A Mathematical Model of Cerebral Cortical Folding Development Based on a Biomechanical Hypothesis Sarah Kim

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2015 A Mathematical Model of Cerebral Cortical Folding Development Based on a Biomechanical Hypothesis Sarah Kim

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COLLEGE OF ARTS AND SCIENCES

A MATHEMATICAL MODEL OF CEREBRAL CORTICAL FOLDING DEVELOPMENT

BASED ON A BIOMECHANICAL HYPOTHESIS

SARAH KIM

A Dissertation submitted to the Department of Mathematics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

2015

Sarah Kim defended this dissertation on October 29, 2015. The members of the supervisory committee were:

Monica K. Hurdal Professor Directing Dissertation

Oliver Steinbock University Representative

Richard Bertram Committee Member

Nick Cogan Committee Member

The Graduate School has veriﬁed and approved the above-named committee members, and certiﬁes that the dissertation has been approved in accordance with university requirements.

ii Soli Deo Gloria

iii ACKNOWLEDGMENTS

First of all, I would like to express my deepest gratitude to my advisor, Dr. Monica Hurdal, for her excellent guidance, support, and devotion to this research. She always opened the door to listen to my ideas, and meetings with her in and out of her oﬃce always inspired me. She helped me navigate the various and unfamiliar territories I experienced when conducting my research, which will be a great asset for my future career. Her letters of recommendation for scholarships always gave me great opportunities. She has been a tremendous mentor and supporter for me. I am so blessed to have her as my advisor at FSU. I would like to thank my committee members: Dr. Richard Bertram, Dr. Nick Cogan, and Dr. Oliver Steinbock. Thank you for your time, brilliant comments and suggestions, and encouraging words throughout my doctoral journey. I would like to thank the FSU Math Department professors, especially, Dr. Xiaoming Wang, Dr. Giray Okten, Dr. Penelope Kirby, and Dr. Steve Bellenot, for providing great lectures, being supportive, and opening new opportunities throughout my graduate career. I would like to thank my FSU Math Department friends who have spent their precious time with me during my academic journey in a strange place, especially, Sevgi, Angela, Daozhi, Mao, Chaoxu, Diego, Justin, Tim, Arij, Qiuping, and Yingyun. Thank you for your friendship, help, and caring. I also would like to thank my seniors, Debbie and Greg, for their work on this topic. Your contributions were a great guide and formed the basis for my study. I would like to thank my family in South Korea, my father, mother, and younger brother, Joseph, for their sincere prayers and encouragement during this journey. Your prayers remind me that he who began a good work in us will carry it on to completion, especially when I am struggling with my weaknesses. Last but not least, I would like to thank my husband, Yoonhwak, and daugther, Claire Taehee, for their love, support, and prayers. Running together with you makes me keep going for this marathon ﬁnish line willingly and joyfully. In addition, I was really thankful for researching on the development of the fetal brain with the life inside of me.

iv TABLE OF CONTENTS

ListofTables...... viii ListofFigures ...... ix ListofSymbols...... xiii List of Abbreviations ...... xiv Abstract...... xv

1 Introduction 1

2 Biology Background 4 2.1 Neuroanatomy and Corticogenesis ...... 4 2.1.1 BrainStructure...... 4 2.1.2 Early Development and Ventricular System ...... 5 2.1.3 Neurons and Glial Cells ...... 7 2.1.4 Anatomical References ...... 7 2.2 Cortical Folding Development ...... 12 2.2.1 Magnetic Resonance Imaging of the Brains of Preterm Infants ...... 14 2.2.2 Summary of Human Fetal Brain Folding Development ...... 15 2.3 Biological Hypotheses ...... 21 2.3.1 Biochemical Hypothesis: Intermediate Progenitor Model (IPM) ...... 21 2.3.2 Biomechanical Hypothesis: Axonal Tension Hypothesis (ATH) ...... 25 2.3.3 Biomechanical Hypothesis: Diﬀerential Growth Hypothesis (DGH) ...... 25 2.3.4 Debating Hypotheses ...... 28 2.4 Conclusions...... 29

3 The Proposed Mathematical Model of Cerebral Cortical Folding Development 30 3.1 ModelGeometry ...... 30 3.2 Directions of the Applied Axonal Tension Forces ...... 31 3.3 Magnitudes of the Applied Axonal Tension Forces ...... 33 3.3.1 Turing Reaction-Diﬀusion System ...... 35 3.3.2 IPC Self-ampliﬁcations to the Axonal Tension Forces ...... 37 3.4 Displacements...... 38 3.4.1 Theory of Elasticity ...... 39 3.4.2 Finite Element Formulation ...... 43 3.5 Conclusions...... 47

4 Numerical Results 48 4.1 Parameters and Gyriﬁcation Index ...... 48 4.1.1 Tissue Elasticity and Geometric Size of the Brain ...... 49 4.1.2 Direction of the Applied Axonal Tension Force ...... 49 4.1.3 Strength of the Applied Axonal Tension Force ...... 50

v 4.1.4 GyriﬁcationIndex(GI) ...... 51 4.2 DomainSize...... 54 4.2.1 Bigger vs. Smaller Brains ...... 54 4.2.2 Thickervs.ThinnerCortices ...... 57 4.2.3 Summary ...... 57 4.3 Directions of the Applied Axonal Tension Forces ...... 60 4.3.1 Summary ...... 62 4.4 Strengths of the Applied Axonal Tension Forces ...... 65 4.4.1 Weaker vs. Stronger Pulling Tension Forces ...... 66 4.4.2 Distance between the Applied Forces ...... 67 4.4.3 Uneven Magnitudes of the Applied Forces ...... 70 4.4.4 Summary ...... 76 4.5 Conclusions...... 77

5 Applications to Human Cerebral Cortical Folding Malformations 78 5.1 Decreased Proliferation ...... 79 5.1.1 Microcephaly and Microlissencephaly ...... 79 5.1.2 Modeling Microcephaly and Microlissencephaly ...... 80 5.2 Increased Proliferation ...... 81 5.2.1 Hemimegalencephaly ...... 82 5.2.2 Modeling Hemimegalencephaly ...... 83 5.3 Undermigration ...... 87 5.3.1 Classic (Type I) Lissencephaly ...... 88 5.3.2 Modeling Classic Lissencephaly ...... 88 5.4 Deranged Organization ...... 90 5.4.1 Polymicrogyria ...... 90 5.4.2 Modeling Polymicrogyria ...... 92 5.5 Conclusions...... 94

6 Conclusions 96 6.1 Summary ...... 96 6.2 Ongoing and Future Work ...... 98 6.2.1 Three-dimensionalModel ...... 98 6.2.2 Dynamic Model ...... 100 6.2.3 Drug Use Model ...... 101 6.3 ConcludingRemarks ...... 101

Appendices A Derivations 103 A.1 MethodsofWeightedResidual ...... 103 A.2 Linear Quadrilateral Element ...... 104 A.3 Gauss-Legendre Quadrature Rule ...... 106 A.4 DerivingTuringConditions ...... 107 A.4.1 Turing Criterion: Linear Stability in the Absence of Diﬀusion ...... 107 A.4.2 Turing Criterion: Diﬀusion-Driven Instability ...... 109

vi B Figures 112 B.1 DomainSize...... 113 B.1.1 Bigger vs. Smaller Brains ...... 113 B.1.2 Thickervs.ThinnerCortices ...... 116 B.2 Directions of the Applied Axonal Tension Forces ...... 118 B.3 Strengths of the Applied Axonal Tension Forces ...... 123 B.3.1 Weaker vs. Stronger Pulling Tension Forces ...... 123 B.3.2 DistancebetweentheAppliedForces ...... 125 B.4 Preliminary numerical results of a three-dimensional model ...... 126

References...... 128 BiographicalSketch ...... 138

vii LIST OF TABLES

4.1 Brain size and the GI when n0=5...... 59

4.2 Force direction and the GI when n0=3...... 61

4.3 Force direction and the GI when n0=5...... 62

4.4 Proﬁles of the applied forces based on the irregular Turing patterns when Cu = 625 . 73

5.1 Causes of cortical folding malformations and simulation parameters ...... 79

5.2 Proﬁles of the forces applied in Figures 5.4 and 5.5 ...... 86

A.1 n-point Gauss-Legendre quadrature weights and points ...... 107

viii LIST OF FIGURES

2.1 Sixmajorlobes...... 5

2.2 Gyriandsulci ...... 6

2.3 Serial MR imaging of normal brain growth ...... 8

2.4 Major gyri and sulci (lateral view) ...... 9

2.5 Major gyri and sulci (medial view) ...... 9

2.6 Sixembryoniclayers...... 10

2.7 Cerebralventricles ...... 10

2.8 Volumetric changes of the fetal brain, germinal matrix , and lateral ventricles . . . . . 11

2.9 Neuron ...... 11

2.10 Radial glial cells ...... 12

2.11 Anatomical references regarding direction ...... 13

2.12 Sections of the human brain ...... 13

2.13 MR images of a preterm infant born at the 25th week of GA ...... 16

2.14 MR images of a preterm infant born at the 26th week of GA ...... 16

2.15 MR images of a preterm infant born at the 28th week of GA ...... 17

2.16 MR images of a preterm infant born at the 30th week of GA ...... 18

2.17 MR images of a preterm infant born at the 32nd week of GA ...... 19

2.18 MR images of a preterm infant born at the 34th week of GA ...... 20

2.19 MR images of a preterm infant born at term (approximately 40 weeks of GA) . . . . . 20

2.20 Radial unit hypothesis ...... 22

2.21 Intermediate progenitor hypothesis and model ...... 23

2.22 Subventricular zone thickness and cortical folding ...... 24

2.23 Axonal tension hypothesis ...... 26

2.24 Compactness of cortical wiring and cortical folding ...... 27

ix 2.25 Camera lucida drawings of brain layers ...... 27

3.1 Representation of the human brain ...... 31

3.2 Components of the applied force on the semi-circular model cortex ...... 32

3.3 The angle φ ...... 34

3.4 The angle φ′ = π φ ...... 35 − 3.5 Free body diagram of a two-dimensional, inﬁnitesimal element with dimensions dx by dy ...... 40

3.6 Two-dimensional geometric strain deformation ...... 41

3.7 Shape functions of the linear quadrilateral element ...... 46

4.1 Turing patterns and strengths of the applied axonal tension forces ...... 52

4.2 Gyriﬁcationindex ...... 53

4.3 GI calculation in our simulations ...... 53

4.4 Changing gray matter radius r2 when n0=5...... 55

4.5 Eﬀects of changing gray matter radius r2 when n0=5...... 56

4.6 Graphs of r2 versus GA and the GI versus r2 ...... 56

4.7 Changing cortical thickness, thick = r r , when n0=5...... 58 2 − 1 4.8 Graph of GI versus thick ...... 59

4.9 Changing the directions of the applied forces when n0 = 3 in Case 1 ...... 63

4.10 Graphs of the GI versus CTR ...... 64

4.11 Graph of the number of sulci versus CTR ...... 65

4.12 Changing the magnitudes of the applied forces when n0 = 5 ...... 68

4.13 Graphs of the GI versus the average of the magnitudes of all applied forces and Cfm . 69

4.14 Graphs of the GI versus the magnitude of each force ...... 70

4.15 Changing distance between the applied pulling forces when f = 4.5 N ...... 71 | | 4.16 Irregular Turing patterns and the applied forces ...... 72

4.17 Cortical folding patterns resulting from applied forces due to the Turing pattern in Figure 4.16a ...... 74

x 4.18 Cortical folding patterns resulting from applied forces due to the Turing pattern in Figure 4.16b ...... 75

4.19 Cortical folding patterns resulting from applied forces due to the Turing pattern in Figure 4.16c ...... 76

5.1 Microcephaly and microlissencephaly ...... 81

5.2 Modeling microcephaly and microlissencephaly ...... 82

5.3 Hemimegalencephaly ...... 84

5.4 Modeling hemimegalencephaly with an enlarged right hemisphere ...... 85

5.5 Modeling hemimegalencephaly with an enlarged left hemisphere ...... 85

5.6 Classiclissencephaly...... 89

5.7 Modelingclassiclissencephaly...... 89

5.8 Diagrammatic representation of a cerebral cortex with polymicrogyria...... 90

5.9 Polymicrogyria with abnormal brain size ...... 91

5.10 Unilateral polymicrogyria ...... 92

5.11 Modeling polymicrogyria ...... 93

6.1 Preliminary numerical result of a three-dimensional model ...... 99

6.2 Diﬀerent views and skeletons of Figure 6.1b ...... 99

6.3 More preliminary numerical results of a three-dimensional model ...... 100

A.1 Mappingbetweencoordinatesystems ...... 105

A.2 Legendre polynomials ...... 107

B.1 Changing gray matter radius r2 when n0 = 3 ...... 113

B.2 Changing gray matter radius r2 when n0 = 10 ...... 114

B.3 Eﬀects of changing gray matter radius r2 when n0 = 3 ...... 115

B.4 Eﬀects of changing gray matter radius r2 when n0 = 10 ...... 115

B.5 Changing cortical thickness, thick = r r , when n0 = 3 ...... 116 2 − 1 B.6 Changing cortical thickness, thick = r r , when n0 = 10 ...... 117 2 − 1 B.7 Changing the directions of the applied forces when n0 = 5 in Case 1 ...... 118

xi B.8 Changing the directions of the applied forces when n0 = 3 in Case 2 ...... 119

B.9 Changing the directions of the applied forces when n0 = 3 in Case 3 ...... 120

B.10 Changing the directions of the applied forces when n0 = 5 in Case 2 ...... 121

B.11 Changing the directions of the applied forces when n0 = 5 in Case 3 ...... 122

B.12 Changing the magnitudes of the applied forces when n0 = 3 ...... 123

B.13 Changing the magnitudes of the applied forces when n0 = 10 ...... 124

B.14 Changing distance between the applied pulling forces when f = 4.0 N ...... 125 | | B.15 Changing distance between the applied pulling forces when f = 5.0 N ...... 125 | | B.16 Diﬀerent views and skeletons of Figure 6.3a ...... 126

B.17 Diﬀerent views and skeletons of Figure 6.3b ...... 126

B.18 Diﬀerent views and skeletons of Figure 6.3c ...... 127

B.19 Diﬀerent views and skeletons of Figure 6.3d ...... 127

xii LIST OF SYMBOLS

The following list of symbols are used throughout this dissertation, that I tried to use consistently.

r1 radius of white matter r2 radius of gray matter thick cortical thickness r r 2 − 1 ni i-th node on r1 lR radial line through ni lT tangent line touching ni θ angle between a diameter line and lR φ angle between lT and the line on which f is located f applied force vector fx and fy x and y components of f, respectively fT tangential force fR radial force fT magnitude of fT having the same sign as x component of fT fR magnitude of fR having the same sign as y component of fR CcTR constant controlling force direction Ccu scaling constant for force magnitude Cfm constant controlling the scaling constant Cu σx and σy stresses in the x and y directions, respectively τxy shear stress ǫx and ǫy extensional strains in the x and y directions, respectively γxy shear strain udisp and vdisp displacements in the x and y directions, respectively [D] material property matrix [Ke] elementstiffnessmatrix E Young’s modulus ν Poisson’s ratio u and v concentrations of activator and inhibitor morphogens, respectively du and dv diffusion coefficients of the activator and inhibitor, respectively d ratio of du to dv p and q reaction kinetics of the activator and inhibitor equations, respectively δ coefficient that is inversely proportional to a domain scaling n0 wave number α,β, and γ kinetic parameters for linear terms kq kinetic parameter for quadratic interaction terms kc kinetic parameter for cubic interaction terms tn final time step GI ratio of the total contour to the outer contour, gyrification index Nsul number of sulci

xiii LIST OF ABBREVIATIONS

ATH axonal tension hypothesis AVG average BVAM Barrio-Varea-Aragon-Maini CP cortical plate CV cerebralvesicle DGH diﬀerential growth hypothesis E embryonicday GA gestational age GI gyriﬁcation index GM germinal matrix IPC intermediate progenitor cell IPH intermediate progenitor hypothesis IPM intermediate progenitor model IZ intermediate zone LV lateral ventricle MR magnetic resonance MZ marginal zone RGC radial glial cell RSD relative standard deviation RUH radial unit hypothesis SD standard deviation SP subplate SVZ subventricular zone VZ ventricular zone

xiv ABSTRACT

The cerebral cortex is a thin folded sheet of neural tissue forming the outmost layer of the cerebrum (brain). Several biological hypotheses have suggested different mechanisms involved the development of its folding pattern into sulci (inward valleys) and gyri (outward hills). One hypothesis suggests that mechanical tension along cortico-cortical connections is the principal driving force for cortical folding development. We propose a new mathematical model based on the tension-based hypothesis surrounding the 26-week gestational age when the human brain cortex noticeably begins to fold. In our model, the deformation of a two-dimensional semi-circular domain is analyzed through the theory of elasticity. The governing coupled partial differential equations are implemented computationally using a finite element formulation. Plausible brain tissue elasticity parameters with realistic brain domain size parameters were used in our simulations. Gyrification index, which is a measure of cortical folding, is employed to compare the degree of folding between simulation results. Our proposed model combines two different biological hypotheses as the magnitude of the applied tension force is determined from a previous mathematical model of cortical folding based on a biochemical hypothesis. Our model is able to explain the mechanisms behind disorders occurring in all stages of development. In addition, the ability to freely set the directions and magnitudes of the applied forces allows analysis of various abnormal cortical foldings by comparing MR imaging features of human brain cortical disorders. Our simulation results show the unveiled mechanisms underlying the abnormal cortical folding development are well captured by our proposed model.

xv CHAPTER 1

INTRODUCTION

The cerebral cortex forms the outmost layers of neural tissue of the cerebrum. It is also called the gray matter and surrounds the white matter. The principle characteristic of the geometry of the cerebral cortex is its folding, forming gyri (outward hills) and sulci (inward valley). The development of cerebral cortical folding involves three stages: cell proliferation, cell migration, and cortical organization. These major stages are not completely separated [1,2]. For example, even after cell migration begins, from the ventricular zone to the cortex, the differentiation of stem cells into neurons and glial cells continues. In humans, out of the forty gestational weeks, the cortex begins to fold during either the 16th or the 17th week [3–5]. Some studies on dates for cortical development events indicate the first groove on the brain appears at the 8th week of gestational age (GA) [6]. Around the 26th week of gestation, obvious changes occur. These changes countine until the early postnatal period [6,7]. The process of cortical folding continues and reaches the maximum degree of folding at an age of about 23 years [4]. Disorders of cortical formations have been classified according to abnormal development in the three stages [1, 2]. In the first stage, abnormally decreased or increased proliferation, or abnormal proliferation in specific brain areas are reasons that cause the disorders of cortical formation. For example, microlissencephaly and microcephaly result from decreased proliferation. They are considered lissencephaly-type disorders which are characterized by an abnormally smooth cortical surface. In the second stage, cell migration to a wrong place is a reason for cortical disorders. Classic (Type I) lissencephaly results from undermigration while cobblestone (Type II) lissencephaly occurs due to overmigration. On the other hand, polymicrogyria, which is characterized by an excessive number of small gyri, results from deranged cortical layer organization. There have been three leading biological hypotheses that attempt to explain the development of cortical folding. The most recent hypothesis [8] suggests that a chemical morphogen activates cell proliferation in specific regions. Therefore the folding patterns on the cerebral cortex are

1 caused by irregular distributions of cell populations throughout the surface. The other hypotheses are mechanical hypotheses. Van Essen [9] suggests that mechanical tension along axons in white matter is a major inducing force for cortical convolution. Richman et al. [10] explain a mechanical buckling followed by differential growth of the outer and inner cortical layers leads to cortical folding. Based on these biological hypotheses, mathematical models of cortical folding have been proposed. A Turing system [11] has been often used to formulate the models representing chemical morphogen concentration gradients [12–14]. Elastic mechanical models of cortical folding have been proposed based on the axonal tension-based hypothesis [15–17]. Other mathematical models [18,19] have explained the idea suggested by the hypothesis of differential growth in layers. However, all of these mathematical models exhibit inconsistencies in determining the major mechanism for inducing cortical folding patterns, a mechanism that is not understood explicitly yet. We propose a new axonal tension-based mathematical model based on the biomechanical hypothesis [9] to elucidate the mechanisms underlying cortical folding development. In our model, the deformation of a two-dimensional semi-circular domain representing the cerebral cortex is analyzed through the theory of elasticity. The governing coupled partial differential equations are implemented computationally using a finite element formulation. During the earlier stages of development, the cortico-cortical connections are established on the cerebral cortex. We adopt one of the previous mathematical models using a Turing system to establish numerically the concentration of neurons forming the cortico-cortical connections. The concentration of neurons will determine the magnitude of the applied axonal tension force in our model. The adopted mathematical model [12] is based on the biochemical hypothesis [8]. Chapter 2 discusses neuroanatomy and the process by which the cerebral cortex is created and folded. The three leading biological hypotheses of cortical folding development are presented. Debates on the major mechanisms underlying cortical folding among the biological hypotheses and previous mathematical models based on these hypotheses are also discussed. Chapter 3 presents our proposed two-dimensional semi-circular model of cerebral cortical folding. To explain how we adopt the previous mathematical model to determine the magnitude of the tension force applied in our model, the key properties and characteristics of a Turing system are presented. The driving equations used for controlling the direction of the tension force are pre-

2 sented. Lastly, the key properties of the theory of elasticity as well as its finite element formulation are presented. In Chapter 4, numerical simulations are conducted to visualize deformed configurations of the semi-circular model cortex due to the applied tension forces. We discuss how the degree of folding changes depending on different domain sizes, directions of the pulling tension forces, and strengths of the forces on the semi-circular domain. In order to compare the degree of folding among the simulation results, we compute the gyrification index (GI) of each deformed configuration. Chapter 5 utilizes the proposed mathematical model in order to model several brain malformations. We describe the characteristics of malformed cerebral cortical folding associated with microlissencephaly, hemimegalencephaly, polymicrogyria, and so on. Magnetic resonance (MR) images are presented to decribe the characteristics. We also explore the major causes of the disorders. We model the malformations of the cerebral cortex based on the unveiled reasons. Comparing the results from our model with the MR images demonstrates the ability of our model to capture various cerebral cortical folding malformations. Chapter 6 concludes the investigation for developing mathematical models of cortical folding development by discussing challenges as well as presenting our ongoing and future work. Notably, preliminary numerical results of a three-dimensional model which is an extended version of the proposed model are presented. Overall, our proposed tension-based mathematical model takes a new approach for investigating the connections between two different biological hypotheses. Our model explains the development of cortical folding malformations occurring in all stages. In addition, the ability to freely set the directions and magnitudes of the applied pulling tension forces allows analysis of various abnormal cortical foldings.

3 CHAPTER 2

BIOLOGY BACKGROUND

Since ancient Egypt people have been studying the anatomy of the human brain [20]. Even though the developmental process of the brain is intricate and many things are still unknown, neuroanatomy and corticogenesis have been well documented from the microscopic to macroscopic level. In addition, the developmental process occurs in a predictable sequence. In this chapter, some of the key elements of the brain, which are involved in cerebral cortical development, are introduced in Section 2.1. The process in which the cerebral cortex is folded is described with MR imaging of the brain taken between the 25th week of gestational age (GA) and full-term in Section 2.2. Lastly, three leading biological hypotheses that explain possible mechanisms underlying the process of cerebral cortical folding are addressed in Section 2.3.

2.1 Neuroanatomy and Corticogenesis

Knowing the major features of the brain which are introduced in this section will help to understand the following sections regarding the development of cerebral cortical folding. The brain from the macroscopic level to the microscopic level is described in Sections 2.1.1–2.1.3. In addition, anatomical references regarding direction and sectional planes of the human brain are illustrated in Section 2.1.4.

2.1.1 Brain Structure

The human brain consists of the cerebrum, cerebellum, and brain stem. The cerebrum is separated into right and left halves by a deep groove called the longitudinal ﬁssure [21]. Other major sulci (grooves) separate the two hemispheres into six major lobes: frontal lobe, temporal lobe, occipital lobe, parietal lobe, insular lobe, and limbic lobe (see Figure 2.1) [21, 22]. Each lobe has specialized functions. The cerebral cortex is a thin folded sheet of neural tissue forming the outermost layer of the cerebrum. Also called gray matter, the cerebral cortex is approximately 1–4.5 mm thick with an

4 Figure 2.1: Six major lobes. (a) Frontal lobe, temporal lobe, occipital lobe, parietal lobe, and insular lobe. Figure adapted from [23]. (b) Limbic lobe. Figure adapted from [24]. overall average thickness of approximately 2.5 mm [25,26]. It is a six-layered structure [10,27] and surrounds the white matter. The surface area of the adult human cortex is about 1, 692 cm2 [28]. The highly convoluted shape of the cortex enables the large surface area to be ﬁtted within the relatively small size of the cortical volume which is about 458 cm3 [9, 28]. Each human brain cortical folding pattern is unique. The cortex begins to fold during the 16th or 17th week of GA in humans (out of 40 gestational weeks) [3–5], forming gyri (outward hills) and sulci (inward valleys) (see Figure 2.2). Clearly visible changes occur surrounding the 26th week of GA [6, 7] and continue until after birth (see Figure 2.3). Primary sulci are most consistent in position across individuals while secondary sulci, which appear later in cortical development, are highly variable in position and appearance (see Figure 2.4 and 2.5) [26]. The developmental process of cerebral cortical folding is described in more detail in Section 2.2.

