time-frequency maps; (b) Classification pro cedures where an SVM mo del w as trained using the original 17 patien ts in the previous study and the epileptogenic zone w as predicted for the 24 patien ts in the curren t study; (c) In terp olation of prediction scores on to patien ts’ individual MR images. Chapter 3. Learning to Define An Biomark er of the Epileptogenic Zone 63 Figure 3.2: An example of the seizure clustering pro cedure for Sub ject 112. (a) Ictal time-series illustrating v ariations in ictal patterns: Isolated pre-ictal spiking only in c hannel T’ in seizure 1 and con tin uous pre-ictal spiking sync hronous b et w een c hannel T’ and R’ in seizure 2 and 5; (b) Prediction scores obtained individually for eac h seizure; (c) Cross-correlations of the prediction scores across the electro de arra y b et w een all pairs of seizures (fiv e seizures in total for this sub ject); (d) Man ual clustering of seizures according to cross-correlation matrix. 64 3.2. Metho ds con tact b eing in the epileptogenic zone and a negativ e score indicates a higher probabilit y that the con tact is not in the epileptogenic zone, while zero score sho ws a maximal uncertain t y ab out the prediction. F or a giv en sub ject, w e computed the cross-correlations of the prediction scores across the electro de arra y b et w een all pairs of seizures. Figure 3.2 (c) sho ws an example of the cross-correlation matrix among all 5 seizures for Sub ject 112 . W e man ually group the seizures in to clusters based on the cross-correlation matrix. F or most patien ts, determining ho w man y clusters and ho w w e should group the seizures is v ery straigh tforw ard. F or example, for Sub ject 112 where the cross-correlation matrix is sho wn in Figure 3.2 (c), fiv e seizures naturally fell in to t w o clusters: seizure 1 and 3 b elonged to one group and seizure 2 , 4 and 5 b elonged to another group (Fig- ure 3.2 (d)). The clustering pro cedure w as p erformed for all of our patien ts. W e then ev aluated the prediction p erformance b y iden tifying the epileptogenic zone con tacts using the EZF separately for eac h cluster and then taking the union of the prediction results, i.e., a con tact is finally predicted to b e in the epileptogenic zone if it is iden tified as an epileptogenic-zone con tact in an y of the clusters of seizures. See Figure 3.1 (b). T o address the question of whether fast activit y alone can differen tiate epileptogenic-zone con- tacts from non-epileptogenic-zone con tacts, as is commonly practiced in SEEG, w e further compared the p erformance using the fingerprin t vs fast activit y alone as a biomark er of the epileptogenic zone. W e predicted the epileptogenic zone (follo wing the exact same training and testing pro cedure as ab o v e) using features from fast activit y only (top-righ t “fast activit y” sub-blo c k in Figure 3.1 (a) and compared the prediction lab els with the resection lab els. Note that the features w e used for fast activit y are based on our o wn c haracterization of the narro w-band fast activit y as describ ed in Section 2.3 rather than the energy of en tire gamma band. In total there are 17 n umerical descriptors extracted for fast activit y , suc h as n um b er of narro w bands, the maxim um/minim um frequency , the starting/ending time, the orien tation of the fast activit y bands. See Section 2.4.2.1 for details. W e dev elop ed an extension of the EZF metho d that incorp orates anatomical information in to the prediction b y mapping the discrete prediction lab els on to the patien ts’ MR image, hence offering a complete automated end-to-end classification pip eline that starts from the prepro cessing of the ra w SEEG signals and yields an individualized prediction and visualization of the epileptogenic Chapter 3. Learning to Define An Biomark er of the Epileptogenic Zone 65 zone in the end. W e ac hiev ed suc h a goal b y first co-registering the p ost-implan tation CT image to the patien t’s pre-op erativ e MR image, so that the electro de con tacts w ere aligned to the MRI space; Then, w e extracted a mask of the brain (cerebrum and cereb ellum only) from the MR image using the skull stripping to ol pro vided in BrainSuite [ 162 ]; T o minimize computational complexit y w e set the v alue of the brain’s b oundary v o xels to the minimal predicted scores across all con tacts and the v alue of the nearest v o xel to eac h con tact lo cation to b e the corresp onding predicted v alue for that con tact. Finally , w e in terp olated the v alues for a ll other v o xels inside the brain mask using linear in terp olation b et w een the con tact lo cations and brain b oundary v alues. In terp olation w as based on Delauna y triangulation [ 5 ] pro vided b y the “scatteredIn terp olan t” function in MA TLAB (the Math w orks, Inc., Natic k, MA, USA). See Figure 3.1 (c). F ollo wing the same in terp olation pro cedure, w e also linearly mapp ed the maxim um frequency (one of the 17 n umerical descriptors in feature extraction step) of the fast activit y on to the patien ts’ MRI space for visualization and comparison purp ose. T o statistically test the difference of the predicted epileptogenic zone (in the MRI space) b et w een the seizure-free group and the non-seizure-free group, w e calculated the kurtosis of the distribution of in terp olated predicted epileptogenic zone scores o v er the en tire mask ed b rain for eac h sub ject. “Kurtosis” is a measure of the “tailedness” of probabilit y distributions [ 205 ] and sensitiv e to outliers. The higher the kurtosis, the more lo calized the predicted epileptogenic zone is and the higher prediction score within the predicted epileptogenic zone. W e also compared the anatomical prediction results with surgical outcomes (Engel score). F or eac h patien t, w e first predicted the epileptogenic zone using the extended EZF pip eline. Then w e man ually examined the relationship b et w een the predicted epileptogenic zone and the resected area (hereafter the “relationship”), determining eit her that the predicted epileptogenic zone w as fully resected, partially resected, or not resected. 3.3 Results In total, 24 sub je cts met the patien t selection criteria (Figure A.2 ), 11 of them b ecame seizure- free and the remaining 13 had seizure recurrence after surgery , i.e., non-seizure-free. T able 3.1 sho ws 66 3.3. Results Figure 3.3: Statistics of the maxim um frequency and minim um frequency of fast activit y . Ex- amples of iden tified fingerprin t pattern with gamma activit y and b eta activit y are sho wn on the righ t. the clinical profiles of the sub jects included in this study . W e analyzed all a v ailable seizures, except in cases where patien ts had more than 10 consecutiv e seizures, then only the first 10 seizures w ere analyzed. Nine sub jects had additional seizures without initial fast activit y that w ere not included in t he study (additional information for these seizures is sho wn in T able E.1 in App endix E ). All 24 selected sub jects had seizure onset from L VF A, Figure 3.3 sho ws the statistics of the maxim um and minim um frequency of fast activit y . The median of the maxim um frequency of fast activit y w as 74 Hz with an IQR 25 Hz, and the median of the minim um frequency of fast activit y w as 45 Hz with an IQR 22 Hz. Out of 11 seizure-free patien ts, the full fingerprin t pattern w as observ ed from the time-frequency plots in 9 patien ts and partially observ ed, i.e., only one or t w o out of the three comp onen ts (fast activit y , suppression, pre-ictal spik es) w ere observ ed, in the other 2 patien ts. On the other hand, out of the 13 non-seizure-free patien ts, the full fingerprin t pattern w as observ ed in 10 patien ts, partially observ ed in 1 patien t and missed in the other 2 patien ts. T able 3.2 ( a ) sho ws classification results using the EZF metho d on the consecutiv e series of patien ts from 2015 . These results are based on analysis in whic h seizures are clustered for eac h Chapter 3. Learning to Define An Biomark er of the Epileptogenic Zone 67 T able 3.1 : Clinical profiles of the patien ts Sub ject ID * Age (y ears) Epilepsy duration (y ears) MRI lesion Surgical pathology Resection/Ablation details Outcome F ollo w-up duration (mon ths) 101 25 11 Normal F o ca l gliosis L. lateral temp oral cor- texectom y SF 27 102 17 7 Normal F CD t yp e 1 L. temp oral p olar and am ygdala resection SF 36 103 30 12 Normal F CD t yp e 1 R An terior temp oral lob ectom y SF 36 106 17 9 Normal F CD t yp e 2B R SMA/cingulate resec- tion SF 28 108 37 32 Susp ected F CD F CD t yp e 2B R Sub cen tral resection SF 20 111 48 6 Normal F CD T yp e 1 L an terior temp oral lob ectom y SF 28 112 21 18 Normal No due to laser surgery L insu- lar/temp oral/fron tal op erculum laser ablation SF 31 113 24 17 Susp ected F CD F CD t yp e 1 R an terior temp oral lob ectom y SF 29 116 11 7 Prior resec- tion, other- wise normal Gliosis R Insular/ fron to- parietal and temp oral op erculum SF 22 118 33 13 Normal Gliosis R pre-fron tal resection SF 19 140 39 3 Normal F o cal p eriv as- cular gliosis An terior temp oral lob ec- tom y SF 21 215 † 35 4 PNH No due to laser surgery Laser ablation, p eriv en- tricular no dule Seizures 219 ‡ 30 5 Normal No due to laser surgery Laser ablation, L cingu- late/SMA 1-y ear SF then seizure recurred 220 38 22 Normal F CD t yp e 1 R P osterior basal temp o- ral resection Seizures 221 41 39 Normal Gliosis R lateral temp oro- parietal resection Seizures 222 24 14 Normal F CD t yp e 1 R Basal p osterior tem- p oral resection 1-y ear SF then seizure recurred 223 6 2 Normal Inflammation, F CD t yp e1 L An terior lateral tem- p oral resection Seizures 226 24 18 Normal F CD t yp e 1 L pre-fron tal resection Seizures 228 † 25 12 Multiple ar- eas of gliosis Gliosis R p arieto-o ccipital resec- tion Seizures 231 ‡ 34 34 Normal No due to laser surgery Laser ablation, L fron tal op erculum Seizures 232 † 10 10 Bilateral o ccipital lesion Ulegyria, in- flammation L P arieto-o ccipital resec- tion Seizures 233 29 10 Heterotopic grey matter F CD t yp e 1 R T emp oro o ccipital re- section Seizures 237 20 16 Normal No due to laser surgery Laser ablation R angular gyrus Seizures 238 35 35 PMG No due to laser surgery Laser ablation L fron to- parietal op erculum, sub- cen tral gyrus Seizures * Sub jects f rom 101 to 140 w ere seizure-free after surgery and sub jects from 215 to 238 w ere non-seizure-free. † P atien ts had seizures initi ated from differen t area other than fast activit y , whic h influenced the surgery planning. ‡ Sparse implan tation with inadequate sampling of the epileptogenic zone. L - left; R - righ t; SMA - supplemen tal motor area; F CD - fo cal cortical dysplasia; PNH - p eriv en tricular no dular heterotopia; PMG - p olymicrogyria; SF - seizure-free 68 3.3. Results T a ble 3.2: Epileptogenic zone fingerprin t prediction results and comparison with prediction results using fast activit y only (a) Prediction results using epileptogenic zone fingerprin t Seizure-free Non-seizure-free Prediction T rue Prediction F alse Statistics Prediction T rue Prediction F alse Statistics Inside Resection 42 (TP * ) 267 (FN * ) 38 (TP * ) 104 (FN * ) Outside Resection 5 (FP * ) 838 (TN * ) 0.006 (FPR) 104 (FP * ) 1276 (TN * ) 0.075 (FPR) Statistics 0.894 (PPV) 0.268 (PPV) (b) Prediction results using features from fast activit y only Seizure-free Non-seizure-free Prediction T rue Prediction F alse Statistics Prediction T rue Prediction F alse Statistics Inside Resection 181 (TP * ) 128 (FN * ) 147 (TP * ) 94 (FN * ) Outside Resection 281 (FP * ) 562 (T N * ) 0.333 (FPR) 693 (FP * ) 687 (TN * ) 0.502 (FPR) Statistics 0.392 (PPV) 0.175 (PPV) * TP/FP/TN/FN are with resp ect to the resected region rather than the actual epileptogenic zone. sub ject to determine the epileptogenic zone as describ ed ab o v e. F or the seizure-free group, w e w ere able to ac hiev e a similar p erformance to that in our original rep ort (T able 2.2 ). Our EZF iden tified the epileptogenic zone in 8 out of 11 seizure-free patien ts. In total, only 5 electro de con tacts from2 sub jects (Sub ject 102 and 113) w ere iden tified as epileptogenic-zone con tacts outside the resected area (in the other 6 sub jects, the epileptogenic zone w as predicted strictly inside the resection), yielding 89:4 % PPV and0:6 % FPR. F or the 3 remaining sub jects where no epileptogenic- zone con tact w as predicted, the epileptogenic zone fingerprin t w as indeed observ ed for one sub ject (Sub ject 103 in Figure F.1 ( b ) in App endix F ). This is b ecause the mac hine-learning-based EZF pip eline w as tailored closely to the original dataset and failed to extract one of the three features Chapter 3. Learning to Define An Biomark er of the Epileptogenic Zone 69 for this sub ject in the new dataset whic h resulted in a missed prediction of the epileptogenic zone. The fingerprin t w as only partially observ ed in the other t w o sub jects (Sub ject 101 and 140 in Figure F.1 ). W e h yp othesize the reason for this is that w e did not ha v e electro de con tacts inside (or close to) the epileptogenic zone. Ho w ev er, as a large surgical resection w as p erformed including the epileptogenic zone, hence these t w o patien ts w ere still seizure-free after the surgery . In con trast, for the non-seizure-free group, 104 out of 142 electro de con tacts w ere iden tified outside the resected area, resulting in a v ery lo w PPV of 26:8 %. The FPR for the non-seizure-free group w as 7:5 %, almost 16 times bigger than that for the seizure-free group (0:47 %). Prediction results for eac h individual patien t are sho wn in T able G.1 . Results without considering seizure clusters separately are sho wn in T able H.1 in App endix H where few er epileptogenic-zone con tacts w ere iden tified inside the resection, although similar PPV and FPR w ere obtained. The (true/false) p ositiv e or negativ e electro de con tacts (TP/FP/TN/FN) w ere lab eled with re- sp ect to the surgically resected area, rather than a clinically estimated epileptogenic zone. There- fore, the large n um b er of FN in b oth groups of patien ts w as exp ected as the resected region is t ypically larger than the actual epileptogenic zone. Ov erall, w e observ ed a large difference in epileptogenic zone prediction results with resp ect to the resected regions b et w een the seizure-free group and the non-seizure-free group, although the same metho d w as applied to b oth groups. Con tacts iden tified b y our EZF w ere mostly lo calized within the resection in the seizure-free group, but a large fraction w ere outside the resection in the non-seizure-free group. As w e recruited patien ts with broader selection criteria, w e sho w ed that the fingerprin t pattern could also b e iden tified in seizures initiated with frequencies as lo w as b eta or ev en alpha band (see statistics in Figure 3.3 ). T able 3.2 ( b ) sho ws the prediction results when only fast activit y features w ere used. A large n u m b e r of con tacts w ere predicted to b e “epileptogenic zone” (when the “epileptogenic zone” is defined based on the fast activit y) but with a lo w PPV of 39:2 %. F or the seizure-free group, 181 “epileptogenic zone” con tacts w ere lo calized inside the resection and 281 w ere outside the resection, suggesting that fast activit y b y itself cannot b e a determinativ e iden tifier of the epileptogenic zone. 70 3.3. Results Figure 3.4: T w o exemplar cases illustrating the epileptogenic zone fingerprin t prediction (b ottom- left) in terp olated on to individual patien ts MRI in comparison with fast activit y (b ottom-righ t) and p ost-op erativ e MRI (b ottom-middle) with corresp onding time-series (top-left) and time-frequency plot (top-righ t) for electro des of in terest. (a) Sub ject 106 from the seizure-free group; (b) Sub ject 220 from the non-seizure-free group. Lo cations of the electro de con tacts are illustrated in p ost- op erativ e MRI. Chapter 3. Learning to Define An Biomark er of the Epileptogenic Zone 71 Fingerprin t-based epileptogenic zone prediction results in terp olated on to the patien ts’ pre- op erativ e MRI are sho wn in Figure 3.4 for Sub ject 106 from the seizure-free group, illustrating a complete resection of the predicted epileptogenic zone and Sub ject 220 from the non-seizure-free group, illustrating an incomplete resection of the predicted epileptogenic zone. Similar results for all other sub jects are sho wn in Figure I.1 in App endix I . A dditionally , T able 3.3 sho ws the rela- tionships b et w een the predicted epileptogenic zone and resected areas (the predicted epileptogenic zone w as either fully resected or partially resected or not resected) and the surgical outcomes. Comparing to the seizure-free patien ts where the predicted epileptogenic zone w as almost alw a ys inside the resected area, hence they b ecame seizure-free after surgery , the predicted epileptogenic zone w as either partially resected or not resected for most of the non-seizure-free patien ts, whic h ma y explain the reason for the surgical failure in those cases. In con trast to EZF, the fast activit y w as often more propagated and extended far b ey ond the epileptogenic zone, although the EZF and the fast activit y are partially co-lo calized (most often EZF is a subset of the fast-activit y-iden tified epileptogenic zone). F or example, in Figure 3.4 (a), the epileptogenic zone w as unilateral and lo calized within the resected region while the fast activit y w as in bilateral homotopic areas. This phenomenon can also b e v erified from the SEEG time-series as w ell as the time-frequency plots, with fast activit y exhibited on b oth electro de N and N’, although the app earance of fast activities differs across con tacts. Since this patien t (Figure 3.4 (a)) w as seizure-free after surgery , the epileptogenic zone w as situated only in the righ t hemisphere. Hence, it is v ery difficult to delineate the epileptogenic zone purely based on fast activit y alone. The epileptogenic zone can b e o v er-estimated when using only fast activit y as the biomark er. Ov erall, anatomical visualization of the predicted epileptogenic zone sho w ed a substan tial differ- ence in app earance b et w een the seizure-free group and the non-seizure-free group. In the seizure-free group, the iden tified epileptogenic zone usually presen ts as a restricted region within the resected area. On the other hand, in the non-seizure-free group the predicted epileptogenic zone is usually m uc h more diffuse and not w ell lo calized. Statistically , Figure 3.5 sho ws b o xplots of the kurtosis across all sub jects in the seizure-free group on the left and the non-seizure-free group on the righ t. The kurtosis for the seizure-free group is significan tly higher than that for the non-seizure-free group 72 3.3. Results T able 3.3: Comparison of the resection/laser ablation and surgical outcomes Sub ject ID Resection of the predicted epileptogenic zone * Outcome (Engel) 101 1A 102 † Complete 1A 103 1A 106 Complete 1A 108 Complete 1A 111 Complete 1A 112 Complete 1A 113 P artial 1A 116 Complete 1A 118 Complete 1A 140 1A 215 Not resected 2A 219 P artial 2A 220 P artial 4 221 3 222 P artial 2 223 Complete 3 226 P artial 2B 228 Not resected 4 231 Not resected 4 232 4 233 Complete 2 237 4 238 P artial 3 * The relationship b et w een the resection and the predicted epileptogenic zone w as determined b y visualization and man ual insp ection of the in terp olated epileptogenic zone prediction scores in patien ts’ MRI space. † The false p ositiv e con tact (only one) predicted in this patien t (see T able 3.2 ( a )) ha v e v ery lo w scores relativ e to the true p ositiv e con tacts, hence do not affect the o v erall estimation of the epileptogenic zone. (See Figure I.1 in App endix I for details). with a one-sided p-v alue of 3:810 3 for a studen t t test with 22 degree of freedom, indicating that Chapter 3. Learning to Define An Biomark er of the Epileptogenic Zone 73 the predicted epileptogenic zone is more constrained, lo calized and has relativ ely higher prediction scores in the seizure-free group than that in the non-seizure-free group. 3.4 Discussion 3.4.1 V alidation of Epileptogenic Zone Biomark ers Ev en though the concept of the “epileptogenic zone” has b een used for more than 50 y ears [ 15 ], its precise definition remains con tro v ersial and a reliable biomark er is still missing. Sev eral algorithms ha v e b een prop osed to assess the exten t of the epileptogenic zone and compare it with either the epileptogenic zone iden tified b y an exp ert clinician or the resected region [ 7 , 20 , 56 , 79 , 80 , 194 ], and some of them sho w ed high concordance b et w een t he t w o. Figure 3.5: Bo xplot of the kurtosis of the in terp olated prediction scores for the seizure-free group on the left and the non-seizure-free group on the righ t. Example of the app earance of the epileptogenic zone fingerprin t prediction that corresp ond to lo w and high kurtosis are sho wn on the righ t. 74 3.4. Discussion Ho w ev er, the lac k of ground truth and precise definition of the epileptogenic zone p oses a significan t c hallenge for us to v alidate the lo calization p erformance of a particular biomark er. First, there can b e a large v ariation in ictal patterns across differen t sub jects. Some w a v eform or time in terv al alterations ma y o ccur ev en across differen t seizures within sub jects (see Figure 3.2 (a)). Those latter alterations ma y not affect visual analysis based on the SEEG time series b ecause an exp ert clinician will treat them as the same t yp e with subtle differences in w a v eform. But they can significan tly affect prediction results using mac hine-based algorithms and lead to underestimation of the epileptogenic zone. Indeed, Andrzejak et al. [ 7 ] compared the p erformance of four earlier prop osed approac hes for automatic iden tification of the epileptogenic zone [ 7 , 20 , 56 , 80 ] and found that these metho ds pro duced highly discordan t results dep ending on particular ictal patterns. Moreo v er, the resected region is not an ideal ground truth for the epileptogenic zone either, b ecause, as w e discussed previously , the resected region is t ypically larger, sometimes m uc h larger, than the actual epileptogenic zone, resulting in a large n um b er of false negativ e predictions when using mac hine-based algorithms (T able 3.2 ), regardless of the seizure freedom of patien ts after surgery . Instead of using the clinically defined epileptogenic zone or the resected region as the ground truth, it w ould b e more meaningful to compare the computer-assisted lo calization or prediction against long-term seizure outcomes after surgical treatmen t as it pro vides an ob jectiv e criterion for ev aluation of the accuracy of the epileptogenic zone iden tification. Ho w ev er, w e are una w are of an y previous study in the literature that sho w correlations b et w een the resection of a predicted epileptogenic zone and the surgical outcomes. In fact, Blau wblomme et al. [ 29 ] has sho wn that the epileptogenicit y of insular cortex, measured with an earlier prop osed algorithm [ 56 ], did not influence the outcome of temp oral lob ectom y , suggesting the difficult y of finding a reliable biomark er of the epileptogenic zone. Therefore, if a biomark er of the epileptogenic zone is a reliable one, the epileptogenic zone should b e iden tified exclusiv ely within the resected region in patien ts that b ecome long-term seizure-free after surgery . On the other hand, the epileptogenic zone should b e either completely missed or only partially resected in non-seizure-free patien ts. The aim of this study w as to v alidate the EZF Chapter 3. Learning to Define An Biomark er of the Epileptogenic Zone 75 metho d on a completely indep enden t set of patien ts that ha v e b een dra wn with broader selection criteria including b oth seizure-free and non-seizure-free sub jects. In order to get a more realistic represen tation of the surgical plan, anatomical information w as incorp orated and the predicted epileptogenic zone scores w ere mapp ed on to the patien ts’ MRI space. Finally , w e compared our fingerprin t-based epileptogenic zone prediction results with surgical outcome. The epileptogenic zone prediction results (b oth the binary classification results and the in terp olated imaging results) sho w ed a substan tial difference b et w een the seizure-free group and the non-seizure-free group. The predicted epileptogenic zone w as w ell lo calized inside the resected area in seizure-free patien ts while it w as not fully resected or completely missed in non-seizure-free patien ts. The con trast b et w een the results for seizure-free patien ts and that for non-seizure-free patien ts correlates with their surgical outcomes th us pro viding a p osteriori explanation of the surgical success or failure in those cases. 