KOSZUL CONNECTIONS OF FLAT EEG BUNDLES FOR DESCRIPTION OF

BRAIN SIGNAL DYNAMICS

NIKI ANIS BIN AB KARIM

A thesis submitted in fulfilment of the

requirements for the award of the degree of

Doctor of Philosophy (Mathematics)

Faculty of Science

Universiti Teknologi Malaysia

MARCH 2016

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THIS WORK IS DEDICATED TO

MY PARENTS

MY UNIVERSITY

MY COUNTRY

iv

ACKNOWLEDGEMENT

I extend deepest thanks to my supervisor, Prof. Dr. Tahir bin Ahmad for

showing great patience, wisdom, and openness during the course of this research. He

has allowed me a lot of independence and personal space to explore ideas and solve

problems, and his guidance has been essential through the years. He is as humble as

he is inspirational, and there are so many reasons for him to be proud in leading the

way for future educators and scientists like me.

I also offer gracious thanks to the thesis examiners, Prof. Dato’ Dr. Abdul

Razak bin Salleh and Assoc. Prof. Dr. Yusof bin Yaacob, and to the Chairman of the

viva voce session, Prof. Dr. Zuhaimy bin Ismail, for their advice, wisdom and

encouraging words.

Thanks also to the staff and educators at the Ibnu Sina Institute and the

Mathematics Department of the Faculty of Science, Universiti Teknologi Malaysia

for the life experiences and lessons, both in and out of the offices and lecture rooms,

and for offering a conducive environment for my studies.

Immense gratitude goes to my parents, for nurturing a love for language and

academia, for giving me a wicked sense of humor, and for being patient with my

quirks and oddities. They are great sources of energy and inspiration.

Praises be to the Prophet Muhammad, for the gift of Revelation. Praises be to

Allah the Wise and Merciful, for the gift of Creation. Verily He is the source of all

things, and all return to Him.

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ABSTRACT

The bio-electromagnetic inverse problem is the identification of

electromagnetic sources based on signals recorded from Electroencephalography

(EEG) or Magnetoencephalography (MEG) and physical equations with a minimum

of priori information. Following initial applications in modelling epilepsy patients'

electrical brain activity, extensive research has been done in processing massive

amounts of signal data from patients while preserving as much information as

possible. Improvements in analyzing EEG/MEG recordings have significant utility

not just in epilepsy treatment and diagnosis, but neuro-cognitive research in general.

The main objective of this research is to build structures holding visualized and

flattened EEG data as differential geometric entities, and to extend these structures as

a dynamical system dealing with the evolution of large amounts of point data about

brain activity. One aspect of existing research involves EEG signal data recorded

across a time interval and processed using fuzzy clustering techniques, resulting in

point data sets representing areas of high electrical activity within the brain.

Concepts in differential geometry are applied to these spaces as a dynamical and

visualized approach to modelling the evolution of signal clusters in the brain over

time. Initially, Flat EEG data sets are shown to be topological spaces, manifolds and

vector spaces. Two vector bundle structures for the Flat EEG space are thus

developed: one analogous to Minkowski space-time and the other based on the

classical notion of spatial change over linear time. From there, Koszul connections

were constructed for both vector bundles, and both are shown to have zero curvature.

Having provided a continuous differential structure to Flat EEG data, the evolution

of signal clusters as a discrete dynamical system is then interpolated into a

continuous form, allowing an enhanced view of the brain's state changes over time.

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ABSTRAK

Masalah songsangan bio-elektromagnet ialah mengenal pasti sumber

elektromagnet berdasarkan isyarat rakaman Elektroensefalografi (EEG) atau

Magnetoensefalografi (MEG) dan persamaan fizis bersama maklumat priori yang

minimum. Berikutan daripada penggunaan awal dalam membina model aktiviti

elektrik otak pesakit epilepsi, penyelidikan telah dilakukan demi memproses

bilangan besar data isyarat pesakit sambil memelihara sebanyak mungkin maklumat

asal. Penambahbaikan analisis rakaman EEG/MEG mempunyai kegunaan bukan

sahaja dalam rawatan dan diagnosis epilepsi, malah juga penyelidikan neuro-kognitif

secara am. Objektif utama penyelidikan ialah membina struktur menempatkan data

EEG yang berbentuk visual dan rata sebagai objek geometri keterbezaan, dan

melanjutkan struktur tersebut sebagai sistem dinamik berkenaan evolusi sejumlah

besar data titik aktiviti otak. Satu aspek penyelidikan kini melibatkan data isyarat

EEG yang dirakam pada suatu selang masa dan diproses menggunakan teknik

kelompok kabur, menghasilkan set data titik mewakili kawasan aktiviti elektrik yang

tinggi di dalam otak. Konsep geometri keterbezaan digunakan terhadap ruang set

tersebut sebagai pendekatan dinamik dan visual bagi permodelan evolusi semasa

kelompok isyarat dalam otak. Sebagai permulaan, data set EEG Rata dibuktikan

sebagai ruang topologi, manifold dan ruang vektor. Dengan itu dua struktur berkas

vektor untuk ruang EEG Rata dibina: satu setara dengan ruang-masa Minkowski dan

satu lagi berdasarkan idea klasik perubahan ruang melawan masa linear. Dari situ,

kaitan Koszul dibangunkan untuk kedua-dua berkas vektor, yang juga terbukti

mempunyai kelengkungan sifar. Setelah menyediakan struktur keterbezaan lancar

bagi data EEG Rata, evolusi kelompok isyarat tersebut dalam bentuk sistem dinamik

diskret kemudian diberi interpolasi menjadi sistem dinamik berterusan,

membolehkan pandangan lebih baik bagi perubahan keadaan otak terhadap masa.