2.1.2 Early Development and Ventricular System

The development process of the cerebral cortex begins at approximately embryonic day (E)10 [34]. The layers formed in early corticogenesis consist of, from bottom to top, the ventricular zone (VZ), subventricular zone (SVZ), intermediate zone (IZ), subplate (SP), cortical plate (CP), and marginal zone (MZ) (see Figure 2.6) [27]. The IZ exists only for a certain period of time and

5 Figure 2.2: Gyri and sulci. Figure adapted from [29]. eventually transforms into white matter [27]. The CP is precursor of the cortex (gray matter). More precisely, the CP transforms into ﬁve of the six layers of the cortex: layer 2 through layer 6, and the MZ transforms into layer 1 that is the most exterior layer of the cortex [27]. The ventricular system of the brain is composed of a series of interconnected cavities (see Figure 2.7) [21]. The central canal of the embryonic forebrain forms the two lateral ventricles (LVs), the interventricular foramina, and the third ventricle. The central canal of the embryonic midbrain and hindbrain form the cerebral aqueduct and the fourth ventricle, respectively [21]. The VZ is along the boundary of the LVs and is a layer of densely packed proliferative neuroepithelial cells [8]. The SVZ is a second proliferative layer that is formed by loosely and randomly arranged cells between the VZ and cortical preplate [8]. Self-ampliﬁcation of intermediate progenitor cells (IPCs), which play an important role in the developing cerebral cortex, occurs in the SVZ [8, 35]. More precisely, a distinct accumulation of tissue called the germinal matrix (GM), which is a key structure in neurogenesis, is produced by proliferating cells in the SVZ [37]. The GM contains progenitors (germ cells) of neurons and glial cells [37, 38]. The GM has been known to appear at the 7th week of GA in histopathological studies [39]. The volumetic change of the GM structure has been investigated using MR images of postmortem fetuses [38]. According to the volumetic analysis, GM volume was found to increase exponentially by the 23rd week of GA while keeping the volume ratio of GM to total brain at about 5% between the 11th and 23rd weeks of GA. The

6 GM volume was found to decrease rapidly after the 25th week of GA. The volume ratio of GM to total brain decreases to 0.7% at the 28th week of GA (see Figure 2.8).

2.1.3 Neurons and Glial Cells

The nervous system is composed of two principal types of cells: neurons and glial cells [40]. The function of neurons is to process and transmit information via electrical impulses (action potentials) [21, 41]. The major structural components of neurons are a soma, multiple dendrites, and an axon (see Figure 2.9). The soma is the cell body that contains the nucleus. The dendrites extrude from the cell body like the branches of a tree. The role of dendrites is to receive incoming impulses from other neurons through dendritic spines, which are tiny membranous protrusions on them, or directly on their surface [21]. Most neurons have only one axon. The axon stretches out from the soma, and the connection is called axon hillock. The axon has side branches called axon collaterals. The function of axons is to transmit nerve impulses to other neurons through the stuctures called synaptic terminals at the ends of the terminal branches. The axon can be insulated by a myelin sheath. The whiteness of myelin, that surrounds axons, affects the color difference between white matter and gray matter of the brain. White matter contains more myelinated axons than gray matter [42, 43]. The other kind of cell, glial cells, support neurons. Glial cells supply nutrients including oxygen to neurons and provide structural support [40]. Radial glial cells (RGCs) play important roles in neurogenesis and corticogenesis [33, 45, 46]. RGCs are the main source of neurons as they produce IPCs that eventually divide to generate neurons [47–49]. In addition, the RGCs establish a one-to- one correspondence between the VZ and CP (see Figure 2.10) [45]. The radial fibers of the RGCs support the neurons that travel up to the CP [45, 46]. The cell migration process along the RGCs will be further discussed in Section 2.3.1.

2.1.4 Anatomical References

There are special terms of direction, from Latin or Greek, describing the structure of the brain. The terms of direction separate the brain into front and back, top and bottom, or sides [51]. The front of the brain is anterior or rostral (rostrum, “beak of a bird”). The back of the brain is posterior or caudal (cauda, “tail”). The top of the brain is superior. The term dorsal (dorsum, “back”) sometimes also mean the top of some areas of the brain except for the brain stem and

7 Figure 2.3: Serial MR imaging of normal brain growth. The MR images are obtained from the brain of one normal female preterm infant. She was born at the 25th week GA. Figure from [30].

8 Figure 2.4: Major gyri and sulci (lateral view). The surface of the cerebrum shows its gyri and sulci such as angular gyrus, central sulcus, and lateral (Sylvian) sulcus. Figure from [31].

Figure 2.5: Major gyri and sulci (medial view). The medial wall of the cerebrum sectioned along a sagittal plane shows the gyri and sulci. Major gyri and sulci including superior frontal gyrus, cingulate gyrus, subcallosal gyrus, paracentral lobule, precuneus, cuneus, lingual gyrus, parahippocampal gyrus, cingulate sulcus, parieto-occipital sulcus, calcarine sulcus, collateral sulcus, and hippocampal sulcus are shown. Figure from [32].

9 Figure 2.6: Six embryonic layers. (a) Cerebral vesicle (CV). (b) Coronal section across the occipital lobe at the level indicated by a vertical broken line in (a). (c) A block of the tissue dissected from the upper bank of calcarine ﬁssure. At this early stage, six embryonic layers from the ventricular surface (bottom) to the pial surface (top) can be recognized: ventricular zone (VZ); subventricular zone (SVZ); intermediate zone (IZ); subplate zone (SP); cortical plate (CP); and marginal zone (MZ). Figure adapted from [33].

Figure 2.7: Cerebral ventricles. The human cerebral ventricular system is composed of two lateral ventricles, third ventricle, and fourth ventricle. (a) Anterior view. (b) Left lateral view. Figure adapted from [36].

10 Figure 2.8: Volumetric changes of the fetal brain, germinal matrix (GM), and lateral ventricles (LVs). (a) Volumetric changes from 7 weeks gestational age (GA) to 28 weeks GA. The brain surfaces (upper row), GM (middle row, orange), and LVs (lower row, blue) of human fetal brain were reconstructed by surface rendering. The volume of the GM increased until 23 weeks GA and decreased rapidly at 28 weeks GA. (b) The fetal brain (open triangles), GM (open circles), and LVs (solid circles). Increasing fetal brain volume (gray curve) has an exponential relationship (r2 = .963) to GA, reaching 132.5 cm3 at the 28th week of GA. The GM also increases exponentially, reaching a volume of 2.3 cm3 at the 23rd week of GA, then decreases rapidly after the 25th week of GA. In contrast to the GM, the volume of the LVs gradually increases, up to 2.6 cm3 at the 23rd week of GA. Figure adapted from [38].

Figure 2.9: Neuron. The major structural components of neurons are a soma, multiple dendrites, and an axon. Figure from [44].

11 Figure 2.10: Radial glial cells (RGCs). The radial ﬁbers and somas of RGCs establish a one-to-one correspondence between the ventricular zone (VZ) and cortical plate (CP). (a) Figure adapted from [45]. (b) Figure from [3]. spinal cord. The bottom of the brain is inferior. The term ventral (venter, “belly”) sometimes means the bottom of some areas of the brain as well except for the brain stem and spinal cord (see Figure 2.11) [21]. There are also useful terms of brain slices cut along speciﬁed orientations. A sagittal plane is a vertical plane passing from anterior to posterior (see Figure 2.12a) [52]. The midsaggital plane is referred to as the median plane, and it divides the brain into right and left halves [51]. A transverse plane is a horizontal plane dividing the brain into superior and inferior parts. It is perpendicular to the sagittal plane and it is also known as the horizontal, axial, or transaxial plane (see Figure 2.12b) [51]. Lastly, a coronal plane is a vertical plane separating the brain into anterior and posterior parts [51]. It is perpendicular to both the sagittal and transverse planes and it is also called the frontal plane (see Figure 2.12c) [21, 51].

2.2 Cortical Folding Development

In this section, we describe how the fetal brain develops its folding with MR images of preterm infant brains obtained from [6]. The majority of the MR images were originally taken from preterm

12 Figure 2.11: Anatomical references regarding direction. (a) Longitudinal axes of the brain. (b) Terms of direction. Figure from [50].

Figure 2.12: Sections of the human brain. The human brain sectioned along (a) a sagittal plane, (b) a transverse (horizontal) plane, and (c) a coronal (frontal) plane. Figure adapted from [51].

13 infants born at the Hammersmith hospital in England.

2.2.1 Magnetic Resonance Imaging of the Brains of Preterm Infants

Seven sets of MR images of preterm infants who were born during the 25th, 26th, 28th, 30th, 32nd, and 34th weeks of GA and term are presented in this section. Each set has different views of a newborn’s brain sectioned along different planes. Anatomical references regarding direction and sectional planes of the human brain were illustrated in Section 2.1.4. The squares colored with orange, blue, and yellow on the left top of each MR image presented in this section indicate the brain slices sectioned along a sagittal plane, a transverse plane, and a coronal plane, respectively (see Figure 2.12). In the transverse plane, the development of the central sulcus as well as pre- and postcentral sulci can be observed. In addition, at a lower level of the transverse plane and in the coronal plane, the Sylvian fissure narrows and closes as the fetal brain becomes more mature. In the sagittal plane, the calcarine sulcus, the parieto-occipital sulcus, and cingulate sulcus are visible. The location and appearance of these gyri and sulci in a mature brain are illustrated in Figures 2.4 and 2.5. Note that the term fissure is sometimes used instead of sulcus to indicate a deeper groove. For example, the lateral (Sylvian) sulcus, which separates the frontal and parietal lobes from the temporal lobe (see Figure 2.4), is also called the Sylvian fissure.

Cortical Folding at the 25th week of GA. At the 25th week of GA, the rudimentary central sulcus is visible in some, but not all, preterm newborns (see Figure 2.13a) [6]. At this time point, the insular lobe is not wrapped since the Sylvian fissure is widely open (see Figure 2.13b). The distance between the Sylvian fissure’s walls differ depending on individuals [6]. The rudimentary calcarine sulcus is also visible and it begins near the parieto-occipital sulcus (see Figure 2.13c).

Cortical Folding at the 26th week of GA. At the 26th week of GA, the central sulcus is deeper than that of the 25th week of GA, and the precentral sulcus is also visible as a rudimentary shallow valley (see Figure 2.14a). The distance between the Sylvian ﬁssure’s walls is narrower than that of the 25th week of GA but it is still open (see Figure 2.14b). The calcarine sulcus is also deeper than that of the 25th week of GA and the parieto-occipital ﬁssure is clearly seen (see Figure 2.14c).

14 Cortical Folding at the 28th week of GA. At the 28th week of GA, the central sulcus is more developed and its walls are mostly attached to each other. The precentral sulcus is also slightly deeper than that of the 26th week of GA. In addition, the postcentral sulcus begins to show as a rudimentary shallow valley (see Figure 2.15a). The Sylvian ﬁssure is also almost closed as seen in the coronal MR image (see Figure 2.15b). In the sagittal plane, the calcarine sulcus is longer than in previous images. Also, the cingulate sulcus begins to appear (see Figure 2.15c).

Cortical Folding at the 30th week of GA. At the 30th week of GA, all of the three sulci, central, precentral, and postcentral sulci, are clearly seen but their shapes are not yet complex (see Figures 2.16a, c, d). The slightly curvy cingulate sulcus is also observed (see Figure 2.16d). The Sylvian ﬁssure’s walls get much closer as the brain develops (see Figure 2.16b). In addition, both the parieto-occipital and calcarine sulci are well formed at this stage (see Figure 2.16c).

Cortical Folding at the 32nd week of GA. At the 32nd week of GA, the curvy central sulcus is detected (see Figure 2.17a). Also, the secondary branches from the cingulate sulcus are observed (see Figure 2.17d). The Sylvian ﬁssure continues to close (see Figure 2.17b). The parieto- occipital and calcarine sulci are clearly seen demonstrating the rotated Y-shape together (see Figure 2.17d). Any particular sulci in the areas of the frontal and anterior temporal lobes have still not developed (see Figures 2.17a, c).

Cortical Folding at the 34th week of GA and term. The secondary branches from the calcarine ﬁssure appear and the secondary branches from the cingulate sulcus become more complex (see Figures 2.18c and 2.19c). The overall appearance of the brain is getting more complex. The brain becomes more folded and the sulci become curved as well as have secondary branches (see Figures 2.18 and 2.19).

2.2.2 Summary of Human Fetal Brain Folding Development

The overall appearance of cerebral cortical folding becomes more complex as the brain grows. The number of sulci increases with GA. Each of the rudimentary sulci becomes curved. Secondary branches are also extruded from some of the sulci such as the cingulate sulcus. In the transverse plane at a supraventricular level, the central sulcus is ﬁrst seen at the 25th week of GA in this MR imaging study (see Figure 2.13a) [6]. In addition to the provided MR images, from the 25th week of GA to term, in this study, the central sulcus is known to be seen

15 Figure 2.13: MR images of a preterm infant born at the 25th week of GA. The MR images are taken (a) at a supraventricular level in the transverse plane, showing the central sulcus (arrow), (b) at a mid-ventricular level in the transverse plane, showing the Sylvian fissure (arrow), and (c) in the sagittal plane, showing the calcarine fissure (longer arrow). The shorter arrow heading down in (c) points out the corpus callosum which is a broad band of nerve fibers connecting the two hemispheres. Figure adapted from [6].

Figure 2.14: MR images of a preterm infant born at the 26th week of GA. The MR images are taken (a) at a supraventricular level in the transverse plane, showing the central sulcus (lower arrow) and the precentral sulcus (upper arrow), (b) at a lower level in the transverse plane, showing the Sylvian fissure (arrow), and (c) in the sagittal plane, showing the parieto-occipital fissure (shorter and upper arrow) and the calcarine fissure (longer and lower arrow). Figure adapted from [6].

16 Figure 2.15: MR images of a preterm infant born at the 28th week of GA. The MR images are taken (a) at a supraventricular level in the transverse plane, showing the central sulcus (middle of the two arrows), the precentral sulcus (upper and longer arrow), and the postcentral sulcus (shorter and lower arrow), (b) in the coronal plane, showing the Sylvian fissure (arrow), and (c) in the sagittal plane, showing the calcarine sulcus (lower arrow) and the cingulate sulcus (upper arrow). Figure adapted from [6]. from the 20th week of GA [6]. The precentral sulcus is formed anteriorly to the central sulcus at the 26th week of GA (see Figure 2.14a), and then the postcentral sulcus appears behind the central sulcus from the 28th week of GA (see Figure 2.15a). From the 32nd week of GA, the curved central sulcus is seen (see Figure 2.17a). All of the sulci becomes longer with GA. At a lower level in the transverse plane, the converging process of the Sylvian fissure’s walls has been observed (see Figure 2.13–2.19b). The Sylvian fissure is opened wide at the earlier developmental stages, exposing the insular lobe. Individuals differ on the distance between the walls of the Sylvian fissure [6]. The Sylvian fissure begins forming as a shallow groove on the smooth cortical surface at the 14th week of GA. At this stage, the insular lobe is not exposed. When the Sylvian fissure becomes deeper, the insular lobe begins to be visible under the groove around the 19th week of GA [6]. In the sagittal plane, we have oberved that the rudimentary calcarine sulcus (see Figure 2.13c) gets longer and more complex. It develops near the parieto-occipital sulcus, and the two joined sulci show the rotated Y shape together (see Figure 2.17d).

17 Figure 2.16: MR images of a preterm infant born at the 30th week of GA. The MR images are taken (a) at a supraventricular level in the transverse plane, showing the central sulcus (arrow), (b) at a lower level in the transverse plane, showing the Sylvian ﬁssure, (c) in the sagittal plane, showing the parieto-occipital sulcus (near arrow) and the calcarine sulcus (near arrow), and (d) in the sagittal plane, showing the cingulate sulcus. Figure adapted from [6].

18 Figure 2.17: MR images of a preterm infant born at the 32nd week of GA. The MR images are taken (a) at a supraventricular level in the transverse plane, showing the central sulcus (arrow), (b) in the coronal plane, showing the Sylvian ﬁssure (arrow), (c) in the sagittal plane, showing the relatively smooth frontal and anterior temporal lobes (arrow), and (d) in the sagittal plane, showing the cingulate sulcus (upper and longer arrow) and the parieto-occipital sulcus (short middle arrow) and the calcarine sulcus (short bottom arrow). Figure adapted from [6].

19 Figure 2.18: MR images of a preterm infant born at the 34th week of GA. The MR images are taken (a) at a supraventricular level in the transverse plane (b) at a lower level in the transverse plane, showing the Sylvian ﬁssure, and (c) in the sagittal plane, showing the cingulate sulcus (shorter and upper arrow) and the calcarine sulcus (longer and lower arrow). Figure adapted from [6].

Figure 2.19: MR images of a preterm infant born at term (approximately 40 weeks of GA). The MR images are taken (a) at a supraventricular level in the transverse plane (b) at a lower level in the transverse plane, and (c) in the sagittal plane, showing the cingulate sulcus (arrow). Figure adapted from [6].

20 In addition, the development of the cingulate sulcus is also seen from the 28th week of GA (see Figure 2.15c). The appearance of the cigulate sulcus is slightly curved at the 30th week of GA (see Figure 2.16d) and the secondary branches emerge at the 32nd week of GA (see Figure 2.17d and 2.18–2.19c). In sum, the rudimentary shallow grooves on the smooth cerebral cortical surface become deeper and steeper. In the process, the widely separated side walls of the primary sulci progressively move closer. In addition, the shapes of the sulci become complex with secondary sulci representing not only straight but also split ends. Lastly, the frontal and anterior temporal lobes fold slowly as these lobes remain relatively smooth while the other lobes fold sooner.

2.3 Biological Hypotheses

In this section, we describe three leading biological hypotheses that suggest major mechanisms underlying the developmental process of cerebral cortical folding. Even though these biological hypotheses insist upon different perspectives of the major mechanisms, they each address some deficiencies which the other hypotheses cannot explain. In particular, the biochemical hypothesis describes the earlier development stages from cell proliferation to cell migration. On the other hand, the other biomechanical hypotheses are based on the later development stages during or after cells migrate to the cortex. Section 2.3.1 describes the Intermediate Progenitor Model (IPM) which explains cortical folding with an irregular cell population across the cortex [8]. Section 2.3.2 explains the Axonal Tension Hypothesis (ATH) which suggests that mechanical tension along axons is a major inducing force generating cortical folding [9]. In Section 2.3.3, the Differential Growth Hypothesis (DGH) is described. The DGH explains cerebral cortical folding with a mechanical buckling due to different growth rates of the outer and inner cortical layers [10].

2.3.1 Biochemical Hypothesis: Intermediate Progenitor Model (IPM)

To explain the IPM, we introduce two other biochemical hypotheses [48,49] that describe how the brain layers, from the VZ to the CP, is populated with neurons. The Radial Unit Hypothesis (RUH) [48] explains the initial development process for the lower layer formation, and the Intermediate Progenitor Hypothesis (IPH) [49] explains the upper layer formation. The IPM [8] stands on the bases of these hypotheses.

21 Figure 2.20: Radial unit hypothesis. (a) In the symmetric division stage, each RGC divides into two RGCs per each round of cell division. (b) In the asymmetric division stage, each RGC divides into one replacement RGC and one neuron during each round of the sequential multiple cell divisions. The multiple asymmetric divisions can occur in the same location in the VZ, forming a column of neurons. A newer neuron locates on the outer. Figure adapted from [8].

Radial Unit Hypothesis (RUH). The RUH [48] describes how the founding population of RGCs is generated, how the RGCs produce neurons, and how cortical columns are created between the VZ and the CP. The RUH explains these processes in two stages: the symmetric division1 stage before embryonic day (E) 40, followed by the asymmetric division2 stage [33, 53]. The founding population of RGCs is built up during the symmetric division stage. RGCs within the VZ divide into two RGCs. Each division doubles the population, increasing exponentially (see Figure 2.20a). Next, in the asymmetric division stage, each RGC generated in the previous stage divides into one neuron and one replacement RGC. The neuron travels up to the CP along the radial fiber of the RGC. The newly generated RGC can divide again during this stage. Multiple asymmetric divisions, producing a linear increase of the neuron population, can occur in the same location in the VZ. The newly produced neurons travel up along the same radial fiber of the RGC, and then stack outwardly; the newer neuron stacks on the outer (see Figure 2.20b). This column of neurons along the radial fiber forms a one-to-one connection between the VZ and the CP and it is called an ontogenetic column (see Figure 2.10) [33].

1Symmetric division: a cell division that produces two cells with identical fate potential [8]. 2Asymmetric division: a cell division that produces two cells with diﬀerent fate potential [8].

22 Figure 2.21: Intermediate progenitor hypothesis and model. (a) In the asymmetric division stage, each RGC divides into one replacement RGC and one IPC per each round of cell division. In the SVZ, the produced IPCs divide symmetrically into either two neurons (left panel) or two IPCs, each of which eventually divides into two neurons (right panel). The neurons form the ontogenetic columns. (b) The intermediate progenitor model suggests that the folding patterns on the CP are caused by the irregular cell population. Figure adapted from [8].

Intermediate Progenitor Hypothesis (IPH). The IPH [49,54] explains there is an indirect production of cortical neurons through IPCs, which form upper layer cortical neurons, after lower cortical layer neurons are created directly from the RGCs as described by the RUH [48]. The IPH states that each RGC in the VZ divides asymmetrically into a replacement RGC and an IPC. The IPCs migrate into the SVZ, and then they divide symmetrically into either pairs of neurons or pairs of IPCs [35]. The newly generated pairs of IPCs might repeat the symmetric division multiple times [35]. The neurons then travel up along the radial ﬁber of their originating ‘self-renewed’ RGC and stack outwardly, generating the columnar distribution (see Figure 2.21a).

Intermediate Progenitor Model (IPM). The IPM [8] suggests the IPCs play a major role in developing cortical folding. The IPM insists that the symmetric division of the IPCs in the SVZ enables the surface area of the upper cortical layer to increase by generating many neurons while the size of the VZ is constant, leading to cortical folding (see Figure 2.21b).

23 Figure 2.22: Subventricular zone thickness and cortical folding. (a) Sagittal sections of the developing macaque monkey brain. A thickened SVZ (indicated by arrows under 1, left panel) leads to gyral formation that is observed 16 embryonic days later (1∗, right panel) while a thinner SVZ (arrows under 2, left panel) leads to sulcal formation (2, right panel). (b) Coronal sections of the developing human brain. Areas of thickened SVZ (indicated by brackets under 1, 3, and 5, left panel) lead to gyral formations that are observed four weeks later (1∗, 3∗, and 5∗, right panel). Areas of thinner SVZ (indicated by brackets under 2 and 4, left panel) lead to sulcal formation (2 and 4, right panel). Figure from [8].

Comparing the sizes of embryonic SVZ across several species, it was found that species with a larger embryonic SVZ have a more highly folded brain. This is because a bigger SVZ would hold more IPCs than a smaller one. Furthermore, the IPM suggests that the non-uniform SVZ thickness due to regional diversity of IPC population gives rise to regional patterning of cortical folding. Cortical areas corresponding to thicker SVZ develop into gyri because the high population of IPCs in the SVZ produces plenty of neurons. On the other hand, cortical areas above relatively thin SVZ regions develop into sulci since less numbers of IPCs generate fewer neurons (see Figure 2.22). Although it is not yet fully discovered how IPCs are spatially and temporally distributed along the SVZ, it has been shown that the self-amplification of IPCs is sensitively regulated by mutations of some genes such as Pax6, Ngn2 and Id4 in mice [35]. Many diffusible morphogens also have been found to affect cellular proliferation [35, 55, 56].

24 2.3.2 Biomechanical Hypothesis: Axonal Tension Hypothesis (ATH)

After or even while migrating to the cortex, neurons send axons to create cortico-cortical connec- tions3 [9]. The ATH [9], published about 10 years before the IPM, suggests that mechanical axonal tension along the cortico-cortical connections is the principal driving force for cortical folding development. According to the ATH, cortical areas densely populated by neurons eventually become gyri since the cortical walls are pulled together by strongly interconnected cortico-cortical connections which generate tangential force components. Sulci are formed in the weakly interconnected or unconnected cortical areas due to lack of axonal tension (see Figure 2.23). The ATH [9] is supported by studies on cortico-cortical connections in the macaque monkey cortex [58]. The studies compare connection strength and folding polarity and show strongly connected regions become gyri while weakly connected regions become sulci (see Figure 2.24). Experimental studies have also proven that axons must be under tension and axons are clustered into bundles of ﬁbers [18, 59–63]. Quantitative data by [60], in particular, provides experimental evidence of the ATH. They investigate the overall pattern of cortico-cortical connections of the prefrontal cortex of adult rhesus monkeys using photomicrograph. To trace the neuronal connections, a tracing chemical, HRP-WGA4, is injected. The experimental studies showed the neuronal connections under gyri have straight trajectories while curved trajectories were found below sulci. These results coincide with the ATH prediction (see Fig. 2.25). Additionally, mechanical tension within axons has been found to be essential for neurons’ elec- trophysiological functions such as synaptic signaling [62, 63]. Tension in axons has been found to be able to activate axonal elongation as well [61].