3.4.2 F ast A ctivities Yield Epileptogenic Zone Blurred Image F ast activit y has b een increasingly used as a p oten tial biomark er of the epileptogenic zone since gro wing use of SEEG in presurgical ev aluation. Ho w ev er, previous studies did not p erform time-frequency analysis so they could not differen tiate b et w een broad-band and narro w-band fast activities. The former w as frequen tly measured as the p o w er of gamma activities that is m uc h less discriminativ e than the latter in terms of epileptogenic zone lo calization. F or example, the epileptogenicit y index [ 20 ] emphasizes on the analysis of fast activit y and is calculated as the energy ratio b et w een the fast frequencies (b eta + gamma) and slo w frequencies (theta + alpha). Ho w ev er, our time-frequency analysis sho w ed that b eta and lo w gamma frequencies w ere often suppressed and the p o w er in fast activit y w as highly v ariable. Also the epileptogenicit y index mixes these t w o phenomena (high frequency oscillations and lo w frequency suppression) and do es not accoun t for the fact that frequency range and duration can v ary significan tly b oth across sub jects and b et w een epileptogenic-zone and non-epileptogenic areas. Da vid et al. [ 56 ] iden tified the epileptogenic zone based on the maximal frequency of fast activit y during early seizure spread. Gnatk o vsky et al. [ 79 , 80 ] measured the p o w er of fast activit y in a high gamma range and found that maxim um p o w er of high gamma activit y correlated with a slo w p olarizing shift and less with 76 3.4. Discussion EEG flattening. Ho w ev er, these studies either used fast activit y solely or treated fast activit y indep enden tly from other features. T o address the question of whether narro w-band fast activit y , one of the three comp onen ts of the fingerprin t, can b e an equally discriminativ e factor in lo calization of the epileptogenic zone, w e n umerically classified the epileptogenic zone using features from fast activit y only and mapp ed fast activit y on to patien ts’ MRI space using the same EZF pip eline (Figure 3.1 ). Classification results sho w ed a large n um b er of false p ositiv e con tacts outside the resected region (T able 3.2 ( b )). On the other hand, imaging results sho w ed that the fast activit y could extend/propagate to areas far b ey ond the epileptogenic zone (Figure 3.4 ). Both results confirmed that the epileptogenic zone w as v ery difficult to lo calize and significan tly o v er-estimated when (ev en narro w-band) fast activit y only is used as a biomark er. Moreo v er, as opp osed to the previous studies where fast activit y w as usually defined as gamma activities or b ey ond, our analysis sho w ed that the fingerprin t patterns can b e iden tified with fre- quency activities as lo w as b eta or ev en alpha band (Figure 3.3 ), indicating the difficult y of epilep- togenic zone lo calization when using fast-activit y-based biomark ers with con v en tional but heuristic c hoice of frequency range. 3.4.3 A utomated Epileptogenic Zone Fingerprin t Classification Pip eline The prop osed EZF pip eline consists of four ma jor parts: data pre-pro cessing, fingerprin t feature extraction, SVM-based classification, MRI in terp olation (Figure 3.1 ). F or the b est of our kno wl- edge, it is the first complete end-to-end epileptogenic zone prediction/estimation pip eline that w as v alidated based on long-term seizure outcome on a represen tativ e series of patien ts. This automated mac hine-learning-based EZF classification pip eline is crucial for successful ap- plication of the fingerprin t metho d to delineate epileptogenic zone from other areas b ecause the fingerprin t pattern ma y v ary significan tly across sub jects, sometimes ev en across differen t seizures within sub jects. Moreo v er, in man y cases, it is v ery difficult to distinguish epileptogenic-zone con- tacts from non-epileptogenic-zone con tacts only b y visual insp ection of the time-frequency plots. F o r example, the fingerprin t pattern is clearly sho wn on the electro de N in Figure 3.4 (a), while Chapter 3. Learning to Define An Biomark er of the Epileptogenic Zone 77 the time-frequency plots of electro de N’ seem to ha v e a blurred/v ague v ersion of the fingerprin t. Based on the visualization of suc h t yp e of time-frequency plots, it’s not easy to den y the h yp othesis that con tacts on N’ b elong to the epileptogenic zone (while in fact N’ is outside the epileptogenic zone). This is b ecause w e are often lo oking for the existence of the fingerprin t pattern during visual insp ection, but lac k a prop er ev aluation of the t ypicalit y of the pattern (i.e., the narro w bands of fast activit y are blurred and the lo w er frequencies are not w ell suppressed on N’ relativ e to N). In con trast, b y taking all asp ects of the fingerprin t in to accoun t (i.e., all differen t t yp es of features extracted from the time-frequency plot), the SVM-based classification system is able to distinguish the pattern on N apart from that on N’, hence yielding an accurate prediction of the epileptogenic zone. 3.4.4 P oten tial Pitfalls in Application of the Epileptogenic Zone Fingerprin t Metho d Among all steps in the epileptogenic zone fingerprin t pip eline, feature extraction is of particular imp ortance, due to the high complexit y in the fingerprin t patterns across sub jects. As a result, the originally designed feature extraction steps migh t ha v e b een biased to w ards the set of 17 seizure-free patien ts used in our previous study (Chapter 2 ), in the sense that the parameters are tuned based on that set of patien ts and ma y not generalize w ell to other patien ts. F or example, in Figure F.1 , the time-frequency plot for Sub ject 103 sho ws a clear fingerprin t pattern with a do wn-c hirping narro w- band fast activit y . Ho w ev er, the slop e of this banding is to o steep to fall in to the reasonable range (45 ◦ a w a y from p ositiv e x-axis) that w as designed to rule out false p ositiv e detections in the original dataset. Therefore, this fast activit y w as not successfully iden tified resulting in a missing prediction of the epileptogenic zone. Although the fingerprin t pattern (the com bination of pre-ictal spik e, narro w-band fast activit y and suppression) is v ery consisten t across sub jects, the individual features ma y v ary substan tially across sub jects as w ell as across differ en t seizures within s ub jects. Therefore, prediction-score-based seizure clustering is a v ery imp ortan t step in the pip eline to ha v e go o d prediction outcomes. This is b ecause our original EZF mo del w as v ery conserv ativ e, so an epileptogenic-zone con tact predicted 78 3.4. Discussion from a single cluster of seizures will yield no prediction w hen taking all seizures in to accoun t in the case where there is inconsistency across seizures. Results when considering differen t seizure clusters sho w ed impro v ed p erformance of epileptogenic zone prediction o v er that without clustering. Here w e use man ual clustering. A systematic approac h to clustering seizures and in v estigating the pathoph ysiological meaning b ehind differen t clusters ma y b e of in terest for future researc h. Another p ossible pitfall of the fingerprin t metho d ma y o ccur when the SEEG electro des are not optimally placed. In that case it is almost imp ossible to delineate epileptogenic zone from other areas based on the fingerprin t metho d as the fingerprin t pattern do es not presen t or only partially presen ts if the electro de con tact is not to o distan t from the core epileptogenic zone (e.g., Sub ject 101 and 140 in Figure F.1 ). Nev ertheless, in these cases, large resections ma y still include the epileptogenic zone, hence patien ts ma y b ecome seizure-free after surgery ev en though no fingerprin t has b een iden tified. In fact, b oth the accuracy and the densit y of electro de implan tations strongly affect and can b e the b ottlenec k limiting the p erformance of the application of the fingerprin t metho d to the epileptogenic zone iden tification problem. Therefore, iden tifying a relativ ely accurate estimate of the epileptogenic zone using pre-surgical non-in v asiv e data, th us pro vide go o d guidance to electro de implan tation, p oses a c hallenging problem for the future. This study sho ws that the time-frequency pattern describ ed as a fingerprin t of the epileptogenic zone is a real mark er of the cortical region whic h m ust b e ablated in order to ac hiev e seizure freedom for a giv en patien t. F ast activit y , as it spreads out of the epileptogenic zone, cannot b e utilized separately from the other t w o elemen ts of the fingerprin t pattern for an accurate lo calization. Suc h in ter-dep endence reinforces the pathoph ysiological significance of the mark er. It fits w ell with the mo del prop osed b y A v oli and de Curtis [ 11 , 57 ]: progressiv e sync hronized activ ation of p yramidal cells (lo w-frequency comp onen t of the pre-ictal spik es) that excites disinhibited fast somatic inhibitory in terneurons (FSIN) (fast-frequency comp onen t of the pre-ictal spik es) leading to a sustained in terneuronal activ ation (ictal fast activit y) with p yramidal silencing as a consequence (suppression of lo w frequencies) [ 11 , 65 , 75 , 81 , 187 , 200 ]. A t the end of the sustained FSIN disc harge, reb ound activ ation of p yramidal cells and in terneurons can o ccur b efore seizure stops. Ho w the fast activities resonate in a more extended net w ork remains to b e elucidated. P art I I I Robust Iden tification of Dynamic Brain Net w orks 79 Chapter 4 Scalable and Robust Sequen tial T ensor Decomp osition of Sp on taneous Stereotactic EEG Data , ; , — 4.1 In tro duction Exploring functional connectivit y in resting brain signals is a ric h approac h to studying brain net w orks [ 74 ]. Of particular recen t in terest is the dynamic nature of functional connectivit y [ 148 ]. The most commonly used strategy for deco ding dynamic functional connectivit y is to compute correlation or coherence using a sliding windo w [ 46 , 91 ]. Ho w ev er, using a long temp oral windo w to obtain robust functional connectivit y estimates inevitably leads to o v er-smo othing of dynamic c hanges [ 99 , 146 ]. T o o v ercome this difficult y , PCA-based and ICA-based approac hes ha v e b een prop osed. Although they do not in tro duce temp oral smo othing, a limitation of those metho ds is that either the time series of eac h net w ork is required to b e indep enden t (temp oral ICA) [168 ] 80 Chapter 4. T ensor Decomp osition of Sp on taneous SEEG Data 81 or the spatial mo des of the net w orks are disjoin t (spatial ICA) [ 23 , 42 ] or orthogonal (PCA) [ 73 ], whereas real net w orks can o v erlap and b e correlated in b oth space and time [ 104 ]. T ensors are a generalization of matrices in whic h w e can represen t data with more than t w o indices. F or example, he re w e use a third-order tensor to represen t SEEG data in terms of space, time and frequency (in the follo wing “tensors” refers to tensors of order 3 or higher). As with matrices, tensors can b e represen ted as a sum of rank-1 comp onen ts. Structured data often then admit to lo w rank mo dels with resp ect to this tensor represen tation. One of the t w o p opular tensor mo dels is the canonical p oly adic (CP) form [ 97 ], whic h has b een sho wn [ 49 , 108 , 112 ] to also b e equiv alen t to parallel factors analysis [ 92 ] and the canonical decomp osition [ 44 ]. Canonical p oly adic form is a mo del that can capture structure inheren t in m ultidimensional data when represen ted in a lo w rank tensor of order 3 or higher. Con v ersely , standard 2-dimensional (2D) matrix decomp osition metho ds, suc h as PCA or ICA, applied to matricized or unfolded ten- sors [ 49 ] are t ypically not able to capture this structure using a similar lo w-rank mo del. Moreo v er, CP decomp osition has a unique solution under less restrictiv e conditions than t he orthogonalit y or indep endence assumptions implicit in PCA or ICA [ 115 , 164 ]. This latter prop ert y is particu- larly app ealing when analyzing SEEG or other brain data that can b e represen ted as a third or higher-order tensor, since w e can a v oid the restrictiv e assumptions of orthogonalit y or indep endence b et w een comp onen ts. Among the man y algorithms for computing the CP decomp osition, alternating least square (ALS) [ 44 , 92 ] is the most widely used b ecause of its simplicit y relativ e to alternativ es [ 49 , 69 , 112 , 185 ]. Comparisons in these pap ers sho w that, in general, ALS pro vides solutions of similar qualit y to other algorithms. While some are more robust to o v er-factoring than ALS, esp ecially in ill-conditioned cases, the exp ense is higher computational complexit y in b oth memory and time [ 49 ]. ALS-based CP decomp osition has b een widely used in EEG analysis b y transforming the ra w EEG recordings to a time-frequency represen tation using short-time F ourier or Morlet w a v elet transforms and applying a 3 -w a y CP decomp osition (c hannel time frequency) [ 50 ]. Mö c k [ 135 ] applied CP to ev en t-related p oten tials. Miw ak eic hi et al. [ 134 ] analyzed b oth sp on taneous and 82 4.1. In tro duction ev ok ed EEG recordings and sho w ed that theta activit y w as predominan t during a task condi- tion, while alpha activit y w as observ ed con tin uously during b oth rest and task conditions. CP decomp osition has also b een applied to ictal EEG recordings from patien ts with epilepsy . The extracted comp onen ts ha v e b een used to lo calize the seizure onset zone [ 1 , 58 , 59 ] as w ell as to remo v e artifacts [ 1 , 58 ]. When applied to group data analysis, CP decomp osition and the extracted comp onen ts/features can b e used for a v ariet y of purp oses: classification [ 119 ], cross-mo dalit y comparison [ 193 ], and h yp othesis testing [ 138 ]. The application of CP dec omp osition to fMRI data has also b een explored, usually at the group- lev el with the goal of finding common factors among sub jects [ 112 ]. Individual analysis using CP is rarely p erformed b ecause fMRI data lac ks the ric h sp ectral information presen t in EEG data. Andersen et al. [ 6 ] applied CP decomp osition to finger-tapping fMRI data. Bec kmann et al. [ 22 ] prop osed a v ariation of CP whic h extends probabilistic ICA (PICA) to higher orders b y adding an indep endence constrain t in the spatial dimension. While originally applied to task fMRI data, Damoiseaux et al. [ 55 ] applied tensor-PICA to rfMRI data and iden tified 10 net w orks consisten t with previous findings. In previous studies where C P decomp osition w as applied to either EEG or fMRI data [ 1 , 6 , 18 , 22 , 58 , 59 , 119 , 134 , 137 , 193 ], the size of the datasets is relativ ely small (the largest dimension has an order of 10 4 elemen ts or less). T o explore dynamic functional connectivit y , esp ecially large-scale dynamics, w e need to explore m uc h larger datasets. There are t w o issues that ha v e limited studies of this kind: scalabilit y and robustness. Scalabilit y: The ma jorit y of the studies cited ab o v e either truncate the data in to short temp oral segmen ts (e.g., in EEG ictal and ev en t related p oten tial recordings) or hea vily do wn-sample the data in the spatial domain (e.g., in fMRI studies) or som etimes b oth. The degrees of freedom, appro ximately reflected b y the largest dimension in the CP mo del for a t ypical 10 -min ute SEEG recording used in our exp erimen ts is larger ( 10 5 ) than in these previous studies. Moreo v er, as w e will sho w in Section 4.4.1 b elo w, the computational complexit y increases appro ximately quadratically as the degrees of freedom increase when using ALS to estimate rank. A dditionally , to ac hiev e similar qualit y of solutions relativ e to our algorithm, ALS has to b e applied with m ulti-start Chapter 4. T ensor Decomp osition of Sp on taneous SEEG Data 83 as sho wn in Section 4.4.1 , resulting in an ev en higher complexit y . Therefore, in order to compute CP decomp ositions on data of this size, a fast and efficien t algorithm is required. Robustness (against lo cal minima): It is w ell kno wn that the ALS algorithm on a CP mo del is not guaran teed to con v erge to a global minim um or a stationary p oin t, ev en when m ulti-start is applied during the optimization [ 49 , 112 ]. The lo cal minim um problem b ecomes more sev ere as the n um b er of comp onen ts increases. P erformance is further compromised when a larger n um b er of comp onen ts than necessary are fit to the data (o v er-factoring), resulting in splitting rank- 1 comp onen ts in to t w o or more factors. Sev eral tec hniques ha v e b een explored to impro v e the robustness and efficiency of the ALS algorithm. F or example, Ra jih et al. [ 150 ] added a line searc h after eac h ma jor ALS iteration. Na v asca et al. [ 142 ] applied Tikhono v regularization on eac h sub-problem in ALS iteration. Ho w- ev er, similar to the ALS alternativ es review ed ab o v e, these mo difications result in significan tly higher computation cost limiting their practical utilit y , particularly for large scale problems. In 2D matrix scenarios, Haldar et al. [ 90 ] prop osed an incremen ted-rank P o w erF actorization (IRPF) approac h to solv e minim um-rank matrix reco v ery problems, where higher-rank solutions w ere obtained recursiv ely using lo w er-rank results as w arm initializations, resulting in a substan- tially impro v ed p erformance compared to the standard con v ex optimization approac h. P erformance w as also theoretically c haracterized later in [ 101 ]. In this w ork, w e extend IRPF to higher-order tensors, with the goal of resolving the scalabilit y and robustness issues discussed ab o v e. W e refer to our approac h as “scalable and robust sequen tial CP decomp osition” (SRSCPD). As w e sho w b elo w, this algorithm is more robust than ALS and can b e extended to large-scale problems. 4.2 Notation and Preliminaries W e first define some necessary notation and review the ALS algorithm whic h w e use as part of our SRSCPD framew ork in Section 4.3 . W e largely follo w the notational con v en tions and definitions of K olda and Bader [ 112 ]. 84 4.2. Notation and Pre liminaries 4.2.1 Scalar, V ector, Matrix and T ensor A scalar is denoted b y a lo w ercase letter, e.g., x ; a v ector b y a b old lo w ercase letter, e.g., x ; A matrix b y a b old upp ercase letter, e.g., X ; and a tensor b y a b old script letter, e.g., X . The n um b er of the dimensions is called the or der and eac h dimension is referred as a mo de . W e use a third-order tensor X 2R IJK in the follo wing with individual elemen ts denoted b y x i;j;k . The notation and algorithms extend naturally to higher-order tensors. A rank-1 tensor can b e expressed as the outer pro duct of v ectors. i.e., X =a◦b◦c where◦ represen ts the v ector outer pro duct. W e use the norm: ∥X∥ = v u u t I ∑ i=1 J ∑ j=1 K ∑ k=1 x 2 i;j;k (4.1) 4.2.2 Matricization T ensors can b e unfolded or “matricized” in to matrix form [ 112 ]. Matricization along dimension n is denoted b yX (n) so that a third-order tensorX2R IJK can b e matricized in toX (1) 2R IJK orX (2) 2R JIK orX (3) 2R KIJ . 4.2.3 Kronec k er, Khatri-Rao and Hadamard Pro duct W e use the definitions of the follo wing matrix pro ducts defined in [ 112 ] and rep eated here for con v enience. The Kronec k er pro ductX Y of matrixX2R IJ andY 2R KL is defined as X Y = 2 6 6 6 6 6 6 6 4 x 11 Y x 12 Y x 1J Y x 21 Y x 22 Y x 2J Y . . . . . . . . . . . . x I1 Y x I2 Y x IJ Y 3 7 7 7 7 7 7 7 5 (4.2) where x ij is the (i;j) th elemen t ofX . Chapter 4. T ensor Decomp osition of Sp on taneous SEEG Data 85 The Khatri-Rao pro ductX⊙Y of matrixX2R IK andY 2R JK is defined as the column- wise Kronec k er pro duct ofX andY X⊙Y = [ x 1 y 1 x 2 y 2 x K y K ] (4.3) wherex i is the i th column ofX . The Hadamard pro ductXY of matrixX2R IJ andY 2R IJ is defined as the elemen t-wise matrix pro duct XY = 2 6 6 6 6 6 6 6 4 x 11 y 11 x 12 y 12 x 1J y 1J x 21 y 21 x 22 y 22 x 2J y 2J . . . . . . . . . . . . x I1 y I1 x I2 y I2 x IJ y IJ 3 7 7 7 7 7 7 7 5 (4.4) In the follo wing, w e use the prop ert y of the Khatri-Rao pro duct [ 112 ]: (X⊙Y ) y = (X T XY T Y ) y (X⊙Y ) T (4.5) whereX y represen ts the Mo ore-P enrose pseudo-in v erse ofX . 4.2.4 Canonical P oly adic Decomp osition Canonical p oly adic mo del decomp oses a tensor in to a sum of rank-1 tensors or comp onen ts. F or a third-order tensorX2R IJK X = R ∑ r=1 a r ◦b r ◦c r +E (4.6) wherea r 2R I ,b r 2R J ,c r 2R K , R is the r ank or the n um b er of comp onen ts andE is the error tensor. If w e group the comp onen ts in eac h mo de in to a matrix, i.e., letA = [a 1 a 2 a R ]2R IR and similarly forB2R JR andC2R KR , then the CP decomp osition can b e expressed as [ 112 ] X (1) =A(C⊙B) T +E (1) (4.7) 86 4.2. Notation and Pre liminaries or X (2) =B(C⊙A) T +E (2) (4.8) or X (3) =C(B⊙A) T +E (3) (4.9) whereA;B;C are called the lo ading matric es for the three mo des, resp ectiv ely . 4.2.5 Computation of CP Decomp osition and the ALS Algorithm Supp ose w e w an t to find the b est rank R appro ximation ofX2R IJK via min ^ X X ^ X +g( ^ X) (4.10) where ^ X = ∑ R r=1 r a r ◦b r ◦c r , r represen ts the scale of comp onen t r and a r , b r and c r ha v e unit norm. g( ^ X) = 1 g 1 (A) + 2 g 2 (B) + 3 g 3 (C) is a regularizing function with ( 1 ; 2 ; 3 ) the corresp onding regularization parameters. The ALS algorithm solv es this problem in an alternating fashion. W e first solv e for A withB andC fixed, then solv es for B withA andC fixed, and so on. This pro cedure is rep eated un til some con v ergence criterion is satisfied. Note that, for quadratic regularizers, eac h sub-problem reduces to an ordinary least square. Sp ecifically , assume B andC are fixed and w e are solving for A . Using the equiv alen t matrix expression discussed ab o v e, w e can write the optimization problem as ^ A = argmin A X (1) A(C⊙B) T F + 1 g 1 (A) (4.11) The solution with 1 = 0 (without regularization) reduces to a regular least square solution: ^ A =X (1) [(C⊙B) T ] y (4.12) Using the prop ert y in Equation ( 4.5 ), w e can rewrite as ^ A =X (1) (C⊙B)(C T CB T B) y (4.13) Chapter 4. T ensor Decomp osition of Sp on taneous SEEG Data 87 This expression is almost alw a ys preferable to Equation ( 4.12 ) b ecause it ac hiev es a m uc h lo w er computational complexit y b y only calculating the pseudo-in v erse of an RR matrix. Finally , w e normalize eac h comp onen t and set r equal to the normalization factor for the r th comp onen t, r = 1;:::;R . F or the case 1 ̸= 0 , the solution in Equation ( 4.12 ) is replaced b y the solution to Equation ( 4.11 ), whic h will b e closed form if g 1 (A) is quadratic, but ma y require iterativ e solution in other cases. The full ALS algorithm is sho wn in Algorithm V . Algorithm V Alternating least square function CP-ALS (X;R;fA ;B ;C ; g ) Initialize * A2R IR ;B2R JR ;C2R KR ;2R R while not con v erged † do A argmin A X (1) A(C⊙B) T F + 1 g 1 (A) B argmin B X (2) B(C⊙A) T F + 2 g 2 (B) C argmin C X (3) C(B⊙A) T F + 3 g 3 (C) NormalizeA;B;C suc h that eac h column has unit norm and set r equal to the normal- ization factor for the r th comp onen t, r = 1;:::;R end while returnA;B;C and end function * The initialization is t ypically p erformed using either random matrices or the R leading singular v ectors of the matricizedX . W e define sp ecific initializations for our SRSCPD algorithm. † Algorithm con v ergence is determined when the mean of the absolute difference of the loading matrices b et w een t w o adjacen t iterations o v er all mo des is less than some small constan t, e.g., 10 5 . 