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TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION

DEDICATION

ACKNOWLEDGEMENT

ABSTRACT

ABSTRAK

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

LIST OF SYMBOLS/NOTATIONS

LIST OF APPENDICES

ii

iii

iv

v

vi

vii

x

xi

xiii

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1 INTRODUCTION 1

1.1 Introduction 1

1.2 Background of Research 1

1.3 Problem Statement 2

1.4 Research Objectives 2

1.5 Research Scope 3

1.6 Significance of Research 3

1.7 Thesis Outline 4

1.8 Context of Research 5

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2 LITERATURE SURVEY 6

2.1 Epilepsy 6

2.2 Inverse Problem and FTTM 8

2.3 Differential Geometry and Seizures as Dynamical

Systems 10

2.4 Research in Context 12

3 DIFFERENTIAL GEOMETRY ON MANIFOLDS 15

3.1 Fibre Bundles 15

3.2 Vector Bundles 17

3.3 Connections 18

3.4 Dynamical Systems 20

3.5 Minkowski Space 22

4 VECTOR BUNDLES ON FLAT EEG 24

4.1 Flat EEG as Base Space 24

4.2 Tangent Bundle of MEEG 34

4.3 Flat EEG as Vector Bundle 37

4.4 Differences Between Tangent Bundle TMEEG and

Vector Bundle VEEG 40

4.5 Conclusion 40

5 CONNECTIONS FOR VECTOR BUNDLES OF

FLAT EEG 41

5.1 Connection for Tangent Bundle TMEEG 41

5.2 Connection for Vector Bundle VEEG 44

5.3 Physical Interpretation of XY∇ and TV∇ 46

5.4 Associated Connection between Bundles 47

5.5 Curvature of Connections 50

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6 FLAT EEG BUNDLES AS DYNAMICAL SYSTEMS 53