2.3.3 Biomechanical Hypothesis: Diﬀerential Growth Hypothesis (DGH)

The DGH [10] suggests that convolutional development of the brain is due to differential growth of the cortical layers. It also explains that a significant difference in the elastic modulus of each cortical layer leads to cortical folding.

3Cortico-cortical connection: axonal connection to dendrites in the cortex, either in the same hemisphere or in the opposite hemisphere across the corpus callosum. cf. Cortico-thalamic connection: axonal connection to the thalamus located above the brainstem [57]. 4HRP-WGA: horseradish peroxidase conjugated wheat germ agglutinin. It can be used for both anterograde and retrograde tracing [64].

25 Figure 2.23: Axonal tension hypothesis. (a) Neurons (black) migrate to the CP along RGCs (red). Internal pressure (arrows) would cause surface tension, which increases the surface area of the CP. (b) While migrating to the cortex, neurons send axons to create cortico-cortical connections (thin black curves). Tension along axons pull strongly interconnected regions together (long arrows). In the process, weakly interconnected regions start to drift apart (thin small arrows). (c) Gyri and sulci are formed above the highly interconnected and weakly interconnected regions, respectively, by the pulling axonal tension forces. (d) Enlarged view of gyrus and sulcus. Figure adapted from [9].

This hypothesis is supported by a comparison among two extreme cases of abnormal brains and a normal brain. In the normal brain, the outer cellular layers grow at a slightly faster rate than the inner layers. In microgyric cortex, the outer cortical layers grow at a slightly greater rate as compared to normal cortex while the inner cortical layers grow much slower (less than half) than in normal cortex. Yet, in lissencephalic cortex, there is no significant differential growth rate between cortical layers. In addition, the cellular patterns in the two extreme abnormal cortices differ from the six-layered cortex of the normal brain (see Figure 2.25).

26 Figure 2.24: Compactness of cortical wiring and cortical folding. (a) Lateral view of the right hemisphere of the macaque monkey brain. (b) A sagittal slice through occipital cortex representing selected corresponding points between (a) and (c). (c) A cortical ﬂat map. The crests of gyri and sulci are indicated by white curves and black curves, respectively. Strongly connected areas lead to gyral formation while weakly connected areas lead to sulcal formation. Figure adapted from [9].

Figure 2.25: Camera lucida drawings of brain layers. (a) Lissencephalic, (b) normal, and (c) microgyric cerebral cortex. Figure adapted from [10].

27 2.3.4 Debating Hypotheses

In this section, we discuss debates on the major mechanisms underlying cerebral cortical folding among the biological hypotheses and previous mathematical models based on the hypotheses.

Debating between IPM and ATH. Van Essen, who proposes the ATH [9], contends that the IPM [8] is a problematic hypothesis and expresses his concerns in a corresponding article [65]. In the article, Van Essen insists that uneven thickness of the SVZ forms not because of irregular populations of cells, which causes cortical folding according to the IPM, but as a direct result of cortical folding. He points out there is not enough evidence to support the IPM. However, Kriegstein et al. reply to the corresponding paper with emphasis on new ﬁndings of the impact of IPCs from multiple laboratories [66]. They mention that “It is not our intention to dismiss previous hypotheses, but to suggest that a revision that incorporates recent ﬁndings might be appropriate [66].”

Debating between ATH and DGH. Geng et al. [15] simulate the axonal tension forces at the lateral interior cortical surfaces using a stress-strain model, and their numerical results are consistent with the tension-based hypothesis. However, Xu et al. [18] argue against the tension- based hypothesis by providing experimental data of developing ferret brains. According to their dissection data, Xu et al. claim that the location of subcortical white matter, where tension is present, is too deep to aﬀect cortical folding. Xu et al. support the diﬀerential cortical growth hypothesis of [10] with their computational models and experimental data.

Debating DGH. Bayly et al. [67] also support differential growth, where tangential growth of the cortex is a mechanism driving cortical convolution. In contrast to Richman’s differential growth hypothesis [10], Bayly et al. suggest cortical folding can be explained by differential growth even if the stiffness of the cortex and interior regions of the brain are similar. Other mechanical models of cortical folding pattern development, such as [16, 17, 68], suggest a major mechanism inducing cortical folding patterns to be tangential cortical surface growth without the differential growth rate of the outer and inner cortical layers.

28 2.4 Conclusions

In this biology background chapter, we reviewed some neuroanatomy and corticogenesis involved in the development of cerebral cortical folding. The process of cerebral cortical folding from the 25th week of GA to term was also described with a series of sequential MR images of the brains of preterm infants. Lastly, the three major biological hypotheses that attempt to explain mechanisms inducing the development of cerebral cortical folding were presented. The three leading biological hypotheses have diﬀerent perspectives, and most previous mathematical models are based on only one of the biological hypotheses. The debates among the three biological hypotheses were discussed in Section 2.3.4. Nevertheless, the biological hypotheses are rounding out the incomplete explanations of the others. In particular, the ATH [9] suggests mechanical tension along axons but it does not explain how and where the axons are developed. The earlier development stages from cell proliferation to migration can be explained by the IPM [8]. In addition, these both two hypotheses, the IPM and ATH, coincide on the conclusion that regions highly populated by neurons develop into gyri while regions populated by fewer neurons form sulci. Our proposed mathematical model assumes that the major mechanism causing cerebral cortical folding is the pulling tension forces as suggested by the ATH [9]. We also assume that distribution on neuronal population forming the cortico-cortical connections is explained by the IPM [8]. Our model is presented in Chapter 3.

29 CHAPTER 3

THE PROPOSED MATHEMATICAL MODEL OF CEREBRAL CORTICAL FOLDING DEVELOPMENT

In this chapter, we describe our proposed mathematical model of cerebral cortical folding development based on the axonal-tension hypothesis [9], which was described in Section 2.3.2. We introduce the geometry of the model cortex in Section 3.1. In Section 3.2, we describe how the directions of the applied forces, corresponding to the pulling axonal tension forces, are determined on the model cortex. In Section 3.3, we describe how the magnitudes of the applied forces are obtained. We assume that, for simplicity, the magnitudes of the applied forces are proportional to the populations of neurons forming the cortico-cortical connections. In addition, we assume that the populations of neurons are proportional to the concentrations of an activator which regulates self-amplifications of IPCs. A previous mathematical model presented in [69] using a two-equation activator-inhibitor Turing reaction-diffusion system is adapted to obtain the concentrations of the activator throughout a one-dimensional domain representing the VZ. In our model, the deformations of the model cortex due to the pulling axonal tension forces are analyzed through the theory of elasticity. The governing coupled partial differential equations are implemented computationally using a finite element formulation. We describe the theory of elasticity and finite element formulation in Section 3.4.

3.1 Model Geometry

We propose a two-dimensional, semi-circular model geometry representing the initial shape of the cerebral cortex (see Figure 3.1). The outmost layer of the model cortex, corresponding to gray matter, has radius r2, and the boundary between white matter and gray matter has radius r1. We assume a uniform cortical thickness given by thick = r r . 2 − 1

30 Figure 3.1: Representation of the human brain. (a) Coronal section of the adult human brain taken perpendicular to the anterior-posterior axis. Figure adapted from [70]. (b) Computational model of the cerebral cortex. Quadrilateral elements, where (xi,yi) represents the Cartesian coordinates of the ith node, are used. The radius of the gray matter, white matter, and cortical thickness are given by r , r , and thick = r r respectively. 2 1 2 − 1

In order to compute the displacements of the model cortex numerically, the semi-circular model cortex is discretized into a mesh of 200 quadrilateral elements (a schematic diagram is shown in Figure 3.1b). To avoid the inﬁnite displacements, the semi-circular model cortex is restrained by the four nodes on the bottom line so that the four constrained nodes do not move in any direction during simulations. In Section 2.1, we described that the radial ﬁbers and somas of RGCs establish a one-to-one correspondence between the VZ and CP (see Figure 2.10) [33,45]. In our model, each node on the boundary between gray matter and white matter of the semi-circular model cortex, representing the CP, has a corresponding node on a one-dimensional domain, representing the VZ, which will be introduced in Section 3.3.

3.2 Directions of the Applied Axonal Tension Forces

In order to model axonal tension force pulling together on the semi-circular domain, we deﬁne a vector f as a force (or a tension-force). The force f loaded at a node ni can be separated into a tangential force fT and a radial force fR at the node (see Figure 3.2). The magnitude of f is given, and its direction is controlled by combinations of the two components, fT and fR, according to a

31 Figure 3.2: Components of the applied force on the semi-circular model cortex. A vector f (red arrow) represents the applied force at a node ni. The force f is the sum of a tangential force fT and a radial force fR. The components of the force vector f are fx and fy and are in the directions of x- and y-axes respectively. A radial line lR is from the center of the bottom side through the node ni, which is perpendicular to the tangent line lT touching the node ni. The angle θ is measured from the positive x-axis to lR. The angle φ is measured from lT to the line on which the force f is located.

constant between 0 and 1. We deﬁne the magnitude of fT , which is one of the two components of f, from the given magnitude of f, f , as Equation (3.1a), and then compute the magnitude of the | | other component fR by Equation (3.1b).

fT = CTR f , (3.1a) | | | | 2 2 fR = f fT , (3.1b) | | | | −| | p where CTR [0, 1] is the scaling constant. ∈ The vector components of fT and fR in the directions of x- and y-axes are as follows:

fT =(fT sin θ, fT cos θ) , − fR =(fbR cos θ, fRbsin θ) ,

b b where fT is the magnitude of fT having the same sign of x-component of fT while fR is the magnitude of f having the same sign of y-component of f . The angle θ is measured from the b R R b positive x-axis to the radial line lR (see Figure 3.2). By adding the vectors fT and fR, the force

32 vector f can be represented by x- and y-components such as

f =(fx, fy)=(fT sin θ + fR cos θ, fT cos θ + fR sin θ) . (3.3) −

In the following simulation resultb ﬁgures, web denote φ bto representb the angle between the tangent line lT and the line on which the force f is located (see Figure 3.2). We assume that φ rotates clockwise from lT to f when fx > 0 (see Figure 3.3a, c) while it rotates counterclockwise when fx < 0 (see Figure 3.3b, d). Then, the angle φ can be found as follows.

If fy > 0, arctan ( fy / fx ) π/2+ θ if fx > 0 , − | | | | − φ = θ if f = 0 , (3.4)  x arctan ( fy / fx )+ π/2 θ if fx < 0 ,  − | | | | − if fy < 0,  arctan ( fy / fx ) π/2+ θ if fx > 0 (see Figure 3.3a, c) , | | | | − φ = θ if f = 0 , (3.5)  x arctan ( fy / fx )+ π/2 θ if fx < 0 (see Figure 3.3b, d) ,  | | | | − if fy = 0,  π/2+ θ if fx > 0 , − φ = 0 if f = 0 , (3.6)  x π/2 θ if fx < 0 ,  − where fx and fy are the components off as deﬁned in Equation (3.3). When φ is an obtuse angle, we denote φ′ = π φ instead (see Figure 3.4). −

3.3 Magnitudes of the Applied Axonal Tension Forces

We strongly agree with the opinion that existing hypotheses must be more sophisticated in their incorporation of new findings and data. In addition, we believe the biochemical hypothesis IPM [8] does not contradict the biomechanical hypothesis ATH [9] as Kriegstein et al. indicated in [66]. The IPM is rather contributing to a deficiency in the ATH by providing a specific explanation about earlier cortical developmental stages up to creation of neurons. Furthermore, both hypotheses coincide on the conclusion that regions highly populated by neurons develop into gyri while regions populated with fewer neurons form sulci [8, 9]. The biochemical hypotheses described in Section 2.3.1 illustrate the earlier phase of cortical folding development. According to the RUH [48], the founding population of RGCs in the VZ

33 Figure 3.3: The angle φ. The angle φ (green shade) is measured from the tangent line lT to the line on which the force f (red arrow) is located, which rotates clockwise when fx > 0 and rotates counterclockwise when fx < 0. The angle ψ is an acute angle between lT and the line lx on which fx locates. Observe ψ = π/2 θ when 0 θ π/2 while − ≤ ≤ ψ = π/2+ θ when π/2 < θ π. (a) 0 θ π/2 and fx > 0 , (b) 0 θ π/2 and − ≤ ≤ ≤ ≤ ≤ fx < 0 , (c) π/2 <θ π and fx > 0 , (d) π/2 <θ π and fx < 0. ≤ ≤ is determined by symmetric division before E40, and then each RGC divides asymmetrically into one replacement RGC and one neuroblast [33, 53]. The IPH [49, 54] explains more concretely the asymmetric division process of RGCs. It illustrates that each RGC produces one replacement RGC and not a neuroblast directly but one IPC. The produced IPCs migrate into the SVZ, and then they divide symmetrically into either pairs of neuroblasts or pairs of IPCs [35]. The symmetric division can be repeated by the newly produced IPCs until the IPCs divide ultimately into pairs of neuroblasts [35]. The IPM [8] suggests that regional diﬀerences in the thickness of the SVZ, which is determined by the population of IPCs occupying the region, lead to cortical folding patterns. The cellular proliferation of IPCs is genetically regulated [35]. In addition, in the SVZ, the self-ampliﬁcation of IPCs produces a densely cellular and highly vascularized region called the GM. The volumetric growth of the GM has been investigated by MR imaging studies [38, 39, 71]. The GM appears at the 7th week of GA and increases exponentially

34 Figure 3.4: The angle φ′ = π φ. In the following figures taken from the simulation − results, we denote φ′ (orange shade) instead when φ (green shade) is greater than π/2. (b’) We assumed that φ rotates counterclockwise from lT to f when fx < 0. (c’) We assumed that φ rotates clockwise when fx > 0. by the 23rd week of GA. Then, the GM volume decreases rapidly after the 25th week of GA. The period of exponential GM volume growth can be interpreted as a prosperous period of the IPC self-amplifications. The time-point of a rapid decline of GM volume also coincides with the time when the cerebral cortex begins to fold ostensibly after finishing the production of cortical neurons. Therefore, we have assumed that the magnitudes of the applied forces are proportional to the concentrations of neurons forming cortico-cortical connections. According to the biochemical hypothesis IPM discussed in Section 2.3.1, the production of cortical neurons is determined by the self-amplifications of IPCs. There are mathematical models such as [13, 14, 69] which describe pre-patterns of cortical folding based on the IPM using a Turing system. We adopt [69], one of the mathematical models, which employs the Barrio-Varea-Aragon-Maini (BVAM) system as the reaction kinetics of the Turing system on a one-dimensional domain. In Section 3.3.1, we describe the Turing system and the BVAM system. In Section 3.3.2, we describe how the magnitudes of the applied forces are determined by the Turing patterns in our proposed model.

3.3.1 Turing Reaction-Diﬀusion System

The Turing system, which was introduced by Alan M. Turing in 1952 [11], has been widely applied in mathematical modeling of developmental biology [72–75]. A Turing system is constituted

35 of two partial diﬀerential reaction-diﬀusion equations as follows:

∂U 2 = du U + p(U, V ) , ∂t ∇ (3.7) ∂V 2 = dv V + q(U, V ) , ∂t ∇ where U(x,t) and V (x,t) are concentrations of an activator morphogen U and an inhibitor morphogen V , respectively, at spatial position x and time t. Diﬀusion coeﬃcients of the activator and the inhibitor are du and dv, respectively. The functions p and q represent the reaction kinetics.

A spatially-uniform steady state of System (3.7) is (U0,V0) such that p(U0,V0) = q(U0,V0) = 0. Under certain conditions, which are so-called Turing conditions, the system is linearly stable in the absence of diﬀusion. To model the genetic regulation of self-ampliﬁcation of IPCs, the nondimensionalized Turing system [75] is applied as follows:

∂u = dδ 2u + p(u,v) , ∂t ∇ (3.8) ∂v = δ 2v + q(u,v) , ∂t ∇ where u = U U and v = V V , and System (3.8) has the uniform stationary solution (0, 0) when − 0 − 0 p and q are zeros [76]. The diffusion coefficient d = du/dv (0, 1) is the ratio of du to dv. Note that ∈ 0 < du < dv which is a required condition to generate spatially inhomogeneous patterns [75, 77]. The coefficient δ > 0 arose from nondimensionalization and is inversely proportional to a domain scaling [69, 75]. We select the BVAM system [76, 78] for the reaction kinetics p and q, which considers all the possible nonlinear interactions of the two morphogens up to cubic terms. The BVAM system [76] is expressed as:

2 p(u,v)= αu 1 kcv + v(1 kqu) , − − αk (3.9) q(u,v)= βv 1+ cuv + u (γ + k v) , β q where α,β, and γ are kinetic parameters for linear terms, and kc and kq are kinetic parameters corresponding to cubic uv2 and quadratic uv interaction terms, respectively. The BVAM system has been employed for mathematical modeling of cortical folding pattern formation of the brain [12, 41, 69] because it is appropriate for systems involving little known biological mechanisms [79].

36 Two criteria must be satisfied for the generic reaction-diffusion equations (3.8) to be a Turing system. Although diffusion is generally considered to drive a stabilizing process, Turing looked at it from the opposite point of view. The first Turing criterion is that u and v tend to a linearly stable spatially uniform steady state (u0,v0) in the absence of diffusion. In constrast, the other Turing criterion states that the system tends to be unstable and drives heterogeneous spatial patterns in the presence of diffusion. The diffusion-driven instability is possible because of the unequal rates of the two diffusion coefficients which satisfy du

tr A = pu + qv < 0 , (3.10a)

det A = puqv pvqu > 0 , (3.10b) −

pu + dqv > 0 , (3.10c) 1 det A< (p + dq )2 , (3.10d) 4d u v where p p A = u v . qu qv (u0,v0) Note from conditions (3.10a) and (3.10c) require d = 1 and puqv < 0. In addition, since d (0, 1), 6 ∈ the kinetics parameters must be selected to satisfy pu > 0 and qv < 0. The Turing system (3.8) with the BVAM kinetics in (3.9) is numerically solved using a forward- time central-space finite difference scheme on the domain 0 x L. Periodic boundary conditions ≤ ≤ are employed. The initial conditions of u and v are chosen by a random number generator in range of [ 0.5, 0.5] at a point xinit (0,L). − ∈ 3.3.2 IPC Self-amplifications to the Axonal Tension Forces

We assume that the activator morphogen concentration u is proportional to a repetition of IPC self-ampliﬁcations until the IPC ﬁnally divides into a pair of neuroblasts. In addition, we assume

37 that the pulling tension along axons is proportional to the concentration of the neurons. Therefore, the concentration of the activator u is ultimately assumed to be proportional to the magnitude of the axonal tension force. The following equations (3.11) and (3.12) describe these assumptions.

Let χi (0,L) be the values of x such that u(χi) = 0, and let the number of χi be j. Then, fi ∈ is the tension force that will be applied on a node corresponding to χi on the boundary between gray matter and white matter of the semi-circular domain. The number of the applied forces is then also j. The magnitude of fi is determined by following equations. If the periodic boundary values are positive,

χi+1 Cu fi = fi = udx, i = 1, 3,...,j 1 , (3.12) | | | +1| 2 − Zχi where Cu > 0 is a constant of proportionality of the forces to the amount of u in the given intervals. Note that we do not consider the cases when the boundary values are zeros.

3.4 Displacements

The theory of elasticity is employed to establish a mathematical model to compute the deformed shape of the initial semi-circular model cortex under the influence of external forces. The applied forces, corresponding to the axonal tension forces, are defined in Section 3.2 and Section 3.3. In this section, we describe the theory of elasticity and its finite element formulation. In Section 3.4.1, kinetics, kinematics, and constitution, which are the basic ideas associated with the development of any theory in solid mechanics, are discussed in order to explain the theory of elasticity. In Section 3.4.2, two principal fundamentals of the finite element method, weak formulation and discretization of the domain, are discussed.

38 3.4.1 Theory of Elasticity

In this dissertation, we assume that the behavior of the two-dimensional, semi-circular model cortex is regulated and deformed by linear elasticity. We calculate the static response of the shape of the model cortex to the applied forces in a two-dimensional Cartesian coordinate system. The governing coupled partial diﬀerential equations of elasticity are established by three ideas as follows: kinetics, kinematics, and constitution [80]. In this section, we describe each of these ideas in order to derive the governing vector equations of elasticity with displacements udisp(x,y) and vdisp(x,y) in the x- and y-directions, respectively, as dependent variables.

Kinetics or balance of force. In a static problem, the kinetics or balance of force refers to the equations of equilibrium [80]. The internal forces applied on an inﬁnitesimal element are illustrated in Figure 3.5. Summation of the forces in the horizontal and vertical axes yields the following equations:

∂σx ∂τxy Fx =(σx + dx)dy σxdy +(τxy + dy)dx τxydx + fxdxdy = 0 , (3.13) ∂x − ∂y − X and

∂τxy ∂σy Fy =(τxy + dx)dy τxydy +(σy + dy)dx σydx + fydxdy = 0 , (3.14) ∂x − ∂y − X where σx, σy, and τxy are stresses in the x- and y-directions, and the shear stress, respectively, and fx and fy are the external forces per unit area in the x- and y-axes, respectively, and they are assumed to be positive when acted along the positive axes [81]. Simplyfying Equations (3.13) and (3.14) yields the equations of equilibrium that can be stated as: ∂σ ∂τ x + xy + f = 0 , ∂x ∂y x ∂τ ∂σ (3.15) xy + y + f = 0 . ∂x ∂y y

Kinematics or strain-displacement. The kinematics of material deformation refers to a description about strains which are the responsive deformations due to the applied forces [80]. An extensional or normal strain is deﬁned as a ratio of a change in length to the original length. Shear strain refers to distortion as a result of changes in angles usually between two originally orthogonal directions [80, 82].

39 Figure 3.5: Free body diagram of a two-dimensional, inﬁnitesimal element with dimensions dx by dy. Stresses in the x- and y-directions are σx and σy respectively, and τxy and τyx are the shear stresses. A balance of angular momentum produces τxy = τyx [80]. All stresses shown are positive. Figure adapted from [81].

We describe how to derive the equations of strains by referring to Figure 3.6 illustrating the deformation of a rectangular element whose original side lengths are dx and dy [82]. Vertices of the reference element are A, B, C, and D. In the ﬁgure, A is located at (x,y), and the corresponding displacements of A are udisp(x,y) and vdisp(x,y) in the x- and y-directions, respectively. Vertex B is located at (x + dx,y). Then, the corresponding displacements of B are written as udisp(x + dx,y) and vdisp(x + dx,y). Since we have assumed a small linear elastic deformation, the higher-order terms can be neglected after expansions. Thus,

∂udisp udisp(x + dx,y) udisp(x,y)+ dx , ≈ ∂x ∂vdisp vdisp(x + dx,y) vdisp(x,y)+ dx . ≈ ∂x

The other displacement terms of the vertices are derived in the same manner. The deﬁnition of the extensional strain yields the following equation of extensional strain in the x-direction, A′B′ AB ǫ = − . x AB

40 Figure 3.6: Two-dimensional geometric strain deformation. Figure adapted from [82].

To ﬁnd the distance between A′ and B′, we consider the right triangle whose length of the hy- ′ ′ potenuse is A B (see Figure 3.6). The length of the base of the right triangle is dx + udisp(x + ∂u dx,y) u , and it can be stated as dx + disp dx by applying the expansion. The length of − disp(x,y) ∂x ∂vdisp the height is vdisp(x + dx,y) vdisp(x,y), and it can be also stated as dx. According to the − ∂x Pythagorean theorem and ignoring higher-order terms,

∂u 2 ∂v 2 A′B′ = dx + disp dx + disp dx ∂x ∂x s ∂u ∂u 2 ∂v 2 ∂u = 1 + 2 disp + disp + disp dx 1+ disp dx . ∂x ∂x ∂x ≈ ∂x s

Applying AB = dx yields ∂u ǫ = disp . x ∂x Similarly, the extensional strain in the y-direction is calculated as

∂v ǫ = disp . y ∂y

41 According to the deﬁnition of shear strain, the shear strain of the element in Figure 3.6 can be stated as π ′ ′ ′ γxy = 6 D A B = α + β tan α + tan β . 2 − ≈ Ignoring higher-order terms yields

∂u ∂vdisp disp dy ∂x dx ∂y ∂udisp ∂vdisp γxy = + + . ∂udisp ∂vdisp ≈ ∂y ∂x dx + ∂x dx dy + ∂y dy

Provided that α, β, and the partial derivatives of udisp and vdisp with respect to x or y are small. In summary, for the two-dimensional linear theory, the appropriate linear strains are ∂u ǫ = disp , x ∂x ∂v ǫ = disp , (3.16) y ∂y ∂u ∂v γ = disp + disp , xy ∂y ∂x where ǫx and ǫy are the extensional strains in the x- and y-directions, respectively, and γxy is the shear strain.