4.3 Materials and Metho ds 4.3.1 SRSCPD F ramew ork The b est rank-r appro ximation of a matrix with resp ect to the F rob enius norm is giv en b y the leading r factors of the singular v alue decomp osition (SVD). This is not the case for CP decomp osition of a higher-order tensor. K olda [ 111 ] sho w ed an example where the b est rank-1 appro ximation is not part of the b est rank-2 appro ximation of a tensor. As a result, comp onen ts in 88 4.3. Materials and Metho ds the CP decomp osition for a giv en desired rank should b e found sim ultaneously . Smilde et al. [ 166 ] (example 4.3) sho w ed that the naiv e sequen tial CP , in whic h a rank-1 tensor is fit to the residue at eac h iteration, failed to extract the correct comp onen ts ev en when the data are kno wn to b e p erfectly trilinear. In terestingly , this greedy sequen tial approac h is still frequen tly used, simply b ecause it is the most tractable approac h to fitting tensor mo dels to large datasets [ 113 , 211 ]. The determination of tensor rank is NP-hard [ 93 ]. Man y metrics ha v e b een prop osed to help find the correct rank, e.g., the core consistency diagnostic (COR CONDIA) [ 34 ], difference in fit [ 183 ] and automatic relev ance determination [ 139 ]. All these metrics require a set of decomp osition results for all ranks up to the maxim um rankR . Obtaining suc h a set of solutions using CP decomp osition is quadratically more complex than finding a rank- 1 appro ximation, as w e need to compute the decomp ositions for eac h rank r = 1;2;:::;R separately (1+2++R =O(R 2 )) . This represen ts a significan t c hallenge to use of higher rank tensor mo dels and w as the primary motiv ation for our dev elopmen t of the SRSCPD framew ork. The SRSCPD framew ork is built on the original ALS algorithm. Our goal is to compute a rank-recursiv e set of decomp ositions from rank 1 to rank R . Our approac h uses the result for rank r to initialize the decomp osition for rank r + 1 . Initialization for the additional comp onen t is found b y fitting a rank- 1 tensor to the residual from the rank r fit. In con trast, the original ALS algorithm do es not use an y information from rank r when fitting a mo del of rank r+1 . This “w a rm start” greatly impro v es con v ergence sp eed relativ e to the standard ALS algorithm except in v ery lo w rank cases (see sim ulation result in Figure 4.4 ). The w arm start ma y also help to a v oid p o or lo cal minima in this non-con v ex optimization problem. As a result, w e are able to address the problems with robustness and scalabilit y for large-scale datasets. The full SRSCPD framew ork is sho wn in Algorithm VI , for a third-order tensor example. The inputs of the algorithm are a tensor X 2 R IJK and the desired maxim um rank R . F or eac h iteration r , a rank-r appro ximation is calculated using the original CP-ALS algorithm with initializations fA ;B ;C ; g . The initializations are formed b y concatenating the solutions fA r1 ;B r1 ;C r1 ; r1 g from the previousr1 recursion with the rank-1 appro ximationfa ′ ;b ′ ;c ′ ; Chapter 4. T ensor Decomp osition of Sp on taneous SEEG Data 89 ′ g of the residue tensorX res , whereX res is obtained b y subtracting the reconstructed tensor using fA r1 ;B r1 ;C r1 ; r1 g from the original data tensorX . SRSCPD is flexible in the sense that tec hniques that ha v e b een prop osed to impro v e the ALS algorithm can b e directly incorp orated. F or example, one can add a line searc h along the estimated gradien t descend direction for eac h mo de at the end of eac h ma jor iteration of ALS [ 150 ]. Moreo v er, constrain ts and regularization terms can b e applied to eac h of the ALS sub-problems, e.g., non- negativit y , sparsit y , and smo othness. Algorithm VI Scalable and robust sequen tial canonical p oly adic decomp osition function SRSCPD-ALS (X;R ) a 1 ;b 1 ;c 1 ; 1 CP-ALS(X;1 ) X res X T ensor-Recon (a 1 ;b 1 ;c 1 ; 1 ) fa ′ ;b ′ ;c ′ ; ′ g CP-ALS(X res , 1 ) A [a 1 a ′ ] ;B [b 1 b ′ ] ;C [c 1 c ′ ] ; [ 1 ′ ] for r = 2;3;:::;R do A r ;B r ;C r ; r CP-ALS(X;r;fA ;B ;C ; g ) X res X T ensor-Recon (A r ;B r ;C r ; r ) fa ′ ;b ′ ;c ′ ; ′ g CP-ALS(X res , 1) A [A r a ′ ] ;B [B r b ′ ] ;C [C r c ′ ] ; [ r ′ ] end for return a set of solutions:fa 1 ;b 1 ;c 1 ; 1 g;fA 2 ;B 2 ;C 2 ; 2 g;:::;fA R ;B R ;C R ; R g end function 4.3.2 Sim ulation W e sim ulated SEEG data [ 15 ] with 100 c hannels, 200 Hz sampling rate, 2 second duration for ranks from R = 1 to 10 . In eac h comp onen t a total of N c hannels w ere co-activ ated where N w as c hosen randomly b et w een 2 and 10 . F or eac h of the activ e c hannels for eac h comp onen t w e generated a time series to represen t a blo c k activ ation pattern with the signal switc hing on and off, resp ectiv ely , in activ e and inactiv e blo c ks. The n um b er of activ e blo c ks o v er the 2 -second p erio d 90 4.3. Materials and Metho ds Figure 4.1: An example of the sim ulated data with 5 comp onen ts. Eac h comp onen t is represen ted b y a distinct color in all three mo des. F rom left to righ t: The c hannel (spatial) mo de sho ws the activ ated c hannels that participate in eac h net w ork; The time (temp oral) mo de sho ws the blo c k activ ation pattern for eac h net w ork; The sp ectrum (sp ectral) mo de sho ws the frequency sp ectrum for eac h net w ork. w as selected randomly b et w een 2 and 5 and b oth the minim um blo c k length and the minim um in terv al b et w een an y adjacen t activ ated blo c ks w as set to 0:1 second. Within eac h activ e blo c k, the signals in eac h comp onen t w ere unit amplitude sin usoids with frequencies c hosen randomly b et w een 10 and 80 Hz. Finally , w e added white Gaussian noise to the sim ulated data with a range of SNRs. The third-order tensorX w as generated b y calculating the magnitude squared of the complex Morlet w a v elet transform co efficien ts of the sim ulated data matrix with ce n ter frequency 1 Hz, time resolution FWHM of 2 seconds [ 178 ] in a linearly spaced frequency range from 1 to 100 Hz with in terv al 1 Hz. Th us, the final tensor X has the dimensions ofR IJK , where R = 100;J = 400;K = 100 . An example of the mo del used to sim ulate the data is sho wn in Figure 4.1 . Note that o v erlaps b et w een comp onen ts ma y o ccur in an y of the three mo des. W e first compared the robustness of the decomp osition using the SRSCPD framew ork against ALS using 1 , 2 and 5 random initializations (generated from a standard uniform distribution in the in terv al (0;1) using MA TLAB (The Math W orks, Inc., Natic k, MA, USA) function “rand”). The same con v ergence criterion w as used for b oth algorithms and in all cases, w e computed solutions from rank 1 to R . In b oth algorithms w e used a non-negativit y constrain t on all loading matrices, b ecause the squared magnitude of the w a v elet co efficien ts are naturally non-negativ e and the con- strain t helps a v oid degeneracy [ 129 ]. LetA2R IR ;B2R JR ;C2R KR b e the loading matrix Chapter 4. T ensor Decomp osition of Sp on taneous SEEG Data 91 in eac h of the three mo des as describ ed in Section 4.2.4 . Then in eac h sub-problem of the ALS, w e used the follo wing cost function forA (lik ewise forB andC ) ^ A = argmin A X (1) A(C⊙B) T F s.t. A⪰ 0 (4.14) where “⪰ ” denotes the elemen t-wise inequalit y . Since w e kno w the ground truth under the sim ulated settings, w e assessed the qualit y of the solutions using the a v eraged congruence pro duct (A CP) [ 184 ]. A CP is a measure of correlations b et w een comp onen ts. Sp ecifically , let A;B;C b e the column-wise normalized ground truth loading matrices and ^ A; ^ B; ^ C their estimated cou n terparts. Then the A CP is defined p er [ 184 ] as A CP = max P tr((A T ^ AB T ^ BC T ^ C)P) (4.15) whereP is a p erm utation matrix accoun ting for the am biguit y of the ordering of the solutions [ 92 ] and tr(X) indicates the trace ofX . W e ev aluated the A CP of the solutions obtained from b oth ALS and SRSCPD as a function ofR for SNR = 10 . F or eac hR , w e ran 100 Mon te Carlo trials and b o xplots of A CP w ere generated. F or eac h sim ulated tensor, w e rep eated ALSM times, whereM = 1;2 and 5 , eac h time using a differen t random initialization. The final solution w as selected as that whic h has the lo w est cost. W e also b o x-plotted the F rob enius norm error as sho wn in Equation ( 4.10 ) for eac h trial. A dditionally , w e recorded the run time for eac h of the metho ds. W e then rep eated the ab o v e study , but instead of v arying R w e conducted the exp erimen t as a function of SNR with R = 5 . 4.3.3 Application to In-viv o SEEG Dataset W e p erformed retrosp ectiv e analysis of patien t data collected under an IRB appro v ed proto- col for SEEG ev aluation and monitoring in the Epilepsy Cen ter, Clev eland Clinic, OH, USA. The SEEG ev aluation p erforms in v asiv e pre-surgical electro-ph ysiological mapping for patien ts who ha v e pharmaco-resistan t fo cal epilepsy . F or eac h patien t, the implan tation w as p erformed using m ulti- lead depth electro des, with eac h electro de comprising t ypically 10 con tacts spaced a few millimeters 92 4.3. Materials and Metho ds T able 4.1: Summary of the patien t data Sub ject 1 Sub ject 2 Num b er of Channels 69 113 Epilepsy T yp e P osterior cingulate epilepsy Left fron to-parietal Data Segmen t Time 09:23:00 - 09:33:00 Same Da y: 13:22:02 - 13:32:02 21:50:30 - 22:00:30 Next Da y: 08:35:20 - 08:45:20 Implan tation Sc heme apart (A dT ec h, Racine, Wisconsin; In tegra, Plainsb oro, New Jersey; or PMT, Chanhassen, Min- nesota). The electro de lo cations w ere determined after a m ultidisciplinary patien t managemen t conference where the h yp otheses ab out the epileptogenic zone w ere dra wn based on a v ailable non- in v asiv e data: clinical history , video EEG, MRI, PET, ictal SPECT and MEG. The electro des w ere implan ted according to the T alairac h stereotactic metho d using orthogonal or oblique tra jec- tories [ 84 ]. The implan tation sc hemes are sho wn in T able 4.1 . The SEEG signals w ere recorded using a common reference on a Nihon K ohden EEG system with a sampling rate of 1000 Hz. W e c hose t w o 10 -min ute data segmen ts a minim um of 4 hours apart (see T able 4.1 ) for eac h patien t. The segmen ts w ere selected using annotated video of the patien ts for p erio ds of ph ysical inactivit y (e.g., reading, w atc hing TV). Using co-registration of the p ost-implan t X-ra y CT to the patien ts’ MR image, w e selected the subset of the SEEG con tacts that w ere in gra y matter. F or eac h data segmen t w e applied the Morlet w a v elet transform with cen ter frequency 1 Hz and time resolution FWHM of 2 seconds in a linearly spaced frequency range of 1 to 100 Hz with in terv al 1 Hz. W e computed the squared magnitude of the w a v elet co efficien ts and temp orally do wnsampled the resulting en v elop e data in eac h frequency band b y a factor of 5 , resulting in a new en v elop e sampling rate of 200 Hz. W e p erformed a flattening of the p o w er sp ectrum to comp ensate for Chapter 4. T ensor Decomp osition of Sp on taneous SEEG Data 93 its “1/f ” c haracteristics to emphasize the higher frequency comp onen ts. The resulting data w ere then represen ted as a third-order tensor X 2 R IJK , where I is the n um b er of c hannels (see T able 4.1 ), J = 120;000 and K = 100 . W e applied SRSCPD to eac h tensor with three additional constrain ts: non-negativit y on all three mo des due to the non-negativ e squared magnitude of the w a v elet co efficien ts; a sparsit y constrain t on the spatial (c hannel) mo de as w e assumed that in eac h net w ork only a small set of c hannels w ould b e in v olv ed; and a smo othness constrain t on the sp ectral mo de reflecting the limited frequency resolution of the Morlet w a v elet transform. Let A2 R IR ;B2 R JR ;C2 R KR b e the loading matrix in eac h of the three mo des. Then in the sub-problems of the ALS, w e use the follo wing cost functions: ^ A = argmin A X (1) A(C⊙B) T 2 F + 1 ∑ i ∥A i ∥ l 1 s.t. A⪰ 0 (4.16) ^ B = argmin B X (2) B(C⊙A) T 2 F ; s.t. B⪰ 0 (4.17) ^ C = argmin C X (3) C(B⊙A) T 2 F + 3 ∑ k ∥∇C k ∥ 2 l 2 s.t. C⪰ 0 (4.18) whereX i denotes the i th column ofX , 1 and 3 are the regularization parameters whic h w ere set to 0:2 empirically .∥∥ l 1 and∥∥ l 2 denotes the l 1 norm and l 2 norm resp ectiv ely .∇ is the finite difference op erator on the columns of C . W e solv ed eac h of the con v ex sub-problems ( 4.16 ) - ( 4.18 ) using A uslender and T eb oulle’s single-pro jection algorithm [ 10 ] in the T emplates for First-Order Conic Solv ers to olb o x [ 21 ]. Finally , the rank of the tensor w as estimated based on the decomp osition results using COR- CONDIA [ 34 ]. This rank metric used the fact that the trilinearit y of comp onen ts starts decreasing in t he case of o v er-factoring (the n um b er of fitted comp onen ts is greater than the actual rank). 94 4.4. Result 4.4 Result 4.4.1 Sim ulation Figure 4.2 sho ws p erformance of ALS vs SRSCPD as a function of rank R . F or small R all results are similar. Ho w ev er, for larger ranks w e see that the ALS results are strongly dep enden t on initialization and that p erformance forM = 5 is significan tly b etter than for M = 2 andM = 1 . SRSCPD b enefits from using the results of the lo w er rank as an initialization, resulting in o v erall impro v ed p erformance (higher median A CP) relativ e to all three v ersions of ALS. Figure 4.3 sho ws the corresp onding F rob enius norm error. As with Figure 4.2 , ALS withM = 1 and 2 sho ws larger error than M = 5 , with the difference increasing with rank. In con trast to the A CP metric in Figure 4.2 , the error for ALS with M = 5 in Figure 4.3 is v ery similar to that for SRSCPD and sometimes smaller for higher ranks. Closer examination of these results rev ealed that in cases where this o ccurs, ALS fails to find one or more of the w eak er comp onen ts that SRSCPD do es find. Instead, part of the noise in the data is fit to one of the tensor comp onen ts. This in turn Figure 4.2: Sim ulation results. Bo xplots of A CP o v er 100 Mon te Carlo trials are sho wn as a function of R . M denotes the n um b er of random initializations when using the original ALS algorithm. Chapter 4. T ensor Decomp osition of Sp on taneous SEEG Data 95 Figure 4.3: Sim ulation results. Bo xplots of the F rob enius norm error o v er 100 Mon te Carlo trials are sho wn as a function of R . M denotes the n um b er of random initializations when using the original ALS algorithm. leads to a lo w er squared error in the fit ev en though the extracted comp onen ts are a p o orer fit to the ground truth as measured with A CP . Figure 4.4 sho ws the run time as a function of R (the run time w as measured using MA TLAB with Dell Precision T3610 computer, In tel Xeon E5-1650 v2 CPU). As exp ected, the ratios of the run time among the ALS metho ds are appro ximately prop ortional t o M , the n um b er of differen t initializations. The cost of SRSCPD is significan tly lo w er than that for ALS with M = 2 and 5 restarts. As the rank increases (R > 4 ), the cost for SRSCPD is ev en lo w er than that for ALS without restart, M = 1 . The reason for this is that the w arm start in SRSCPD pro duces a b etter initialization that not only results in impro v ed p erformance (Figure 4.2 ) but also faster con v ergence of the ALS sub-problems as sho wn in Figure J.1 in App endix J . Figure 4.5 sho ws that as the SNR increases, A CP also impro v es for all metho ds. SRSCPD sho ws generally similar p erformance to ALS with M = 5 restarts and is substan tially b etter than results for M = 1 . Ho w ev er, for lo w er SNRs, the p erformance of ALS with M = 5 restarts is sup erior to SRSCPD. 96 4.4. Result Figure 4.4: Sim ulation results. Bo xplots of the run time in seconds o v er 100 Mon te Carlo trials are sho wn as a function of R . M denotes the n um b er of random initializations when using original ALS a lgorithm. T op-left panel sho ws the zo omed-in results for a b etter comparison for lo w er rank data. Figure 4.5: Sim ulation results. Bo xplots of the A CP o v er 100 Mon te Carlo trials are sho wn as a function of SNR. M denotes the n um b er of random initializations when using the original ALS algorithm. Chapter 4. T ensor Decomp osition of Sp on taneous SEEG Data 97 4.4.2 In-viv o SEEG Dataset 4.4.2.1 Estimation of Rank Figure 4.6 sho ws plots of the COR CONDIA rank metric as a function of R for the t w o sessions for b oth sub jects. P er recommendations in [ 34 ], rank should b e c hosen so that the COR CONDIA v alue is higher than 0:9 . Based on the plots in Figure 4.6 , w e selected R = 3 and 4 for the t w o sessions of Sub ject 1 and R = 5 and 3 for the t w o s essions of Sub ject 2. Figure 4.6: COR CONDIA rank metric are sho wn as a function of rank R for t w o sessions of b oth sub jects. 4.4.2.2 In tra-sub ject Net w ork Comparison W e found corresp onding consisten t comp onen ts across the t w o sessions as the pair of comp onen ts that had the largest pro duct of spatial congruence and sp ectral congruence, i.e., for eac h comp onen t (a 1 i ;b 1 i ;c 1 i ) in the first session, w e found (a 2 j ;b 2 j ;c 2 j ) in the second session suc h that j = argmax j tr((a 1 i T a 2 j )(c 1 i T c 2 j ));i = 1;2;:::;R (4.19) wherea k i ;b k i ;c k i represen t the i th spatial, temp oral and sp ectral comp onen t of thek th (k = 1;2 ) ses- sion, resp ectiv ely . Note that w e do not exp ect temp oral congruence across sessions. F or Sub ject 1, w e found three consisten t comp onen ts with large congruence pro duct (> 0:6 ). Figure 4.7 sho ws t w o of these comp onen ts, the third is sho wn in App endix K , Figure K.1 ( a ). The congruence w as 0:892 (spatial mo de) and 0:997 (sp ectral mo de) for comp onen t ( a ) and 0:984 (spatial mo de) and 0:967 (sp ectral mo de) for ( b ). 98 4.4. Result (a) Default mo de net w ork in the alpha band. (b) Motor net w ork in the b eta band. Figure 4.7: T w o consisten t comp onen ts for Sub ject 1. F or eac h pair of consisten t comp onen ts, w e sho w mo des for session 1 in red and session 2 in blue in the top ro w. F rom left to righ t: The c hannel (spatial) mo de sho ws the activ ated c hannels that participate in eac h net w ork; The time (temp oral) mo de sho ws the dynamic v ariations of eac h net w ork (only the first 10 seconds is sho wn for b etter visualization); The sp ectral mo de sho ws the frequency-dep enden t comp onen t of the tensor. In the b ottom ro w the left and middle sub-figures sho w the spatial distribution of the activ ated c hannels mapp ed on to the sub ject’s smo othed cortical surface. F or visualization purp oses, a con tact or c hannel is defined as activ ated if the v alue of the (normalized) c hannel mo de at that con tact exceeds a threshold of 0:05 in b oth sessions. The righ t sub-figure sho ws the W elc h p o w er sp ectrum of the temp oral mo de. Chapter 4. T ensor Decomp osition of Sp on taneous SEEG Data 99 The first net w ork for Sub ject 1 includes con tacts in angular gyrus, mid-temp oral gyrus, and precuneus all consisten t with activ ation in the DMN [ 38 ]. Electro des w ere not presen t in this sub ject for other regions that are t ypically included in the DMN, suc h as the sup erior fron tal gyrus (SF G). The sp ectrum for this comp onen t is dominated b y alpha rh ythms, whic h is consisten t with studies of the DMN in the EEG/MEG literature [ 35 , 82 , 157 ]. The second net w ork w e found for this sub ject con tains con tacts mainly in the somatosensory and motor cortex with a p eak in the b eta band, whic h is consisten t with sensorimotor rh ythms as rep orted in the EEG/ECoG literature [ 13 , 212 ]: b eta activit y app ears most strongly in b oth motor and somatosensory cortex in mo v emen t preparation p erio ds and steady con traction p erio ds follo wing a mo v emen t. These sub jects w ere monitored during p erio ds of natural inactivit y when w e migh t exp ect some amoun t of motor activit y , for example page-turning while reading. The temp oral mo de sho ws the p o w er en v elop e of the SEEG signals, i.e., the dynamic v ariation in p o w er across time of the sp ectral mo del. The b ottom righ t sub-figure sho ws the p o w er sp ectrum of this en v elop e estimated using the W elc h metho d [ 201 ] after high-pass filtering with a cutoff frequency 0:02 Hz to remo v e DC drift. F or b oth comp onen ts, the p o w er sp ectral densit y p eaks at a frequency of appro ximately 0:1 Hz (0:05 Hz - 0:15 Hz), whic h is similar to the dominan t frequency found in rfM RI BOLD oscillations [ 28 ]. F or Sub ject 2, w e also found three consisten t comp onen ts. Figure 4.8 sho ws t w o of the three, the remaining one is sho wn in Figure K.1 ( b ). Again, the similarit y b et w een sessions w as high with congruence 0:652 (spatial mo de) and 0:959 (sp ectral mo de) for the DMN and 0:799 (spatial mo de) and 0:957 (sp ectral mo de) for the somatomotor net w ork. In this sub ject there are electro des in the SF G, unlik e the first sub ject, and w e no w observ e that the DMN do es indeed include SF G. As with the first sub ject w e see a second strong comp onen t in the somatosensory and motor cortex. Ho w ev er, unlik e the first sub ject, the signal in this case is predominan tly m u rather than b eta. Again, this is consisten t with p erio ds of natural inactivit y , where m u rh ythms are t ypically observ ed in somato motor cortex in parallel with alpha activit y in the visual cortex during resting [ 157 ]. 100 4.4. Result (a) Default mo de net w ork in the alpha band. (b) Motor net w ork in the m u band. Figure 4.8: T w o consisten t comp onen ts for Sub ject 2. Details as for Figure 4.7 . 4.4.2.3 In ter-sub ject Net w ork Comparison When comparing the results b et w een Sub ject 1 and Sub ject 2, w e found that in b oth cases the DMN activit y is dominated b y alpha activit y while the motor net w ork is predominan tly b eta or m u, despite the fact the t w o sub jects had differen t electro de implan tation sc hemes and differen t lo cations of their epileptogenic zones (see T able 4.1 ). Chapter 4. T ensor Decomp osition of Sp on taneous SEEG Data 101 4.4.2.4 Artifact Detection SRSCPD not only can b e used to iden tify brain net w orks, it can also detect artifacts. F or example, in App endix L , Figure L.1 ( a ) sho ws a comp onen t that w as mismatc hed (i.e., lo w spatial and sp ectral congruence) b et w een the t w o sessions for Sub ject 1. Similarly , Figure L.1 ( b ) sho ws one of the comp onen ts that w as mismatc hed b et w een sessions for Sub ject 2. These mismatc hed comp onen ts are lik ely artifacts as they mostly con tain a burst or bursts of activit y on a v ery limited n u m b e r of c hannels and the time courses do not lo ok ph ysiological in nature. 4.4.2.5 Comparison to Results Using ALS Algorithm W e also applied the traditional ALS algorithm to the same datasets and compared the comp o- nen ts obtained from the t w o metho ds. ALS did not find as man y functionally distinct net w orks as SRSCPD did from eac h individual session based on the same rank selection criterion. The rank w as estimated to b e R = 2 and 3 for the t w o sessions of Sub ject 1 and R = 3 and 3 for the t w o sessions of Sub ject 2 using the ALS algorithm as sho wn in Figure M.1 in App endix M . Moreo v er, w e only found one consisten t comp onen t (a DMN sho wn in Figure N.1 ) b et w een the t w o sessions in one of the sub jects. No other consisten t comp onen ts w ere found as the maxim um congruence pro duct (Equation ( 4.19 )) w as less than 0:15 b et w een all other pairs of comp onen ts. These results sho w that ALS is not as robust as the SRSCPD algorithm esp ecially when SNR is lo w. See App endix N (Figure N.1 - Figure N.10 ) for details. 