6.1 Advantages of Structuring Data as Dynamical Systems 53

6.2 MEEG as Discrete Dynamical System 54

6.3 VEEG as Discrete Dynamical System 55

6.4 Interpolation and Approximated Continuous

Dynamical System 56

7 CONCLUSION 61

7.1 Summary of Research 61

7.2 Future Research 62

7.3 Conclusion 63

REFERENCES 64

APPENDIX A 67

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LIST OF TABLES

TABLE NO. TITLE PAGE

4.1 Positions of Sensors and Potential Differences Detected

on a Flattened EEG Plane for Model of a Patient’s Head

with Mean Radius of 8.3 Centimeters 27

4.2 Sample Data of Five Clusters Generated by FCM

Clustering at Time t 28

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LIST OF FIGURES

FIGURE NO. TITLE PAGE

1.1 Thesis Outline 4

1.2 Context of Research 5

2.1 Woman Undergoing EEG 7

2.2 Typical EEG Readouts 7

2.3 FTTM Version 1 Algorithms 8

2.4 FTTM Version 2 Algorithms 9

2.5 Electrical Potential Readings from Sensors and Signal

Clusters of Neural Activity on Flat EEG 10

2.6 Time Series of Sensor Readings and Signal Clusters

of Neural Activity on Flat EEG 11

2.7 Comparison of Big Bang Model with Neuroelectrical

Brainstorm 13

2.8 Signal Cluster Partitions and Their Centroids 14

3.1 Fibre Bundle 15

3.2 Fibre Bundle Relations 16

3.3 Tangent Bundle of S 2 17

3.4 Parallel Transport as Connection for Tangent Bundle 18

3.5 State Transition of Dynamical System 20

3.6 Chapman-Kolmogorov Law 21

3.7 Orbit of a Dynamical System 22

3.8 Minkowski Space Depicting World Line Boundary

Light Cone 23

4.1 Construction of space for EEG signal data, W 25

4.2 Set of signal cluster centers at a time t 28

4.3 Combining W and C to Generate MEEG 29

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LIST OF FIGURES

FIGURE NO. TITLE PAGE

4.4 An MEEG Section with Real Recorded Data. 29

4.5 Tangent space TpMEEG 35

4.6 Construction of fibre tEEGV for vector bundle VEEG 38

5.1 Elements of XY∇ 42

5.2 Elements of TV∇ 44

5.3 Mapping of Flat EEG and Connection Elements to S 2 46

5.4 Permutations of Orbits for Cluster Centroids in MEEG 47

5.5 Relationship Diagram between Vector Bundles 48

6.1 Determinism of MEEG 55

6.2 Determinism of VEEG 56

6.3 Linear Interpolation from Single-Point State to

Single-Point State 57

6.4 Linear Interpolation between Two Time-Parametrised

Points in MEEG 56

6.5 Orbits for Cluster Centers in MEEG Using Linear

Interpolation 57

7.1 Summary of Research 62

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LIST OF SYMBOLS/NOTATIONS

M – Manifold ∇ – Connection on a manifold Lx – Lie differential operator E – Vector bundle on a manifold Tp(M) – Tangent space on a point p in manifold M Hom(X,Y) – Set of homomorphisms from X to Y L(X,Y) – Set of linear transforms from X to Y End(X) – Set of endomorphisms from X to itself S2 – Unit sphere manifold

xiv

LIST OF APPENDICES

APPENDIX TITLE PAGE

A Publications 67

CHAPTER 1

INTRODUCTION

1.1 Introduction

The bio–electromagnetic inverse problem is the identification of

electromagnetic sources based on signals recorded from Electroencephalography

(EEG) or Magnetoencephalography (MEG) and physical equations with a minimum

of priori information. Going from initial applications in modelling epilepsy patients'

electrical brain signals, extensive research has been done in processing and

transforming massive amounts of signal data from patients while preserving as much

information as possible. Improvements in analysing EEG/MEG records have

significant utility not just in epilepsy treatment and diagnosis, but neuro–cognitive

research in general.

1.2 Background of Research

The field of signal processing presents the challenge of preserving data

integrity as they are transformed into spaces with varying dimensions. One example

of this in the field of medicine is the visualization of EEG signal data into two-

dimensional arrays to aid diagnosis. Previous research were dependent on data

recorded over instances within an interval. The data set is essentially discretised over

time, even though EEG signal data is continuous. This is due to transformations

required for the clustering process.

2

The question of the movement of clusters within space should involve

differential geometrical structures. Vector spaces are simple, well-understood objects

useful for modelling the spatial path of information over time. Neuroelectrical

activity in the brain also exhibits the properties of a dynamical system.

1.3 Problem Statement

Analysis of Flat EEG signals has depended on discrete instances recorded

over an interval. In order to aid the visualization and ease of interpretation of these

massive sets of data by professionals, the dynamics of Flat EEG signals should be

described as a set of vectors and thus rendered as an evolving system.

1.4 Research Objectives

The objectives of this research include

i. To construct vector bundles based on Flat EEG space.

ii. To construct connection(s) on the bundles.

iii. To express these geometric objects as dynamical systems.

iv. To approximate orbits of Flat EEG cluster data.

3

1.5 Research Scope

This research utilises mathematical theory based on differential geometry and

dynamical systems on bioelectromagnetic field data of the brains of epilepsy patients

during seizures, in order to be considered in conjunction with the Fuzzy Topographic

Topological Mapping (FTTM) model.

1.6 Significance of Research

A differential structure for post-clustering Flat EEG is needed to stitch

together disparate sets of instantaneous information into a continuous whole. This is

helpful in order to aid the detection of epileptogenic foci by modelling the paths of

the strong electrical signals in the brain.

4

1.7 Thesis Outline

Figure 1.1 shows a flowchart outlining all the chapters in this thesis and their

main topics of interest.

KOSZUL CONNECTIONS OF FLAT EEG BUNDLES FOR

DESCRIPTION OF BRAIN SIGNAL DYNAMICS

BACKGROUND OF RESEARCH

METHODOLOGY Constructing Bundles and Connections

IMPLEMENTATION Dynamical Systems & Interpolation

CONCLUSION

Figure 1.1 Thesis Outline

Chapter 1 Introduction

Chapter 2 Literature Survey Flat EEG/MEG

Chapter 3 Differential Geometry

on Manifolds

Chapter 4 Vector Bundles

on Flat EEG

Chapter 5 Connections for

Bundles of Flat EEG

Chapter 7 Future Research and Conclusion

Chapter 6 Flat EEG Bundles

as Dynamical Systems

5

1.8 Context of Research

Figure 1.2 shows the relationship between the research in this thesis and other

work previously done in similar fields. Further details and elaborations on this topic

can be found in Chapter 2.

Figure 1.2 Context of Research

Tahir Ahmad (2000)

EEG Signal Sources and Processing Via FTTM

Fauziah Zakaria (2008)

EEG Signal Sourcing and Topologised Spaces

Nazihah Ahmad (2009)

Digitization of Flat EEG

Tan Lit Ken (2010)

Topology and Stability of Flat EEG as

Dynamical System

Vinod Ramachandran (2012)

High-Accuracy Statistical Model / Probability

Distribution for Flat EEG Data

Noraini Ismail (2013)

Maxwell’s Equations and Einstein Field Equation

in Relation to EEG Source Localization

Niki Anis Ab Karim (2015)

Differential Geometric Structure for Flat EEG

Clusters

fuzzy c-means clustering

Flat EEG dynamical systems

spacetime mechanics

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