Constitution or stress-strain. The constitutive equations, which are also referred to as stress-strain equations, deﬁne the response of a material to external stimuli [80]. In this dissertation, the model cortex is considered as an isotropic elastic material that has two independent constants in its stiﬀness and compliance matrices. For a material property matrix [D] that makes a connection between the stresses and the strains, we assume the plane stress condition is applied as follows: 1 ν 0 E [D]= 2 ν 1 0 , (3.17) 1 ν  1−ν  − 0 0 2   where E is the Young’s modulus and ν is Poisson’s ratio. The linear constitutive equations are

σ =[D] ǫ , (3.18) { } { }

′ ′ where σ = σx, σy, τxy and ǫ = ǫx, ǫy, γxy . The constitutive equations (3.18) provided by { } { } { } { } the stress-strain relations complete the establishment of eight equations (ﬁve equations in Equations

(3.15) and (3.16) and three equations in Equation (3.18)) with eight unknowns: udisp, vdisp, σx, σy,

τxy, ǫx, ǫy, and γxy.

42 Combination. Solving for the strains in Equation (3.18) yields

1 ǫx = (σx νσy) , E − 1 ǫy = (σy νσx) , (3.19) E − 2(1+ ν) γ = τ . xy E xy

Equating Equation (3.19) with Equation (3.16) and solving for the stresses yields

E ∂u ∂v σ = disp + ν disp , x (1 ν2) ∂x ∂y − E ∂u ∂v σ = ν disp + disp , (3.20) y (1 ν2) ∂x ∂y − E ∂u ∂v τ = disp + disp . xy 2(1+ ν) ∂y ∂x Substituting Equation (3.20) into Equation (3.15) results in E ∂ ∂u ∂v E ∂ ∂u ∂v disp + ν disp + disp + disp + f = 0 , (1 ν2) ∂x ∂x ∂y 2(1+ ν) ∂y ∂y ∂x x − E ∂ ∂u ∂v E ∂ ∂u ∂v (3.21) disp + disp + ν disp + disp + f = 0 . 2(1+ ν) ∂x ∂y ∂x (1 ν2) ∂y ∂x ∂y y −

The set of coupled partial diﬀerential equations in Equation (3.21) are satisﬁed by udisp(x,y) and vdisp(x,y) in the interior of the region Ω. In addition, the displacements udisp(x,y) and vdisp(x,y) must also satisfy the given boundary conditions on the boundary Γ. In this dissertation, the essential boundary conditions are employed as follows:

udisp(Γ) = 0 ,

vdisp(Γ) = 0 .

In general, the boundary conditions can be essential (also called geometric or displacement), natural (also called traction or stress), or mixed types [80].

3.4.2 Finite Element Formulation

In order to solve the coupled partial differential equations (3.21) numerically, the finite element method is employed. The finite element method is characterized by variational or weak formulations and discretization of the domain.

43 The methods of weighted residual, which are explained with a simple example for clarification in Appendix A.1, are classified according to how the test functions are determined [81]. The Galerkin method of weighted residuals has the advantages that the test functions are obtained directly from the chosen trial functions, and it is most widely used for obtaining the global stiffness matrix in the finite element method [81, 83, 84]. In this section, we describe a weak formulation using the Galerkin method of weighted residuals. We discretize the two-dimensional, semi-circular domain using linear quadrilateral elements which are illustrated in Appendix A.2. In this section, we explain how the displacements udisp and vdisp are interpolated using the shape functions of the linear quadrilateral element. The shape functions correspond to the trial functions in the finite element method [81]. Substituting the shape functions for the test functions and displacements results in obtaining the stiffness matrix for elasticity.

Weak Formulation. The weak form is equivalent to the strong form of the coupled governing partial differential equations (3.21) established in Section 3.4.1. The strong form (3.21) requires that the unknown displacements udisp and vdisp are continously differentiable until at least the second- order partial derivatives. The weak formulation, which conducts the method of integration by parts in the process, reduces the order of differentiability by transforming one of the partial derivatives into the test functions, which are also called the weight functions. It relaxes the requirements of choosing basis functions by reducing the order of continuity. The process of weak formulation, using the Galerkin method of weighted residuals, can be described as follows. Applying the weighted average of the residual over the domain Ω to the kinetic equations (3.15) yields

∂σx ∂τxy ω1 ∂x + ∂y ω f dΩ+ 1 x dΩ = 0 , (3.22) ∂τxy ∂σy Ω  ω +  Ω ω2fy Z  2 ∂x ∂y  Z where Ω denotes the domain and ω1 and ω2are the weight functions. Applying integration by parts to the terms in the ﬁrst integral in Equation (3.22) yields

∂ω1 ∂ω1 ∂x σx + ∂y τxy ω1fx ω1tx ∂ω2 ∂ω2 dΩ+ dΩ+ dΓ = 0 , (3.23) − τ + σ ω2fy ω2ty ZΩ ( ∂x xy ∂y y ) ZΩ ZΓn where Γn is the boundary for natural conditions and tx and ty are x- and y-components of the stress vector on the natural boundary. The weight functions vanish on all essential boundaries.

44 Note that we have assumed that the essential boundary conditions are employed in our model so the last term in Equation (3.23) can be neglected. In addition, the equation can be rewritten as

∂ω1 ∂ω1 σx ∂x 0 ∂y ω1fx ∂ω2 ∂ω2 σy dΩ= dΩ . (3.24) Ω " 0 ∂y ∂x #   Ω ω2fy Z  τxy  Z Substituting the constitutive equation (3.18) for the stresses in Equation (3.24) gives

∂ω1 ∂ω1 ǫx ∂x 0 ∂y ω1fx ∂ω2 ∂ω2 [D] ǫy dΩ= dΩ . (3.25) Ω " 0 ∂y ∂x #   Ω ω2fy Z  γxy  Z Lastly, substituting the kinematic equation (3.16) for the strains in Equation (3.25) results in

∂udisp ∂ω1 ∂ω1 ∂x ∂x 0 ∂y ∂vdisp ω1fx ∂ω2 ∂ω2 [D] ∂y dΩ= dΩ . (3.26) 0   ω2fy ZΩ " ∂y ∂x #  ∂udisp ∂vdisp  ZΩ  ∂y + ∂x 

Discretization and Interpolation. The semi-circular model cortex is discretized into a mesh of 200 quadrilateral elements (see Figure 3.1b). The shape functions of the linear quadrilateral elements are bilinear functions, 1 N = (1 + ξ ξ)(1+ η η) , i 4 i i where (ξ,η) is a point in the natural coordinate system, and Ni = 1 at the corner (ξi,ηi) of the four-sided element and Ni = 0 at the other corners (see Figure 3.7). A more speciﬁc description of the linear quadrilateral elements is given in Appendix A.2.

The displacements udisp and vdisp in Equation (3.26) are interpolated using the shape functions

Ni as follows:

4 4 udisp = Niudisp i , vdisp = Nivdisp i . i i X=1 X=1 These displacements can be rewritten as

u N 0 N 0 N 0 N 0 disp = 1 2 3 4 d [N] d , (3.27) vdisp 0 N1 0 N2 0 N3 0 N4 { }≡ { } ′ where d = udisp ,vdisp ,udisp ,vdisp ,udisp ,vdisp ,udisp ,vdisp is the nodal displacement { } { 1 1 2 2 3 3 4 4} vector and we deﬁne a matrix [N] as above.

45 Figure 3.7: Shape functions of the linear quadrilateral element.

The displacements in the kinematic equations (3.16) can be also expressed as a form of interpolation using Equation (3.27), and we deﬁne a matrix [B] which includes the partial derivatives of the shape functions Ni with respect to x or y. That is,

∂udisp ∂N1 ∂N2 ∂N3 ∂N4 ∂x ∂x 0 ∂x 0 ∂x 0 ∂x 0 ∂vdisp 0 ∂N1 0 ∂N2 0 ∂N3 0 ∂N4  ∂y  =  ∂y ∂y ∂y ∂y  d [B] d . (3.28) ∂u ∂v ∂N1 ∂N1 ∂N2 ∂N2 ∂N3 ∂N3 ∂N4 ∂N4 { }≡ { }  disp + disp  ∂y ∂x  ∂y ∂x ∂y ∂x ∂y ∂x ∂y ∂x      The Galerkin method chooses the weight functions ω1 and ω2 from the chosen trial function Ni (see Appendix A.1). Therefore, on the region of the element, the left-hand-side of Equation (3.26) becomes

∂udisp ∂ω1 ∂ω1 ∂x 0 ∂v ∂x ∂y [D] disp dΩ= [B]′[D][B]dΩ d , 0 ∂ω2 ∂ω2  ∂y  { } ZΩe " ∂y ∂x #  ∂udisp ∂vdisp  ZΩe  ∂y + ∂x 

  ′ where Ωe is the domain of the linear quadrilateral element and [B] is the transpose of the matrix [B]. The right-hand-side of Equation (3.26) can be also expressed as follows:

ω f ω 0 f f 1 x dΩ = 1 x dΩ= [N]′ x dΩ . ω2fy 0 ω2 fy fy ZΩe ZΩe ZΩe

46 Correspondingly, Equation (3.26) becomes f [B]′[D][B]dΩ d = [N]′ x dΩ (3.29) { } fy ZΩe ZΩe in the element domain Ωe. The element stiﬀness matrix, Ke, for elasticity can be obtained from Equation (3.29), which is a form of [the stiﬀness matrix][displacement vector] = [force vector], as follows:

[Ke]= [B]′[D][B]dΩ . (3.30) ZΩe We can solve for the global nodal displacement vector after assembling the element stiffness matrices into the global stiffness matrix, applying the boundary conditions, and then multiplying both sides by the inverse of the global stiffness matrix. In addition, we can compute the integrations in the physical coordinate system as integrations in the natural coordinate system using a mapping between the two coordinate systems. For example, the integration in Equation (3.30) is mapped to be 1 1 [B]′[D][B]dΩ= [B]′[D][B] J dξdη , | | ZΩe Z−1 Z−1 where J is the determinant of the Jacobian matrix [J] defined in Equation (A.8) (see Appendix | | A.2). The two-point Gauss-Legendre quadrature rule is employed to compute approximate solutions to the definite integrals (see Appendix A.3).

3.5 Conclusions

This chapter presented the proposed mathematical model of cerebral cortical folding development. First the geometry of the two dimensional, semi-circular model cortex was introduced. Then, the directions and magnitudes of the forces applied on the model cortex were defined. Lastly, the governing coupled partial differential equations of elasticity were derived in order to compute the deformations of the model cortex due to the applied forces. The Turing reaction-diffusion system was also described in Section 3.3.1. The Turing system has often been employed to create prepatterns of activation of IPC self-amplification in the SVZ in previous mathematical models of cerebral cortical folding such as [12, 13, 69]. In our model, the magnitudes of the applied forces are obtained from the Turing patterns. Simulation results using this model are presented in the next chapter.

47 CHAPTER 4

NUMERICAL RESULTS

Numerical simulation results presented in this chapter are carried out to visualize the development of cerebral cortical folding due to the applied axonal tension forces on the semi-circular domain. The governing coupled partial differential equations of elasticity are numerically implemented using the finite element method coded in MATLAB to obtain the deformed configurations of the model cortex. The two-dimensional, semi-circular model cortex is discretized into a mesh of N = 200 linear quadrilateral elements. Element nodes are labeled counter-clockwise beginning from the bottom right inner semi-circle (see Figure 3.1b). Each element has four nodes, and each node has two degrees of freedom. We solve for the horizontal and vertical components of displacement at each node. To avoid infinite displacements, the four nodes on the bottom are constrained so that they do not move in any direction during simulations. In order to compare the degrees of folding among the simulation results, we compute the gyrification index (GI) of each deformed model cortex. The GI is a well established measure of cortical folding used in the field of brain mapping [85–87]. Section 4.1.4 describes the definition of the GI as well as how we compute the GI in the numerical simulations. Parameter values used in the simulations are also described in Section 4.1. Section 4.2 exhibits how the GI changes depending on different domain sizes obtained by varying the values of the brain radius r2 and the cerebral cortical thickness thick. In Section 4.3, we also explore how the changing directions of the pulling tension forces affect the overall shape of cortical folding such as the number of sulci and the GI. Lastly, we investigate the effects of the strengths of forces in Section 4.4.

4.1 Parameters and Gyriﬁcation Index

In this section, we introduce the parameter values applied in the numerical simulations. Pa- rameters for elastic properties of the brain, geometric sizes, and the direction of the applied force

48 and its strength are described. We also describe what the GI is and how we compute the GI of each deformed model cortex.

4.1.1 Tissue Elasticity and Geometric Size of the Brain

Brain tissue elasticity parameters are not readily available for a developing brain and thus we assume they are similar to those of a mature brain. In order to apply plausible elasticity parameters, we choose values of Young’s modulus E and Poisson’s ratio ν from previous experimental and statistical studies [88–90]. In the simulations conducted in this dissertation, E = 9, 210.87 Pa and ν = 0.458344. They are the root-mean-square values obtained from the computational results of [88].

In Section 4.2, we change the gray matter radius and cortical thickness parameters, r2 and thick. The selected increasing values of r2 are obtained from the data provided by [4, 68], which are the radii of the human fetal brain taken at the 11th, 21st, 28th, 30th, 35th, and 37th weeks of the 40 week gestational period. For the values of thick, we adopt the data provided by [25]. To observe a wider range of cortical thicknesses, we also use two additional values, 2.0 mm and 4.5 mm.

Unless stated otherwise, the gray matter radius is taken to be r2 = 40.464 mm which is the radius of the human brain at 28 weeks’ GA [4,68]. For cortical thickness, thick = 2.5 mm which is the average thickness of the human cerebral cortex [25] is applied in the simulations.

4.1.2 Direction of the Applied Axonal Tension Force

The direction of the applied force f starting from a node on the inner semi-circular domain having radius r is determined by the constant CTR [0, 1] in Equation (3.1a). When CTR = 0, f 1 ∈ is a radial force fR which points to the center of the bottom side of the domain; when CTR = 1, f is a tangential force fT . The value CTR determines how much the force vector departs from the semi-circular domain, and so corresponds to the angle φ, which is the angle between the tangent line lT and the line on which the force f is located (see Equations (3.4)–(3.6) and Figures 3.3 and

3.4). When CTR = 0, f becomes fR with φ = π/2 = 1.5708 (unit in radians).

49 4.1.3 Strength of the Applied Axonal Tension Force

We have assumed that the magnitude of the applied force f, corresponding to the pulling axonal tension force, is regulated by the population of neurons forming the cortico-cortical connections. The Turing reaction-diffusion system (3.8) is employed to set up the population of neurons. The Turing system determines the distribution of the concentration of the activator morphogen u on the domain 0 x 1. The value of u affects the rate of the IPC self-amplification, which determines ≤ ≤ the population of neurons. We integrate u in an interval where u> 0 and then multiply the scaling constant Cu to obtain f (see Equations (3.11) and (3.12)). | | The one-dimensional Turing system domain represents the SVZ of the brain is mapped to the boundary having radius r1 on the semi-circular model cortex. This correspondence is based on the fact that the RGCs establish a one-to-one correspondence between the VZ and CP (see Figure 2.10) [45]. The Turing system is implemented numerically using a forward-time central-space finite difference scheme coded in MATLAB with periodic boundary condition. The initial values of the activator and inhibitor morphogens, u and v, are random numbers between 0.5 and 0.5 on the − center of the domain (at x = 1/2), unless otherwise specified. The selected magnitudes of the time and space steps are ∆t = 0.004 and ∆x = 1/200, respectively. The number of iterations tn is 150, 000 unless otherwise specified. To obtain the concentration of neurons forming the cortico-cortical connections, three cases of Turing patterns are generated by the following parameter sets and are shown in Figure 4.1. The diffusion coefficient d of the Turing system is selected to be d = 0.516. The coefficient δ, which is inversely proportional to the domain scaling, is chosen as:

0.00109765 if n0 = 3 , δ = 0.00039515 if n0 = 5 ,   0.00009879 if n0 = 10 , where n0 is a wave number. In addition, the parameter values for the BVAM kinetics (3.9) were selected as follows: α = 0.899, β = 0.91, γ = α, kc = 3.5, and kq = 0.0, which are traditional − − values used for the BVAM kinetics [76]. Figure 4.1 also shows the initial conﬁgurations of the semi-circular model cortices. Each model cortex displays a set of red arrows which represent the applied axonal tension forces. The magnitude

50 2 of each force vector is computed by multiplying the scaling constant Cu = 25 n0 Cfm where × × Cfm = 1 in the ﬁgure. Diﬀerent values of the constant Cfm are applied in some simulations. In Figure 4.1, the averages of the magnitudes of the applied forces are 1.9381 newtons (N), 3.2244 N, and 6.3069 N with small relative standard deviations (RSDs) when n0 = 3, n0 = 5, and n0 = 10, respectively. We apply these three sets of the forces in the following simulations unless indicated otherwise.

By reducing the ﬁnal time step tn and changing the locations of the random initial values of u and v, we are able to generate irregular Turing patterns. From those Turing patterns, we obtain several sets of the force magnitudes with large RSDs. In Section 4.4, we present simulation results using forces generated with these irregular Turing patterns.

4.1.4 Gyriﬁcation Index (GI)

The GI is a measure of the degree of cortical gyriﬁcation [85–87]. It is a ratio of the total contour length of the cortex, including sulci, to an outer contour length that includes the superﬁcially exposed surface [86]. A more convoluted cortex has a higher GI value (see Figure 4.2).

In our simulations, the nodes on the outer cortex which had an initial radius of r2 are used to compute the GI. Note that tension forces are applied to the nodes on the inner cortex which forms the boundary between white matter and gray matter. In order to obtain the length of the complete contour including sulci, we sum up all the lengths between two adjacent nodes on the outer cortex. To obtain the length of the superficial contour, we select the node of each gyrus crest and surrounding nodes as well as a node on each end of the bottom line of the model cortex. In cases wherein gyral crests are found on the bottom line, we select fourteen additional nodes near the two nodes to enhance accuracy (see Figure 4.3a, b). Then, all lengths between the chosen nodes on the outer contour are summed. Figure 4.3 shows three examples of the GI values computed in the simulations. The convoluted cortex configurations in this figure are obtained by using the applied forces shown in Figure 4.1.

51 n0 = 3 r = 40.464 mm, r −r = 2.5 mm, C = 0.99 2 2 1 TR (a) 0.1 40 30

u 0 20 −0.1 10 0 0 0.2 0.4 0.6 0.8 1 −40 −20 0 20 40 C = 225 x u n0 = 5 (b) 0.1 40 30

u 0 20 −0.1 10 0 0 0.2 0.4 0.6 0.8 1 −40 −20 0 20 40 C = 625 x u n0 = 10 (c) 0.1 40 30

u 0 20 −0.1 10 0 0 0.2 0.4 0.6 0.8 1 −40 −20 0 20 40 C = 2500 x u

Figure 4.1: Turing patterns and strengths of the applied axonal tension forces. The magnitudes of the applied forces on the semi-circular model cortices (right panel) are obtained from the Turing patterns (left panel) when the wave number n0 is (a) 3, (b) 2 5, or (c) 10. The scaling constant Cu in each case is 25 n0 . The averages of the × magnitudes of the applied forces (red arrows, right panel) are (a) 1.9381 N 2.4032%, ± (b) 3.2244 N 3.2404%, and (c) 6.3069 N 0.0521%, and the distributions are symmetric ± ± on the domain.

52 Figure 4.2: Gyriﬁcation index (GI). The GI is a ratio of the length of total contour of the cortex (solid curve) to the length of outer contour which includes the superﬁcially exposed cortex (dotted curve). Figure from [86].

n0=3 n0=5 n0=10 (a) (b) (c) 40 40 40

20 20 20

0 0 0 −40 −20 0 20 40 −40 −20 0 20 40 −40 −20 0 20 40 GI = 1.1213 GI = 1.0948 GI = 1.0919

Cerebral cortex Outer contour

Figure 4.3: GI calculation in our simulations. The deformed conﬁgurations representing cerebral cortex (red curves) are obtained using the applied forces shown in Figure 4.1. Each GI value is the ratio of the length of the outer cortex (outer red curve) to the length of outer contour (blue dotted curve). (a) GI = 1.1213, (b) GI = 1.0948, and (c) GI = 1.0919.

53 4.2 Domain Size

We can infer that the degree of folding increases as the cortex becomes bigger or thinner. In this section, we vary the values of two parameters, r2 and thick, to prove this expectation and investigate how diﬀerent values of gray matter radius (and hence brain size) and cortical thickness aﬀect cortical folding.

4.2.1 Bigger vs. Smaller Brains

Simulation results shown in Figure 4.4 change the value of r2 while other parameters, such as cortical thickness and elastic parameters as well as the directions and magnitudes of the applied forces are fixed, as defined in Figure 4.1b. The ten force vectors are derived from the Turing pattern with wave number n0 = 5. Regarding the directions of the applied forces, each force has the constant CTR = 0.99, and eight of the ten forces located in the middle constitute four pairs that pull together. When CTR = 0.99, then φ = 0.1415 radians is an acute angle between the tangent line lT through the starting point of the vector f and the line on which each f locates. The scaling 2 constant for force strength Cu is 25 5 . The average of the magnitudes of the forces is 3.2244 N × 3.2404%. The set of the ten forces is applied on the semi-circular domain having a thickness of ± 2.5 mm. The increasing values of r2 are the radii of the human fetal brain at 11, 21, 28, 30, 35, and 37 weeks’ GA [4, 68]. Some of these results are displayed together to illustrate the effects of increasing r2 (see Figure 4.5).

In addition, we computed the changing GI values depending on r2 using wave numbers n0 = 3 and n0 = 10. All parameters except r2 are as in Figure 4.1a (n0 = 3) and c (n0 = 10). The results are displayed in Figures B.1–B.4. We collate all the results in Figure 4.6. The GI increases with increasing values of r2 in all cases. The slope of each line segment in each graph becomes steeper as r2 becomes bigger in all cases. The simulation results show that a smaller brain leads into a smaller GI, which means less folding, while a larger brain brings out a larger GI, meaning more folding. These results can be interpreted as more cortical convolutions will develop as the brain grows agreeing with Figure 2.3. An alternative interpretation to these results is that species with smaller cortices will tend to have relatively smoother brains. On the other hand, species with larger brains will tend to have

54 r −r = 2.5 mm, C = 0.99, n0=5 2 1 TR (a) (b) 40 20 30 15 20 10 5 10 0 0 −20 −10 0 10 20 −20 0 20 r = 18.351 mm, GI = 1.0112 r = 31.303 mm, GI = 1.0499 2 2

40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 r = 40.464 mm, GI = 1.0948 r = 42.966 mm, GI = 1.1093 2 2

(e) 60 (f) 60

40 40

20 20

0 0 −50 0 50 −50 0 50 r = 48.88 mm, GI = 1.1462 r = 51.088 mm, GI = 1.161 2 2

Figure 4.4: Changing gray matter radius r2 when n0 = 5. The radius of gray matter is taken at (a) 11 weeks GA (r2 = 18.351 mm), (b) 21 weeks GA (r2 = 31.303 mm), (c) 28 weeks GA (r2 = 40.464 mm), (d) 30 weeks GA (r2 = 42.966 mm), (e) 35 weeks GA (r2 = 48.880 mm), and (f) 37 weeks GA (r2 = 51.088 mm) [4,68]. All other applied forces and parameters are as in Figure 4.1b. The average of the magnitudes of the ten applied forces is 3.2244 N 3.2404%. The GI increases with as values of r increase. ± 2 highly convoluted cortices when the same set of forces are applied. This interpretation agrees with that of [9].

55 r −r = 2.5 mm, C = 0.99, n0=5 2 1 TR 60 (a) (b) 50 (c) (f) 40

0 −60 −40 −20 0 20 40 60

Figure 4.5: Eﬀects of changing gray matter radius r2 when n0 = 5. Figures 4.4a–c, f are displayed together to illustrate the eﬀects of increasing r2.

55 1.25 r n0=3 2 50 n0=5 n0=10 1.2 45

40 1.15

35 GI (mm) 2 r 1.1 30

25 1.05 20

15 1 10 15 20 25 30 35 40 10 20 30 40 50 60 GA (week) r (mm) 2

Figure 4.6: Graphs of r2 versus GA and the GI versus r2. The values of r2 are obtained from the data in [4, 68] (left panel). The GI increases with the increasing values of r2 in all cases regardless of wave number. The slope of each line segment in each graph of the GI becomes steeper as r2 increases with GA (right panel, shown in orange when n0 = 3, green when n0 = 5, and blue when n0 = 10).

56 4.2.2 Thicker vs. Thinner Cortices

Simulation results in Figure 4.7 change the cortical thickness parameter, thick, when the wave number is n0 = 5. The cortical thicknesses reported by [25] are used in our simulations. Those values are the averages of cortical thicknesses in overall, medial, inferior, and lateral areas and ranged from 2.5 to 3.5 mm. We use two additional values, 2.0 mm and 4.5 mm, to observe a wide range of cortical thickness. All other applied forces and parameters are as in Figure 4.1b. We also applied force vectors derived from Turing patterns with wave numbers n0 = 3 and n0 = 10 while changing the parameter thick. All parameters except thick are as in Figure 4.1a (n0 = 3) and c (n0 = 10). The results and GI values are displayed in Figures B.5–B.6. Figure 4.8 shows that the GI decreases exponentially with the increasing values of thick in all cases, regardless of wave number. The simulation results show that the GI reduces as the cortex becomes thicker. In other words, the GI increases as the cortex becomes thinner. These results are consistent with the simulation results of [17]. They can be also interpreted from another point of view. To obtain the same GI from cortices generated with the same wave numbers but having diﬀerent thicknesses requires less force strength for the thinner cortex than the strength required for thicker ones.