4.5 Conclusion W e ha v e describ ed a no v el framew ork for decomp osition of electro-ph ysiological data using a third-order tensor. Our SRSCPD approac h is based on the original ALS algorithm, using a w arm- start in a rank-recursiv e searc h to b oth impro v e the qualit y of results and reduce computation cost relativ e to con v en tional ALS with m ulti-start. The SRSCPD framew ork is scalable to large datasets due to its use of the w arm start. W e ha v e sho wn its application to SEEG data in t w o sub jects with epilepsy and found t w o consisten t brain net w orks across differen t sessions of recordings sev eral 102 4.5. Conclusion hours apart and in t w o differen t sub jects. In con trast, ALS found only one consisten t net w ork (the default mo de) in one sub ject. This consistency sho ws promise for the use of SRSCPD for robust iden tification of sp on taneous brain net w ork activit y from in v asiv ely recorded EEG in individual sub jects. Chapter 5 Robust Brain Net w ork Iden tification F rom Multi-Sub ject Async hronous fMRI Data , ; , — 5.1 In tro duction F unctional MRI is a p o w erful to ol for non-in v asiv ely recording in-viv o h uman brain activities. In the past t w o decades, fMRI has b een greatly p opularized for its abilit y to study t he spatio-temp oral organizations of brain activities via exploring functional connectivit y [ 74 ]. F unctional connectivit y is defined as the statistical in ter-dep endencies b et w een differen t spatial lo cations of the brain and t ypically measured using the P earson correlation b et w een pairs of time-series. Analysis of functional connectivit y can help iden tify coheren t brain activities across distributed and repro ducible brain net w orks. Recen tly , p eople ha v e sho wn that functional connectivit y tends to fluctuate o v er time [ 46 ], indicating that the stationarit y prop ert y of fMRI signals assumed b y most of the previous studies 103 104 5.1. In tro duction could not capture the dynamic nature of the brain [ 148 ]. Since then, a large b o dy of fMRI researc h ha v e re-orien ted themselv es to disco v er the dynamic functional connectivit y [ 99 , 146 ], aiming at iden tifying not only the spatial patterns (e.g., correlations b et w een differen t brain regions), but also the ev olving temp oral c hanges in resp onse to in trinsic (net w ork in teractions) or extrinsic (task- induced) factors. Indep enden t comp onen t analysis has b een widely used for brain net w ork iden tification and dynamic functional connectivit y exploration. Calhoun et al. [ 41 ] and Esp osito et al. [ 68 ] p erformed ICA on eac h individual sub ject then com bined the comp onen ts together. Other group ICA metho ds differ from eac h other in ho w data is organized b efore applying ICA. T emp oral concatenation [ 40 , 89 ] allo ws unique time-series for eac h sub ject but shared spatial maps. On the other hand, spatial concatenation [ 156 , 177 ] assumes common time-series but unique spatial maps. Although meaningful comp onen ts ha v e b een found using these ICA-based approac hes [ 43 ], they require either spatial or temp oral indep endence, whic h ma y not b e realistic as brain net w orks can o v erlap and b e correlated in b oth space and time [ 104 ]. F urther, it has b een sho wn that stabilit y or robustness of the solutions is a w ell-kno wn issue asso ciated with ICA [ 96 ], i.e., differen t indep enden t comp onen ts ma y b e obtained across ev en m ultiple runs, r esulting in an ev en w orse in terpretabilit y of the extracted net w orks from fMRI data. Higher-order tensor decomp osition is a generalization of matrix factorization. Application of tensor decomp osition, esp ecially the canonical p oly adic (CP) mo del [ 49 , 112 ] (also kno wn as parallel factors analysis [ 92 ] or canonical decomp osition [ 44 ]) to fMRI data for brain net w ork iden tification has previously b een explored. Generally , the decomp osition is p erformed using the ALS algorithm on a group-lev el fMRI study , to find common net w orks among sub jects. F or example, Andersen and Ra y ens [ 6 ] applied a third-order CP decomp osition to finger-tapping tfMRI data. Bec kmann and Smith [ 22 ] extended ICA to higher-order tensors b y imp osing an indep endence constrain t in the spatial dimension. Instead of adding an indep endence constrain t as [ 22 ], Sen and P arhi [ 160 ] imp osed an orthogonalit y constrain t in the spatial dimension as with PCA. Ho w ev er, CP decomp osition on fMRI data is not as p opular as other metho ds b ecause there are sev eral issues limit its applicabilit y to fMRI studies: Chapter 5. T ensor Decomp osition of Async hronous fMRI Data 105 Multi-sub ject group analysis on async hronous fMRI data: Almost all fMRI studies us- ing CP decomp osition w ere p erformed on tfMRI data, hoping that the temp oral dynamics w ere sync hronized across sub jects due to the sync hronous stim uli from the task designs. The temp oral sync hron y across m ultiple sub jects is a strict requiremen t for CP comp osition to w ork w ell with lo w-rank mo dels. Ho w ev er, this assumption ma y not b e satisfied ev en when an iden tical task design is used across all sub jects b ecause individual resp onses to tasks ma y differ (sometimes significan tly for higher-lev el cognitiv e tasks) in their latencies. Lo w-rank CP decomp osition will certainly fail when using differen t task designs or no task is giv en as in the case of rfMRI. Moreo v er, an y brain pro cesses indep enden t of the task will not b e iden tified using a CP decomp osition b ecause the task-indep enden t activit y will not b e sync hronized. Robustness against lo cal minima and scalabilit y to large dataset: It has b een sho wn that the ALS algorithm is not guaran teed to con v erge to a global minim um or a stationary p oin t for a CP mo del, ev en with m ulti-start [ 49 , 112 ]. Although adding additional constrain ts, suc h as indep endence [ 22 ] or orthogonalit y [ 160 ], ma y , to some exten t, help a v oid lo cal minima, those constrain ts ma y not b e ph ysiologically reasonable for brain net w ork iden tification as w e discussed ab o v e. Indeed, sp ecific concerns [ 95 , 174 ] ha v e b een raised against imp osing those constrain ts when applying the CP mo del to fMRI data. Moreo v er, the naiv e ALS algorithm do es not scale up w ell to large datasets. As w e ha v e sho wn in [ 124 , 126 ], the computational complexit y is appro ximately quadratically prop ortional to the largest dimension of the tensor. In fact, most of the studies cited ab o v e hea vily do wn-sampled the data in the spatial domain in order to ha v e a tractable CP decomp osition. Besides, if the robustness o f the solutions is strongly desired, whic h is usually the case for fMRI data where the SNR is v ery lo w, a m ulti-start strategy needs to b e emplo y ed, resulting in an ev en higher complexit y . In this pap er, w e describ e a metho d to robustly iden tify common brain net w orks, (i.e., ob- tain b oth spatial maps and temp oral dynamics sim ultaneously) across m ultiple sub jects from their async hronous fMRI data, but without imp osing an y unrealistic constrain t on the net w orks. W e approac h this problem b y incorp orating the Nadam metho d [ 64 ] in to the SRSCPD framew ork 106 5.2. Preliminaries (Chapter 4 ) [ 124 , 126 ], whic h w as designed for brain net w ork iden tification in EEG data, and com- bine it with the BrainSync [ 103 ] algorithm, whic h uses a temp oral orthogonal transform to align time-series across sub jects. W e refer to our tensor decomp osition algorithm as Nadam-accelerated scalable and robust CP decomp osition (NSR CPD). W e sho w that spatially o v erlapp ed and temp o- rally correlated brain net w orks can b e successfully iden tified from m ulti-sub ject tfMRI data using NSR CPD and the solutions are far more robust than those using the group ICA metho d [ 40 ]. 5.2 Preliminaries In this section, w e first define some mathematical notation, mostly follo wing the con v en tions in [ 112 ]. W e use a lo w ercase letter to represen t a scalar, e.g., x ; a b old lo w ercase letter for a v ector, e.g., x ; a b old upp ercase letter for a matrix, e.g., X and a b old script letter for a tensor, e.g., X . The n um b er of dimensions is called the or der and eac h dimension is referred as a mo d e . A third-order tensor X 2 R IJK will b e used in the follo wing section as an example with x i;j;k denoting its individual elemen t. Ho w ev er, notation and algorithms can b e extended naturally to higher-order tensors. A norm of a tensor X is, analogous to the F rob enius norm for a matrix, defined as: ∥X∥ = v u u t I ∑ i=1 J ∑ j=1 K ∑ k=1 x 2 i;j;k (5.1) A tensor can b e metricized in to a matrix along the n th dimension, denoted b yX (n) . Therefore, for a third-order tensor,X (1) 2R IJK orX (2) 2R JIK orX (3) 2R KIJ . 5.2.1 CP Decomp osition, the ALS Algorithm and SRSCPD The CP decomp osition appro ximates a third-order tensor X 2 R IJK as a sum of rank-1 tensors with the follo wing ob jectiv e function: f = min r;ar;br;cr 1 2 X R ∑ r=1 r a r ◦b r ◦c r 2 (5.2) Chapter 5. T ensor Decomp osition of Async hronous fMRI Data 107 where r represen ts the scale of the r th comp onen t,a r 2R I ,b r 2R J ,c r 2R K , ha v e unit norm, “◦ ” is the v ector outer pro duct and R is the r ank or the n um b er of comp onen ts. If w e concatenate thea r together, forming a matrixA = [a 1 a 2 a R ]2R IR and similarly forB2R JR and C2R KR , then Equation ( 5.2 ) c an b e re-written as one of the follo wing three equations [ 112 ]: f = min A;B;C 1 2 X (1) A(C⊙B) T 2 F (5.3) or f = min A;B;C 1 2 X (2) B(C⊙A) T 2 F (5.4) or f = min A;B;C 1 2 X (3) C(B⊙A) T 2 F (5.5) where “⊙ ” is the Khatri-Rao pro duct b et w een t w o matrices. The ALS algorithm solv es this problem iterativ ely: It first solv es for A withB andC fixed, then solv es for B withA andC fixed, and so on un til con v ergence. Sp ecifically , supp ose w e fix B and C and solv e for A (with p ossible regularization): ^ A = argmin A 1 2 X (1) A(C⊙B) T 2 F + 1 2 g 1 (A) (5.6) where g 1 (A) is a regularizing function on the loading matrixA and 1 is the corresp onding regu- larization parameter. The solution of Equation ( 5.6 ) reduces to a least square solution if 1 = 0 : ^ A =X (1) [(C⊙B) T ] y (5.7) When 1 ̸= 0 , it has a closed form expression if g 1 (A) is quadratic but ma y require an iterativ e solution in other cases. The SRSCPD framew ork w as originally designed for tensor decomp ositions of EEG data [ 124 , 126 ]. The robustness and scalabilit y w ere ac hiev ed b y using the results from lo w er-rank solutions as w arm initializations for higher-rank decomp osition. A t eac h iteration r , a rank-r fit is computed using an y CP decomp osition algorithm with initializationfA ;B ;C ; g , whic h is formed b y concatenating the resultsfA r1 ;B r1 ;C r1 ; r1 g from the previous iterationr1 with the rank-1 108 5.2. Preliminaries appro ximationfa ′ ;b ′ ;c ′ ; ′ g of the residue tensorX res , whereX res is obtained b y subtracting the reconstructed tensor usingfA r1 ;B r1 ;C r1 ; r1 g from the original data tensorX . Details are giv en i n [ 124 ]. 5.2.2 Gradien t of the CP Mo del, A dam and Nadam If w e treat the v ariables in a CP mo del as a high-dimensional v ector lying in the space ofx2R N , where N = IR +JR +KR , then ob jectiv e function f(A;B;C) in Equation ( 5.2 ) can b e though t as a scalar-v alued cost function f(x) :R N !R . Therefore, the solutions can b e obtained using a gradien t-based optimization metho d. The partial gradien t of the ob jectiv e function f with resp ect to the loading matrixA , without an y regulariza tion, is: ∇ A f =X 1 (C⊙B)+A(C T CB T B) (5.8) and lik e-wise forB andC . Pro of and detailed deriv ations are giv en in [ 2 ]. A gradien t-based searc h on the unregularized cost function will not pro duce a unique solution b ecause all solutions in the form off 1 A; 2 B; 3 Cg with 1 2 3 = 1 are equiv alen t to eac h other. T o resolv e this am biguit y , w e u se the Tikhono v regularizer: ^ A = argmin A 1 2 X (1) A(C⊙B) T 2 F + 1 2 ∥A∥ 2 F (5.9) and similarly forB andC . In this regularized case, the corresp onding gradien t b ecomes [ 2 ]: ∇ A f =X 1 (C⊙B)+A(C T CB T B)+ 1 A (5.10) and lik e-wise forB andC . A daptiv e momen t estimation (A dam) [ 109 ] is a p opular first-order solv ers used in the deep learning comm unit y [ 153 ]. Its sup erior p erformance w as ac hiev ed b y using the momen tum-based acceleration together with an adaptiv e learning rate, whic h allo ws small step size for parameters Chapter 5. T ensor Decomp osition of Async hronous fMRI Data 109 with large accum ulativ e gradien ts and large step size for parameters with small accum ulativ e gradi- en ts. Recen tly , Dozat [ 64 ] describ ed a mo dified algorithm, Nadam, in whic h Nestero v acceleration is incorp orated in to A dam. The gradien t-based up date rules are sho wn in [ 64 ] and used in the follo wing sections to compute the CP decomp osition. The default v alues for the parameters are c hosen to b e = 0:001 , 1 = 0:9 , 2 = 0:999 and ϵ = 10 8 p er Kingma and Ba [ 109 ]. 5.2.3 BrainSync F unctional MRI time series from t w o differen t sub jects are often not directly comparable. This is clearly the case for rfMRI studies where sp on taneous activit y v aries o v er sub jects and time. Ev en in ev en t-related studies, brain activit y can v ary due to differing latencies in resp onse. But to p erform group analysis based directly on time series, temp oral sync hronization across sub jects is necessary . Recen tly , Joshi et al. [ 103 ] dev elop ed a sync hronization tec hnique for fMRI data called BrainSync that addresses this problem. BrainSync assumes that the fMRI data b et w een an y t w o sub jects or sessions ha v e b een mapp ed on to a tessellated represen tation of the mid-cortical la y er of the cortex and non-rigidly aligned and resampled on to a common mesh. LetX andY b e the matrices represen ting the cortically mapp ed fMRI data for t w o sub jects, eac h of size TV , where T is the n um b er of time p oin ts andV is the n um b er of v ertices withV ≫T , whic h is t ypically true in fMRI data. BrainSync finds an orthogonal transform O S that minimizes the o v erall squared error: O S = argmin O2O(T) ∥XOY∥ 2 (5.11) whereO(T) represen ts the group of TT orthogonal matrices. The problem is w ell-p osed giv en the fact that V ≫T and the solution is computed from the SVD of the temp oral cross-correlation matrix [ 103 ]. After applying this transform, the time series at homologous lo cations in the t w o sub jects will b e aligned in the sense that they will b e highly correlated, as demonstrated in [ 103 ]. 110 5.3. Materials and Metho ds Algorithm VI I Nadam-accelerated scalable and robust canonic al p oly adic decomp osition 1: function NSR CPD (X;R ) 2: a 1 ;b 1 ;c 1 ; 1 CP-ALS(X;1 ) 3: X res X T ensor-Recon (a 1 ;b 1 ;c 1 ; 1 ) 4: fa ′ ;b ′ ;c ′ ; ′ g CP-ALS(X res , 1) 5: A [a 1 a ′ ] ;B [b 1 b ′ ] ;C [c 1 c ′ ] ; [ 1 ′ ] 6: for r = 2;3;:::;R do 7: Scale the i th comp onen ts ofA ,B ,C b y 3 √ i 8: A r ;B r ;C r Nadam(f ,fA ;B ;C g ) 9: Normalize the i th comp onen ts ofA r ,B r ,C r and store the norm pro duct in to r 10: X res X T ensor-Recon (A r ;B r ;C r ; r ) 11: fa ′ ;b ′ ;c ′ ; ′ g CP-ALS(X res , 1) 12: A [A r a ′ ] ;B [B r b ′ ] ;C [C r c ′ ] ; [ r ′ ] 13: end for 14: return a set of solutions:fa 1 ;b 1 ;c 1 ; 1 g;fA 2 ;B 2 ;C 2 ; 2 g;:::;fA R ;B R ;C R ; R g 15: end function 5.3 Materials and Metho ds 5.3.1 NSR CPD W e dev elop ed a NSR CPD algorithm whic h incorp orates Nadam in to the SRSCPD framew ork. As with SRSCPD, for eac h rank, NSR CPD uses w arm initializations from lo w er rank solutions to impro v e its scalabilit y and robustness. Ho w ev er, unlik e SRSCPD, where for eac h rank the solutions w ere obtained using the ALS algorithm, NSR CPD uses Nadam to up date all mo des sim ultaneously . It has b een sho wn that optimization-based CP algorithms are more robust than ALS but at the exp ense of a m uc h higher computational cost [ 49 , 112 ], making them exp ensiv e to scale to large datasets. Incorp orating Nadam in to the SRSCPD framew ork allo ws us to further impro v e the robustness relativ e to ALS but without sacrificing scalabilit y . Chapter 5. T ensor Decomp osition of Async hronous fMRI Data 111 The NSR CPD algorithm is outlined in Algorithm VI I . W e use Nadam to solv e the main decom- p osition problem at eac h rank from 2 to R with w arm initializationsfA ;B ;C g , where f in line 8 is the Tikhono v regularized ob jectiv e with gradien t with resp ect toA sho wn in Equation ( 5.10 ), with similar forms for B and C . Note that the comp onen ts returned from the ALS algorithm ha v e unit norms and the correct scaling factors are represen ted b y while Nadam optimizes all (un-normalized) comp onen ts sim ultaneously . Therefore, in order to ha v e Nadam start from the correct w arm initialization p oin t, w e normalize and then re-scale the comp onen ts b efore and after the Nadam pro cedure as sho wn in line 9 and 7 . 5.3.2 Sim ulation W e sim ulated third-order tensors X 2 R 20108 of rank R = 1;:::;10 from outer pro duct of factors randomly sampled factors from a standard normal distribution. W e then added Gaussian white noise to the sim ulated tensorX with a SNR of 2 . W e p erformed a CP decomp osition with desired rank R on X using the ALS algorithm, the original SRSCPD algorithm (with the ALS inside) and the NSR CPD algorithm. F or a fair comparison, w e generated and used the same random initializations for all three algorithms. W e assessed the qualit y of the solutions using the a v eraged congruence pro duct (A CP) [ 184 ]. A CP is a measure of correlation b et w een comp onen ts defined as A CP = max P tr((A T ^ AB T ^ BC T ^ C)P) (5.12) whereA;B;C b e the column-wise normalized ground truth loading matrices, ^ A; ^ B; ^ C their esti- mated coun terparts,P is a p erm utation matrix accoun ting for the am biguit y of the ordering of the solutions [ 92 ] and tr(X) is the trace ofX . It has b een sho wn that this metric is sup erior to the F rob enius norm error [ 124 ]. W e ev aluated the A CP of the solutions for eac h of the three metho ds as a function of the rank R . F or eac h rank R , w e ran 100 Mon te Carlo trials and the corresp onding b o xplots w ere generated. 112 5.3. Materials and Metho ds 5.3.3 Application to In-viv o Language T ask fMRI Data W e p erformed NSR CPD on 40 sub jects (2 sessions for eac h sub ject) of minimally pro cessed language tfMRI data, represen ted in the gra y ordinate format [ 78 ], from the HCP [ 192 ]. T ask fMRI w as used here, instead of rfMRI, b ecause the task designs and the results from the generalized linear regression (GLM) mo del [ 17 ] can b e used for v alidation purp ose. Ho w ev er, w e note that through use of BrainSync alignmen t, NSR CPD can b e applied to a range of m ulti-sub ject fMRI recording paradigms including rfMRI and self-paced ev en t-related fMRI studies. The tfMRI data w ere resampled on a cortical surface extracted from sub ject T1-w eigh ted MRI scans and co-registered to a common surface atlas. Eac h session w as represen ted as aVT matrix, where V 22K is the n um b er of v ertices across the t w o hemispheres and T = 316 is the n um b er of time p oin ts. The language task w as selected b ecause it consists of sev eral spatially o v erlapp ed net w orks that span a substan tial fraction of the cortical surface. The order of task blo c ks v aries across sessions. W e therefore applied the BrainSync algorithm to align all sessions of tfMRI datasets to the first session of the first sub ject (this reference w as HCP sub ject 100307). The temp orally aligned tfMRI data w ere then com bined along third dimension, forming a third-order data tensor X2R VTS , where S = 80 is the n um b er of sub jects (40 ) b y sessions (2 ). Analogous to rank-reduction prepro cessing metho ds used in ICA, w e p erformed a greedy CP decomp osition [ 3 ] to the tensorX to reduce its rank to 20 . Sp ecifically , w e recursiv ely fit a rank- 1 comp onen t to the data tensor and then subtracted this from the residual data tensor un til w e had found20 comp onen ts in total. Leta i 2R V ,b i 2R T ,c i 2R S b e thei th normalized spatial, temp oral and session comp onen t found b y the greedy CP decomp osition and i b e the corresp onding norm. Then w e reconstructed the rank-reduced tensor as Y = ∑ 20 i=1 i a i ◦b i ◦c i . Next, w e applied the NSR CPD algorithm to the rank-reduced tensor Y to extract brain net w orks with a desired rank of 20 . The rank 20 here is c hosen to matc h the rank used in the group ICA metho d [ 40 ] as a fair comparison b elo w. W e used a non-negativit y constrain t on the session mo de since w e assumed eac h sub ject could either participate or not participate in a net w ork but could not negativ ely con tribute to the net w ork. Sp ecifically , w e used Chapter 5. T ensor Decomp osition of Async hronous fMRI Data 113 ^ A = argmin A 1 2 X (1) A(C⊙B) T 2 F + 2 ∥A∥ 2 F (5.13) and ^ B = argmin B 1 2 X (2) B(C⊙A) T 2 F + 2 ∥B∥ 2 F (5.14) for the spatial and temp oral mo de and the follo wing ob jectiv e function for the session mo de within the Nadam up date: ^ C = argmin C 1 2 X (3) C(B⊙A) T 2 F + 2 ∥C∥ 2 F ; s.t. C⪰ 0 (5.15) where⪰ represen ts the elemen t-wise inequalit y , is c hosen to b e 0:001 exp erimen tally based on the data for all three mo des. 5.3.4 Comparison of the Solutions and Stabilit y with Group ICA F ollo wing the pro cedures describ ed in [ 40 ], group ICA w as applied to the same language tfMRI dataset as a comparison, where the (temp orally) PCA-denoised (to rank 40 ) individual language tfMRI data w as temp orally concatenated, then the temp oral dimensionalit y w as further reduced to 20 using PCA again and a s patial group ICA p erformed to extract indep enden t comp onen ts [ 40 ]. As noted ab o v e, the stabilit y or robustness is a ma jor issue with ICA-based metho ds. Therefore, here w e in v estigated the stabilit y of net w orks obtained from the language tfMRI data using our NSR CPD algorithm relativ e to those obtained using the group ICA metho d. W e ran b oth the NSR CPD and the group ICA on the same dataset 100 times, with differen t random initializations at eac h run. Th us, appro ximately 20 100 = 2000 spatio-temp oral comp onen ts w ere obtained in total for eac h metho d (the n um b er of extracted comp onen ts is sligh tly less than 2000 in the ICA case as ICA ma y not alw a ys con v erge and find all 20 comp onen ts ev ery time). Then w e pro jected the obtained spatial maps non-linearly on to a 2D plane using the curvilinear comp onen t analysis (CCA) algorithm [ 61 ] pro vided b y the ICASSO soft w are [ 96 ] for visualization and comparison purp oses. F or our NSR CPD algorithm, w e color-co ded the pro jected spatial maps with the n um b er of participating sub jects for eac h net w ork. A sub ject is defined to b e in v olv ed in a particular 114 5.4. Results net w ork if an y of the t w o normalized session mo des exceeded 0:05 for that sub ject. The threshold w a s c hosen heuristically based on the o v erall histogram of the session mo des from all decomp osition results. Moreo v er, w e also b o otstrapp ed the data b y randomly resampling the data in the session mo de with replacemen t and rep eating the decomp ositions using b oth the NSR CPD metho d and the group ICA metho d. 5.3.5 The Imp ortance of BrainSync and Nadam T o demonstrate that b oth the optimization-based CP decomp osition algorithm (Nadam in our c hoice) and the sync hronization pro cedure using BrainSync are necessary to successfully extract brain net w orks, w e rep eated our exp erimen t as describ ed ab o v e three more times: 1) using the orig- inal SRSCPD (with ALS inside) [ 124 ] directly on all 80 sessions (40 sub jects 2 sessions/sub ject) of BrainSync-ed dataset; 2) using NSR CPD on all 80 sessions of un-sync hronized (no BrainSync) dataset; 3) using NSR CPD only on the first sessions of the 40 sub jects’ un-sync hronized (no Brain- Sync) dataset. 5.3.6 Application to In-viv o Resting fMRI Data W e also p erformed NSR CPD (with BrainSync) on the same set of 40 sub jects’ (there are also 2 sessions for eac h sub ject) of minimally pro cessed rfMRI data from HCP . The exact same pro cessing pip e line, as with the language tfMRI data, w as applied to the rfMRI dataset, except that for eac h sub ject the length of rfMRI recordings w as truncated to the first 5 min utes for computational tractabilit y . Hence, in this case the final tensor X2R VTS , where V 22K , T = 417 (300 sec 0:72 sec/sample) and S = 80 (40 sub jects 2 sessions/sub jects). 5.4 Results 5.4.1 Sim ulations Figure 5.1 sho ws the b o xplots of the A CP o v er 100 trials as a function of the rank R for eac h of the three metho ds: ALS, SRSCPD with ALS inside, and NSR CPD. WhenR is small, they p erform Chapter 5. T ensor Decomp osition of Async hronous fMRI Data 115 Figure 5.1: Sim ulation results. Bo xplots of A CP o v er 100 Mon te Carlo trials are sho wn as a function of R almost equally w ell. Ho w ev er, asR increases, NSR CPD outp erforms the SRSCPD with ALS solv er as w ell as the original ALS algorithm b y a margin that increases with rank, indicating robustness of the NSR CPD algorithm. 