4.2.3 Summary

We determined that the GI increases exponentially as the cortex becomes bigger or thinner (see Figures 4.6 and 4.8). Table 4.1 summarizes the simulation results quantitatively when n0 = 5.

However, if both r2 and thick change, then no simple conclusion can be drawn. For example, in order to compare the simulation results across the two cases: one of changing values of r2 and the other of changing values of thick, ratios of brain radius to cortical thickness are also arranged in the table. We construct the ratio of brain radius to cortical thickness, which is also shown in Table

4.1. For example, when we compare Figures 4.4f and 4.7a, the ratio of r2 to thick of Figure 4.4f is greater than that of Figure 4.7a. However, the GI of Figure 4.4f is less than the GI of Figure 4.7a. Both cases were simulated with the same set of ten force vectors as well as same elastic parameters. In each case, when the ratio decreases, the GI also decreases, providing a clearer relationship GI, brain radius and cortical thickness. However, it is not guaranteed when we compare the GI values across the two diﬀerent cases.

57 r = 40.464 mm, C = 0.99, n0=5 2 TR (a) (b) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 r −r = 2 mm, GI = 1.2396 r −r = 2.5 mm, GI = 1.0948 2 1 2 1

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 r −r = 2.7 mm, GI = 1.0657 r −r = 3 mm, GI = 1.038 2 1 2 1

(e) (f) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 r −r = 3.5 mm, GI = 1.0156 r −r = 4.5 mm, GI = 1.0031 2 1 2 1

Figure 4.7: Changing cortical thickness, thick = r r , when n0 = 5. Cortical thickness 2 − 1 values are from [25] and (a) thick = 2.0 mm, (b) thick = 2.5 mm (overall average), (c) thick = 2.7 mm (medial cortex), (d) thick = 3.0 mm (inferior cortex), (e) thick = 3.5 mm (lateral cortex), and (f) thick = 4.5 mm. All other applied forces and parameters are as in Figure 4.1b. The average of the magnitudes of the ten applied forces is 3.2244 N 3.2404%. The GI decreases exponentially with the increasing values of thick. ±

1.3 n0=3 n0=5 n0=10 1.25

1.2

1.15 GI

1.1

1.05

2 2.5 3 3.5 4 4.5 r −r (mm) 2 1

Figure 4.8: Graph of GI versus thick. The GI decreases exponentially with the increasing values of thick in all cases (shown in orange when n0 = 3, green when n0 = 5, and blue when n0 = 10).

Table 4.1: Brain size and the GI when n0 = 5. Note that Figure 4.7b has the same parameters and GI as those of Figure 4.4c.

r thick Figure 2 r /thick GI (unit: mm) 2 (a) 18.351 7.3404 1.0112 (b) 31.303 12.5212 1.0499 (c) 40.464 16.1856 1.0948 Figure 4.4 2.5 (d) 42.966 17.1864 1.1093 (e) 48.880 19.5520 1.1462 (f) 51.088 20.4352 1.1610 (a) 2.0 20.2320 1.2396 (b) 2.5 16.1856 1.0948 Figure 4.7 (c) 2.7 14.9867 1.0657 40.464 (d) 3.0 13.4880 1.0380 (e) 3.5 11.5611 1.0156 (f) 4.5 8.9920 1.0031

59 4.3 Directions of the Applied Axonal Tension Forces

We assess the effects of the directions of the applied forces by changing the constant CTR [0, 1] ∈ defined in Equation 3.1a. As CTR approaches zero, the force direction tends to head to the center of the bottom line of the semi-circular model cortex. In the following simulations, we change CTR from 0.99 to 0.01. In this section, the numerical simulations are conducted using the two sets of applied forces defined in Figures 4.1a (n0 = 3) and 4.1b (n0 = 5). Each of the Turing patterns is symmetric and the concentration u is distributed regularly throughout the domain with the small RSD. The number of the applied forces is double the wave number: six forces when n0 = 3 and ten forces when n0 = 5. Each set of forces is applied on three semi-circular domains having different sizes. One case

(Case 1) has a radius r2 of 40.464 mm and cortical thickness thick of 2.5 mm as shown in Figure

4.1. The other cases change one of the two parameter values: Case 2 has r2 = 40.464 mm and thick = 3.5 mm; Case 3 has r2 = 51.088 mm and thick = 2.5 mm. The simulation results conducted with the diﬀerent cases are arranged in Table 4.2 when n0 = 3 and Table 4.3 when n0 = 5. In the tables, the value φ corresponds to the constant CTR to represent quantitatively how much the force vectors depart from the semi-circular model cortex, where φ is the angle between the tangent line lT and the line on which the force f is located (see Figure 3.2).

As CTR approaches zero, the heads of the force vectors drift apart and become radial forces with φ = π/2 = 1.5708 radians. The GI and the number of sulci are also described. Figures 4.9, B.7, and B.8–B.11 display results when CTR = 0.9, 0.6, 0.3, and 0.1.

When n0 = 3, as CTR approaches zero, the number of sulci increases up to six. In Case 1 and

3, the GI decreases until the number of sulci reaches its maximum when CTR = 0.4. After that point, the GI increases. Also the GI values when CTR = 0.01 are less than those of CTR = 0.99. In Case 2, the minimum value of the GI appears when the number of sulci increases for the ﬁrst time from three to four when CTR = 0.6. After that point, the GI increases to more than CTR = 0.99. In all cases, the overall shape of the semi-circular model cortex becomes ﬂatter (see Table 4.2 and Figures 4.9, B.8, and B.9).

When n0 = 5, as CTR approaches zero, the number of sulci increases up to ten. In Case 1 and

3, the GI decreases until the number of sulci becomes close to its maximum when CTR = 0.4 and

60 Table 4.2: Force direction and the GI when n0 = 3. The value of φ is the angle between the tangent line lT and the line on which the force f is located. The set of the six forces is applied on three diﬀerent semi-circular domains. Case 1: r2 = 40.464 mm and thick = 2.5 mm. Case 2: r2 = 40.464 mm and thick = 3.5 mm. Case 3: r2 = 51.088 mm and thick = 2.5 mm. The other parameters as well as the magnitudes of the applied forces are as in Figure 4.1a. As CTR approaches zero, the GI decreases up to (*) and then increases. The number of sulci (Nsul) increases as the force vectors become radial forces. See Figures 4.9, B.8, and B.9 for visualization results.

Case 1 Case 2 Case 3 n0 = 3 CTR φ Figure 4.9 Figure B.8 Figure B.9 Figure (unit:radians) GI Nsul GI Nsul GI Nsul 0.99 0.1415 1.1213 3 1.0191 3 1.2138 3 (a) 0.9 0.4510 1.1022 3 1.0135 3 1.1819 3 0.8 0.6435 1.0828 3 1.0094 3 1.1512 3 0.7 0.7954 1.0659 3 1.0068 3 1.1247 3 (b) 0.6 0.9273 1.0523 4 1.0056 (*) 4 1.1037 4 0.5 1.0472 1.0432 4 1.0058 4 1.0901 4 0.4 1.1593 1.0394 (*) 6 1.0076 6 1.0855 (*) 6 (c) 0.3 1.2661 1.0416 6 1.0110 6 1.0909 6 0.2 1.3694 1.0502 6 1.0161 6 1.1067 6 (d) 0.1 1.4706 1.0656 6 1.0228 6 1.1329 6 0.01 1.5608 1.0854 6 1.0302 6 1.1654 6

CTR = 0.5, respectively. After that point, the GI increases but the GI values when CTR = 0.01 are less than those of CTR = 0.99. In Case 2, the minimum value of the GI appears before the number of sulci increases for the first time from five to nine when CTR = 0.6. After that point, the GI increases up to more than that of CTR = 0.99. In all cases, the overall shape of the semi-circular model cortex becomes flatter (see Table 4.3 and Figures B.7, B.10, and B.11).

As described in Section 4.2, when CTR = 0.99 the GI increased as the ratio of r2 to thick increased (see Table 4.1). Among the three cases presented in this section, Case 3 has the greatest ratio of r2 to thick and Case 2 has the lowest ratio. For all simulation results, the highest ratio

(Case 3) generates the highest GI and the lowest ratio (Case 2) generates the lowest GI when CTR is kept constnat. Figure 4.11 graphs the number of sulci versus CTR.

61 Table 4.3: Force direction and the GI when n0 = 5. The value of φ is the angle between the tangent line lT and the line on which the force f is located. The set of the ten forces is applied on three diﬀerent semi-circular domains. Case 1: r2 = 40.464 mm and thick = 2.5 mm. Case 2: r2 = 40.464 mm and thick = 3.5 mm. Case 3: r2 = 51.088 mm and thick = 2.5 mm. The other parameters as well as the magnitudes of the applied forces are as in Figure 4.1b. As CTR approaches zero, the GI decreases up to (*) and then increases. The number of sulci (Nsul) increases as the force vectors become radial forces. See Figures B.7, B.10, and B.11 for visualization results.

Case 1 Case 2 Case 3 n0 = 5 CTR φ Figure B.7 Figure B.10 Figure B.11 Figure (unit:radians) GI Nsul GI Nsul GI Nsul 0.99 0.1415 1.0948 5 1.0156 5 1.1610 5 (a) 0.9 0.4510 1.0929 5 1.0135 5 1.1561 5 0.8 0.6435 1.0807 5 1.0111 5 1.1358 5 0.7 0.7954 1.0683 5 1.0095 5 1.1161 5 (b) 0.6 0.9273 1.0590 7 1.0089 (*) 5 1.1016 9 0.5 1.0472 1.0541 7 1.0094 9 1.0944 (*) 9 0.4 1.1593 1.0539 (*) 9 1.0110 9 1.0945 10 (c) 0.3 1.2661 1.0579 10 1.0136 10 1.1011 10 0.2 1.3694 1.0656 10 1.0172 10 1.1134 10 (d) 0.1 1.4706 1.0761 10 1.0217 10 1.1302 10 0.01 1.5608 1.0879 10 1.0264 10 1.1486 10

4.3.1 Summary

We explored the eﬀects of the directions of the applied forces with the parameter CTR. As

CTR approaches zero, the model cortex is aﬀected by the radial component of each force pulling radially more than its tangential component. Therefore, the overall shape of the model cortex becomes ﬂatter and the number of sulci increases. After the number of sulci reaches its maximum, the newly generated sulci become deeper as CTR approaches zero. However, each of the sulci when

CTR = 0.01 is shallower than each of those is when CTR = 0.99. The GI values are presented quantitatively in Tables 4.2 and 4.3 as well as visually in Figures 4.9, B.7, and B.8–B.11.

In addition, Figure 4.10 displays the graphs of the GI versus the decreasing CTR and Figure

4.11 shows the graph of the number of sulci versus CTR. The graphs show the GI decreases until the number of sulci becomes close or equal to the maximum number that can occur due to the wave number that generates the applied forces. The maximum number of sulci is same as the number

62 r = 40.464 mm, r −r = 2.5 mm 2 2 1 50 (a) 40

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.9 GI = 1.1022 TR 50 (b) 40

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.6 GI = 1.0523 TR 50 (c) 40

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.3 GI = 1.0416 TR 50 (d) 40

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.1 GI = 1.0656 TR

Figure 4.9: Changing the directions of the applied forces when n0 = 3 in Case 1. The set of six forces is applied on the semi-circular domain whose radius of gray matter is 40.464 mm and cortical thickness is 2.5 mm (left panel). The directions of the forces are determined by the constant CTR [0, 1] which controls the ratio between the magnitudes ∈ of the tangential force fT and the radial force fR. (a) CTR = 0.9, (b) CTR = 0.6, (c) CTR = 0.3, and (d) CTR = 0.1. As CTR decreases, the number of sulci increases and the overall shape of the deformed configuration becomes flatter (right panel). See Table 4.2 for additional results using different values of CTR. The other parameters as well as the magnitudes of the applied forces are as in Figure 4.1a.

63 of the applied forces (six when n0 = 3 and ten when n0 = 5). In Case 1 and 3, the maximum values of the GI appear when CTR = 0.99. In Case 2, the maximum values of the GI appear when

CTR = 0.01.

Among the three cases, Case 3 has the greatest ratio of r2 to thick and its GI is highest for all

CTR values. Case 2 has the smallest ratio of r2 to thick and has the lowest GI. Note that the other parameters including the magnitudes of the applied forces are same among the cases if they have the same wave number.

Case 1: r = 40.464 mm, r −r = 2.5 mm Case 2: r = 40.464 mm, r −r = 3.5 mm Case 3: r = 51.088 mm, r −r = 2.5 mm 2 2 1 2 2 1 2 2 1 1.25 1.04 Case 1: n0=3 Case 2: n0=3 Case 1: n0=5 Case 2: n0=5 Case 2: n0=3 Case 2: n0=5 1.035 Case 3: n0=3 1.2 Case 3: n0=5

1.03

1.15 1.025

1.02 GI GI

1.1

1.015

1.05 1.01

1.005 1

1 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 C C TR TR

Figure 4.10: Graphs of the GI versus CTR. As CTR approaches zero, the GI decreases until the number of sulci becomes close or equal to the maximum number that it can have with the numbers of the applied forces (six when n0 = 3 shown in orange and ten when n0 = 5 shown in green). After the point, the GI increases. The maximum values of the GI appear when CTR = 0.99 in Case 1 (solid dots on the solid curves) and Case 3 (solid dots on the dotted curves) while the maximum values appear when CTR = 0.01 in Case 2 (punched bigger dots on the solid curves). The graphs in Case 2 are displayed separately because they are in a relatively narrow range of the GI (right panel).

64 Case 1: r =40.464 mm, r −r =2.5 mm Case 2: r =40.464 mm, r −r =3.5 mm Case 3: r =51.088 mm, r −r =2.5 mm 2 2 1 2 2 1 2 2 1 11

# of sulci 5

4 Case 1, 3: n0=3 3 Case 1: n0=5 Case 2: n0=3 2 Case 2: n0=5 Case 3: n0=5 1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 C TR

Figure 4.11: Graph of the number of sulci versus CTR. The maximum numbers of sulci are same as the number of the applied forces: six when n0 = 3 (orange) and ten when n0 = 5 (green). As CTR approaches zero, the number of sulci increases while the GI decreases and reaches its minimum when CTR = 0.6 in Case 2 of n0 = 3 and n0 = 5 (diamonds), when CTR = 0.5 in Case 3 of n0 = 5 (square), and when CTR = 0.4 in the other cases (stars). Note that the three cases for n0 = 3 exhibit the same pattern for number of sulci versus CTR (orange curves). Case 1: r2 = 40.464 mm and thick = 2.5 mm (solid dots on the solid curves), Case 2: r2 = 40.464 mm and thick = 3.5 mm (bigger dots on the solid curves), and Case 3: r2 = 51.088 mm and thick = 2.5 mm (solid dots on the dotted curves).

4.4 Strengths of the Applied Axonal Tension Forces

In this section, we discuss how the magnitudes of the applied pulling tension forces have an effect on cortical folding. Section 4.4.1 displays how the GI increases as the magnitudes of the applied forces are intensified. In addition, we characterize the change in cortical folding shapes with the increasing strengths of the forces. Section 4.4.2 compares the degrees of folding among the model cortices when the distances between the applied forces are different. In Section 4.4.3, three irregular Turing patterns are introduced and we simulate cortical folding with four sets of the applied forces with different magnitudes based on each of the Turing patterns.

65 4.4.1 Weaker vs. Stronger Pulling Tension Forces

It is obvious that the GI increases as the cortex is loaded by the stronger pulling forces. We can observe the visible changes in folding shapes as well as the rate of increase of the GI by changing 2 the parameter Cfm > 0 where Cfm is a factor of the scaling constant Cu = 25 n0 Cfm. The × × scaling constant Cu is the multiplication parameter of proportionality of the magnitude of the force to the amount of the activator concentration u in the given interval of the Turing pattern (see Equations 3.11 and 3.12).

All simulation parameters other than the scaling constant Cu are as in Figure 4.1a–c. Thus, three sets of the forces are applied on the semi-circular model cortex with gray matter radius 40.464 mm and cortical thickness 2.5 mm. The directions of the applied forces are determined by the constant CTR = 0.99. All three sets of the forces are based on the regular Turing patterns, thus the magnitudes of the applied forces are spread on the domain with low RSDs (< 3.25%). The magnitudes are varied by changing Cfm. We need to be consider that the averages of the magnitudes of the forces as well as the number of forces are diﬀerent across simulations when comparing results. For example, in Figure 4.1, Cfm = 1 and the averages of the magnitudes of the applied forces are 1.9381 N 2.4032%, 3.2244 N ± ± 3.2404%, and 6.3069 N 0.0521% in the cases of n0 = 3, n0 = 5, and n0 = 10, respectively. As ± wave number n0 increases, the average of the strength of each applied force becomes higher and the number of the applied forces is twice the wave number. Figure 4.13a shows the graphs of the GI versus the average of the magnitudes of the applied forces for the three cases with diﬀerent wave numbers. The average of the forces varies according to the parameter Cfm which changes from 0.25 to 2.5 when n0 = 3 and 5 or changes from 0.25 to

1.5 when n0 = 10. In every case, The graphs increase exponentially before Cfm = 1. After that point, the rate of increase grows logistically in cases of n0 = 5 and 10 or the rate starts to decrease slightly in case of n0 = 3 (see Figure 4.13a). The rate of increase of the GI is further discussed in Section 4.4.2.

Figure 4.12 displays changes in the shape of cerebral cortical folding as the parameter Cfm increases from 0.25 to 2.5 when n0 = 5. The rudimentary sulci become deeper and steeper as Cfm increases. In addition, the overall thickness of the model cortex becomes thicker except the crests of both sulci and gyri. Figures B.12 and B.13 show similar phenomena for n0 = 3 and n0 = 5. In

66 particular, when n0 = 10, the overall shape of the semi-circular domain becomes ﬂat and the side walls of the sulcus located in the center progressively move closer. We observed the development of cortical folding visually through MR images in Section 2.2. The concentration of neurons, that has been assumed to be proportional to the strengths of the axonal tension forces, increases with GA. The progressive changes of cortical folding shapes with increasing GA are well captured by our simulation results.

4.4.2 Distance between the Applied Forces

The distance between the applied forces decreases as wave number n0 increases. In Figure 4.1a–c, several pairs of the force vectors that pull together are applied on the semi-circular domain whose radius of gray matter is r2 = 40.464 mm and cortical thickness is thick = 2.5 mm. The starting points of the applied force vectors lie on the boundary of gray matter and white matter, whose radius is r = r thick. We compute the arc length between the two starting points of each 1 2 − pair of the forces pulling together by multiplying r1 and the central angle of the arc in radians. The arc lengths between each pair of the pulling forces are 19.6805 mm, 11.3323 mm, and 5.3681 mm when n0 = 3, 5, and 10, respectively. Among the three cases having diﬀerent wave numbers, it is found that the steepest slope of the increase of the GI in the graphs in Figure 4.13a is obtained in the case of n0 = 3 even though it has the smallest number of the applied forces. The lowest slope of increase of the GI is obtained when n0 = 10 in spite of it having the largest number of applied forces. All three graphs are displayed together in Figure 4.13a.

The slopes of the lines in the graphs of the GI versus the parameter Cfm are relatively similar each other (see Figure 4.13b). However, the increasing rate when n0 = 10 is still lower than those of the others. The graphs of the GI versus Cfm are increasing exponentially before Cfm = 1.0. After the point, the slopes of the graphs when n0 = 5 and 10 increase more slowly than those do before Cfm = 1.0 and the slope of the graph when n0 = 3 decreases slightly. Since the sets of the applied forces of the three cases shown in Figure 4.13 have diﬀerent averages of magnitudes, we present more comparisons among the three cases when the applied forces have the same magnitude with the RSD of 0%. The magnitude of each applied force is a constant of 4, 4.5, or 5 N in all cases having diﬀerent wave numbers. The graphs of the GI versus the magnitude

67 n0 = 5, RSD = 3.2404% (a) (b) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.25 , AVG = 0.80611 , GI = 1.0018 C = 0.5 , AVG = 1.6122 , GI = 1.0158 fm fm

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 1 , AVG = 3.2244 , GI = 1.0948 C = 1.5 , AVG = 4.8367 , GI = 1.2185 fm fm

(e) (f) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 2 , AVG = 6.4489 , GI = 1.3652 C = 2.5 , AVG = 8.0611 , GI = 1.5218 fm fm

Figure 4.12: Changing the magnitudes of the applied forces when n0 = 5. The magnitudes 2 of the applied forces are determined by the scaling constant Cu = 25 n0 Cfm where × × n0 = 5 and (a) Cfm = 0.25, (b) Cfm = 0.5, (c) Cfm = 1.0, (d) Cfm = 1.5, (e) Cfm = 2.0, and (f) Cfm = 2.5. The average (AVG) of the magnitudes of the applied forces increases Cfm times as much as in Figure 4.1b. The other parameters are as in Figure 4.1b. The rudimentary shallow grooves on the smooth model cortical surface become deeper and steeper as Cfm increases.

68 (a) 1.6 (b) 1.6

1.5 1.5

1.4 1.4

1.3 1.3 GI GI

1.2 1.2

1.1 1.1

n0 = 3 1 1 n0 = 5 n0 = 10

0 2 4 6 8 10 0.5 1 1.5 2 2.5 Average of the magnitudes of all forces (N) C fm

Figure 4.13: Graphs of the GI versus the average of the magnitudes of all applied forces and Cfm. The arc lengths between the pulling forces are 19.6805 mm when n0 = 3 (orange), 11.3323 mm when n0 = 5 (green), and 5.3681 mm when n0 = 10 (blue). (a) Graphs of the GI versus the average of the magnitudes of all forces. (b) Graphs of the GI versus Cfm. The slopes of (a) vary considerably whereas the slopes of (b) are relatively similar. of the force are shown in Figure 4.14. The deformed conﬁgurations are shown in Figure 4.15, B.14, and B.15. Figure 4.14 also shows the steepest slope of the GI increase is obtained when n0 = 3, although the total magnitude of the applied forces is the lowest of the three cases as the number of applied forces is smallest. As n0 increases, the total magnitude of the applied forces increases but the slope becomes less steep. We can conclude from the graphs in Figures 4.13 and 4.14 that the rate of increase of the GI becomes higher as wave number decreases so the distances between the pulling forces become wider. In other words, stronger forces are required to obtain a particular GI value as the distances between the pulling forces become narrower.

69 1.5

1.45

1.4 n0 = 3 n0 = 5 1.35 n0 = 10 1.3

GI 1.25

1.2

1.15

1.1

1.05

1 4 4.5 5 Magnitude of each force (N)

Figure 4.14: Graphs of the GI versus the magnitude of each force. In each simulation, every force has constant magnitude, 4, 4.5, or 5 N. All parameters except the strengths of the forces are as in Figure 4.1a–c. The arc length between each pair of the pulling forces becomes narrower as wave number n0 increases. n0 = 3 (orange), n0 = 5 (green), and n0 = 10 (blue).

4.4.3 Uneven Magnitudes of the Applied Forces

We have presented numerical results applied by the forces whose magnitudes came from the evenly folded Turing patterns so far. In this section, we introduce three irregular Turing patterns which are obtained by reducing number of iterations from tn = 150, 000 to tn = 15, 000. We also change the position of the initial conditions from the center at x = 0.5 to the right side at x = 0.75 or 0.9 in order to obtain two of the three irregular Turing patterns. The other parameters are as in Figure 4.1b when n0 = 5. The crooked Turing patterns are presented in the left panel of Figure 4.16a–c. The magnitudes of the applied forces are determined by the integrals of u and the scaling 2 constant Cu = 25 n0 Cfm where n0 = 5 (see Equations (3.11) and (3.12)). The lengths of × × the red arrows on the right panel of Figure 4.16 are proportional to the magnitudes of the applied

70 The magnitude of each force = 4.5 (a) 60 (b) 60 (c) 60

40 40 40

20 20 20

0 0 0 −50 0 50 −50 0 50 −50 0 50 n0 = 3, GI = 1.4337 n0 = 5, GI = 1.1911 n0 = 10, GI = 1.0441

Figure 4.15: Changing distance between the applied pulling forces when f = 4.5 N. On | | the initial semi-circular domain, the arc lengths between each pair of the pulling forces are (a) 19.6805 mm when n0 = 3, (b) 11.3323 mm when n0 = 5, and (c) 5.3681 mm when n0 = 10. The magnitude of each force is 4.5 N. All parameters except the strengths of the forces are as in Figure 4.1a–c.

force vectors. We present four simulation results having diﬀerent values of Cfm with each of the three irregular Turing patterns (see Figures 4.17–4.19).

The directions of the forces are determined by the constant CTR = 0.99. The number of the applied forces is ten in all simulation sets. Note that the number of the forces is again twice the wave number n0 = 5 as we have seen in previous numerical results dealing with the evenly folded Turing patterns. By choosing more or less numbers of iterations, we can increase or reduce the number of forces.

Table 4.4 displays profiles of the applied forces when Cu = 625. The first column of the table indicates the corresponding figures. It also includes the averages (AVGs) and relative standard deviations (RSDs) of the magnitudes of the applied forces. The value n is the node number located on r1, on which the starting point of the applied force is located. The corresponding location of the nth node on the one-dimensional domain is x(n)=(n 1)∆x where ∆x = 1/200. The corresponding − angle θ, which is the angle from the bottom line to the radial line lR, is θ(n)=(n 1)∆θ where − ∆θ = π/200. The fifth column of the table presents the magnitudes of the forces unit in newtons (N). The remaining columns display the components of the forces, where fx and fy are x- and y-components of the force f, fT is a tangential component of f and we have assumed that the sign of fT is same as

71 the sign of the x-component of fT , and lastly, fR is a radial component of f and we have assumed that the sign of fR is same as the sign of the y-component of fR.

I.C.: x = 0.5 r = 40.464 mm, r −r = 2.5 mm, C = 0.99 (a) 0.05 2 2 1 TR 40

u 0 20

0 −0.05 −40 −20 0 20 40 0 0.5 1 x I.C.: x = 0.75 (b) 0.05 40

u 0 20

0 −0.05 −40 −20 0 20 40 0 0.5 1 x I.C.: x = 0.9 (c) 0.05 40

u 0 20

0 −0.05 −40 −20 0 20 40 0 0.5 1 x

Figure 4.16: Irregular Turing patterns and the applied forces. The irregular Turing patterns (left panel) are obtained by reducing the number of iterations from tn = 150, 000 to tn = 15, 000. In addition, the position of the initial conditions is changed from x = 0.5 to (b) x = 0.75 or (c) x = 0.9. The other parameters are as in Figure 4.1b when wave number n0 = 5. The red arrows on the semi-circular model cortices represent the applied forces (right panel). In the ﬁgures, the magnitude of the force is proportional to the length of the red arrow.

72 Table 4.4: Proﬁles of the applied forces based on the irregular Turing patterns when Cu = 625. The value n is the node number on which the starting point of the applied force is located. The corresponding location of the nth node on the left panel of Figure 4.16 and the corresponding angle θ on the right panel of the ﬁgure are shown in the next columns. Also, the table displays the magnitude of each force vector and its components: fx, fy, fT , and fR. Note that the sign of fT is same as the sign of the x-component of fT ; and the sign of fR is same as the sign of the y-component of fR.

Figure 4.16 n x θ f fx fy fT fR | | 11 0.050 0.1571 0.1446 0.0022 -0.1446 0.1432 -0.0204 30 0.145 0.4555 0.3412 -0.1919 0.2822 -0.3378 -0.0481 53 0.260 0.8168 0.3412 0.2133 -0.2663 0.3378 -0.0481 70 0.345 1.0838 0.6144 -0.5781 0.2080 -0.6082 -0.0867 (a) 92 0.455 1.4294 0.6144 0.5900 -0.1715 0.6082 -0.0867 110 0.545 1.7122 0.6144 -0.5900 -0.1715 -0.6082 -0.0867 AVG( f ): 0.4112 N 132 0.655 2.0577 0.6144 0.5781 0.2080 0.6082 -0.0867 | | RSD( f ): 46.3438% 149 0.740 2.3248 0.3412 -0.2133 -0.2663 -0.3378 -0.0481 | | 172 0.855 2.6861 0.3412 0.1919 0.2822 0.3378 -0.0481 191 0.950 2.9845 0.1446 -0.0022 -0.1446 -0.1432 -0.0204 23 0.110 0.3456 0.1246 -0.0583 0.1101 -0.1234 -0.0176 40 0.195 0.6126 0.1246 0.0565 -0.1110 0.1234 -0.0176 62 0.305 0.9582 0.1246 -0.1110 0.0565 -0.1234 -0.0176 79 0.390 1.2252 0.1246 0.1101 -0.0583 0.1234 -0.0176 (b) 104 0.515 1.6179 0.2129 -0.2091 -0.0399 -0.2108 -0.0300 118 0.585 1.8378 0.2129 0.2112 0.0266 0.2108 -0.0300 AVG( f ): 0.1999 N 143 0.710 2.2305 0.3246 -0.2258 -0.2331 -0.3213 -0.0458 | | RSD( f ): 38.9023% 159 0.790 2.4819 0.3246 0.2331 0.2258 0.3213 -0.0458 | | 184 0.915 2.8746 0.2129 -0.0266 -0.2112 -0.2108 -0.0300 198 0.985 3.0945 0.2129 0.0399 0.2091 0.2108 -0.0300 12 0.055 0.1728 1.0086 0.0315 -1.0081 0.9985 -0.1423 29 0.140 0.4398 0.5539 -0.3042 0.4629 -0.5484 -0.0781 52 0.255 0.8011 0.5539 0.3394 -0.4377 0.5484 -0.0781 71 0.350 1.0996 0.2277 -0.2155 0.0737 -0.2254 -0.0321 (c) 91 0.450 1.4137 0.2277 0.2176 -0.0670 0.2254 -0.0321 110 0.545 1.7122 0.5539 -0.5319 -0.1546 -0.5484 -0.0781 AVG( f ): 0.6694 N 133 0.660 2.0735 0.5539 0.5182 0.1957 0.5484 -0.0781 | | RSD( f ): 47.2909% 150 0.745 2.3405 0.9725 -0.5959 -0.7686 -0.9628 -0.1372 | | 172 0.855 2.6861 0.9725 0.5468 0.8043 0.9628 -0.1372 190 0.945 2.9688 1.0687 -0.0334 -1.0682 -1.0581 -0.1508

73 I.C.: x = 0.5 (a) (b) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 625, GI = 1.0026 C = 3125, GI = 1.0499 u u

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 6250, GI = 1.1935 C = 8125, GI = 1.2911 u u

Figure 4.17: Cortical folding patterns resulting from applied forces due to the Turing 2 pattern in Figure 4.16a. The applied scaling constants are (a) Cu = 25 5 1 = 625, (b) 2 2 × ×2 Cu = 25 5 5 = 3125, (c) Cu = 25 5 10 = 6250, and (d) Cu = 25 5 13 = 8125. × × × × × × All parameters except the number of iterations (tn = 15, 000) are as in Figure 4.1b when the wave number is n0 = 5.

The simulation results in Figures 4.17–4.19 demonstrate that cortical areas where stronger forces are applied become deeply convoluted as the scaling constant Cu increases. Interestingly, some pairs of forces that are weakly pulling together combine to form a single wide fold. (see Figure 4.18). In addition, one hemisphere is enlarged if the overall magnitude of the applied forces is greater than the forces do on the other side (see Figure 4.19). However, Figure 4.18 shows an enlarged hemisphere on the right side (0 < θ < π/2) in spite of the fact that the right side has smaller magnitudes of the applied forces than the other side. This distribution of the applied forces (see Figure 4.16b) explains this result: there is one pair of pulling forces near the constrained nodes on the left side (θ = π) while there is no pair of forces near such nodes on the right side (θ = 0.0). Also, the arc length between the strongest forces located on the left side is narrow while the four weak forces are distributed widely on the right side. In

74 I.C.: x = 0.75 (a) (b)

40 40

20 20

0 0 −50 0 50 −50 0 50 C = 625, GI = 1.0059 C = 3125, GI = 1.0062 u u

40 40

20 20

0 0 −50 0 50 −50 0 50 C = 6250, GI = 1.0284 C = 8125, GI = 1.047 u u

Figure 4.18: Cortical folding patterns resulting from applied forces due to the Turing 2 pattern in Figure 4.16b. The applied scaling constants are (a) Cu = 25 5 1 = 625, (b) 2 2 × ×2 Cu = 25 5 5 = 3125, (c) Cu = 25 5 10 = 6250, and (d) Cu = 25 5 13 = 8125. × × × × × × All parameters except the number of iterations (tn = 15, 000) and the position of the initial conditions (x = 0.75) are as in Figure 4.1b when the wave number is n0 = 5. addition, the other pair of pulling forces on the left side is located near the center. Therefore, the distribution of the locations of the applied forces and the magnitudes of the forces are critical for cortical folding as well as the magnitudes of the forces. Through simulations dealing with the applied forces from the unevenly folded Turing patterns, we are able to analyze how events such as self-ampliﬁcation of IPCs occurring in earlier stages of development aﬀect cortical folding. Our proposed axonal-tension based model can study possible mechanisms underlying abnormally formed cortical folding occuring in a broad range of developmental stages.

75 I.C.: x = 0.9 (a) 80 (b) 80 60 60 40 40 20 20 0 0 −50 0 50 −50 0 50 C = 625, GI = 1.0046 C = 3125, GI = 1.0387 u u

Figure 4.19: Cortical folding patterns resulting from applied forces due to the Turing 2 pattern in Figure 4.16c. The applied scaling constants are (a) Cu = 25 5 1 = 625, (b) 2 2 × 2 × Cu = 25 5 5 = 3125, (c) Cu = 25 5 6 = 3750, and (d) Cu = 25 5 6.9 = 4312.5. × × × × × × All parameters except the number of iterations (tn = 15, 000) and the position of the initial conditions (x = 0.9) are as in Figure 4.1b when the wave number is n0 = 5.

4.4.4 Summary

We have assumed that the concentration of neurons constituting the cortico-cortical connections is proportional to the strengths of the axonal tension forces. As the brain develops, the number of neurons increases in the process of cell diﬀerentiation. In Section 4.4.1, we observed that rudimentary sulci become deeper and steeper as the strengths of the applied forces become stronger. This is consistent with what we observed through MR images in Section 2.2. In Section 4.4.2, we compared the rates of increase of the GI among the three cases having diﬀerent wave numbers n0 = 3, 5, and 10. As n0 increases, the slope of the graph of the GI versus the strengths of the force decreases. The distance between the applied forces decreases as n0 increases.

76 Lastly, we investigated how unevenly distributed strengths of forces aﬀect cortical folding in Section 4.4.3. The Turing pattern represents the concentration of the activator morphogen u that determines the rate of self-ampliﬁcation of IPCs. The ability to creat and apply a set of axonal tension forces from irregular Turing patterns means our proposed model has the ability to study a larger variety of cortical folding malformations.

4.5 Conclusions

In this chapter, we investigated the effects of domain size, the direction of the applied forces and their magnitude. We compared the degrees of folding of the deformed configurations due to the applied forces by computing the GI. We found that the GI increases exponentially as the model cortex becomes bigger or thinner. In regard to the directions of the applied forces, we observed that the overall shape of the model cortex becomes flatter and the number of sulci increases when the magnitudes of the radial components of the applied forces become greater. As the strengths of the applied forces become stronger, the rudimentary sulci become deeper and steeper. Among the three cases having different wave numbers, the GI increases faster because the distance between the applied forces is wider. We also presented simulation results showing how the unevenly distributed magnitudes of the forces affect cortical folding. In the next chapter, we utilize the simulation results presented in this chapter in order to explore the possible mechanisms involved in disorders of cortical formations.

77 CHAPTER 5

APPLICATIONS TO HUMAN CEREBRAL CORTICAL FOLDING MALFORMATIONS

The development of cerebral cortical folding involves three stages: cell proliferation, cell migration, and cortical organization. These three major stages of cortical formation overlap. The differentia- tions of IPCs continue after neurons begin to migrate from the SVZ to the CP. The migration of neurons continues after neurons begin to construct cortico-cortical connections. Disorders of cortical formations have been classified according to abnormal development occurring during the three stages. In the cell proliferation stage, abnormally decreased or increased proliferation, or abnormal proliferation in specific brain areas are the typical causes of cortical folding malformations. In the second stage, cell migration to a wrong place causes disorders of cortical formations. In the last stage, deranged organization is the reason for the malformations. In this chapter, our proposed mathematical model is employed to model some cases of human cerebral cortical folding malformations. To model decreased or increased proliferation, the scaling constant Cu or the irregular Turing patterns are utilized. When the process of cell migration is arrested, the cerebral cortex becomes thick due to the arrested cells. We model the thickened cortex using the parameter thick. To model deranged organization, we utilize the parameter CTR to change the directions of the applied forces. The majority of abnormal cortical folding is often accompanied by abnormal size of the brain. We utilize the parameter r2 to model the size of the brain. Section 5.1 describes microcephaly and microlissencephaly based on MR imaging features and presents simulation results that capture the characteristics of the malformations using our model. Microcephaly and microlissencephaly result from decreased proliferation. In Section 5.2, two irregular Turing patterns are newly presented in order to model the increased degree of cell proliferation in part of the brain. The irregular Turing patterns are utilized to model hemimegalencephaly. In Section 5.3, we model classic (type I) lissencephaly which is due to undermigration. Lastly, in Section 5.4 we model polymicrogyria which is due to deranged organization. Table 5.1 summarizes

78 Table 5.1: Causes of cortical folding malformations and simulation parameters. Table adapted from [91].

Changing Simulation Cause Disorder Section parameters results Microcephaly Figure 5.2c Decreased proliferation Section 5.1 C , r , thick Microlissencephaly u 2 Figure 5.2d Figure 5.4 Increased proliferation Hemimegalencephaly Section 5.2 t , I.C., thick n Figure 5.5 Abnormal proliferation Focal cortical dysplasia Undermigration Type I lissencephaly Section 5.3 thick Figure 5.7 Overmigration Type II lissencephaly Ectopic migration Heterotopia Polymicrogyria Section 5.4 C , r , C Figure 5.11 Deranged organization TR 2 u Schizencephaly

the classiﬁcation of disorders of cortical formation as well as the changing parameters used in the simulations presented in this chapter.

5.1 Decreased Proliferation

Either abnormally decreased cell production or increased apoptosis, which means programmed cell death, can occur during the ﬁrst stage of the development of cerebral cortical formation. The abnormally small population of neuroblasts produced during the early stage of development causes lissencephalic brains which are characterized by lack of convolutional patterns in the cerebral cortex. Lissencephalic microcephaly and microlissencephaly are typical disorders that result from a decreased number of neurons [91]. In this section, we describe the disorders and present simulation results modeling the malformations.

5.1.1 Microcephaly and Microlissencephaly

The word microcephaly describes its characteristic: micro means small and cephaly comes from the Greek word which means head [92]. When a child is diagnosed with microcephaly, his/her head circumference signiﬁcantly smaller, 2-3 SD below the mean, than that of other children of the

79 same age and sex [2,91,92]. About 25,000 children are diagnosed with microcephaly in the United States of America each year [92]. Microcephaly is a congenital disease but it can occur later during infancy. Depending on the severity, children with microcephaly may encounter learning disabilities and other complications. The children may suffer from developmental delays such as in speech, move- ment, and height, difficulties with coordination and balance, facial distortions, hyperactivity, mental retardation, and seizures [93]. Neurological care and physical therapy can help to manage the symptoms but there is generally no treatment to cure microcephaly [92]. Microcephaly can be associated with other kinds of cortical malformations such as lissencephaly or polymicrogyria [92,94]. Lissencephaly is characterized by an abnormally smooth cortical surface. The Greek roots of the word lissencephaly describe the characteristic: lissos means smooth and enkaphalos means brain [95]. On the other hand, polymicrogyria is characterized by an excessive number of small gyri. Poly means many, micro means small, and gyria means folds in the brain [94]. We discuss, in this section, lissencephalic microcephaly which presents with a few number of cortical folds as well as a small brain size. This cortical malformation results from the abnormally small population of neurons produced due to the low rate of self-amplification of IPCs or increased apoptosis during the cell proliferation stage. Microlissencephaly which is characterized by the thickened agyria (no gyri) cortex is a severe form of the lissencephalic microcephaly [91]. Microlissencephaly has a thickened cortex (> 3 mm) while microcephaly usually has a normal cortical thickness (see Figure 5.1).

5.1.2 Modeling Microcephaly and Microlissencephaly

To model microcephaly with a small brain due to decreased cell proliferation, our model systems utilize the parameters Cu and r2. For the sake of comparison, we model a normal brain using 2 Cu = 25 n0 Cfm where n0 = 5 and Cfm = 2.0 and r = 40.464 mm (see Figure 5.2a). The × × 2 folding shape when only Cu is decreased half is displayed in Figure 5.2b. When we reduce the value of r2 to r2 = 31.303 mm as well as Cu, the folding shape becomes much shallower (see Figure 5.2c). To model microlissencephaly, the value of thick is increased from thick = 2.5 mm to 4.5 mm (see Figure 5.2d). The generated patterns exhibit smooth cortex, representing lissencephalic microcephaly and microlissencephaly.

80 Figure 5.1: Microcephaly and microlissencephaly. (a) MR images of the brain of a one-year old child who has been diagnosed with lissencephalic microcephaly. The images sectioned along the sagittal plane (left) and the transverse plane (right) show decreased number of sulci and broad gyri. Figure adapted from [96]. (b) MR images of the brain with microlissencephaly. The sagittal (left) and transverse (right) images show the thickened agyria cortex. Figure adapted from [91].

5.2 Increased Proliferation

Abnormally increased proliferation due to either excessive cell production or decreased apoptosis also causes malformations of cerebral cortical folding. In this section, we describe and model

81 C = 0.9 TR (a) 40 (b) 40 30 30

20 20

10 10

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 2, r = 40.464 mm, r −r = 2.5 mm, GI = 1.4519 C = 1, r = 40.464 mm, r −r = 2.5 mm, GI = 1.0929 fm 2 2 1 fm 2 2 1

20 20

10 10

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 1, r = 31.303 mm, r −r = 2.5 mm, GI = 1.0492 C = 1, r = 31.303 mm, r −r = 4.5 mm, GI = 1.0017 fm 2 2 1 fm 2 2 1

Figure 5.2: Modeling microcephaly and microlissencephaly. Changing parameters are Cfm, r2, and thick (r2 r1). The magnitudes of the applied forces are determined by the − 2 scaling constant Cu = 25 n0 Cfm where n0 = 5. The constant for determining the × × directions of the forces is CTR = 0.9 and the angle between the tangent line lT and each force vector is φ = 0.4510 radians. The other parameters are as in Figure 4.1b when n0= 5. (a) Modeling normal brain. Cfm = 2.0, r2 = 40.464 mm, thick = 2.5 mm. (b) Modeling decreased proliferation by reducing Cfm to 1.0. (c) Modeling microcephaly by reducing r2 to 31.303 mm as well as Cfm (see Figure 5.1a). (d) Modeling microlissencephaly by increasing thick as well as reducing Cfm and r2 (compare to Figure 5.1b). hemimegalencephaly which results from the excessive number of neurons [91].

5.2.1 Hemimegalencephaly

The averages of normal adult and newborn human brain weights are 1, 300-1, 400 grams and 350- 400 grams, respectively [97]. Megalencephaly is characterized by abnormally heavy brain weight exceeding 1, 800 grams or 2.5-3 SD above the mean than that of other children of the same age and sex [2, 98].

82 Patients with hemimagalencephaly show one cerebral hemisphere or parts of it to be markedly enlarged [2, 91]. A brain with hemimagalencephaly grows asymmetrically so it is suspected with an enlarged head circumference or an asymmetrical head shape [99]. Prevalence estimates of hemimegalencephaly are not available since it is a very rare disorder [99]. The cerebral cortex of hemimegalencephaly may be normal but it can be associated with cortical dysplasia such as polymicrogyria or lissencephaly [2, 91]. When hemimegalencephaly is associated with cortical dysplasia, patients suﬀer from severe psychomotor retardation; for example, a slowing- down of thought and a reduction of physical movements, and seizures. Autism is the most prevalent neurodevelopment disorder linked with the malformation [100]. Hemimegalencephaly without cortical dysplasia also results in developmental delay and mild cognitive abnormalities in learning, memory, perception, or problem solving [2]. Like microcephaly and microlissencephaly, there is currently no treatment to cure hemimegalencephaly. However, the abnormally increased proliferation of neurons or decreased apoptosis can be inhibited by using anti-epileptic drugs which control epilepsy what is called seizure disorder [98,101]. MR images of brains with hemimegalencephaly in Figure 5.3 demonstrate the markedly enlarged right (a) or left (b) cerebral hemisphere. The right or left LV is also enlarged so the large size of the LV may contain more IPCs than the normal brain does, producing an excessive number of neurons. This is consistent with the IPM discussed in Section 2.3.1. In addition, the MR images show a thinkened cortex which results from migration arrest [102] or because of a large population of neurons. Cortical dysplasia is also shown in the cerebral cortex of the englarged hemisphere but the opposite hemisphere looks normal in appearance in both cases (see Figure 5.3).

5.2.2 Modeling Hemimegalencephaly

To model hemimegalencephaly due to asymmetrically increased proliferation, we utilize the irregular Turing patterns. As discussed in Section 4.4.3, the irregular Turing patterns of the concentration of activator u with large RSD are numerically obtained by reducing the iteration number tn and moving the position of the initial condition. When tn = 150, 000, the evenly folded Turing pattern results in 3.2404% RSD of the magnitudes of the applied forces (see Figure 4.1b when n0 = 5). We reduce tn to tn = 10, 000 and the Turing patterns generated with the reduced iteration number lead to applied forces that have magnitudes with large RSDs of more than 30.0% (see Table 5.2).

83 Figure 5.3: Hemimegalencephaly. (a) MR images of the brain of a four-year old female child who has been diagnosed with hemimegalencephaly. The transverse images show an enlarged right cerebral hemisphere with thickened cortex. The right side of the brain is labelled with R. Figure adapted from [102]. (b) MR images of a diﬀerent patient with hemimegalencephaly on the other side of the brain. The coronal images show an enlarged left cerebral hemisphere as well as a thickened cerebral cortex. The left side of the brain is labelled with L. Figure adapted from [91].

84 t = 10000, I.C.: x = 0.25, n0 = 5 n r = 40.464 mm, C = 0.7, C = 0.9 (a) 0.04 2 fm TR 0.02 (b) 40 0 30 u −0.02 20 −0.04 10

−0.06 0 −40 −20 0 20 40 0 0.2 0.4 0.6 0.8 1 x

20 20

0 0 −50 0 50 −50 0 50 r −r = 2.5 mm, GI = 1.0357 r −r = 3.5 mm, GI = 1.0203 2 1 2 1

Figure 5.4: Modeling hemimegalencephaly with an enlarged right hemisphere. (a) An irregular Turing pattern is obtained using the same parameters as in Figure 4.1b when n0 = 5 except the iteration number tn and the location of the initial conditions. (b) Four pairs of the forces which are pulling together (red arrows) are applied on the semi- circular domain. The length of the arrow is proportional to the magnitude of the force. (c) Deformed conﬁguration due to the forces shown in (b). (d) Modeling hemimegalencephaly with an enlarged right hemisphere and thickened cortex (compare to Figure 5.3a).

t = 10000, I.C.: x = 0.75, n0 = 5 n r = 40.464 mm, C = 0.7, C = 0.9 (a) 0.04 2 fm TR 0.02 (b) 40 0 30 u −0.02 20 −0.04 10

−0.06 0 −40 −20 0 20 40 0 0.2 0.4 0.6 0.8 1 x

20 20

0 0 −50 0 50 −50 0 50 r −r = 2.5 mm, GI = 1.04 r −r = 3.5 mm, GI = 1.0216 2 1 2 1

Figure 5.5: Modeling hemimegalencephaly with an enlarged left hemisphere. (a) An irregular Turing pattern is obtained using the same parameters as in Figure 4.1b when n0 = 5 except the iteration number tn and the location of the initial conditions. (b) Four pairs of the forces which are pulling together (red arrows) are applied on the semi- circular domain. The length of the arrow is proportional to the magnitude of the force. (c) Deformed conﬁguration due to the forces shown in (b). (d) Modeling hemimegalencephaly with an enlarged left hemisphere and thickened cortex (compare to Figure 5.3b).

85 The initial concentrations of the activator u and inhibitor v were located in the middle of the domain, x = 0.5, in the evenly folded Turing patterns. The location of the initical condition is set to x = 0.25 (see Figure 5.4) or x = 0.75 (see Figure 5.5) to model hemimegalencephaly with an enlarged right or left hemisphere, respectively. When the location of the initial condition is x = 0.25, the highest peak of the folding of the Turing pattern appears at x = 0.25 but the longest distance between the values of x which satisfy u(x) = 0.0 appears on the opposite side of the domain. Note that the function is not continuous but discrete in numerical computation, so we choose two smallest values of x that satisfy u(x) > 0.0 in each interval where u > 0 to compare the distances between the two points. In Figure 5.4, the two nodes having the longest distance between themselves are 125 and 177 (see Table 5.2). The force vectors which start from the two nodes and pull together have the largest magnitudes as well as the widest distance. The resulting deformed conﬁguration due to the four pairs of the forces which pull together shows an enlarged right hemisphere (see Figure 5.4c and compare with Figure 5.3a MR image). The enlarged right hemisphere is generated by the pair of strongest forces coming from the large population of neurons due to increased proliferation. In addition, the simulation result is also consistent with what we discussed in Section 4.4.2; cortical folding is generated easily if the distance between a pair of the pulling forces is wider. Increasing the value of thick reduces the asymmetric development but the numerical result still shows an enlarged right hemisphere (see Figure 5.4d). Modeling hemimegalencephaly with an enlarged left hemisphere shows similar numerical results (see Figure 5.5).

5.3 Undermigration

As the cortex becomes thicker, the degree of folding decreases (see Table 4.1) and it results in a smooth cortex. The major reason for cortex thickening is undermigration of neurons; the radial columns are surrounded by the under-migrated cells so the cortex becomes thick. In this section, we describe classic lissencephaly, which is also known as complete or type I lissencephaly, and model the malformation.