5.4.2 Application to In-viv o Language T ask fMRI Data Fifteen plausible net w orks w ere iden tified b y the NSR CPD metho d. Figure 5.2 sho ws 8 recog- nizable brain net w orks out of the 15 . F or eac h net w ork, the left column sho ws the spatial map, the middle column sho ws the temp oral dynamics o v erlaid with the color-co ded task design blo c ks (math tasks sho wn in red and story tasks sho wn in blue) f or the reference sub ject and the righ t column sho ws the a v eraged session mo de o v er t w o sessions, roughly reflecting the participation lev el for eac h sub ject. Recall that task-timing v aries across sessions. The results here are aligned to the timing for the first (reference) sub ject. Applying the appropriate in v erse BrainSync transform w e can find the corresp onding net w ork dynamics for all other sub jects. The temp oral dynamics of Figure 5.2 ( a ), ( e ), ( f ), and ( g ) sho w p eaks at the b eginning of eac h task blo c k, whic h visually correlate w ell with the cues for eac h task, with a short dela y consisten t 116 5.4. Results (a) (b) (c) (d) (e) (f ) (g) (h) Figure 5.2: NSR CPD results on language tfMRI data: (a) F ron to-parietal atten tional con trol net w ork (FP A CN); (b) Extended language net w ork (Lang); (c) Default mo de net w ork (DMN); (d) Respiration activit y (Resp); (e) Reading net w ork (RN); (f ) A uditory net w ork (AN); (g) Visual net w ork (VN); (h) Sensorimotor activit y near the tongue area (T). with the effects of the hemo dynamic resp onse function. The spatial pattern of these net w orks indicates a fron to-parietal atten tional con trol net w ork (FP A CN) [ 98 ] for ( a ); a reading net w ork Chapter 5. T ensor Decomp osition of Async hronous fMRI Data 117 (RN) [ 70 ] for ( e ); an auditory net w ork (AN) for ( f ) and a visual net w ork (VN) for ( g ), all reflecting the sub jects’ resp onses to the task cues. Figure 5.2 ( b ) sho ws an extended language net w ork (Lang) reflecting the sub jects’ resp onse to the language task and is consisten t with the result sho wn in Figure 8 in [ 17 ], whic h w as obtained using the GLM mo del from 77 sub jects. Ho w ev er, unlik e the GLM mo del, the net w ork obtained via NSR CPD w as naturally extracted from the data without using an y prior information regarding the task designs. The spatial map for Figure 5.2 ( c ) sho ws a t ypical DMN. The DMN w as first kno wn as a task- negativ e net w ork [ 149 ] and in fact a strong negativ e correlation b et w een the temp oral mo de of ( c ) and the task blo c ks can b e clearly observ ed (dips within task blo c ks and p eaks in b et w een task blo c ks). Figure 5.2 ( d ) sho ws a spatially global, temp orally relativ ely fast ( 0:3 Hz) and non-task-related activit y , suggesting that it ma y represen t a residual respiration (Resp) effect common across sub- jects. Figure 5.2 ( h ) also sho ws a non-task-related sensorimotor activit y around the tongue (T) area that can b e v erified b y insp ection of the spatial mo de. Note that it is the sync hronization of non- task-related activit y across sub jects using BrainSync that allo ws us to iden tify these comp onen ts (( d ) and ( h )) from the third-order tensor in addition to the task-related net w orks. Finally , the sub ject mo des for all iden tified 15 net w orks are appro ximately uniformly distributed across 40 sub jects, indicating that these net w orks are indeed common brain net w orks across all sub jects. Other 7 un-recognized brain net w orks also exhibited smo oth spatial maps whic h seem to b e plausible. One example is sho wn in Figure O.1 ( a ) in App endix O where homologous regions, suc h as the p osterior cingulate gyrus and part of the precuneus, on b oth hemispheres are highly activ e in this net w ork. The w eigh ts along the sub ject mo de are also appro ximately equally distributed, indicating that these 7 net w orks are common brain net w orks across all 40 sub jects as w ell. F urther insp ection of the remaining 5 net w orks rev ealed that their sub ject mo des had a high v alue for one sub ject and w ere appro ximately zero for all other sub jects, indicating that these 5 comp onen ts 118 5.4. Results reflect net w orks particular to a single sub ject. Figure O.1 ( b ) sho ws one suc h net w ork out of the 5 as a n example in App endix O . T able 5 .1: Summary of brain net w orks iden tified in language task fMRI data NSR CPD Group ICA SRSCPD NSR CPD NSR CPD BrainSync Y es Not Applicabl e Y es No No Num b er of sessions 80 80 80 80 40 * Iden tifiable or Recognizable 8 F ron to-parietal atten tional con trol net w ork 2 6 3 † 2 Extended language net w ork Extended language net w ork Extended language net w ork ‡ † Extended language net w ork ‡ Default mo de net w ork Respiration Respira- tion Reading net w ork Extended language net w ork A uditory net w ork A uditory net w ork A uditory net w ork A uditory net w ork A uditory net w ork ‡ Visual net w ork Visual net w ork Visual net w ork Motor net w ork Motor net w ork Plausible but unrecognized 7 8 4 2 0 Unkno wn (sub ject- sp ecific or noise) 5 10 10 15 18 T otal 20 20 20 20 20 * Only the firs t sessions from all 40 sub jects w ere used, resulting in 40 sessions in total. † T w o extended language net w orks w ere found, one for eac h session. Ho w ev er, these t w o net w orks exhibited incomplete spatial patterns of the extended language net w ork. ‡ Only a subset of sub jec ts sho ws participation in this net w ork. 5.4.3 Comparison of the Solutions and Stabilit y with Group ICA T en plausible brain net w orks w ere iden tified using the group ICA metho d, only 2 of whic h can b e recognized as visual and auditory net w orks as sho wn in Figure P .1 in App endix P , similar to net w orks ( g ) and ( f ) found using the NSR CPD metho d. Other net w orks found using the NSR CPD metho d, suc h as FP A CN, DMN, language net w ork, and etc., w ere not ob viously iden tifiable in the Chapter 5. T ensor Decomp osition of Async hronous fMRI Data 119 ICA results. W e b eliev e the reason for this is that those net w orks spatially o v erlap across large cortical regions (e.g., high spatial similarit y w as observ ed b et w een the DMN and the language net w ork in Figure 5.2 ), breaking the assumption of spatial indep endence required b y the group ICA metho d. A summary of brain net w orks obtained using NSR CPD (with BrainSync) and group ICA is sho wn in T able 5.1 (2 nd and 3 rd column). Figure 5.3 ( a ) and ( b ) sho w scatter plots of the pro jected spatial maps obtained using the group ICA metho d and the NSR CPD metho d, resp ectiv ely , with differen t random initializations. The coun terparts sho wn in ( c ) and ( d ) are for b o otstrapp ed data, again with differen t random initialization in eac h case. Eac h dot represen ts a single comp onen t. The lo cations of the iden tified brain net w orks as sho wn ab o v e (8 net w orks for NSR CPD and 2 net w orks for group ICA) are mark ed with blue text b eside the comp onen ts. First of all, regarding the spatial relationship among the comp onen ts, the tigh ter a cluster is, the more similar the comp onen ts are within that cluster and therefore the more robust the algorithm is. With differen t initializations, the comp onen ts found b y the group ICA metho d sho wn in ( a ) tend to form m ultiple clusters, but with s ubstan tial differences among the comp onen ts within clusters. In con trast, the comp onen ts obtained from the NSR CPD metho d sho wn in ( b ) form v ery dense clusters. Closer examination rev ealed that in ( b ) the clusters that con tained the 8 iden tified brain net w orks from the original non-b o otstrapp ed language tfMRI dataset (those with blue text aside) had exactly 100 individual comp onen ts, i.e., ev ery single run of NSR CPD pro duced those 8 common brain net w orks. An example is sho wn on the b ottom-righ t corner in ( b ). The difference b et w een the group ICA metho d and the NSR CPD metho d b ecomes more appar- en t in the b o otstrap analysis. It is difficult to visually separate the comp onen ts in to more than 3 or 4 clusters in the group ICA case as sho wn in ( c ), whereas the comp onen ts in the NSR CPD case sho wn in ( d ) still can b e clearly group ed in to m ultiple clusters. There are 8 clusters with dense cen troids that corresp ond to the 8 net w orks in Figure 5.2 , indicating that they are robustly iden- tified in eac h of the b o otstrap runs. F urthermore, the red color indicates that almost all sub jects exhibited that net w ork in eac h of the b o otstrap runs. Con v ersely the blue clusters corresp ond to 120 5.4. Results (a) (b) (c) (d) Figure 5.3: Stabilit y testing results: (a) T he scatter plot of the CCA pro jected spatial maps ob- tained using the group ICA metho d with differen t random initializations only; (b) The coun terpart to (a) when using NSR CPD; (c) Same as (a) but on the b o otstrapp ed dataset; (b) The coun terpart to (c) when using NSR CPD. The pro jected lo cations of the iden tified brain net w orks ( 2 net w orks for ICA and 8 net w orks for NSR CPD) are sho wn with blue text aside. F or (b) and (d), for eac h comp onen t (a single dot), the color represen ts the n um b er of sub jects in v olv ed in that comp onen t. The big circle on the b ottom-righ t corner in (b) sho ws the zo omed-in v ersion of an example cluster. Chapter 5. T ensor Decomp osition of Async hronous fMRI Data 121 net w orks that are sp ecific to a single sub ject and repro duced for that sub ject in m ultiple b o otstrap runs. 5.4.4 The Imp ortance of BrainSync and Nadam T able 5.1 (4 th column) sho ws the results using SRSCPD (with ALS inside) on the sync hronized full dataset. The FP A CN and DMN w ere not found using SRSCPD, although other iden tified net- w o rks visually iden tical to those found using NSR CPD (with BrainSync), indicating the impro v ed robustness using the Nadam solv er relativ e to the ALS algorithm. When NSR CPD w as used on the full dataset but without BrainSync, as sho wn in T able 5.1 (5 th column), only the language net w ork with incomplete spatial patterns and the auditory net w ork w ere iden tified on a subset of sub jects. This is exp ected as lo w-rank CP mo dels will not w ork prop erly when using differen t task designs across sessions without sync hronization, as w e discussed early . Ho w ev er, ev en when iden tical task design w as used, without sync hronization, net w orks w ere not all successfully iden tified. As sho wn in T able 5.1 (last column) where only the first sessions from 40 sub jects w ere used (in HCP data, task design in a single session w as iden tical across all sub jects), only the language and the auditory net w ork w ere found but on a subset of sub jects due to the differing latencies of the sub jects’ resp onses to task designs. Moreo v er, brain net w orks that are indep enden t of the tasks, suc h as the respiration and the sensorimotor activit y , w ere not iden tified without BrainSync on either the full dataset (T able 5.1 , 5 th column) or the reduced dataset (T able 5.1 , 6 th column), b ecause the task-indep enden t activit y will not b e sync hronized. Therefore, the comparison results in T able 5.1 sho ws that it’s the usage of b oth the optimization- based NSR CPD and the sync hronization across sub jects via BrainSync algorithm that allo ws us to robustly iden tify brain net w orks from async hronous fMRI data. 5.4.5 Application to In-viv o Resting fMRI Data When our NSR CPD w as applied to rfMRI data with the same desired rank of 20 , w e successfully found 12 comp onen ts (sub-net w orks) that constitute 5 recognizable large-scale net w orks when they 122 5.5. Discussion (a) (b) (c) (d) (e) (f ) Figure 5.4: NSR CPD results on rfMRI data: (a) Default mo de net w ork (DMN); (b) Executiv e con trol net w ork (ECN); (c) Sensorimotor net w ork (SMN); (d) Visual net w ork (VN); (e) A uditory net w ork (AN). are prop erly iden tified and com bined as sho wn in Figure 5.4 . The spatial maps of these 5 large-scale net w orks sho w a DMN in ( a ); an executiv e con trol net w ork [ 159 ] in ( b ); a sensorimotor net w ork (SMN) in ( c ); a visual net w ork in ( d ) and an auditory net w ork in ( e ). Individual comp onen ts (sub-net w orks) are sho wn in Figure Q.1 in App endix Q . Finally , the uniform sub ject mo de of these net w orks suggests that they are indeed common net w orks across all 40 sub jects in the rfMRI data. 5.5 Discussion Using NSR CPD with BrainSync, w e iden tified 8 spatially o v erlapp ed and temp orally correlated common net w orks across m ultiple sub jects: fiv e task-related (related to language task, visual cues, Chapter 5. T ensor Decomp osition of Async hronous fMRI Data 123 atten tional) net w orks, a DMN, respiration effect and a sensorimotor activit y in the language tfMRI data. Although w e did not use an y prior information regarding the task designs, our results not only replicated the task timing, but also sho w ed exp ected differences in the temp oral dynamics of those net w orks. These net w orks w ere not all found using the group ICA metho d or when either BrainSync or Nadam w as not used. The b o otstrapping results sho w that NSR CPD is m uc h more robust in iden tifying net w orks in comparison with the group ICA metho d. Finally , when our metho d w as applied to the rfMRI data, fiv e large-scale net w orks w ere successfully iden tified, whic h are common across all the sub jects. A limitation of this study is that only 40 HCP sub jects w ere used and the time series w ere truncated to5 min utes in the rfMRI cases (the recordings are shorter than5 min utes in tfMRI cases). Using more sub jects and longer recordings will further impro v e the robustness of the estimates of brain net w orks, ho w ev er, the high c omputational complexit y in b oth memory requiremen t and pro cessing time is still a c hallenging problem for us to scale up to larger dataset, alb eit our NSR CPD is m uc h faster than the naiv e ALS algorithm. This page in ten tionally left blank P art IV F unctional-Boundary-Preserving Noise Reduction and Filtering 125 Chapter 6 P arameter Selection for Optimized Non-lo cal Means Filtering , — 6.1 In tro duction Dynamic fMRI indirectly reflects neuronal activit y b y measuring BOLD c hanges in image con- tract. Strong temp oral correlations are eviden t in these data b et w een ph ysiologically related corti- cal regions, ev en during p erio ds of rest [ 28 ]. Quan tifying correlations b et w een m ultiple brain areas forms the basis for brain net w ork iden tification from rfMRI [ 167 ]. While the role and analysis of tfMRI are quite differen t, the BOLD time-series in these data will similar ly sho w strong correlations within and b et w een regions that resp ond to a particularly cognitiv e c hallenge [ 37 ]. BOLD-related con trast c hanges are small and noise in the data limit o ur abilit y to reliably iden tify regions of sp on taneous or task-related activit y . Both resting and task-related fMRI are t ypically spatially filtered to reduce noise, in addition to other prepro cessing steps. Ev en after extensiv e prepro cessing, iden tification of brain net w orks from correlation patterns in rfMRI and iden tification of activ e regions using a GLM in tfMRI can still b e c hallenging. 126 Chapter 6. P arameter Selection for Optimized Non-lo cal Means Filtering 127 Spatial smo othing is applied to the fMRI data either v olumetrically with 3D isotropic Gaussian k ernel [ 169 ], or on data mapp ed on to a 2D represen tation of the cortical surface using the LB op erator [ 8 ]. Both metho ds suffer from the common problem that in addition to reducing noise, they also inevitably spatially mix signals b et w een adjacen t regions of functional sp ecialization. Recen tly , NLM filtering has b een applied for structural-preserving denoising of anatomical MRI [ 53 , 131 ], fMRI [ 24 , 214 ] and diffusion MRI [ 206 ]. All of these applications compute the NLM k ernels based on spatial similarit y measures similar to that in the pap er that originally describ ed this approac h [ 36 ]. Recen tly , Bh ushan et al. [ 26 ] describ ed a v arian t on NLM that filters spatio- temp oral data based on measures of similarit y in the time series rather than spatial similarit y . W e refer to this as tNLM and demonstrated its effectiv eness in denoising of rfMRI data. By filtering based on temp oral similarit y , tNLM will reduce noise b y a v eraging o v er regions in the image that ha v e similar functional roles without blurring across functional b oundaries b et w een regions with distinct temp oral activit y . P erformance of tNLM is dep enden t on a parameter h that determines the form of the mapping from temp oral correlation to the filter k ernel w eigh ts. T o o small a c hoice will cause inadequate smo othing and less SNR impro v emen t. On the other hand, to o large a v alue will result in o v er- smo othing and blurring b et w een functional regions. Buades et al. [ 36 ] empirically suggested setting h = 10 , where is the SD of the noise. In application to MRI, Manjon et al. [ 131 ] exhaustiv ely searc hed the parameter space and found h = 1:2 to b e the b est c hoice. Since the optimal h is not only a function of the noise lev el, but also a function of the blo c k size o v er whic h filtering is p erformed, Coup e et al. [ 53 ] dev elop ed a metho d for selecting h automatically b y normalizing the l 2 distance and estimating the noise v ariance. They found that an optimal c hoice of h = p 2^ , where is a man ually-tuned constan t. None of these approac h extends directly to tNLM where the temp oral nature of the data needs to b e accoun ted for. F urther these metho ds are not optimal for the k ey application considered here: preserv ation of discrete functional regions while also reducing noise. 128 6.2. Metho ds Here w e describ e a new approac h to selecting the tNLM parameterh to optimize differen tiation of regions that are functionally connected from those that are not. W e describ e the approac h b elo w and presen t ev aluation based on sim ulated and exp erimen tal tfMRI data. 6.2 Metho ds 6.2.1 T emp oral NLM Filtering W e apply tNLM filtering to fMRI data defined on the v ertices of a tessellated represen tation of the mid-cortical surface. Let s(i;t) b e the time series data at v ertex i and time t . LetN(i) denote the set that con tains i and all of its k -hop neigh b ors. Then tNLM filtering is defined as s ′ (i;t) = 1 ∑ j2N(i) w(i;j) ∑ j2N(i) s(j;t)w(i;j) (6.1) where w(i;j) is the w eigh t applied when a v eraging across v ertices. This w eigh t dep ends on a temp oral measure of similarit y , whic h w e define as w(i;j) = exp 0 @ 1 T ⃗ s(i) ∥⃗ s(i)∥ ⃗ s(j) ∥⃗ s(j)∥ 2 h 2 1 A (6.2) where ⃗ s(i) = [s(i;1);:::;s(i;T)] T is the v ector represen tation of the time series at v ertex i with length T and h is the scalar parameter that con trols the degree of smo othing. As noted in [ 26 ], the distance 1 T ∥⃗ s(i)/∥⃗ s(i)∥⃗ s(j)/∥⃗ s(j)∥∥ 2 b et w een an y pair of v ertices in Equation ( 6.2 ) can b e expressed as 22 ^ r(⃗ s(i);⃗ s(j)) where ^ r is the sample correlation co efficien t b et w een ⃗ s(i) and⃗ s(j) . W e can rewrite tNLM filtering in matrix form. Let X2 R NT b e the data matrix with N v ertices and T time samples. Then the w eigh ting matrixW can b e expressed as W(i;j) = 8 > > < > > : exp ( 2(1A(i;j)) h 2 ) ;j2N(i) 0 ;j̸2N(i) (6.3) Chapter 6. P arameter Selection for Optimized Non-lo cal Means Filtering 129 whereA =XX T 2R NN is the data correlation matrix. W e can further define the degree matrix D to b e a diagonal matrix whose diagonal elemen t d ii = ∑ j W i;j . Then the tNLM filtered signal Y can b e written as Y =D 1 WX (6.4) 6.2.2 Optimization of tNLM P arameter h The parameter h in tNLM filtering determines the degree of noise reduction and smo othing. Here w e fo cus on its application to spatio-temp oral fMRI data represen ting brain net w orks. Eac h net w ork is made up of a n um b er of discrete areas of functional sp ecialization in the brain. W e w an t to select the parameter to maximize the SNR in the tNLM-filtered data within eac h net w ork but without mixing the signals b et w een them. T o do this w e assume that the matrixW = f(A) in Equation ( 6.3 ), defines a graph with no des represen ting the v ertices on the cerebral cortex and the edge strength b et w een them giv en b y the elemen ts ofW . Our goal is to select h so that the graphW optimally differen tiates, in terms of edge strength, b et w een pairs of v ertices in the same net w ork, and pairs in differen t net w orks. In this w a y the tNLM filter will lead to impro v ed SNR while minimizing mixing of signals b et w een distinct functional net w orks. Let the observ ed signal b ex i =s i +n i , a sup erp osition of the true signals i and the noisen i at v ertex i . Assume s i and n i are indep enden t with s i N(0; 2 s ) and n i N(0; 2 n ) . F urther assume p erfect correlation within eac h net w ork (H 1 ) and zero b et w een net w orks (H 0 ) with resp ect to the tru e signals i . Then the correlation has the follo wing form: = E[x i x j ] x i x j = 8 > > < > > : 0 ;H 0 : E[s i s j ] 2 s = 0 2 s 2 s + 2 n ;H 1 : E[s i s j ] 2 s = 1 (6.5) where x i represen ts the SD of x i . The sample correlation r will v ary from this exp ected v alue according to the distribution [ 71 ] P(r) = (T2)(T1)(1 2 ) T1 2 (1r 2 ) T4 2 p 2(T 1 2 )(1r) T 3 2 2 F 1 ( 1 2 ; 1 2 ; 2T1 2 ; r +1 2 ) (6.6) 130 6.2. Metho ds Figure 6.1: The distributio n of the elemen ts of the correlation matrixA for zero true correlation (blue) and 0:2 correlation (red), with the k ernel function in Equation ( 6.2 ) ev aluated for differen t v alues of the parameter h . where T is the n um b er of samples and 2 F 1 (a;b;c;z) is the Gaussian h yp ergeometric function. An example of the distribution of the elemen ts of correlation matrix A , for cases with (red curv e) and without (blue curv e) correlation are sho wn in Figure 6.1 . F or large T and small , cases t ypical in fMRI, w e can appro ximate these distributions as normal. T o optimally differen tiate connections within and b et w een net w orks, w e select h to maximize the exp ected v alue ofW(i;j) forH 1 and minimize it forH 0 . T o accoun t for differen t frequencies of o ccurrence of H 0 and H 1 , w e w eigh t these exp ected v alues b y their resp ectiv e probabilities P(H 0 ) and P(H 1 ) . T o ac hiev e this w e solv e the optimization problem ^ h = argmax h E[f tNLM (r;h)jH 1 ]P(H 1 )E[f tNLM (r;h)jH 0 ]P(H 0 ) = argmax h ∫ 1 1 f tNLM (r;h)P H 1 (r)dr P(H 1 ) ∫ 1 1 f tNLM (r;h)P H 0 (r)dr P(H 0 ) = argmax h ∫ 1 1 f tNLM (r;h)[P(H 1 )P H 1 (r)P(H 0 )P H 0 (r)]dr (6.7) where f tNLM (r;h) = exp(2(1r)/h 2 ) . P(H 1 ) and P(H 0 ) are the relativ e frequencies of H 1 and H 0 with P H 1 (r) and P H 0 (r) . F or illustration, w e ha v e o v erlaid the function f tNLM (r;h) for sev eral v alues of h on the sample correlation distributions in Figure 6.1 . Note that to solv e Equation ( 6.7 ) w e m ust first learn the PDF s. W e do this b y fitting a bimo dal Gaussian mixture mo del (GMM) Chapter 6. P arameter Selection for Optimized Non-lo cal Means Filtering 131 to the correlation data inA , rather than using the precise form in Equatio n ( 6.6 ). W e then use a directed line searc h to find the optimal h . This then determines the mappingW =f(A) whic h is used in turn to generate the filtered data according to Equation ( 6.4 ). W e summarize our metho d in A lgorithm VI I I : Algorithm VI I I Optimal selection of parameter h 1: Compute the correlation matrixA =XX T 2: Fit a bimo dal GMM to elemen ts ofA using an EM algorithm to find P H 1 (r) , P(H 1 ) , P H 0 (r) , P(H 0 ) 3: Find the optimal ^ h b y solving Equation ( 6.7 ) using P H 1 (r) , P(H 1 ) , P H 0 (r) , P(H 0 ) obtained from the GMM 6.2.3 Exp erimen ts and P erformance Ev aluation 6.2.3.1 Sim ulation W e sim ulated data matrixX with K = 5 net w orks and 100 v ertices in eac h net w ork, T = 80 time samples and SNR = 2 s / 2 n = 0:25 . These parameters w ere c hosen so the distribution of the data in the correlation matrixA from the sim ulated data visually matc hed that for our fMRI data. The neigh b orho o dN(i) w as c hosen to b e the en tire set of v ertices. The optimal h w as obtained b y running Algorithm VI I I for 1000 Mon te Carlo trials. The v alue of the cost function in Equation ( 6.7 ) w as then ev aluated as a function of h for eac h of the trials as sho wn in Figure 6.2 , whic h yielded an a v erage optimal v alue of h = 0:49 (0:03 SD). T o ev aluate the impact of differen t v alues of h on iden tifying net w orks, w e generated the filtered dataY according to Equation ( 6.4 ), and used the correlation matrixC =YY T to partition the data in to 5 net w orks using a No rmalized Cuts (NCuts) algorithm [ 163 ]. W e then computed a p erformance measure using the A djusted Rand Index (ARI) [ 151 ] b et w een the resulting net w ork lab els and the ground truth. Finally , w e plotted the ARIs as a function of h in Figure 6.3 . The v alue that actually pro duced the highest ARI, a v eraged o v er 1000 trials, w as h = 0:52 (0:06 SD), compared to the a v erage optimal v alue h = 0:49 (0:03 SD) selected b y our algorithm. 132 6.2. Metho ds Figure 6.2: Cost function in Equation ( 6.7 ) ev aluated as a function of h under the sim ulation settings. The ligh t gra y curv es are 1000 individual Mon te Corlo trials, the blue curv e is the mean and t he t w o red curv es are one SD a w a y from the mean for eac h h . Figure 6.3: ARI metric of the clustering results ev aluated as a function of h under the sim ulation settings. The colors of curv es ha v e the same meaning as those in Figure 6.2 Chapter 6. P arameter Selection for Optimized Non-lo cal Means Filtering 133 Figure 6.4: Cost function in Equation ( 6.7 ) ev aluated as a function of h for the LP task (blue) and the SC task (red), resp ectiv ely . 6.2.3.2 Application to T ask fMRI Data Our earlier application of tNLM w as to rfMRI [ 26 ], here w e fo cus on filtering of tfMRI. T o ev aluate p erformance w e used a single sub ject minimally pro cessed tfMRI data set from the HCP [ 192 ]. The minimal prepro cessing pip eline for HCP tfMRI data is describ ed in [ 78 ]. W e briefly summarized here: fMRI data w ere acquired with TR = 720 ms, TE = 33:1 ms, 222 mm v o xels with t w o indep enden t sessions. A cquisition artifacts including head motion and spatial distortion w ere corrected and the data w ere co-registered with the T1 structural image and resampled on to the 32K -v ertex cortical surface. W e applied tNLM to t w o data sets: the Language Pro cessing (LP) task and the So cial Cognition (SC) task [ 17 ]. W e ev aluated p erformance as describ ed b elo w using the con trast of “Story” vs rest for LP and “Random” In teraction vs rest for SC b ecause t hese t w o con trasts sho w ed activ ated adjacen t functional areas, allo wing us to ev aluate whether optimal-tNLM is able to preserv e the separation of these areas after filtering. T o ev aluate the efficacy of our approac h, the LP and SC fMRI data w ere pro cessed separately to find the optimal h for tNLM filtering and analyzed as describ ed b elo w. T o ac hiev e computational tractabilit y , the cortical surface w as do wnsampled to 11K v ertices prior to filtering. W e used the 134 6.2. Metho ds Figure 6.5: Com bined represen tativ e z-score maps ( = 0:05 ) for “Story” con trast in LP (orange) and “Random” con trast in SC (blue). R OI for analysis con taining activ ated regions in b oth con trasts is i ndicated in sup erior temp oral gyrus. data from the righ t hemisphere only to obtain the optimal h based on Algorithm VI I I for LP (h = 0:43 ) and SC (h = 0:45 ) separately . The v alue of the cost function in Equation ( 6.7 ) w as also ev aluated as a function of h as sho wn in Figure 6.4 for the t w o tasks. Eac h data set w as filtered using tNLM (with the neigh b orho o d c hosen as the 11 -hop neigh b ors for eac h v ertex i , as in [ 26 ]) for m ultiple differen t v alues of the parameter h . F or comparison the data w as also isotropically smo othed with FSL ( http://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FSL ) for m ultiple differen t v alues of the FWHM parameter s of the Gaussian smo othing k ernel. W e then applied FSL’s lev el 1 and lev el 2 analyses, whic h fits a GLM to the data [ 78 ]. As a result of this pro cessing, w e obtained a v ertex-wise cortical map of z-scores for the “Story” and “Random” con trasts for m ultiple lev els of Gaussian and tNLM smo othing. The z maps w ere thresholded for t w o differen t uncorrected lev els (0:02 , 0:05 ). The uncorrected v alues w ere used as w e w ere in terested in exploring consistency of and differences in smo othing results across differen t lev els o v er all v ertices. Based on a visual comparison of results for the t w o tasks, w e selected an region of in terest (R OI) in the sup erior temp oral gyrus of the righ t hemisphere, whic h con tains adjacen t regions of Chapter 6. P arameter Selection for Optimized Non-lo cal Means Filtering 135 (a) Alpha = 0:05 (b) Alpha = 0:02 Figure 6.6: Mean z score vs Dice’s co efficien t for differen t smo othing parameters and differen t lev els. The red curv e represen ts the result under tNLM smo othing and the blue curv e represen ts the result under isotropic smo othing. The parameter v alues for h and s are annotated ab o v e the curv es. activ ation from the t w o tasks as sho wn in Figure 6.5 . Let R s denote the region within the R OI exceeding the -threshold for “Story” and R r its coun terpart for “Random” . W e computed the mean z-score o v er the en tire R OI vs the Dice co efficien t [ 63 ] b et w een R s and R r as a function of b oth h and s and for t w o differen t v alues. The results are sho wn in Figure 6.6 . W e can conclude from these figures that 1. F or isotropic linear smo ot hing: with increased smo othing the a v erage z-score (reflecting strength of resp onse) increases along with the Dice co efficien t. Th us, filtering will help detec- tion of activ ation (increased z-score) but at the exp ense of blurring (increased Dice co efficien t) b et w een the t w o distinct but adjacen t func tional areas. 2. F or tNLM smo othing, with increased lev els of smo othing there is a range o v er whic h the a v erage z-score increases without a corresp onding increase in Dice co efficien t. This indicates impro v ed abilit y to detect activ ation, but no w without blurring b et w een functional regions. A t a certain p oin t, there is a knee in the curv es at whic h p oin t the Dice co efficien t starts 136 6.3. Discussion increasing while the mean z-score actually starts decreasing. The part of the curv e ab o v e the knee indicates p o or p erformance of tNLM b ecause the v alue of h is to o large causing blurring b et w een the adjacen t functional areas. 3. In terestingly , the knee in the tNLM curv es w as found to b e 0:45 whic h matc hes w ell the optimal v alues found using Equation ( 6.7 ) for LP (h = 0:43 ) and SC (h = 0:45 ). This represen ts the v alue whic h pro duces the maxim um v alue of a v erage z-score within the R OI without pro ducing an y significan t increase in Dice co efficien t or equiv alen tly , blurring b et w een functional areas. 4. W e found this result consisten t for m ultiple lev els ranging from 0:05 to 0:01 . Only t w o lev els w ere sho wn in Figure 6.6 to a v oid re dundancy . 6.3 Discussion The optimization-based metho d dev elop ed in this section pro vides a means of systematically se- lecting the parameter for tNLM filtering when used as a prepro cessing step for analyzing task-based or resting fMRI. The sim ulation pro duced a v alue of h close to that whic h optimized p erformance in terms of net w ork iden tification when compared against ground truth. F or exp erimen tal task fMRI, the optimal v alue coincided with the p oin t at whic h w e ac hiev ed maxim um enhancemen t in SNR without blurring b et w een distinct functional regions. In this w ork w e assumed a sp ecific form of the k ernel. In the next Chapter (Chapter 7 ), w e will discuss ab out ho w to optimize the k ernel function to further impro v e the filtering effect. Chapter 7 Global PDF-based T emp oral Non-lo cal Means Filtering , ; , — 7.1 In tro duction F unctional MRI (fMRI) is a p o w erful in-viv o neuroimaging mo dalit y that allo ws us to indirectly infer information ab out the neuronal activit y of the brain b y measuring BOLD signal fluctuations [ 28 ]. T emp oral correlations in resting fMRI (rfMRI) BOLD signals across m ultiple spatially distinct brain areas are often used to define functional brain net w orks [ 167 ]. Ho w ev er, BOLD signals inheren tly ha v e lo w SNR. Prepro cessing of fMRI data often includes a spatial smo othing step to reduce noise. Isotropic 3D Gaussian filtering is the most commonly used approac h to smo oth v olumetric rfMRI data [ 169 ], or equiv alen tly , LB smo othing is applied when the data is mapp ed on to a 2D represen tation of the cortical surface [ 8 ]. Both metho ds suffer from a critical common problem as they b oth spatially mix signals b et w een adjacen t functional regions [ 26 ], limiting our abilit y to accurately iden tify connectivit y at the micro-to-meso scale in individual fMRI recordings. 137 138 7.1. In tro duction Non-lo cal means (NLM) filtering is an edge-preserving metho d originally designed for natural image denoising [ 36 ] and has b een adapted to filter anatomical MRI [ 53 , 131 ], fMRI [ 24 ] and diffusion MRI [ 206 ] to preserv e spatial structure in imaging data. Bh ushan et al. [ 26 ] recen tly dev elop ed a v arian t for filtering rfMRI data called temp oral NLM (tNLM) that assigns non-lo cal smo othing k ernel w eigh ts based on temp oral similarities b et w een time series rather than spatial similarities. W e demonstrated tNLM filtering’s abilit y to reduce noise b y using (w eigh ted) a v erages of only those times series that are similar, th us minimizing blurring across functional b oundaries. Here w e iden tify t w o k ey c hallenges in using tNLM filtering as describ ed in [ 26 ]. First, the exp onen tial k ernel function used in computing the w eigh ts is c hosen heuristically . The exp onen t is an affine function of the sample correlation b et w een the t w o time-series. As w e sho w b elo w, this function do es not p erform w ell in terms of optimizing the trade-off b et w een the application of large w eigh ts when the correlations are high and smaller (or near zero) w eigh ts for lo w correlations. A second issue is that almost all NLM-based filtering metho ds, including tNLM, ha v e b een applied o v er a restricted neigh b orho o d around the p oin t to b e filtered, partially b ecause of the high com- putational cost if they are applied globally . Ho w ev er, since net w orks span the en tire brain, global rather than lo cal filtering has the p oten tial for impro v ed results when filtering using tNLM. It has b een suggested previously that the brain has the structure of a small-w orld net w ork [ 39 ] and there- fore most “no des” (or v o xels) in the brain are not strongly correlated with eac h other. As a result, when filtering a particular no de using data from the en tire brain, the fraction of uncorrelated no des is m uc h larger than the p ortion of correlated no des. This can result in an undue influence of the large n um b er of uncorrelated no des on the filtered signal if the filter w eigh ts applied to these no des are not sufficien tly suppressed. W e address eac h of these issue in the metho d describ ed b elo w. Here w e prop ose Global PDF-based tNLM filtering (GPDF): a new k ernel function for tNLM filtering of fMRI data based on the probabilit y densit y function (PDF) of the correlation of the time series b et w een pairs of v o xels. This metho d enables us to p erform glob a l filtering with impr ove d noise r e duction effects while minimizing blurring of adjacen t functional regions. Chapter 7. Global PDF-based T emp oral Non-lo cal Means Filtering 139 7.2 Metho d 7.2.1 NLM-based Filtering and tNLM Let’s assume the fMRI data are represen ted on a 2D tessellation of the mid-cortical surface with V v ertices and T time samples for eac h v ertex. Let s(i;t) b e the time series at v ertex i2 V and time t2 T . Let S i b e the set of v ertices that are used to compute the filtered signal at v ertex i . In the tNLM metho d, S i con tains v ertex i and all its k -hop neigh b oring v ertices, for some k > 0 . Then t NLM filtering is defined as s ′ (i;t) = 1 ∑ j2S i w(i;j) ∑ j2S i s(j;t)w(i;j) (7.1) where the w eigh t w(i;j) is c hosen to b e a temp oral similarit y measure and defined as a function of the sample correlation [ 125 ]: w(i;j) =f(r(i;j);h) (7.2) f tNLM (r;h) = exp ( 2(1r) h 2 ) (7.3) where r(i;j) is the P earson correlation co efficien t b et w een v ertices i and j and h is the parameter whic h con trols the degree of filtering. 7.2.2 Global PDF-based tNLM Filtering Our GPDF filtering differs from the original tNLM filtering in the follo wing t w o w a ys: (i) the spatial range o v er whic h the filtered signal is computed: in GPDF the set S i = S;8i , where S con tains all v ertices on the tessellated brain surface instead of just a lo cal neigh b orho o d; (ii) w e use a differen t k ernel function f in Equation ( 7.3 ). 7.2.2.1 GPDF Kernel F orm ulation Let the observ ed signal b ex i =s i +n i at v ertexi , a sup erp osition of the true signals i and noise n i . Assume thats i andn i are indep enden t withs i N(0; 2 s ) andn i N(0; 2 n ) . Also assume 140 7.2. Metho d some non-zero correlation b et w een s i and s j if i and j are within the same functional net w ork (H 1 ) and zero correlation if they are in differen t net w orks ( H 0 ). Then the correlation b et w een t w o observ ed signals is: ij = E[x i x j ] x i x j = E[s i s j ] 2 s + 2 n = 8 > > < > > : 0 ;H 0 :E[s i s j ] = 0 2 s c 2 s + 2 n ;H 1 : E[s i s j ] 2 s =c (7.4) where c2 [1;0)[(0;1] represen ts some non-zero correlation and x i represen ts the SD ofx i . T o further help a v oid n umerical issues and impro v e the robustness of the algorithm describ ed b elo w, w e form ulate our h yp othesis in a sligh tly relaxed form: ij = E[x i x j ] x i x j 2 8 > > < > > : [;] ;H 0 [1;)[(;1] ;H 1 (7.5) where is a small p ositiv e constan t. The sample correlation distribution is giv en b y the follo wing [ 71 ]: P(rj;T) = (T2)(T1)(1 2 ) T1 2 (1r 2 ) T4 2 p 2(T 1 2 )(1r) T 3 2 2 F 1 ( 1 2 ; 1 2 ; 2T1 2 ; r +1 2 ) (7.6) where T is the n um b er of samples and 2 F 1 (a;b;c;z) is the Gaussian h yp ergeometric function. The parameter T will b e omitted in the follo wing deriv ation as for a giv en fMRI dataset, as T is a fixed constan t. An example is sho wn in Figure 7.1 . where = 0:2 under H 1 (blue curv e) and = 0 under H 0 (red curv e). The histograms of the sample correlations are distributed ab out their means according to Equation ( 7.6 ) due to the finite n um b er of samples. This causes a significan t o v erlap b et w een the red and blue curv es. There is therefore a range of nonzero correlation v alues o v er whic h it is difficult to distinguish H 1 from H 0 giv en an observ ed sample correlation r . But to p erform w ell, tNLM should attac h large w eigh ts only to those time series for whic h H 1 is true. In Figure 7.1 . w e sho w the shap e of the original tNLM k ernel defined in Equation ( 7.3 ) as a function of h (dotted color curv es). The figure sho ws that the k ernel p erforms a p o or job in Chapter 7. Global PDF-based T emp oral Non-lo cal Means Filtering 141 Figure 7.1: The histogram of the correlations under H 1 (blue) and H 0 (red) generated fro m sim ulated data o v erlaid with tNLM k ernel functions for differen t parameter h (dotted) and GPDF k ernel function (blac k solid). differen tiating H 1 from H 0 in the sense that applying significan t w eigh ts for H 1 also results in w eigh ts significan tly greater than zero for H 0 . The blac k curv e sho ws an alternativ e k ernel that, visually at least, do es a b etter job of giving significan tly larger w eigh ts to H 1 while minimizing those for H 0 . W e no w describ e ho w w e select this k ernel and then ev aluate its p erformance. Ba y es theorem tells us the p osterior probabilit y of giv en r is P(jr) = P(rj)P() ∫ P(rj)P()d (7.7) T o b etter differen tiate H 1 from H 0 , w e tak e the ratio b et w een the in tegrated p osterior proba- bilit y under H 1 and the coun terpart u nder H 0 , forming the Ba y es factor [ 105 ] R(r) = ∫ 2H 1 P(rj)P()d ∫ 2H 0 P(rj)P()d (7.8) whereR(r)2 [0;1) . The largerR(r) , the more lik ely b elongs toH 1 giv en that sample correlation r . W e then reform ulate our k ernel function f to b e f GPDF (r;h) = 1 exp ( R(r) h 2 ) (7.9) 142 7.2. Metho d where, similar to the tNLM k ernel in Equation ( 7.3 ), h is a parameter that con trols the degree of smo othing. Replacing the sample correlation in Equation ( 7.3 ) with the Ba y es factor in Equa- tion ( 7.9 ) in tro duces the strong nonlinearit y visible in the blac k curv e in Figure 7.1 . This non- linearit y accoun ts for the fact that the p osterior probabilit y of H 1 vs H 0 can c hange rapidly as a function of r , as reflected in the Ba y es factors. 7.2.2.2 A utomated P arameter Selection In addition to using a differen t k ernel, w e also prop ose an automated metho d for selecting the parameter h . An optimized parameter h for tNLM filtering is crucial b ecause (i) the filtering effect is v ery sensitiv e to the selection of h as sho wn in [ 125 ]; (ii) Our GPDF filtering is a data- dep enden t k ernel so that the parameters can v ary substan tially as differen t scanning proto cols ma y ha v e differen t time series duration and ph ysiological noise sensitivit y . T o select the b est parameter in order to ac hiev e an optimal filtering result, w e maximize the exp ected v alue of the w eigh ting function f GPDF (r;h) under H 1 while con trolling the mean v alue with resp ect to H 0 . Sp ecifically , ^ h = argmax h E H 1 [f GPDF (r;h)]; s.t. E H 0 [f GPDF (r;h)] (7.10) where is the exp ected w eigh t under H 0 , analogous to the false p ositiv e rate in detection theory . Although is another parameter w e need to tune man ually , it is more meaningful and robust than h , b ecause c ho osing the same will generally yield similar filtering result while the in ternal parameter h can ha v e v ery differen t impact for differen t datasets, as a function of the noise lev el, range of correlation v alues and size of the image b eing filtered. W e recommend that b e set conserv ativ ely , e.g. 10 3 or smaller, due to the dominan t fraction of uncorrelated v ertices (H 0 ) in a fMRI dataset. 7.2.2.3 Estimation of the P opulation Correlation Distribution In order to construct the k ernel function in Equation ( 7.9 ) w e need to kno w the Ba y es factor R(r) , whic h requires the conditional distributionP(rj) and the p opulation correlation distribution P() . The sample correlation densit y P(rj) has the analytical solution giv en in Equation ( 7.6 ). Chapter 7. Global PDF-based T emp oral Non-lo cal Means Filtering 143 Therefore, w e need only estimate P() . Let P(r) b e the empirical sample correlation distribution obtained from the fMRI data. LetP ′ (r)2R M ,P ′ (rj)2R MN andP ′ ()2R N b e the discretized v e rsion of the corresp onding v ariables in the con tin uous space, resp ectiv ely . Then P ′ () can b e estimated using a linear regression with non-negativ e constrain ts: ^ P ′ () = argmin P ′ () P ′ (r) ∑ P ′ (rj)P ′ () 2 l 2 ; s.t. P ′ ()⪰ 0 (7.11) This optimization is a w ell-p osed problem as long as M N , i.e., the discretization step for r is smaller than that for , whic h can b e ac hiev ed easily . Also, this problem can b e solv ed efficien tly using t he fast non-negativ e least square metho d [ 33 ]. 7.2.2.4 GPDF Filtering Algorithm W e summarize our GPDF filtering algorithm as follo ws: Algorithm IX GPDF filtering 1: Giv en fMRI dataX2R VT , calculate the correlation matrixA =XX T 2R VV 2: Estimate P ′ (r) from the histogram of the elemen ts ofA 3: Estimate the priors b y solving Equation ( 7.11 ) 4: Optimize the parameter h b y solving Equation ( 7.10 ) 5: Construct the k ernel using Equation ( 7.9 ) 6: Finally filter the signal using Equation ( 7.1 ) 7.3 Exp erimen ts and Results 7.3.1 Sim ulation W e sim ulated a “brain surface” tessellation as t w o 2D blo c ks of sizeVV (V = 32 ) represen ting left and righ t hemispheres. Eac h p oin t in eac h blo c k represen ts a v ertex on the brain surface and has a lab el indicating whic h net w ork it b elongs to. Figure 7.2 (a) sho ws the ground truth lab el blo c ks where eac h color represen ts a distinct lab el. The top and b ottom ro ws ha v e iden tical lab els 144 7.3. Exp erimen ts a nd Results Figure 7.2: P arcellation result of sim ulated data represen ted as a VV matrix for eac h metho d and eac h hemisphere. Columns from (a) to (f ) are indicated b y their titles along upp er ro w. The ro ws represen t the t w o hemispheres. to sim ulate connections b et w een the left and righ t hemispheres (in totalK = 16 unique lab els). F or eac h lab el, w e generated a random time series (white noise) of length T = 200 where p oin ts within the same lab els w ere giv en iden tical time series (p erfectly correlated) in the absence of additional noise. P oin ts with differen t lab els w ere giv en zero correlation indicating that they b elong to differen t net w orks. W e then added Gaussian white noise with SNR = 0:4 to the en tire dataset. T o in v estigate the effects of differen t filtering metho ds, w e applied filtering to the sim ulated data then parcellated the data in to K lab els using NCuts algorithm [ 163 ]. A stable matc hing algorithm [ 76 ] w as applied to matc h lab els b et w een differen t results for easy comparison. Figure 7.2 sho ws the parcellation results for: (b) Gaussian filtering with FWHM appro ximately 8 p oin ts; (c), (d) tNLM filtering with optimized h parameter [ 125 ]; (e), (f ) PDF filtering. T o demonstrate the difference b et w een lo cal filtering and global filtering, w e applied tNLM and PDF filtering b oth lo cally ((c) and (e)) and globally ((d) and (f )). Lo cal filtering pro cessed left and righ t hemispheres separately while global filtering pro cessed them join tly . Gaussian spatial filtering generated lab els along the b oundaries b et w een true lab els not seen in the ground truth. This is most lik ely due to blurring of uncorrelated neigh b oring v ertices. In con trast, b oth tNLM and PDF filtering metho ds preserv ed the blo c ky structures. Ho w ev er, PDF yielded m uc h cleaner results than tNLM b ecause tNLM has a larger con tribution from the uncorrelated v ertices at eac h filtered p oin t as discussed ab o v e. Note that for b oth PDF and tNLM Chapter 7. Global PDF-based T emp oral Non-lo cal Means Filtering 145 the parameter h had b een optimized, in the latter case using [ 125 ], to ac hiev e the b est trade-off. Finally , for b oth tNLM and PDF, lo cal filtering resulted in lab els that w ere mismatc hed b et w een the left and righ t hemispheres. The m y opic p ersp ectiv e of lo cal filtering failed to detect the distal, esp ecially in ter-hemispheric, connections. Quan titativ ely , w e ran this sim ulation for 100 Mon te Carlo trials and calculated the ARI [ 151 ] b et w een eac h parcellation result and the ground truth as a filtering p erformance measure. Results sho w ed that the medians of the ARIs w ere 0:547 , 0:701 , 0:760 , 0:750 , 0:969 resp ectiv ely in cor- resp ondenc e to eac h filtering metho d in Figure 7.2 (b) - (f ), resp ectiv ely , indicating that GPDF outp erformed other filtering metho ds b y a significan t margin. T o in v estigate the robustness of GPDF o v er a v ariet y of sim ulated settings, w e ev aluated the ARI b et w een the parcellation result of filtered data (Gaussian, global tNLM and GPDF) and the ground truth as a function of the time-series length T as w ell as SNR. F or eac h sim ulated dataset, w e ran 100 Mon te Carlo trials and the b o xplots w ere generated for eac h filtering metho d. Figure 7.3 sho ws the result of ARI as a function of T in ( a ) and SNR in ( b ). The p erformance of Gaussian filtering do es not impro v e when T increases as the Gaussian filter applies a pure spatial k ernel to data without using the temp oral information b et w een time series. In con trast, b oth the tNL M and GPDF filter sho w impro v ed p erformance as T increases, but GPDF outp erforms tNLM o v er the en tire range of T due to it s b etter design of the k ernel function. When SNR v aries with fixed T = 100 , the ARI for the Gaussian filtered case increases sligh tly but not m uc h b ecause of the inevitable blurring effects across b oundaries of differen t functional areas. Whereas, similar to ( a ), b oth tNLM and GPDF yield b etter parcellation results as SNR increases and higher ARI is obtained using GPDF-filtered data than the coun terpart using tNLM filtered data. 7.3.2 Application to In-viv o Resting fMRI Dataset 7.3.2.1 Dataset and Filtering 40 sub jects with minimally prepro cessed rfMRI datasets (2 sessions, 2 phase enco dings; 160 sessions total) w ere obtained from HCP [ 192 ]. The data w ere acquired with TR = 720 ms with 146 7.3. Exp erimen ts a nd Results (a) ARI b et w een parcellation result of filtered data and the ground truth as a function of time series length T with fixed SNR = 0:3 . (b) ARI b et w een parcellation result of filtered data and the ground truth as a function of SNR with fixed T = 100 . Figure 7.3: Robustness comparison of results using Gaussian filter, global tNLM filter with optimized parameter and GPDF. Chapter 7. Global PDF-based T emp oral Non-lo cal Means Filtering 147 resolution 222 mm and had b een carefully prepro cessed using the pip eline describ ed in [ 78 ], where only minimal (2 mm FWHM) Gaussian smo othing w as applied. Then the data w ere co- registered on to a common atlas and do wnsampled on to a 32K -v ertex cortical surface. W e further do wnsampled eac h data to 11K v ertices for computational tractabilit y . W e then filtered eac h dataset using LB with a range of v alues of smo othing parameter (the SD of the Gaussian k ernel) and GPDF with a range of v alues of the parameter defined in Equation ( 7.10 ). W e p erformed three exp erimen ts: (i) seeded correlation; (ii) exploration of cor- relation, comm unit y structure and mo dularit y; and (iii) comparison with task fMRI. In eac h case w e compared GPDF with LB b oth qualitativ ely and quan titativ ely . W e did not include tNLM filtering results here b ecause tNLM had b een extensiv ely studied and compared with LB on real datasets in [ 26 , 125 ]. In the first exp erimen t w e demonstrate