86 5.3.1 Classic (Type I) Lissencephaly

Lissencephaly comes from the Greek word and literally means smooth brain [95]. A brain with lissencephaly has no or a few number of convolutions and it has thick and broad gyri (see Figure 5.6). Lissencephaly is classified into classic (type I), cobblestone (type II), and other types of lissencephaly such as microlissencephaly which is accompanied with a small brain as discussed in Section 5.1. Major symptoms of lissencephaly are developmental delays, irregular appearance of the face, malformed toes and figures, swallowing problems, sudden involuntary muscular contractions, psychomotor retardation, and complex seizures [91,103,104]. In the past, the life expectancy of children with lissencephaly was about two years because they may encounter serious respiratory problems as well [105]. However, survival rates are increasing because of developing treatments for respiratory disorders and seizures [98, 101]. The symptoms vary depending on the severity of lissencephaly. Classic lissencephaly occurs due to undermigration [2,91]. The stopping of the migration process causes the radial fibers of RGCs to be surrounded by the produced neurons [91]. Therefore, the cortex becomes thick. We model lissencephaly to observe how the thickened cortex becomes smooth in the following section.

5.3.2 Modeling Classic Lissencephaly

The average thickness of a normal human cerebral cortex is 2.5 mm, and the cortical thickness varies according to brain region [25]. The literature is unclear as to whether the thickness measures are from fetal brains or fully developed brains. Regardless, the average is bigger or equal to that of developing brains. To model classic lissencephaly, we employ bigger values of thick than those of normal brains on the semi-circular model cortex. A normal gray matter radius is assumed to be r2 = 40.464 mm which is the radius of the human brain at 28 weeks GA [4, 68]. Figure 5.7 displays the deformed conﬁgurations when thick = 4.0 mm or thick = 7.0 mm. Both cases demonstrate smooth cortices; one cortex with thick = 4.0 mm shows a small number of sulci and the other with thick = 7.0 mm shows the agyria cortex. As discussed in Section 4.2.2, the GI decreases as the cortex gets thicker.

87 Figure 5.6: Classic lissencephaly. (a) MR image of the brain of a three-month old female child who has been diagnosed with classic (type I) lissencephaly. The transverse image illustrates a few number of sulci and the markedly thickened cerebral cortex. Figure from [106]. (b) The transverse image of the brain of a diﬀerent patient with classic lissencephaly shows the markedly thickened agyria cortex with the shallow Sylvian ﬁssure. Figure from [91].

C = 0.9, C = 1, r = 40.464 mm, TR fm 2 (a) 40 (b) 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 r −r = 4 mm, GI = 1.0053 r −r = 7 mm, GI = 1.003 2 1 2 1

Figure 5.7: Modeling classic lissencephaly. The cortical thickness parameter is changed to (a) thick = 4.0 mm (compare to Figure 5.6a). (b) thick = 7.0 mm (compare to Figure 5.6b).

88 Figure 5.8: Diagrammatic representation of a cerebral cortex with polymicrogyria. Figure from [108].

5.4 Deranged Organization

During the last stage of cortical development, abnormal cortical organization, in particular, derangement of the normal six-layer lamination of the cortex or abnormally formed cortico-cortical connections, can occur due to prenatal infections, an inadequate blood supply to organs, or exposure to toxins, or chromosomal abnormality [91]. The deranged organization results in polymicrogyria which is one of the most common malformations of cerebral cortex [91, 94]. In this section, we describe and model polymicrogyria.

5.4.1 Polymicrogyria

Polymicrogyria is characterized by an excessive number of small and fused gyri (see Figures 5.8–5.10) [107,108]. Poly means many, micro means small, and gyria means folds in the brain [94]. The sulci of the brain with polymicrogyria are shallow. The cerebral cortex of polymicrogyria has been known to be slightly thickened [108, 109]. However, it is also found that some regions of the brain with polymicrogyria have reduced cortical thickness although the cortical thickness appears visually normal in MR images [109]. In regard to the size of the brain, polymicrogyria can accompany with macrocephaly or microcephaly (see Figure 5.9) [110, 111]. The abnormal features of polymicrogyria can be found in one hemisphere (unilateral, see Figure 5.10) or the entire brain (bilateral) [94, 107, 112]. In 60-70% of brains with polymicrogyria, the abnormal features show around the sylvian ﬁssure [94]. Approximately 70% of cases are found around the frontal lobe, approximately 38% of cases are found around the temporal lobe, and approximately 7% of cases are found around the occipital lobe [94].

89 Figure 5.9: Polymicrogyria with abnormal brain size. (a) MR images of the brain of an eleven-month old infant who has been diagnosed with polymicrogyria on the macrocephalic hemispheres. The sagittal image (left) shows an abnormally large head. Figure adapted from [113]. (b) The transverse image displays asymmetric microcephalic hemispheres with marked polymicrogyria (red arrowheads). Figure from [114]. (c) The sagittal image shows signiﬁcant polymicrogyria (black arrowheads) and open-lip schizencephaly which is characterized by abnormal slits, or clefts, in the cerebral hemispheres of the brain (red arrowhead). Figure from [114].

90 Figure 5.10: Unilateral polymicrogyria. MR image sectioned along the transverse plane shows an excessive number of gyri on the left hemisphere. The left side of the brain is labelled with L. Figure adapted from [115].

Depending on the particular brain regions with polymicrogyria and its severity, patients are subjected to brain functional issues related to speech, vision, sensory, motor, hearing, and cognitive problems [116]. Unilateral focal polymicrogyria which aﬀects a small part of one hemisphere is the mildest form, and symptoms associated with it are mild or even non-existent [94,112,116]. Bilateral generalized polymicrogyria which aﬀects the entire brain is the most severe form, and it results in severe seizures that are out of control with or without medicine and also causes severe physical and cognitive disabilities [94].

5.4.2 Modeling Polymicrogyria

To model polymicrogyria, we consider the process of deranged organization which is the major reason of the malformation. The cortico-cortical connections are distorted. We thus expect that the axonal tension forces pulling together tangentially are also moved so the radial components of the pulling forces are increased in the process. We model this change of direction of the applied forces by reducing the value of CTR from 0.9 to 0.4. When CTR = 0.9, the acute angle between the tangent line lT and each force vector is φ = 0.4510 radians while the angle is φ = 1.1583 radians when CTR = 0.4.

91 r −r = 2.5 mm 2 1 (a) (b) 40 40

20 20

0 0 −50 0 50 −50 0 50 C = 0.9, r = 40.464 mm, C = 0.8, GI = 1.0512 C = 0.4, r = 40.464 mm, C = 0.8, GI = 1.0308 TR 2 fm TR 2 fm

20 20

0 0 −50 0 50 −50 0 50 C = 0.4, r = 51.088 mm, C = 0.8, GI = 1.0569 C = 0.4, r = 31.303 mm, C = 0.8, GI = 1.017 TR 2 fm TR 2 fm

(e) (f) 40 40

20 20

0 0 −50 0 50 −50 0 50 C = 0.4, r = 51.088 mm, C = 1.1, GI = 1.1161 C = 0.4, r = 31.303 mm, C = 1.1, GI = 1.0392 TR 2 fm TR 2 fm

Figure 5.11: Modeling polymicrogyria. The changing parameters are CTR, r2, and Cfm. The magnitudes of the applied forces are determined by the scaling constant 2 Cu = 25 n0 Cfm where n0 = 5. The cortical thickness is thick = 2.5 mm. The other × × parameters are as in Figure 4.1b when n0 = 5. (a) Modeling normal brain. CTR = 0.9. (b) Modeling deranged organization. The portion of the radial components of the applied force vectors increases because of the reduced value of CTR from 0.9 to 0.4. (c) Modeling polymicrogyria with macrocephalic hemispheres (compare to Figure 5.9a). (d) Modeling polymicrogyria with microcephalic hemispheres (compare to Figure 5.9b, c). (e) and (f) Modeling increased proliferation due to enlarged LVs on the macrocephalic and microcephalic hemispheres, respectively. The parameter Cfm is increased from 0.8 to 1.1.

92 To capture the abnormal sizes of the brains with polymicrogyria, we also apply three diﬀerent values of r2: 40.464 mm for normal brain, 51.088 mm for macrocephalic polymicrogyria, and

31.303 mm for microcephalic polymicrogyria. In addition, we apply two diﬀerent values of Cfm which is the scaling constant for the magnitudes of the applied forces. This is because some cases of polymicrogyria with either macrocephaly or microcephaly have enlarged LVs [110, 111, 117]. According to the IPM discussed in Section 2.3.1, enlarged ventricles may contain more IPCs that produce neurons. The generated folding pattern on the model cortex exhibits an increased number and decreased size of folding separated by shallower sulci when CTR = 0.4 (see Figure 5.11b) relative to the generated pattern when CTR = 0.9 (see Figure 5.11a). When we change the value of r2 which is gray matter radius, the overall shape is maintained but the GI changes (compare Figure 5.11b with

Figures 5.11c and 5.11d). As Cfm increases from 0.8 to 1.1, the sulcus located in the middle becomes deeper as shown in Figure 5.8. Additionally, the GI increases with the strengthened forces (compare Figure 5.11c with Figure 5.11e; compare Figure 5.11d with Figure 5.11f). The characteristics of cerebral cortical folding of polymicrogyria are well captured in the simulation results.

5.5 Conclusions

In this chapter, we described the characteristics of cerebral cortical folding observed in several disorders of the brain. We also explored the major causes of the malformations of the cerebral cortex. Section 5.1 discussed lissencephalic microcephaly and microlissencephaly which result from decreased proliferation, whereas Section 5.2 discussed hemimegalencephaly which results from increased proliferation during the ﬁrst stage of cortical development. Classic lissencephaly caused by undermigration is investigated in Section 5.3. Lastly, polymicrogyria which is due to deranged organization is discussed in Section 5.4. Based on the major causes, we utilized our proposed mathematical model in order to model the malformations. In order to model decreased or increased proliferation, the scaling constant Cu, which determines the magnitudes of the applied forces, was employed in Section 5.1 and Section 5.4. Partially increased proliferation in part of the brain was modeled by using the irregular Turing patterns in Section 5.2. We modeled the change in direction of the applied forces by reducing the parameter CTR, thus simulating the distortions in the cortico-cortical connections in Section 5.4.

93 Table 5.2: Proﬁles of the forces applied in Figures 5.4 and 5.5. The value n is the node number on which the starting point of the applied force is located. The corresponding location of the nth node on the irregular Turing patterns and the corresponding angle θ on the semi-circular domain are shown in the next columns. Also, the table displays the magnitude of each force vector and its components: fx, fy, fT , and fR. Note that the sign of fT is same as the sign of the x-component of fT ; and the sign of fR is same as the sign of the y-component of fR.

Figure n x θ f fx fy fT fR | | 3 0.01 0.0314 0.1411 -0.0655 0.1250 -0.1270 -0.0615 17 0.08 0.2513 0.1411 -0.0280 -0.1383 0.1270 -0.0615 Figure 5.4 45 0.22 0.6912 0.2034 -0.1850 0.0845 -0.1830 -0.0886 57 0.28 0.8796 0.2034 0.0845 -0.1850 0.1830 -0.0886 85 0.42 1.3195 0.1411 -0.1383 -0.0280 -0.1270 -0.0615 AVG( f ): 0.1925 N 99 0.49 1.5394 0.1411 0.1250 -0.0655 0.1270 -0.0615 | | RSD( f ): 32.6913% 125 0.62 1.9478 0.2845 -0.1924 -0.2095 -0.2560 -0.1240 | | 177 0.88 2.7646 0.2845 0.2095 0.1924 0.2560 -0.1240 24 0.115 0.3613 0.2978 -0.2162 0.2048 -0.2680 -0.1298 78 0.385 1.2095 0.2978 0.2048 -0.2162 0.2680 -0.1298 Figure 5.5 103 0.510 1.6022 0.1537 -0.1362 -0.0713 -0.1384 -0.0670 117 0.580 1.8221 0.1537 0.1507 -0.0305 0.1384 -0.0670 145 0.720 2.2619 0.2205 -0.0916 -0.2005 -0.1984 -0.0961 AVG( f ): 0.2064 N 157 0.780 2.4504 0.2205 0.2005 0.0916 0.1984 -0.0961 | | RSD( f ): 30.7422% 185 0.920 2.8903 0.1537 0.0305 -0.1507 -0.1384 -0.0670 | | 199 0.990 3.1102 0.1537 0.0713 0.1362 0.1384 -0.0670

The thickened cortices and the enlarged or reduced size of the brain were also simulated with the parameters thick and r2, respectively. The characteristics of malformed cortical folding shown in MR images were well captured in the simulation result ﬁgures. In addition, we explored the symptoms and possible treatments associated with the disorders. In most cases, there is currently no treatment to cure the malformations, but physical therapy and care can help to manage the symptoms. However, by using anti-epileptic drugs, seizures can be controlled if the condition is not too severe. We may be able to extend our mathematical model to explore the eﬀects of anti-epileptic drugs.

94 CHAPTER 6

CONCLUSIONS

6.1 Summary

The development of cerebral cortical folding involves a complex sequence of progressive stages, which are accompanied by numerous influencing factors. As a result, cortical folding in each human brain is unique; it appears highly variable in position and appearance [26]. Nevertheless, the development of cerebral cortical folding occurs in predictable stages, and it has been well documented [6,8–10,48,49,54]. Also, primary sulci are most consistent in position across individuals [26]. This dissertation explained major features in neuroanatomy and corticogenesis. In order to help understand the process in which the cerebral cortex folds, MR images of the brains were provided and discussed. These images were taken between the 25th week of GA and term. We also summarized the three leading biological hypotheses which attempt to describe the underlying mechanisms of cortical folding. The major mechanism involving cerebral cortical folding is still an unsolved problem even though numerous mathematical models have been proposed based on the biological hypotheses. Debates among the hypotheses were discussed in the dissertation as well. We proposed a new mathematical model of cerebral cortical folding development based on the Axonal Tension Hypothesis [9], which is a biomechanical hypothesis. The geometry of the proposed model cortex is a two-dimensional, semi-circle. External forces, corresponding to the axonal tension- forces, are applied on the boundary between gray matter and white matter of the model cortex. The directions and magnitudes of the applied forces can be controlled by parameters. The ability to freely set the directions and magnitudes of the applied forces is an advantage of our proposed model. In addition, for the purpose of capturing effects of chemicals control involving cortical folding as well as those of mechanical tension, we adopted a previous mathematical model [69], which is based on the Intermediate Progenitor Model [8] and employed a Turing reaction-diffusion system [11] with the BVAM reaction kinetics [76,78]. The Turing pattern represents the concentration of an activator

95 that affects the degree of self-amplifications of IPCs, and ultimately determines the concentration of neurons forming the cortico-cortical connections. In our proposed mathematical model, the magnitudes of the applied forces are determined by the Turing patterns. The displacements of the model cortex under the influence of the pulling tension-forces are determined by linear elasticity. During development, cortical folding changes are gradual and occur over a long period of time. Thus, linear constitutive equations are a reasonable choice as the brain is undergoing no large deformations. Furthurmore, Wittek et al. [118] show that the choice of constitutive model of brain tissue has almost no influence on the computed deformation field and suggest that one can use the simplest elastic linear model with any reasonable value of Young’s modulus and Poisson’s ratio. In this dissertation, the theory of elasticity and the finite element formulation of the governing coupled partial differential equations of elasticity were described. We investigated the effects of domain size, directions of the pulling forces, and strengths of the forces on the semi-circular domain with plausible brain parameters. In order to compare the degrees of folding among the simulation results, we computed the gyrification index (GI) of each deformed configuration. The GI is the ratio of the length of complete cortical contour to the length of outer contour and is a well-established measure of cortical folding in the field of brain mapping [85–87]. We investigated the degree to which GI increases as the cortex becomes bigger or thinner. We found that the number of sulci increases as the forces are heading to the center of the domain. Regarding the strengths of the applied forces, we discussed how much the degree of folding increases and described the change in cortical folding shapes when the magnitudes of the forces are intensified. In addition, we analyized the effect of the distance between a pair of applied forces that pull together; the rate of increase of GI becomes higher as the distance between the pulling forces becomes wider. Also, three additional irregular Turing patterns were employed to analyze folding patterns induced by the forces whose strengths are asymmerically and unevenly distributed. Overall, the simulation results are consistent with many previous studies and expectations concerning cortical folding development. In particular, simulations increasing the radius of the brain or the magnitudes of the applied forces due to the increasing number of neurons show that the rudimentary shallow grooves on the smooth cerebral cortical surface become deeper and steeper as the brain develops. These results agree with observations from MR images of the developing cortex.

96 Lastly, we utilized our proposed model in order to model several brain malformations. We explained the classiﬁcation of disorders of cortical formations according to the developmental stages. We described the characteristics of the malformations, and we also explored the unveiled causes of the disorders such as abnormal proliferation, undermigration, and deranged organization. Our modelings well captured the characteristics of malformed cortical folding with parameter values determined by the corresponding causes of the disorders.

6.2 Ongoing and Future Work

In this section, we present some preliminary numerical results of the ongoing work and discuss possible future work. We may extend our proposed model to a three-dimensional model, to a dynamic model, or to a drug use model.

6.2.1 Three-dimensional Model

A possible future extension of the research presented in this dissertation would be developing a three-dimensional version of the proposed mathematical model of cerebral cortical folding development. For this purpose, we have studied the theory of shells, and we present some of the preliminary numerical results (see Figures 6.1–6.3 and B.16–B.19). Shell theory is a simplified method of three-dimensional elasticity with several assumptions, and there are many versions of the theory [81,119]. In the theory of shells, the middle surface of the shell is considered only. Thus, the employed finite elements can be also simplified to be two-dimensional elements. The number of degrees of freedom at each node is six: three translational displacements and three rotational displacements. In our preliminary simulations, we neglected the changes in the rotational degrees of freedom. This led to large and unreasonable deformations. Nevertheless, these incomplete preliminary results show consistent phenomena with those shown in Chapter 4 when we change only one of the parameter values such as the radius of the brain, cortical thickness, or the magnitudes of the applied forces (compare Figures 6.1 and 6.3). There are still many things remaining to study and implement in order to develop the extension of our proposed model into a three-dimensional version.

97 r = 40.464 mm, r −r = 2.5 mm, |f| = 0.04 N 2 2 1 (a) 50 (b) 50

40 40

30 30 z z 20 20

10 10

0 0

50 50

0 0 y −50 0 50 y −50 0 50 x x

Figure 6.1: Preliminary numerical result of a three-dimensional model. (a) The initial shape of the model cortex and the applied forces (red arrows). Some parameter values are r = 40.464 mm, thick = 2.5 mm, and f = 0.04 N. (b) The deformed shape under the 2 | | inﬂuence of the forces.

r = 40.464 mm, r −r = 2.5 mm, |f| = 0.04 N 2 2 1

40 40 z z 20 20

0 0 80 50 50 50 60 0 40 0 20 0 −50 0 −50 y x y x

0 50 0 −50 50 z 0 20 50

40 y 40

30 60 z 20 80 80

60 y 10 40 20 0 0 x −50 0 50 x

Figure 6.2: Diﬀerent views and skeletons of Figure 6.1b.

98 r = 18.351 mm, r −r = 2.5 mm, |f| = 0.04 N r = 51.088 mm, r −r = 2.5 mm, |f| = 0.04 N 2 2 1 2 2 1 (a) (b) 60 20

15 40

z 10 z 20 5

0 0 40 100 20 50 0 0 y −20 −10 0 10 20 y −60 −40 −20 0 20 40 60 x x

r = 40.464 mm, r −r = 4.5 mm, |f| = 0.04 N r = 40.464 mm, r −r = 2.5 mm, |f| = 0.024 N 2 2 1 2 2 1 (c) 50 (d) 50

40 40

30 30

z 20 z 20

10 10

0 0

50 50

0 0 y −50 0 50 y −50 0 50 x x

Figure 6.3: More preliminary numerical results of a three-dimensional model. Changing one parameter as follows: (a) r2 = 18.351 mm, (b) r2 = 51.088 mm, (c) thick = 4.5 mm, (d) f = 0.024 N. The other parameters are as in Figure 6.1. See Figures B.16–B.19 for | | additional views of these ﬁgures.

6.2.2 Dynamic Model

This dissertation employed the equations of equilibrium as the kinetic equations for elasticity (see Equation (3.15)). Another possible future direction would be to change the equations of equilibrium to equations of motion for a dynamic problem. For example, Newton’s second law of motion including the acceleration terms could be an option. The equations of motion based on Newton’s second law are as follows: ∂σ ∂τ x + xy + f = ρu¨ , ∂x ∂y x disp ∂τ ∂σ xy + y + f = ρv¨ , ∂x ∂y y disp where σx,σy, and τxy are stresses, and fx(x,y,t) and fy(x,y,t) are the x- and y-components of the applied forces, respectively, ρ is the mass density, udisp(x,y,t) and vdisp(x,y,t) are the displacements

99 in the x- and y-directions, respectively, and the double dots denote the second partial derivatives with respect to time [80].

6.2.3 Drug Use Model

Another possible area of future work would be to extend our proposed mathematical model into a drug use model. There is generally no treatment to cure the disorders associated with the cerebral cortical malformations. However, seizures or epilepsy are common symptoms related to these malformations, and can be controlled using anti-epileptic drugs [98, 101, 120]. In fact, such anti-epileptic drugs are effective for about 70% of patients who suffer from seizures [120]. We may add mathematical terms representing the effects of anti-epileptic medications in our model to observe how the drugs inhibit increased proliferation of neurons or decreased apoptosis [98, 101]. In addition, these extra terms may enable the optimal use of the drugs to be calculated. Understanding mechanisms underlying cortical folding development will provide better strategies to treat neurological diseases associated with abnormal cortical folding.

6.3 Concluding Remarks

There are a multiplicity of variations involved in cortical folding pattern development. The sulci and gyri are generated by complicated interactions among the various factors. According to the converging conclusion of many studies, for example, thinner cortices tend to become more intensely convoluted with shallower and more numerous gyri [17, 121]. In contrast, MR imaging features also reveal that abnormally smooth brains such as microlissencephaly and classic lissencephaly have thickened cortices [1]. MR images also show a thickened cortex in abnormally highly folded brains such as polymicrogyria. A thickened cortex is noted in 25% of the polymicrogyria cases, and generally in infants older than 18 months [1]. Thus, the cortical thickness is not the only factor inﬂuencing the development of cortical folding. In order to study mechanisms involving the development of cortical folding, the following factors should be considered: the radius of the brain, the cortical thickness, the intrinsic architecture near the cortex in particular, and the directions of the tension forces along cortico-cortical connections, the distances between the forces pulling together, and distribution of the strengths of the forces. The simulation results presented in this dissertation should be stepping stones to understanding

100 each of the potential various factors involved in the development of cortical folding. In addition, we have demonstrated that our proposed model captures the characteristics of cortical folding malformations with different combinations of the factors. Ongoing and future work involving a three-dimensional model, dynamic model, and a drug use model were presented. In addition, we may also extend our proposed model to elucidate the reasons why a significantly higher degree of cortical folding appears in particular areas of an abnormal brain by combining the potential factors. For example, it has been found that the left frontal cortex of children/adolescents who suffer from autism demonstrates a significantly higher GI than that of a normal group [122]. Other research found that the brains of people who suffer from temporal lobe epilepsy have a higher degree of folding in the temporolimbic cortices than that of a normal group [123]. Previous mathematical models of cortical folding have been developed only using a biochemical hypothesis or biomechanical hypothesis. Our proposed biomechanical model employed biochemical factors for determining the magnitudes of the applied forces. Our model also used biomechanical factors for determining the cortical folding deformations. This combination enhances the capability of our model to explain the machanisms involving cortical folding. Our model is the first model to incorporate both biomechanical factors and biochemical features.

101 APPENDIX A

DERIVATIONS

A.1 Methods of Weighted Residual

Weighted residual methods are tools for finding approximate solutions to differential equations. The methods involve the following steps. Select an approximate solution, which is called the trial function, with unknown coefficients that will be determined. The trial solution must satisfy the boundary conditions, and it is expanded in the series. By substituting the trial function into the differential equation, compute the residual. Select a test function which is also called a weight function. Let the weighted average of the residual over the domain be zero, and then solve it for the unknown coefficients. To explain with clarity, we describe the methods of weighted residuals with a simple example provided by [81] as follows: d2u u = x, 0

Step 1: select a trial function which satisﬁes the boundary conditions in (A.1) and contains • unknown coeﬃcients that will be determined later. We choose

u = ax(1 x) . (A.2) − The chosen trial function (A.2) satisﬁes the boundary conditions u(0) = u(1) = 0 and it includes the unknown coeﬃcient a.