the effect of filtering parameters (2f1;2;3;4;5g mm and 2f10 1 ;10 2 ;10 3 ;10 4 ;10 5 g in Section 7.3.2.3 . F or the other t w o exp erimen ts, w e c hose the parameters to b e in the middle of their resp ectiv e ranges, = 3 mm and = 10 3 , to a v oid extreme b eha vior. 7.3.2.2 Seeded Correlation Maps Seed-based metho ds ha v e b een widely used in fMRI data analysis and brain net w ork inference [ 28 , 62 , 180 , 188 ]. T o ev aluate the effects of filtering, w e placed a seed p oin t in the caudal pre-cuneus whic h is part of the DMN (Figure 7.4 ( b ) - ( d )) and calculated the P earson correlation of its time series with those of all other v ertices of the brain, to form a correlation map. Figure 7.4 sho ws seed-p oin t correlation maps for a single sub ject for the ( a ) unfiltered data; ( b ) LB filtered data and ( c ) GPDF filtered data in a common scale ranging from 0:2 to 1 . DMN can b e seen in the unfiltered correlation map ( a ) but in the v ery lo w correlation range due to the rfMRI’s inheren t lo w SNR. Figure 7.4 ( d ) exaggerates the color scale of unfiltered data for easy visualization of the correlation structure. LB and GPDF, in con trast, yield higher correlations due to their abilit y to reduce noise and amplify signal. Ho w ev er, GPDF exhibits a wider range of correlation v alues than LB. 148 7.3. Exp erimen ts a nd Results (a) Unfiltered data (b) LB filtered data (c) GPDF filtered data (d) Unfiltered data re-plotted in its o wn scale Figure 7.4: Seeded correlation map for a single sub ject. Seed p oin t w as selected in the caudal pre-cuneus area sho wn as a y ello w dot in sub-figures (b) - (d). P ositiv ely correlated regions are sho wn in red, uncorrelated regions in white and negativ ely correlated regions in blue. A dditionally , GPDF app ears b etter able to preserv es spa tial delineation of adjacen t regions with opp osite correlation p olarit y relativ e to the seed p oin t, for example t w o adjacen t regions are indicated b y the arro ws in Figure 7.4 . Boundaries are clearly visible in b oth the unfiltered data (exaggerated in ( d )) and GPDF but not in LB. These observ ations are indicativ e of LB’s tendency to spatially blur the b oundaries b et w een distinct adjacen t functional areas. LB sho ws strong connections to the lo cal p oin ts surrounding the seed p oin t while connections to distal areas, esp ecially in ter-hemispherical connections, are atten uated due to the lo calness of the filtering. This atten uation do es not o ccur in GPDF as strong correlations are preserv ed across distal and in ter-hemispheric regions of the DMN. GPDF therefore app ears to help rev eal stronger in tra-net w ork connectivit y than the LB filtering metho d. Chapter 7. Global PDF-based T emp oral Non-lo cal Means Filtering 149 W e also selected p oin ts that are highly correlated with the seed p oin t and explored ho w those correlations w ere altered b y filtering. A p oin t w as defined as b eing highly correlated with the seed p oin t if its v alue in Figure 7.4 ( a ) la y in either of the t w o tails of the n ull distribution H 0 : P(rj = 0;T = 1200) (o v erlaid in Figure 7.5 ( b ) and Figure 7.5 ( c ) with 10 6 significance lev el (the corresp onding cut-off is 0:133 )). Figure 7.5 ( a ) sho ws the spatial lo cations in y ello w of those p oin ts highly correlated with the seed. Figure 7.5 ( b ) and ( c ) sho w the scatter plot b et w een the unfiltered correlation (v alues of those p oin ts in Figure 7.4 ( a )) and the filtered correlation (v alues of those p oin ts in Figure 7.4 ( b ) and ( c )) for LB and GPDF, resp ectiv ely . GPDF amplifies the correlation v alues but retains the sign of the correlation, Figure 7.5 ( c ). This is exp ected since GPDF w as designed to a v erage only similar signals. On the other hand, while LB also amplifies the correlation v alues, after filtering the signs of a significan t fraction of these correlation v alues ha v e b een flipp ed from negativ e to p ositiv e, Figure 7.5 ( b ). This is caused b y the blurring of signals across functional b oundaries, indicating a p oten tial pitfall that ma y o ccur when dra wing conclusion or inference based on LB filtered individual fMRI signals. 7.3.2.3 Unfiltered Correlation Matrix, Comm unit y Structure and Mo dularit y The idea of mo dular brain net w orks has b een p opularized in the last decade [ 172 ] due to the widely adopted small-w orld p ersp ectiv e on brain net w orks [ 39 ]. Comm unit y structure can b e directly visualized using the re-ordered connectivit y matrix (also referred as correlation matrix or asso ciation matrix) based on mo dule detection or parcellation results. (See Figure 2 (i) in [ 172 ]) W e use the same concept here to demonstrate the effect of filtering. F or eac h dataset w e to ok the unfiltered data and computed the v ertex-pairwise full correlation matrix, A2 R VV , as the underlying graph structure. W e then applied the NCuts algorithm [ 163 ] to parcellate the brain in to K net w orks using eac h of the follo wing: the unfiltered data, the LB-filtered data and the GPDF-filtered data, generating parcellation lab els for eac h of the three. W e then used those lab els to re-order the connectivit y matrixA so that v ertices that had same lab el w ere group ed together. 150 7.3. Exp erimen ts a nd Results Figure 7.6 ( a ) - ( c ) sho w the re-ordered unfiltered connectivit y matrix A based on the parcel- lation result (K = 7 ) using the unfiltered data, the LB-filtered data and the GPDF-filtered data, resp ectiv ely . Using the same (re-ordered) unfiltered connectivit y matrix A establishes an un biased comparison of the three partitions. The resulting re-ordered connectivit y matrix indicates ho w w ell eac h filtering metho d group ed the data in to functionally homogeneous regions with resp ect to the original (unfiltered) data. In essence, w e assume that a b etter filtering metho d will giv e us a b etter clustering of no des under a giv en parcellation algorithm in the sense that no des that ha v e the same lab e l (are within the same net w ork) tend to ha v e higher as w ell as consisten t correlation with eac h other than with no des in other net w orks (diagonal blo c ks) and tend to ha v e consisten t correlation (can b e either p ositiv e, zero or negativ e) with no des in other net w orks (off-diagonal blo c ks). The GPDF result in Figure 7.6 ( c ) sho ws a neat grouping of no des forming a clearer blo c ky comm unit y structure and higher correlation in the diagonal blo c ks than the unfiltered case in Figure 7.6 ( a ) and the LB-filtered case in Figure 7.6 ( b ). T o quan titativ ely v erify our assumption, for eac h of the off-diag onal blo c ks in eac h fMRI record- ing session, w e computed the SD of the correlation v alues within that blo c k, yielding a KK compressed SD map. Then w e calculated the median SD o v er the 160 fMRI sessions. Figure 7.6 ( d ) - ( f ) sho w the median SD map for the unfiltered, the LB-filtered and the GPDF filtered case, resp ectiv ely , for K = 7 . The GPDF result sho ws m uc h lo w er SD in the off-diagonal blo c ks, indi- cating higher consistency among the no des in one net w ork with resp ect to the relationship to other net w orks. W e also observ ed similar phenomena for other v alues of K . T o further ev aluate filtering p erformance and its robustness for a range of n um b er of parcels K , for eac h unfiltered correlation matrix A , w e binarized it with a threshold T to form a binary connectivit y matrix A ′ . W e then calculated the mo dularit y [ 143 ] for A ′ using eac h of the three K net w ork partitio ns (unfiltered, LB, GPDF) as a function of T . The analyses w ere p erformed on eac h dataset indep enden tly . Figure 7.7 ( a ) sho ws the median mo dularit y across 160 sessions as a function of T . The GPDF filtering metho d outp erformed LB and the unfiltered case b y a large margin regardless of the T and the n um b er of parcels, indicating that GPDF is pro ducing Chapter 7. Global PDF-based T emp oral Non-lo cal Means Filtering 151 parcellations that sho w stronger within net w ork similarit y with resp ect to the unfiltered data than either the unfiltered case or LB filtering. A dditionally , w e in v estigated ho w the filtering parameters for LB and for GPDF influenced the filtering result. W e computed the mo dularit y as describ ed ab o v e while v arying 2f1;2;3;4;5g mm and 2f10 1 ;10 2 ;10 3 ;10 4 ;10 5 g with fixed n um b er of parcels K = 17 . This n um b er w as selected for parcellation stabilit y as suggested in [ 209 ]. Figure 7.7 ( b ) sho ws the mo dularit y as a function of T for eac h filtering parameter. F or LB filtering, smaller yields results similar to the unfiltered case and p erformance is further and further compromised when increases due to the larger amoun t of blurring and mixing of signals across differen t functional regions. In general, LB filter ing actually p erforms w orse than the unfiltered case, regardless of the filtering parameter, when p erforming individual parcellations, suggesting that LB ma y not optimally preserv e differ- ences b et w een individuals based on a single fMRI recording. In con trast, GPDF outp erforms the unfiltered case, and also the LB-filtered cases, for most of the parameter settings except for large with high T . This is b ecause larger allo ws a significan t fraction of uncorrelated no des to b e in v olv ed in the filtered signal, resulting in w orse p erformance as discussed in the in tro duction. W e found = 10 4 giv es the b est result in this exp erimen t, whic h is the basis of our recommendation for a conserv ativ e in the GPDF filtering metho d. 7.3.2.4 P arcellation Agreemen t with T ask fMRI A ctiv ation Maps P arcellation of rfMRI data is used to elucidate underlying spatial patterns in brain connec- tivit y . Ho w ev er, the lac k of an a v ailable ground truth mak es it difficult to in terpret parcellation results, esp ecially when comparing differen t filtering metho ds. W e tried to address this difficult y b y comparing rfMRI parcellation results obtained from differen t filtering metho ds to the lo calized task-based fMRI (tfMRI) results for eac h individual session. T ask fMRI datasets w ere also a v ailable and obtained from HCP for the same 40 sub jects and they con tained 7 ma jor task domains: motor strip mapping (Motor), language pro cessing (Lan- guage), emotion pro cessing (Emotion), rew ard & decision-making (Gam bling), relational pro cessing (Relational), so cial cognition (So cial) and w orking memory (WM). W e used the prepro cessed (4 152 7.3. Exp erimen ts a nd Results mm Gaussian smo othed) and analyzed tfMRI z-score statistical maps from HCP , including a total of 15 task-pair as describ ed in detail in [ 17 ]. W e selected the follo wing con trasts for this exp er- imen t from all task pairs as w ere used in [ 48 ]: tongue vs a v erage (t_a vg), left hand vs a v erage (lh_a vg), righ t hand vs a v erage (rh_a vg), left fo ot vs a v erage (lf_a vg) and righ t fo ot vs a v erage (rf_a vg) from the Motor task; math vs story (math_story) from the Language task; faces vs shap es (faces_shap es) from the Emotion task; punish vs rew ard (punish_rew ard) from the Gam bling task; ob ject matc hing vs geometrical relationship (matc h_rel) from the Relational task; random mo v e- men t vs in ten tional mo v emen t (random_tom) from the So cial task; 0-bac k vs 2-bac k (0bk_2bk), face vs a v erage (face_a vg), place vs a v erage (place_a vg), to ol vs a v erage (to ol_a vg), b o dy vs a v erage (b o dy_a vg) from the WM task. T o ev aluate the p erformance of differen t filtering metho ds, for eac h individual fMRI session, w e first parcellated the brain (rfMRI data) in to K parcels using a spatially constrained hierarc hical parcellation approac h [ 30 ] for eac h of the filtering metho ds (unfiltered, LB and GPDF). This “region gro wing”-based parcellation metho d is particularly appropriate for this tfMRI comparison purp ose as it w as designed to robustly parcellate the entir e h uman cerebral cortex on a single subje ct basis. It also enforces sp atial c ontiguity of the parcels, whic h allo ws us to obtain a reasonable parcellation result from unfiltered data (see [ 30 ] for details). Qualitativ ely , Figure 7.8 sho ws the maps of the parcel b oundaries for K = 100 parcels o v erlaid on the z-score maps of the Motor task tongue vs a v erage con trast in ( a ) - ( c ) and Emotional task faces vs shap es in ( d ) - ( f ) for a single session of sub ject 100307. Figure 7.8 illustrates impro v ed consistency of the parcellation b oundaries with differen t functional regions (e.g., the tongue area and the fusiform face area) from the unfiltered case using the GPDF filtering. In con trast, in the LB-filtered case, either some b oundaries cross task-activ e areas or some parcels con tain b oth task-p ositiv e and task-negativ e regions. T o quan titativ ely measure the p erformance, for a certain task pair, w e computed the v ariance of the z-scores of that task activ ation map in eac h parcel and the v ariance w as a v eraged o v er all cortical regions as a metric of single task v ariabilit y . W e computed this task v ariabilit y metric for all 15 task con trasts and all 160 individual fMRI sessions with differen t n um b er of parcels K . Since Chapter 7. Global PDF-based T emp oral Non-lo cal Means Filtering 153 there is no ground truth of the correct K that should b e used for the parcellation, w e s elected K = K ′ as the v alue that yielded the lo w est task v ariabilit y in the unfiltered case and compared the task v ariabilities in all three cases (unfiltered, LB and GPDF) using the same K ′ , for eac h fMRI session and eac h task pair. Figure 7.9 sho ws b o xplots of task v ariabilities across 160 fMRI sessions for eac h task pair. The task v ariabilities of the GPDF case are consisten tly lo w er than b oth the unfiltered case and the LB-filtered case for all 15 task pairs, despite that they v ary substan tially from task to task. T able 7.1: Statistics (Wilco xon rank-sum) of the task v ariabilit y difference b et w een GPDF and the unfiltered case (Column 3) as w ell as b et w een GPDF and LB (Column 4) T ask Con trast p-v alue (GPDF < Unfiltered) p-v alue (GPDF < LB) Motor t_a vg 2.99E-22 3.38E-22 lh_a vg 8.48E-24 2.64E-28 rh_a vg 1.28E-20 2.80E-28 lf_a vg 1.91E-22 2.69E-28 rf_a vg 2.44E-20 3.71E-28 Language math_story 1.68E-21 3.13E-28 Emotion faces_shap es 1.31E-5 3.02E-28 Gam bling punish_rew ard 1.04E-26 3.64E-28 Relational matc h_rel 3.90E-22 2.80E-28 So cial random_tom 6.18E-20 3.31E-28 WM 0bk_2bk 7.22E-20 5.61E-28 face_a vg 4.10E-11 2.75E-28 place_a vg 8.60E-11 4.48E-28 to ol_a vg 5.42E-22 2.64E-28 b o dy_a vg 1.19E-17 2.64E-28 W e also applied a Wilco xon rank-sum test to determine if there w as a significan t reduction in task v ariabilit y b y filtering. Lo w er task v ariabilit y indicates higher functional homogeneit y 154 7.4. Discussion within parcels hence b etter alignmen t b et w een parcellation b oundaries and task-activ e regions. T able 7.1 sho ws that GPDF filtering yields significan tly lo w er task v ariabilities when compared with the unfiltered case and the LB-filtered case in all 15 task pairs, whic h confirmed our qualitativ e observ ations. 7.4 Discussion In this section, w e systematically dev elop ed a no v el k ernel function based on the Ba y es factor for global tNLM filtering. W e also pro vided a w a y to automatically tune the parameter in or- der to ac hiev e an optimal filtering result. W e demonstrated b oth qualitativ ely and quan titativ ely using sim ulations as w ell as three exp erimen ts on in-viv o fMRI data that this metho d can sim ulta- neously p erform b etter denoising and preserv e b oundaries b et w een regions of differen t functional sp ecializations than standard linear filtering metho d. The sup erior p erformance of the GPDF filtering o v er the traditional linear filtering comes from the non-linearit y of the k ernel function visible as the blac k curv e in Figure 7.1 . Note that the design of the k ernel is a data-driv en approac h, whic h can b e differen t for differen t datasets. Therefore, it ma y b e particularly useful when inferring brain connectivit y patterns from individual fMRI recordings instead of a group analysis. F urthermore, in some limiting cases, the k ernel function ma y ha v e a v ery sharp transition from zero w eigh t to unit w eigh t, forming a nearly binary k ernel function, where v ertices whose correlation exceed the threshold will b e a v eraged together. Ho w ev er, ev en in this case it can b e view ed as a v alid filtering metho d rather than a parcellation metho d since eac h v ertex has a distinct correlation pattern to all other v ertices of the brain, th us the set of v ertices o v er whic h the time series are a v eraged together can v ary from v ertex to v ertex. Based on our exp erience, in most cases the k ernel function exhibits a smo other non-linear transition rather than a binary thresholding. One of the limitation of our GPDF approac h is that the k ernel function is estimated based on the empirical correlation using the en tire time-series, whic h implicitly assume the stationarit y of the fMRI signals. Extensions of this metho d to p erforming temp orally dynamic filtering is a promising future direction. Chapter 7. Global PDF-based T emp oral Non-lo cal Means Filtering 155 (a) Spatial map of the highly correlated v ertices to the seed p oin t in the unfiltered data. (b) Scatter plot of the unfiltered correlation v alues of those v er- tices in (a) v ersus the LB-filtered correlation v alues o v erlaid with the n ull distribution P(rj = 0) for T = 1200 . (c) Scatter plot of the unfiltered correlation v alues of those v er- tices in (a) v ersus the GPDF-filtered correlation v alues o v erlaid with the n ull distribution P(rj = 0) for T = 1200 . Figure 7.5: Changes of the seeded correlation v alues after filtering. 156 7.4. Discussion (a) (b) (c) (d) (e) (f ) Figure 7.6: Re-ordered unfiltered full correlation matrices based on the parcellation result using the unfiltered data (a), LB-filtered data (b) and GPDF-filtered data (c) for K = 7 . (d) - (f ) sho ws the corresp onding blo c k-wise median SD map o v er 160 sessions for (a) - (c). Chapter 7. Global PDF-based T emp oral Non-lo cal Means Filtering 157 (a) The mo dularit y as a function of the threshold T for differen t K and filtering metho ds with fixed filtering parameter = 3 mm and = 10 3 . (b) The mo dularit y as a function of the thresholdT for differen t filtering metho d and parameters with fixed K = 17 . Figure 7.7: The net w ork mo dularit y plots as a function of the threshold and the filtering param- eters. 158 7.4. Discussion (a) (b) (c) (d) (e) (f ) Figure 7.8: Maps of parcel b oundaries o v erlaid with motor task tongue vs a v erage con trast (a) - (c) and emotional task faces vs shap es (d) - (f ) for a single session of sub ject 100307 using the unfiltered data ((a) and (d)), the LB-filtered data ((b) - (e)) and the GPDF filtered data ((c) - (f )). (Num b er of parcels K = 100 ) Chapter 7. Global PDF-based T emp oral Non-lo cal Means Filtering 159 Figure 7.9: Bo xplots of the mean z-score v ariance of all cortical regions o v er 160 fMRI sessions for all 15 task pairs. Eac h column sho ws the b o xplot for one particular task using the unfiltered data (left), the LB-filtered data (middle) and the GPDF filtered data (righ t). This page in ten tionally left blank P art V Concluding Remarks 161 Chapter 8 Conclusions & F uture W orks — Exploring functional connect ivit y of the brain with fMRI and SEEG offers a ric h approac h to studying large-scale neuronal comm unications b et w een differen t brain regions, or brain net w orks. The fundamen tal question regarding brain net w ork iden tification is whic h part(s) of the brain (the question of “where”) talk to eac h other at whic h giv en time or p erio d (the question of “when”) in what particular w a y (the question of “ho w”). Ho w ev er, it is rather difficult to answ er ev en a single question out the three (where, when and ho w), b ecause these three questions are v ery closely related to eac h other. 8.1 Lo calization of the Epileptogenic Zone and Epileptic Net w ork Analysis In patien ts with in tractable fo cal epilepsy , abnormal epileptic neural activities are generated from the epilepto genic zone and gradually propagate to other areas of the brain, th us triggering seizures. In order to explore suc h epileptic brain net w orks (i.e., to find wher e seizures start and wher e they propagate to), one needs to first ha v e an accurate lo calization of the epileptogenic zone. Ho w ev er, finding a go o d biomark er of the epileptogenic zone requires in v estigation and 162 Chapter 8. Conclusions & F uture W orks 163 precise description of the seizures from ictal SEEG recordings in a v ariet y of asp ects: how differen t frequency comp onen ts of a seizure in teract with eac h other and how seizures ev olv e/c hange from in ter-ictal p erio d to ictal p erio d. In this w ork, w e approac hed the epileptogenic zone lo calization problem b y defining a time- frequency pattern at ictal onset as a reliable biomark er of the epileptogenic zone. W e referred to this time-frequency pattern as an epileptogenic zone “fingerprin t” [ 88 ]. In con trast to traditionally- used broad-band fast activit y whic h w as defined mainly based on the energy in gamma band [ 20 , 56 , 80 ], the fingerprin t describ es a complex phenomenon consisting of pre-ictal sharp spik es preceding narro w-band fast activit y concurren t with suppression of lo w er frequencies as illustrated in Figure 2.2 . This fingerprin t pattern is v ery consisten t across patien ts, including b oth seizure- free and non-seizure-free patien ts [ 87 , 88 ], alb eit individual comp onen ts of the fingerprin t ma y v ary substan tially across patien ts as w ell as across seizures. It also has b een sho wn that whether the iden tified epileptogenic zone regions using the fingerprin t metho d w ere resected or not w as w ell aligned with the surgical outcomes [ 54 , 87 ], indicating that the fingerprin t pattern can b e gen uine biomark er of the epileptogenic zone. W e ha v e also sho wn that when using fast activit y or the features from fast activit y only , the epileptogenic zone is often o v er-estimated (Figure 3.4 ) due to the wide spread of fast activit y to regions far outside the epileptogenic zone, whic h p oten tially leads to unnecessary large surgical ablations if fast activit y is used along as the biomark er of the epileptogenic zone. A further ex- ploration of ho w fast activit y propagates, ho w the propagation of fast activit y differs from that of suppression in a net w ork p oin t of view as w ell as the relationship b et w een the propagation of the comp onen ts and the underlying pathoph ysiology of epilepsy [ 11 , 57 ] is a promising future direction. 164 8.2. Resolving the Spat ial and T emp oral Dynamics of Brain Net w orks 8.2 Resolving the Spatial and T emp oral Dynamics of Brain Net w orks Recen tly , the “where” and “when” question are also raised together in brain net w ork iden tifi- cation scenarios. P eople ha v e sho wn that the connections b et w een differen t brain regions are not stationary [ 148 ], rather, functional connectivit y tends t o fluctuate o v er time [ 46 ]. No w ada ys, the most commonly used approac hes for deco ding dynamic functional connectivit y and brain net w orks are sliding-windo w-based metho ds, where some connectivit y measures, suc h as correlation or coherence, is computed o v er a short-time windo w and the windo w is slid o v er the en tire time courses. Then a k-means clustering [ 4 ], graph analysis [ 25 , 179 , 210 ], ICA [ 208 ], PCA [ 120 ], dictionary learning [ 121 ], etc. can b e p erformed on top of the extracted dynamic measures to iden tify differen t quasi-stationary brain macro-states. This framew ork has b een en th usiastically applied b y the neuroimaging comm unit y to explore ho w functional dynamics of the brain is related to cognitiv e abilities [ 66 , 116 ] or other functional [ 47 , 180 ] or structural [ 128 ] connectivit y measures as w ell as ho w it is affected or altered b y neurological diseases [ 102 , 154 ]. Ho w ev er, the c hoice of windo w length is alw a ys debatable, whic h hence is a p oten tial pitfall for all sliding-windo w-based metho ds, b ecause, on one hand, to o short a windo w tends to in tro duce spurious fluctuations in functional connectivit y measures and on the other hand, to o long a windo w limits our abilit y to detect temp oral c hanges. Indep enden t comp onen t analysis has also b een widely used for brain net w ork iden tification and dynamic functional connectivit y exploration. Indep enden t comp onen ts ha v e b een extracted and com bined for individual sub jects [ 41 , 68 ]. A t group lev el, spatial [ 156 , 177 ] or temp oral concatenation [ 40 , 89 ] w as used b efore application of ICA for net w ork analysis from m ulti-sub ject data. Ho w ev er, all ICA-based metho ds require either spatial or temp oral indep endence, whic h ma y not b e realistic as brain net w orks can o v erlap and b e correlated in b oth space and time [ 104 ]. In this w ork, w e dev elop ed a tensor-based metho d, called the SRSCPD framew ork [ 124 , 126 , 127 ], to sim ultaneously iden tify b oth spatial maps and temp oral dynamics of brain net w orks. Th us, it can a v oid the trade-off in the c hoice of windo w length asso ciated with the sliding-windo w-based Chapter 8. Conclusions & F uture W orks 165 metho ds. Also, it do es not imp ose an y unrealistic constrain t on the net w orks, suc h as indep endence asso ciated with the ICA-based metho ds. This framew ork is v ery flexible in the sense that it can b e applied not only to electro-ph ysiological data, suc h as EEG, SEEG, MEG, when the data is transformed in to a tensor form using w a v elet or short-time F ourier transforms, but also to async hronous task-related, self-paced or resting state fMRI data when the data from m ulti-sub jects are temp orally aligned using the BrainSync [ 103 ] algorithm. Compared to the 2D matrix cases, the SRSCPD tensor framew ork pro vides an additional dimen- sion whic h is the sub ject mo de, that indicates the participation or in v olv emen t lev el of a particular sub ject to a net w ork. This extra information can b e utilized to statistically test differences b et w een differen t group of sub jects. One ma y attempt to answ er questions suc h as “Whic h net w ork(s) in sub- jects with a giv en neurological diseases, suc h as A tten tion-Deficit Hyp eractivit y Disorder, A utism Sp ectrum Disorder, or Alzheimer, differs from their coun terpart(s) in normal sub jects? and ho w do they differ?” Although man y b enefits can b e obtained using this tensor-based SRSCPD framew ork, a limi- tation is that the dimensionalit y of the datasets still needs to b e reduced, to some degree, in one or more dimensions b efore applying this framew ork due to the high computational complexit y in b oth memory requiremen ts and pro cessing time. Ho w ev er, using higher spatial resolution, longer recordings and/or more sub jects can further impro v e the robustness of the estimates of brain net- w orks. Therefore, from a metho dological p ersp ectiv e, designing a tensor decomp osition algorithm that can scale up to larger datasets is a v ery c hallenging y et imp ortan t topic for future w ork. 8.3 T o Filter or Not T o Filter Last but not least, regardless of the metho d or algorithm used to dra w conclusions ab out the brain connectivit y question of “when”, “where” and “ho w”, the answ er can still b e far a w a y from the true underlying connectivit y of the brain due to the inheren t lo w SNR in fMRI signals (Electro- ph ysiological signals suc h as EEG, MEG and SEEG also suffer from this noise corruption problem, but our fo cus on denoising in this dissertation is with resp ect to fMRI data). 166 8.3. T o Filter or Not T o Filter Using fMRI data without spatial filtering ma y not b e a sev ere issue, sometimes is ev en encour- aged (to a v oid blurring), in group analysis where the robustness of solutions can b e ac hiev ed from using m ulti-sub ject data. A reliable inference ab out brain connectivit y from individual sub jects, ho w ev er, is difficult to obtain without prop er denoising b ecause of the hea vy noise corruption in limited data. Th us, a spatial smo othing using a Gaussian or LB k ernel is often emplo y ed to impro v e the SNR. Ho w ev er, b oth Gaussian and LB filtering inevitably spatially mix signals b et w een adjacen t functional regions [ 26 , 122 , 123 ], whic h limits our abilit y to accurately iden tify brain connectivit y at the micro-to-meso scale in individual fMRI recordings. In this w ork, w e describ ed a GPDF filtering tec hnique [ 122 , 123 ] whic h allo ws us to p erform a global filtering with impro v ed noise reduction effects while minimizing blurring of adjacen t func- tional regions. The sup erior p erformance of GPDF o v er the original tNLM filtering as w ell as the linear filtering is ac hiev ed b y using a Ba y er’s-factor-based nonlinear k ernel function, whic h re- inforces connections b et w een no des with higher correlation while suppresses connections b et w een no des with lo w er correlation at the same time (Figure 7.1 ). One of the limitations of GPDF filtering is that GPDF form ulates its k ernel based on the pair- wise correlations calculated using en tire time series. But as w e ha v e discussed ab o v e, the correlation, a measure of the functional connectivit y , ma y v ary dynamically o v er time. Hence, extensions of this metho d to temp orally dynamic filtering ma y b e of our in terest in the future. App endix A P atien t Selection Proto col 167 168 A.1. P atien t Selection in the In itial Fingerprin t Study A.1 P atien t Selection in the Initial Fin gerprin t Study (Chapter 2 ) W e review ed SEEG data from all patien ts who w ere ev aluated with SEEG at the Clev eland Clinic b et w een 2009 and July 2014 (n = 280 ). P atien t selection w orkflo w at the time of recruitmen t (Jan uary-F ebruary 2016 ) is sho wn in Figure A.1 . Our inclusion criteria w ere: 1. T ailored resection or laser ablation guide d b y SEEG. 95 patien ts w ere not surgical candidates after SEEG and w ere not inc luded in the study . 2. No seizures, including auras, after surgery . 103 patien ts w ere not seizure free and another nine w ere lost from follo w up. These patien ts w ere not included in the study . 3. Three or m ore seizures recorded during SEEG that w ere c haracterized b y sustained (three seconds duration or longer) gamma activit y at the onset. Based on this criterion w e did not include: (a) 12 patien ts that had less than 3 seizures recorded or EEG data w ere not a v ailable for analysis. (b) 4 patien ts t hat had resection that w as not tailored and w as not guided b y SEEG due to inadequate sampling. (c) 40 patien ts that had ictal onset patterns with c haracteristics differen t than sustained gamma activit y: rh ythmical spik es without clear transition to sustained fast activit y (n = 2 ); rh ythmical oscillations in alpha/theta/delta range or only EEG flattening ( n = 10 ); fast activit y less than3 seconds duration (n = 9 ); fast activit y in alpha/b eta/gamma range (n = 19 ). App endix A. P atien t Selection Proto col 169 Finally , 16 patien ts met all inclusion criteria. W e also included one patien t (Sub ject 4) who underw en t SEEG-guided laser surgery (“minimal resection”). Figure A.1: P atien t selection w orkflo w (Chapter 2 ) 170 A.2. P atien t Selection in the Extended Fingerprin t Study A.2 P atien t Selection in the Extended Fin gerprin t Study (Chapter 3 ) W e recruited a consecutiv e series of patien ts in y ear 2015 (n = 76 ) with e xclusion of: 1. 15 patien ts that w ere not surgical candidates. 2. 3 patien ts without follo w-up. 3. 34 patien ts did not ha v e L VF A at seizure onset. 4. 1 patien t that w as already included in the previous study (See App endix A.1 ). Among the final group of selected patien ts ( n = 24 ), 11 of them b ecame seizure-free and the remaining 13 had seizures re-o ccurred. Figure A.2: P atien t selection w orkflo w (Chapter 3 ) App endix B Statistics of the F requency of F ast A citivit y and Suppression 171 172 B.1. Statistics of the F requency of F ast A ctivit y B.1 Statistics of the F requency of F ast A ctivit y Figure B.1: Statistics of the frequency of fast activit y . Column 1: The maxim um frequency of fast activit y in epileptogenic-zone con tacts; Column 2: The minim um frequency of fast activit y in epileptogenic-zone con tacts; Column 3: The maxim um frequency of fast activit y in con tacts outside resection region; Column 4: The minim um frequency of fast activit y in con tacts outside resection region. App endix B. Statistics of the F requency of F ast A citivit y and Suppression 173 B.2 Statistics of the F requency of Suppression Figure B.2: Statistics of the frequency of suppression. Column 1: The maxim um frequency of suppression in epileptogenic-zone con tacts; Column 2: The minim um frequency of suppression in epileptogenic-zone con tacts; Column 3: The maxim um frequency of suppression in con tacts outside resection region; Column 4: The minim um frequency of suppression in con tacts outside resection region. App endix C Visual Characteristics of Epileptogenic Zone 174 App endix C. Visual Characteristics of Epileptogenic Zone 175 Figure C.1: Visual c haracteristics of th epileptogenic zone. Con tacts in the epileptogenic zone: one exemplar time series and its corresp onding time-frequency plot is sho wn for eac h patien t. The n u m b e r of eac h time series plot indicates the patien t ID. Eac h plot sho ws 10 seconds prior to onset and 20 seconds after, and the frequencies are logarit hmically spaced from 1 to 200 Hz. Note the c haracteristic asso ciation of pre-ictal sharp transien t, bands of fast activit y and suppression is presen t in eac h of the time frequency plot, despite its v ariations across sub jects. App endix D Classification Result with Lesion Information 176 App endix D. Classification Result with Lesion Information 177 T able D.1: Implan tation maps with sc hematic represen tation of the resection margins (shaded in red) for patien ts with confirmed or susp ected lesions. Bip olar SEEG c hannels inside epileptogenic lesion iden tified (epileptogenic-zone) and not Iden tified (Non-epileptogenic-zone) b y the algorithm as epileptogenic are named for eac h individual pati en t. * ID Map † Epileptogenic- zone Con tacts Inside Lesion Non- epileptogenic- zone Con tacts Inside Lesion ID Map † Epileptogenic- zone Con tacts Inside Lesion Non- epileptogenic- zone Con tacts Inside Lesion 1 R1-R2 R2-R3 R3-R4 X1-X2 X2-X3 X3-X4 X5-X6 6 P ossible MRI lesion P’1-P’2 (not confirmed b y pathology) 3 B1-B2 B2-B3 B3-B4 C2-C3 11 L6-L7 5 L’1-L’2 L’3-L’4 12 K4-K5 K5-K6 K7-K8 K9-K10 * Only patien ts who had lesion(s) are sho wn here. † Electro des on the maps are mark ed: (i) as red if they con tain TP c hannels (p oten tially epileptogenic inside the resection); (ii) as green if they con tain FP c hannels (p oten tially epileptogenic outside the resection); (iii) as blac k if they con tain only TN c hannels (not p oten tially epileptogenic outside the resection); (iv) as blac k in the red-shaded area if they con tain FN c hannels (not p oten tially epileptogenic inside the resection). Boundaries of prior resections are sc hematically shaded in y ello w (only P atien ts 3 and 9 had a previous resection). App endix E A dditional Non-L VF A Seizures 178 App endix E. A dditional Non-L VF A Seizures 179 T able E.1: A dditional n on-L VF A seizures excluded from the study Sub ject ID Num b er of recorded sei zures Time-frequency c haracteristics Concordance in anatomical distribution with L VF A seizures 226 1 Theta with o v erriding fast activit y Concordan t 228 4 Sharply coun tered alpha Discordan t 106 3 Theta/delta with o v erriding fast activit y Concordan t 108 Multiple Rh ythmical spik es Concordan t 215 1 Rh ythmical spik es Concordan t 238 1 Rh ythmical spik es Concordan t 118 1 Rh ythmical spik es Concordan t 222 1 Rh ythmical spik es Concordan t 221 2 Sharply coun tered theta Concordan t App endix F Misiden tified Fingerprin t P attern 180 App endix F. Misiden tified Fingerprin t P attern 181 (a) Sub ject 101 (b) Sub ject 103 (c) Sub ject 140 Figure F.1: One represen tativ e con tact sho wing the epileptogenic zone fingerprin t for Sub ject 101, 103 and 140 where no epileptogenic zone w as predicted using the original EZF pip eline. App endix G Individual Epileptogenic Zone Prediction Results 182 App endix G. Individual Epileptogenic Zone Prediction Results 183 T able G.1: Individual epileptogenic-zone predictions with considering differen t seizure clusters Sub- ject ID T rue P ositiv e F alse P ositiv e 101 102 I’1-I’2, I’2-I’3, I’8-I’9 O’13-O’14 103 106 M1-M2, M2-M3, N1-N2, X1-X2, Y2-Y3 108 R5-R6, R7-R8 111 A’1-A’2, A’2-A’3, B’1-B ’2, B’2-B’3, B’3-B’4, E’1-E’2, E’2-E’3, E’3-E’4, E’4-E’5, E’10-E’11 112 R’6-R’7 113 A8-A9 G12-G13, G13-G14, Q7-Q8, R7-R8 116 R1-R2, R2-R3, R3-R4, R4-R5, R5-R6, S2-S3, S3-S4, S4-S5, S5-S6, S6-S7, S7-S8, S8-S9, T5-T6, U1-U2, U2-U3 118 F’3-F’4, O’5-O’6, Y’1-Y’2, Y’2-Y’3, Y’3-Y’4 140 215 W9-W10, W10-W11, W11-W12, W12-W13, W13-W14, X8-X9, X9-X10 219 G’2-G’3, G’3-G’4, L’5-L’6 G’1-G’2, G’7-G’8, G’9-G’10, L’6-L’7, M’6-M’7, M’7-M’8 220 V10-V11 F1-F2, F2-F3, F3-F4, I9-I10, V5-V6, V9-V10 221 A’1-A’2 222 B7-B8 E3-E4, F4-F5, F5-F6, F6-F7, Z’9-Z’10 223 D’6-D’7, D’7-D’8, T’6-T’7, T’7-T’8 226 F’1-F’2, F’2-F’3, F’3-F’4, F’4-F’5, N’1-N’2, N’2-N’3, N’3-N’4, N’4-N’5, N’5-N’6, N’6-N’7, N’7-N’8, N’8-N’9, N’9-N’10, N’10-N’11, O’3-O’4, O’4-O’5, X’1-X’2, X’2-X’3, Y’1-Y’2, Y’2-Y’3, Y’3-Y’4 F1-F2, F2-F3, F3-F4, F4-F5, F5-F6, F10-F11, F11-F12, F13-F14, I3-I4, I’1-I’2, O1-O2, O2-O3, O3-O4, O7-O8, O11-O12, O’1-O’2, O’2-O’3, X1-X2, X2-X3, X3-X4, Y1-Y2, Y2-Y3, Y8-Y9, Y15-Y16 228 231 M1-M2, M2-M3, M3-M4, M4-M5, M7-M8, M’1-M’2, M’2-M’3, M’3-M’4, M’7-M’8, M’8-M’9, M’9-M’10, N1-N2, N’1-N’2, N’2-N’3, N’6-N’7, X’2-X’3, X’4-X’5, X’6-X’7, Y2-Y3, Y4-Y5, Y5-Y6, Y6-Y7, Y7-Y8, Y8-Y9, Y9-Y10, Y10-Y11, Y11-Y12, Y12-Y13, Y13-Y14, Y’3-Y’4, Y’4-Y’5, Y’7-Y’8, Y’12-Y’13 232 233 D6-D7 237 238 Q’3-Q’4, Q’5-Q’6, R ’3-R’4, R’4-R’5, R’5-R’6, R’7-R’8, R’8-R’9 S’1-S’2, S’2-S’3, S’3-S’4, S’4-S’5, T’3-T’4, T’4-T’5, U’4-U’5, U’6-U’7, U’9-U’10, W’4-W’5, W’5-W’6, W’7-W’8, X’2-X’3, Y’3-Y’4, Y’4-Y’5, Y’5-Y’6, Y’6-Y’7, Y’7-Y’8, Z’3-Z’4, Z’5-Z’6, Z’6-Z’7, Z’7-Z’8 App endix H Epileptogenic Zone Prediction Results without Considering Seizure Clusters 184 App endix H. EZ Prediction Results Without Considering Seizure Clusters 185 T able H.1: Epileptogenic zone prediction results without considering differen t seizure clusters Seizure-free Non-seizure-free Prediction T rue Prediction F alse Statistics Prediction T rue Prediction F alse Statistics Inside Resection 29 (TP * ) 280 (FN * ) 21 (TP * ) 220 (FN * ) Outside Resection 4 (FP * ) 839 (TN * ) 0.005 (FPR) 60 (FP * ) 1320 (TN * ) 0.043 (FPR) Statistics 0.879 (PPV) 0.259 (PPV) * TP/FP/TN/FN are with resp ect to th e resected region rather than the actual epileptogenic zone. App endix I In terp olated Fingerprin t-based Epileptogenic Zone Prediction Results 186 App endix I. In terp olated Fingerprin t-based EZ Prediction Results 187 Figure I.1: Epileptogenic zone prediction score in terp olated on to the patien ts’ pre-op erativ e MRI (left) and its comparison with the p ost-op erativ e MRI (righ t). F or Sub ject 219, t w o epileptogenic zone lo cations w ere predicted on the left and on the righ t side, the first resection via laser ablation is sho wn on the top and the resection is sho wn on the b ottom. App endix J Con v ergence Comparison SRSCPD vs ALS 188 App endix J. Con v ergence Comparison SRSCPD vs ALS 189 Figure J.1: Con v ergence comparison of CP-ALS vs SRSCPD on one sim ulated rank-7 tensor. Eac h sub-figure sho ws the mean of the absolute difference of the loading matrices b et w een the curren t iteration and the previous iterations o v er all mo des (see Algorithm V ) in log scale along the y-axis as a function of n um b er of iterations in the x-axis for ranks from 2 to 7 , where the iteration n u m b e r here indicates ma jor iterations including all sub-problems in ALS. The stopping criterion w as set to 10 5 for all cases. The result for r = 1 w as excluded as the t w o algorithms are iden tical in that case. App endix K A dditional Consisten t Comp onen ts F ound b y SRSCPD 190 App endix K. A dditional Consisten t Comp onen ts F ound b y SRSCPD 191 (a) A dditional consisten t comp onen t for Sub je ct 1 (b) A dditional consisten t comp onen t for Sub ject 2 Figure K.1: A dditional consisten t comp onen ts found b y SRSCPD App endix L A dditional Non-ph ysiological Mismatc hed Comp onen ts F ound b y SRSCPD 192 App endix L. A dditional Non-ph ysiological Comp onen ts Using SRSCPD 193 (a) A dditional non-ph ysiological mismatc hed comp onen t (from session 2) for Sub ject 1 (b) A dditional non-ph ysiological mismatc hed comp onen t (from session 1) for Sub ject 2 Figure L.1: A dditional non-ph ysiological mismatc hed comp onen ts found b y SRSCPD App endix M COR CONDIA Metric for Results Using ALS Algorithm 194 App endix M. COR CONDIA Metric for Results Using ALS Algorithm 195 Figure M.1: COR CONDIA are sho wn as a function of R for t w o sessions of b oth sub jects using CP-ALS algorithm. The rank w as selected to b e R = 2;3;3;3 corresp onding to eac h of the sub- figures ab o v e. App endix N Comp onen ts F ound Using ALS Algorithm 196 App endix N. Comp onen ts F ound Using ALS Algorithm 197 Figure N.1: Consisten t comp onen t for Sub ject 1 (DMN similar to that in Figure 4.7 ( a )) Figure N.2: Mismatc hed comp onen t from session 1 for Sub ject 1 (Motor net w ork similar to that in F igure 4.7 ( b )) 198 Figure N.3: First mismatc hed comp onen t from session 2 for sub ject 1 (Unkno wn comp onen t similar to that in Figure K.1 ( a )) Figure N.4: Second mismatc hed comp onen t from session 2 for Sub ject 1 (Artifact similar to that in F igure L.1 ( a )) App endix N. Comp onen ts F ound Using ALS Algorithm 199 Figure N.5: First mismatc hed comp onen t from session 1 for Sub ject 2 (Mix of DMN and Motor net w ork) Figure N.6: Second mismatc hed comp onen t from session 1 for Sub ject 2 (Artifact similar to that in F igure L.1 ( b )) 200 Figure N.7: Third mismatc hed comp onen t from session 1 for Sub ject 2 (Another mix of DMN and Motor net w ork) Figure N.8: First mismatc hed comp onen t from session 2 for Sub ject 2 (DMN similar to that in Figure 4.8 ( a )) App endix N. Comp onen ts F ound Using ALS Algorithm 201 Figure N.9: Second mismatc hed comp onen t from session 2 for Sub ject 2 (Motor net w ork similar to that Figure 4.8 ( b )) Figure N.10: Third mismatc hed comp onen t from session 2 for Sub ject 2 (Unkno wn comp onen t similar to that in Figure K.1 ( b )) App endix O A dditional NSR CPD Results on Language T ask fMRI Data 202 App endix O. A dditional NSR CPD Results on Language T ask fMRI Data 203 (a) (b) Figure O.1: A dditional NSR CPD results on language tfMRI data: (a) An example of plausible but un-recognized net w ork; (b) An example of sub ject-sp ecific net w ork. App endix P Group ICA Results on Language T ask fMRI Data 204 App endix P . Group ICA Results on Language T ask fMRI Data 205 (a) (b) Figure P .1: T w o recognized net w orks obtained from the language tfMRI data using the group ICA metho d: (a) Visual net w ork; (b) A uditory net w ork. App endix Q NSR CPD Results on Resting fMRI Data 206 App endix Q. NSR CPD Results on Resting fMRI Data 207 Figure Q.1: Individual comp onen ts (sub-net w orks) iden tified from rfMRI data using NSR CPD, whic h constitute iden tifiable w ell-kno wn net w orks: DMN, SMN, ECN and VN. The auditory net- w ork is iden tified b y itself without sub-net w orks, hence omitted here. This page in ten tionally left blank Reference [1] E. A car, C. A ykut-Bingol, H. Bingol, R. 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Abstract (if available)
Abstract
Brain connectivity is modeled as a complex, segregative and integrative network of connections between different brain regions. Studying functional brain connectivity can offer us an effective way to examine how different brain networks relate to human behaviors as well as how those networks may be altered in neurological diseases. However, measuring functional connectivity poses a variety of mathematical, signal processing and neuroscience challenges. First, a good high-level representation of the data is often required in order to obtain an accurate estimation of the functional connectivity, because most of the typically-used linear measures are not capable of capturing the true highly non-linear brain interactions. Second, the temporal stationarity of the time series assumed by most of the studies may not be realistic due to the dynamic nature of the brain. Hence, how to reliably estimate the spatial and temporal dynamics of functional connectivity simultaneously is a key challenge to us. Moreover, signals collected via almost all neuroimaging techniques are heavily corrupted with noise. The inherent low signal-to-noise ratio prevents us from obtaining a robust estimation of functional connectivity. In this work, we present and validate several novel approaches and methods to address some of the challenges in functional connectivity estimation and brain network identification problems. ❧ To address the high-level data representation issue, we defined a bio-electrical marker that can differentiate the epileptogenic zone from areas of propagation in patients with epilepsy. We discovered a specific ictal time-frequency pattern, referred as the “fingerprint”, in the epileptogenic zone which contains a combination of sharp spikes preceding multi-band fast activity concurrent with suppression of lower frequencies. We developed a novel machine learning system that automatically extracts each of these features, classifies electrode contacts as being within the epileptogenic zone or outside the epileptogenic zone and generates individualized epileptogenic zone predictions for each patient based on their anatomical magnetic resonance images. ❧ To address the dynamic brain network identification issue, we developed a rank-recursive scalable and robust sequential canonical polyadic decomposition framework that allows us to robustly discover brain networks which can overlap in both space and time in large-scale datasets. The robustness and scalability were achieved by using lower-rank solutions as the warm start to higher-rank decompositions. This scalable and robust sequential canonical polyadic decomposition framework is flexible in the sense that it is not only applicable to wavelet-transformed electroencephalography data but also to multi-subject asynchronous functional magnetic resonance imaging data if the data is temporally aligned across subjects using the BrainSync algorithm. ❧ To address the noise corruption issue, we described an optimization-based method that provides a means of systematically selecting the parameter for the temporal non-local means filtering. We further developed global PDF-based temporal non-local means, a novel data-driven optimized kernel function based on Bayes factor for the temporal non-local means filtering, which allows us to perform global filtering with improved noise reduction effects but without blurring adjacent functional regions. ❧ Applications of these proposed methods are illustrated using a variety of simulated as well as in-vivo clinical data.
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Asset Metadata
Creator
Li, Jian
(author)
Core Title
Functional connectivity analysis and network identification in the human brain
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
05/09/2019
Defense Date
03/18/2019
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
brain networks,epileptogenic zone,filtering,functional connectivity,functional magnetic resonance imaging,OAI-PMH Harvest,stereotactic electroencephalography,tensor decomposition
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application/pdf
(imt)
Language
English
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Electronically uploaded by the author
(provenance)
Advisor
Leahy, Richard (
committee chair
), Haldar, Justin P. (
committee member
), Rosen, Gary (
committee member
), Wisnowski, Jessica L. (
committee member
)
Creator Email
jli981@usc.edu,silencer1127@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c89-157209
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UC11660187
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etd-LiJian-7429.pdf (filename),usctheses-c89-157209 (legacy record id)
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etd-LiJian-7429.pdf
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157209
Document Type
Dissertation
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Li, Jian
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University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
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The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
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Tags
brain networks
epileptogenic zone
functional connectivity
functional magnetic resonance imaging
stereotactic electroencephalography
tensor decomposition