Step 2: substitute the chosen trial function (A.2) into the diﬀerential equation in (A.1). Then, • the residual R becomes d2u R = u + x = 2a ax(1 x)+ x. (A.3) dx2 − − − −

Step 3: select a test (or weight) function ω. The methods of weighted residual are classiﬁed • according to how the test function is determined [81]. For example, the least square method determines the test function ω from the residual such that ω = dR = 2 x(1 x). The da − − −

102 Galerkin method chooses the test function ω from the chosen trial function u such that ω = du = x(1 x). da − Step 4: Set the weighted average of the residual R over the domain equal to zero as follows: • 1 1 d2u 1 I = ωR dx = ω u + x dx = ω 2a ax(1 x)+ x dx = 0 . (A.4) dx2 − {− − − } Z0 Z0 Z0 Step 5: Substitute the test function ω determined in Step 3 into Equation (A.4) to solve for • the unknown coefficient a. When we use the least squares method, it results in a = 0.2305. Thus, the trial function (A.2) becomes u = 0.2305x(1 x). When we use the Galerkin method, − the coefficient becomes a = 0.2272, then u = 0.2272x(1 x). − The exact solution to the differential equation (A.1) at x = 0.5 is u = 0.0566 [81]. The approximate solutions at x = 0.5 are u = 0.0576 and u = 0.0568 using the least squares method and the Galerkin method, respectively.

A.2 Linear Quadrilateral Element

Linear quadrilateral element has one node at each corner of the element which is a four-sided polygon. Mapping between the physical and natural coordinate systems simpliﬁes the shape (or basis) functions Ni. In the natural coordinate system (ξ,η) (see Figure A.1), the four shape functions of the linear quadrilateral element can be expressed as 1 1 N = (1 ξ)(1 η) , N = (1 + ξ)(1 η) , 1 4 − − 2 4 − 1 1 N = (1 + ξ)(1 + η) , N = (1 ξ)(1 + η) . 3 4 4 4 − 4 At any point inside and on the element, i=1 Ni = 1. The shapes of the bilinear interpolation functions Ni are twisted planes whose heightP is 1 at i-th corner of the element and 0 at the other corners (see Figure 3.7). The following steps will provide step by step directions on how to compute the Jacobian for the change of variables from the physical coordinate to the natural coordinate. In addition, we describe how to express diﬀerentiation and integration in terms of (ξ,η) using the Jacobian matrix. According to the chain rule, ∂N(ξ,η) ∂N(ξ,η) ∂x ∂N(ξ,η) ∂y = + , ∂ξ ∂x ∂ξ ∂y ∂ξ (A.5) ∂N(ξ,η) ∂N(ξ,η) ∂x ∂N(ξ,η) ∂y = + . ∂η ∂x ∂η ∂y ∂η

103 Figure A.1: Mapping between coordinate systems. (a) Quadrilateral element in physical coordinate system. (b) Square element in natural coordinate system. Figure from [124]

Equation (A.5) can be rewritten as

∂N ∂x ∂y ∂N ∂N ∂ξ = ∂ξ ∂ξ ∂x [J] ∂x , (A.6) ∂N ∂x ∂y ∂N ≡ ∂N ( ∂η ) " ∂η ∂η # ∂y ∂y where [J] is a Jacobian transformation matrix. Multiplying both sides of Equation (A.6) by the inverse of the Jacobian matrix yields

∂N ∂N ∂x −1 ∂ξ ∂N =[J] ∂N . (A.7) ∂y ( ∂η )

Recall that the bilinear interpolation functions Ni are written in terms of (ξ,η) in the natural coordinate system as follows: 1 N = (1 + ξ ξ)(1+ η η) , i 4 i i where Ni = 1 at the corner (ξi,ηi). Thus, the partial derivatives of Ni with respect to one of ξ and η are linear functions [80]. The partial derivatives of x and y with respect to ξ or η, which are in the Jacobian matrix

[J], can be obtained from the partial derivatives of Ni with respect to ξ or η. The x and y in the physical coordinate system are interpolated using the basis functions Ψi as follows: 4 4 x = xiΨi , y = yiΨi , i i X=1 X=1 and the basis functions Ψi are isoparametric to the shape functions Ni. Thus,

4 4 4 4 ∂x ∂ i=1 xiΨi ∂ i=1 xiNi ∂ (x N ) ∂N = = = i i = x i , ∂ξ ∂ξ ∂ξ ∂ξ i ∂ξ P P i i X=1 X=1

104 and in the same manner,

4 4 4 ∂x ∂N ∂y ∂N ∂y ∂N = x i , = y i , = y i . ∂η i ∂η ∂ξ i ∂ξ ∂η i ∂η i i i X=1 X=1 X=1 Therefore, 4 ∂Ni 4 ∂Ni xi yi [J]= i=1 ∂ξ i=1 ∂ξ , (A.8) 4 x ∂Ni 4 y ∂Ni " Pi=1 i ∂η Pi=1 i ∂η # P P a a and all entries of the matrix are constants. Recall that the inverse of a matrix [A]= 11 12 a a 21 22 is

1 a a [A]−1 = 22 − 12 , A a21 a11 | | − where A = a a a a is the determinant. | | 11 22 − 12 21 Since the partial derivatives of Ni with respect to ξ or η are linear functions, the partial derivatives of x and y with respect to ξ or η become also linear functions. We can express the partial derivatives of N with respect to x and y in Equation (A.7) in terms of (ξ,η) by using the interpolation. On the other hand, the integrations in the two coordinate systems are as follows:

1 1 dxdy = J dξdη , | | Z ZΩe Z−1 Z−1 where Ωe is the region of the element in the physical coordinate and J is the determinant of [J]. | |

A.3 Gauss-Legendre Quadrature Rule

The Gauss-Legendre quadrature rule is used to compute an approximate solution to a definite integral. Conventionally, the rule is applied after transforming the domain of integration into [ 1, 1]. − 1 n n f(x)dx = lim ωif(xi) ωif(xi) , n→∞ ≈ −1 i i Z X=1 X=1 where ωi are weights and xi [ 1, 1] are the specified points [125]. ∈ − The specified points xi, which are used in the n-point Gauss-Legendre quadrature, are the roots of the Legendre polynomial of order n, Pn(x) [126]. The first six Legendre polynomials are are illustrated in Figure A.2. The points xi and the associate weights ωi of the n-point Gauss-Legendre quadrature rule are given in Table A.1.

105 Table A.1: n-point Gauss-Legendre quadrature weights and points. Data from [126, 127].

n xi ωi 2 1 √3 1 ± 3 0 8 3 9 1 √15 5 ± 5 9 1 525 70√30 1 (18 + √30) 4 ± 35 − 36 1 525 + 70√30 1 (18 √30) ± 35 p 36 − 0 128 p 225 5 1 245 14√70 1 (322 + 13√70) ± 21 − 900 1 245 + 14√70 1 (322 13√70) ± 21 p 900 − p

Figure A.2: Legendre polynomials. Figure from [128].

A.4 Deriving Turing Conditions

Four Turing conditions are driven through linear stability analysis. These are mathematical equations guaranteeing satisfaction of the two Turing criteria described in Section 3.3.1. Each Turing criterion is represented by two mathematical Turing conditions.

A.4.1 Turing Criterion: Linear Stability in the Absence of Diﬀusion

In the absence of diﬀusion, the reaction-diﬀusion system (3.8) becomes

∂u = p(u,v) , ∂t (A.9) ∂v = q(u,v) , ∂t

106 where p and q are nonlinear functions representing the reaction kinetics. We assume the system has a steady state (u0,v0), which is the solution to

0= p(u0,v0) ,

0= q(u0,v0) .

Deﬁne w(x,t) to be a small perturbation from the steady state (u0,v0) such that

u(x,t) u ǫu w(x,t)= − 0 = (A.10) v(x,t) v ǫv − 0 with 0 < ǫu , ǫv 1. Through (A.10), (A.9) can be rewritten as | | | | ≪

ut p(u,v) p(u0 + ǫu,v0 + ǫv) wt = = = . v q(u,v) q(u + ǫ ,v + ǫ ) t 0 u 0 v A Taylor series expansion is performed to linearize the nonlinear kinetic functions p and q as follows: 2 ut = p(u ,v )+ ǫupu(u ,v )+ ǫvpv(u ,v )+ (ǫ ) , 0 0 0 0 0 0 O and similarly for vt. The ﬁrst terms p(u0,v0) and q(u0,v0) in the ut and vt expansions, respectively, will be zeros since they are at the steady state. We can also ignore (ǫ2) and higher terms. Thus, O ut ǫupu(u ,v )+ ǫvpv(u ,v ) , ≈ 0 0 0 0

vt ǫuqu(u ,v )+ ǫvqv(u ,v ) . ≈ 0 0 0 0 Therefore it can be written as

wt = Aw , (A.11) where p p A = u v . qu qv (u0,v0) Solutions to (A.11) can be assumed to be proportional to the exponential function eλt due to the form, where λ will be determined. Since we assumed that the Turing system is linearly stable at the steady state (u0,v0) in the absence of diffusion, the perturbation w(t) must approach zero as t goes to infinite. It occurs when Re(λ) < 0. To find such λ, substitute w = ceλt into (A.11), where c is a vector of constants. By dividing by eλt, we obtain the eigenvalue equation λc = Ac. The characteristic polynomial of A is

λ pu pv det(λI A)= − − = λ2 λ tr A + det A = 0, (A.12) − qu λ qv − − −

107 where tr A = pu + qv and det A = puqv pvqu. Solving for λ using the quadratic formula yields −

1 2 λ , = tr A (tr A) 4det A . 1 2 2 ± − p To guarantee Re(λ) < 0, the following two inequalities must be satisﬁed.

tr A = pu + qv < 0 and (A.13a)

det A = puqv pvqu > 0 . (A.13b) − When these two conditions (A.13a) and (A.13b), which constitute the first two of the four mathematical Turing conditions, are satisfied, it is guaranteed that the reaction-diffusion system (3.8) is linearly stable in the absence of diffusion.

A.4.2 Turing Criterion: Diﬀusion-Driven Instability

When diﬀusion is present, linearizing the reaction-diﬀusion system (3.8) near the steady state

(u0,v0) using the small perturbation w deﬁned in (A.10) derives

2 wt = dM δ w + Aw , (A.14) ∇ where d 0 d = , M 0 1 and remind that d = du/dv (0, 1) is the ratio of diffusion coefficients, δ is the positive constant ∈ that is inversely proportional to the domain scale, and the matrix A is defined in (A.11) which is comprised of the first derivatives of the kinetic functions at (u0,v0). Using a method of separation of variables, we can brake the solution w to (A.14) into a set of spatial and temporal equations such that w(x,t) = X(x)T(t). The temporal solution is obtained of the form T(t)= eλt through Section A.4.1. A method of obtaining the spatial solution form will be described in the following paragraph. The final solution can be written as

λt w(x,t)= ckXk(x)e , (A.15) Xk where ck are Fourier coefficients that will be determined according to given initial conditions. To find the spatial solution form, substitute w = Xeλt into (A.14) and divide by eλt. Then, it is simplified to an equation having only a spatial variable as follows: 1 2X + d−1(A λI)X = 0 , (A.16) ∇ δ M −

108 which is the form of the Helmholtz equation 2X + k2X = 0 where ∇ 1 k2I = d−1(A λI) . δ M −

2 2 Substituting (A.15) into (A.14) and applying Xk = k Xk yields the following homogenous ∇ − system 2 ck λXk AXk + δdM k Xk = 0. (A.17) − k X Nontrivial solutions for Xk in (A.17) will be obtained when ck = 0 and 6

2 λI A + δdM k Xk = 0 . − It directs to the characteristic equation,

2 det λI A + δdM k = 0 , − which can be expanded into

λ2 + (1 + d)δk2 tr A λ + h(k2) = 0 , (A.18) − where 2 2 4 2 h(k )= dδ k (pu + dqv)δk + det A. (A.19) − When k2 = 0, (A.18) is reduced to (A.12) which is the characteristic polynomial in the case assuming absence of diﬀusion. We have already driven the Turing conditions, (A.13a) and (A.13b), that guarantee satisfaction of Re(λ(k2 = 0)) < 0 required for a stable steady state. For the case of diﬀusion-driven instability, mathematical conditions that guarantee satisfaction of Re(λ(k2 > 0)) > 0 are required. Solving (A.18) for λ provides

1 2 1 2 2 2 λ , = (1 + d)δk tr A [(1 + d)δk tr A] 4h(k ). 1 2 −2 − ± 2 − − q Since tr A< 0 from the condition (A.13a), (1 + d)δk2 tr A is always positive. Thus, h(k2) < 0 − is the only way to obtain Re(λ(k2)) > 0. 2 To leave open the possibility of h(k ) < 0, (pu + dqv) must be positive because det A> 0 from the condition (A.13b) and all coeﬃcients are positive as well as k2 > 0 in (A.19). It is only a necessary condition.

109 2 Since h(k ) is concave upward parabola, the minimum hmin must be negative to guarantee h(k2) < 0 for some k. Using technique ﬁnding the vertex of the parabola using diﬀerentiation, we obtain 2 (pu + dqv) hmin = det A , − 4d when (p + dq ) k2 = u v . min 2dδ

Thus, hmin < 0 when (p + dq )2 det A< u v . 4d Therefore, to guarantee Re(λ(k2)) > 0, both of the necessary condition and the condition required for the negative minimum value must be satisﬁed:

pu + dqv > 0 and (A.20a) (p + dq )2 det A< u v . (A.20b) 4d

These two conditions (A.20a) and (A.20b) constitute the remaining two mathematical Turing conditions, which are the constraints for diﬀusion-driven instability.

110 APPENDIX B

FIGURES

In this Appendix, we present ﬁgures that were discussed but not displayed in Chapter 4 and Section

6.2.1. Section B.1 contains simulation results changing gray matter radius r2 or cortical thickness thick. Section B.2 contains simulation results changing the constant CTR. Section B.3 contains simulation results changing Cu. Lastly, some numerical results of the ongoing work for the three- dimensional model are presneted in Section B.4.

111 B.1 Domain Size B.1.1 Bigger vs. Smaller Brains

r −r = 2.5 mm, C = 0.99, n0=3 2 1 TR (a) (b) 40 20 30

20 10 10

0 0 −20 −10 0 10 20 −20 0 20 r = 18.351 mm, GI = 1.0113 r = 31.303 mm, GI = 1.0578 2 2

40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 r = 40.464 mm, GI = 1.1213 r = 42.966 mm, GI = 1.1418 2 2

(e) (f) 60 60

40 40

20 20

0 0 −50 0 50 −50 0 50 r = 48.88 mm, GI = 1.1936 r = 51.088 mm, GI = 1.2138 2 2

Figure B.1: Changing gray matter radius r2 when n0 = 3. The radius of gray matter is taken at (a) 11 weeks GA (r2 = 18.351 mm), (b) 21 weeks GA (r2 = 31.303 mm), (c) 28 weeks GA (r2 = 40.464 mm), (d) 30 weeks GA (r2 = 42.966 mm), (e) 35 weeks GA (r2 = 48.880 mm), and (f) 37 weeks GA (r2 = 51.088 mm) [4,68]. All other applied forces and parameters are as in Figure 4.1a. The average of the magnitudes of the six applied forces is 1.9381 N 2.4032%. The GI increases as values of r increase. ± 2

112 r −r = 2.5 mm, C = 0.99, n0=10 2 1 TR (a) (b) 40 20 30 15 20 10 5 10 0 0 −20 −10 0 10 20 −20 0 20 r = 18.351 mm, GI = 1.0286 r = 31.303 mm, GI = 1.0623 2 2

40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 r = 40.464 mm, GI = 1.0919 r = 42.966 mm, GI = 1.1004 2 2

(e) 60 (f) 60

40 40

20 20

0 0 −50 0 50 −50 0 50 r = 48.88 mm, GI = 1.121 r = 51.088 mm, GI = 1.1288 2 2

Figure B.2: Changing gray matter radius r2 when n0 = 10. The radius of gray matter is taken at (a) 11 weeks GA (r2 = 18.351 mm), (b) 21 weeks GA (r2 = 31.303 mm), (c) 28 weeks GA (r2 = 40.464 mm), (d) 30 weeks GA (r2 = 42.966 mm), (e) 35 weeks GA (r2 = 48.880 mm), and (f) 37 weeks GA (r2 = 51.088 mm) [4, 68]. All other applied forces and parameters are as in Figure 4.1c. The average of the magnitudes of the twenty applied forces is 6.3069 N 0.0521%. The GI increases as values of r increase. ± 2

113 r −r = 2.5 mm, C = 0.99, n0=3 2 1 TR 70 (a) 60 (b) (c) 50 (f)

0 −60 −40 −20 0 20 40 60

Figure B.3: Eﬀects of changing gray matter radius r2 when n0 = 3. Figures B.1a–c, f are displayed together to illustrate the eﬀects of increasing r2.

r −r = 2.5 mm, C = 0.99, n0=10 2 1 TR 60 (a) (b) 50 (c) (f) 40

0 −60 −40 −20 0 20 40 60

Figure B.4: Eﬀects of changing gray matter radius r2 when n0 = 10. Figures B.2a–c, f are displayed together to illustrate the eﬀects of increasing r2.

114 B.1.2 Thicker vs. Thinner Cortices

r = 40.464 mm, C = 0.99, n0=3 2 TR (a) 60 (b) 60

40 40

20 20

0 0 −50 0 50 −50 0 50 r −r = 2 mm, GI = 1.2902 r −r = 2.5 mm, GI = 1.1213 2 1 2 1

40 40

20 20

0 0 −50 0 50 −50 0 50 r −r = 2.7 mm, GI = 1.0836 r −r = 3 mm, GI = 1.0474 2 1 2 1

(e) 60 (f) 60

40 40

20 20

0 0 −50 0 50 −50 0 50 r −r = 3.5 mm, GI = 1.0191 r −r = 4.5 mm, GI = 1.0076 2 1 2 1

Figure B.5: Changing cortical thickness, thick = r r , when n0 = 3. Cortical thickness 2 − 1 values are from [25] and are (a) thick = 2.0 mm, (b) thick = 2.5 mm (overall average), (c) thick = 2.7 mm (medial cortex), (d) thick = 3.0 mm (inferior cortex), (e) thick = 3.5 mm (lateral cortex), and (f) thick = 4.5 mm. All other applied forces and parameters are as in Figure 4.1a. The average of the magnitudes of the six applied forces is 1.9381 N 2.4032%. The GI decreases exponentially with the increasing values of thick. ±

115 r = 40.464 mm, C = 0.99, n0=10 2 TR (a) (b) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 r −r = 2 mm, GI = 1.1866 r −r = 2.5 mm, GI = 1.0919 2 1 2 1

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 r −r = 2.7 mm, GI = 1.0695 r −r = 3 mm, GI = 1.0464 2 1 2 1

(e) (f) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 r −r = 3.5 mm, GI = 1.0246 r −r = 4.5 mm, GI = 1.0079 2 1 2 1

Figure B.6: Changing cortical thickness, thick = r r , when n0 = 10. Cortical thickness 2 − 1 values are from [25] and are (a) thick = 2.0 mm, (b) thick = 2.5 mm (overall average), (c) thick = 2.7 mm (medial cortex), (d) thick = 3.0 mm (inferior cortex), (e) thick = 3.5 mm (lateral cortex), and (f) thick = 4.5 mm. All other applied forces and parameters are as in Figure 4.1c. The average of the magnitudes of the twenty applied forces is 6.3069 N 0.0521%. The GI decreases exponentially with the increasing values of thick. ±

116 B.2 Directions of the Applied Axonal Tension Forces

r = 40.464 mm, r −r = 2.5 mm 2 2 1 (a) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.9 GI = 1.0929 TR (b) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.6 GI = 1.059 TR (c) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.3 GI = 1.0579 TR (d) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.1 GI = 1.0761 TR

Figure B.7: Changing the directions of the applied forces when n0 = 5 in Case 1. The set of ten forces is applied on the semi-circular domain whose radius of gray matter is 40.464 mm and cortical thickness is 2.5 mm (left panel). The directions of the forces are determined by the constant CTR [0, 1]. (a) CTR = 0.9, (b) CTR = 0.6, (c) CTR = 0.3, ∈ and (d) CTR = 0.1. As CTR decreases, the number of sulci increases and the overall shape of the deformed configuration becomes flatter (right panel). See Table 4.3 for additional results using different values of CTR. The other parameters as well as the magnitudes of the applied forces are as in Figure 4.1b.

117 r = 40.464 mm, r −r = 3.5 mm 2 2 1 (a) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.9 GI = 1.0135 TR (b) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.6 GI = 1.0056 TR (c) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.3 GI = 1.011 TR (d) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.1 GI = 1.0228 TR

Figure B.8: Changing the directions of the applied forces when n0 = 3 in Case 2. The set of the six forces is applied on the semi-circular domain whose radius of gray matter is 40.464 mm and cortical thickness is 3.5 mm (left panel). The directions of the forces are determined by the constant CTR [0, 1]. (a) CTR = 0.9, (b) CTR = 0.6, (c) CTR = 0.3, ∈ and (d) CTR = 0.1. As CTR decreases, the number of sulci increases and the overall shape of the deformed configuration becomes flatter (right panel). See Table 4.2 for additional results using different values of CTR. The other parameters as well as the magnitudes of the applied forces are as in Figure 4.1a.

118 r = 51.088 mm, r −r = 2.5 mm 2 2 1 (a) 60 60 40 40 20 20 0 0 −50 0 50 −50 0 50 C = 0.9 GI = 1.1819 TR

(b) 60 60 40 40 20 20 0 0 −50 0 50 −50 0 50 C = 0.6 GI = 1.1037 TR

(d) 60 60 40 40 20 20 0 0 −50 0 50 −50 0 50 C = 0.1 GI = 1.1329 TR

Figure B.9: Changing the directions of the applied forces when n0 = 3 in Case 3. The set of the six forces is applied on the semi-circular domain whose radius of gray matter is 51.088 mm and cortical thickness is 2.5 mm (left panel). The directions of the forces are determined by the constant CTR [0, 1]. (a) CTR = 0.9, (b) CTR = 0.6, (c) CTR = 0.3, ∈ and (d) CTR = 0.1. As CTR decreases, the number of sulci increases and the overall shape of the deformed configuration becomes flatter (right panel). See Table 4.2 for additional results using different values of CTR. The other parameters as well as the magnitudes of the applied forces are as in Figure 4.1a.

119 r = 40.464 mm, r −r = 3.5 mm 2 2 1 (a) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.9 GI = 1.0135 TR (b) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.6 GI = 1.0089 TR (c) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.3 GI = 1.0136 TR (d) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.1 GI = 1.0217 TR

Figure B.10: Changing the directions of the applied forces when n0 = 5 in Case 2. The set of the ten forces is applied on the semi-circular domain whose radius of gray matter is 40.464 mm and cortical thickness is 3.5 mm (left panel). The directions of the forces are determined by the constant CTR [0, 1]. (a) CTR = 0.9, (b) CTR = 0.6, (c) CTR = 0.3, ∈ and (d) CTR = 0.1. As CTR decreases, the number of sulci increases and the overall shape of the deformed configuration becomes flatter (right panel). See Table 4.3 for additional results using different values of CTR. The other parameters as well as the magnitudes of the applied forces are as in Figure 4.1b.

120 r = 51.088 mm, r −r = 2.5 mm 2 2 1 (a) 60 60 40 40

20 20

0 0 −50 0 50 −50 0 50 C = 0.9 GI = 1.1561 TR (b) 60 60 40 40

20 20

0 0 −50 0 50 −50 0 50 C = 0.6 GI = 1.1016 TR (c) 60 60 40 40

20 20

0 0 −50 0 50 −50 0 50 C = 0.3 GI = 1.1011 TR (d) 60 60 40 40

20 20

0 0 −50 0 50 −50 0 50 C = 0.1 GI = 1.1302 TR

Figure B.11: Changing the directions of the applied forces when n0 = 5 in Case 3. The set of the ten forces is applied on the semi-circular domain whose radius of gray matter is 51.088 mm and cortical thickness is 2.5 mm (left panel). The directions of the forces are determined by the constant CTR [0, 1]. (a) CTR = 0.9, (b) CTR = 0.6, (c) CTR = 0.3, ∈ and (d) CTR = 0.1. As CTR decreases, the number of sulci increases and the overall shape of the deformed configuration becomes flatter (right panel). See Table 4.3 for additional results using different values of CTR. The other parameters as well as the magnitudes of the applied forces are as in Figure 4.1b.

121 B.3 Strengths of the Applied Axonal Tension Forces B.3.1 Weaker vs. Stronger Pulling Tension Forces

n0 = 3, RSD = 2.4032% (a) (b) 60 60

40 40

20 20

0 0 −50 0 50 −50 0 50 C = 0.25 , AVG = 0.48453 , GI = 1.0071 C = 0.5 , AVG = 0.96907 , GI = 1.0246 fm fm

40 40

20 20

0 0 −50 0 50 −50 0 50 C = 1 , AVG = 1.9381 , GI = 1.1213 C = 1.5 , AVG = 2.9072 , GI = 1.2436 fm fm

(e) (f) 60 60

40 40

20 20

0 0 −50 0 50 −50 0 50 C = 2 , AVG = 3.8763 , GI = 1.362 C = 2.5 , AVG = 4.8453 , GI = 1.4681 fm fm

Figure B.12: Changing the magnitudes of the applied forces when n0 = 3. The magnitudes 2 of the applied forces are determined by the scaling constant Cu = 25 n0 Cfm where × × n0 = 3 and (a) Cfm = 0.25, (b) Cfm = 0.5, (c) Cfm = 1.0, (d) Cfm = 1.5, (e) Cfm = 2.0, and (f) Cfm = 2.5. The average (AVG) of the magnitudes of the applied forces increases Cfm times as much as in Figure 4.1a. The other parameters are as in Figure 4.1a. The rudimentary shallow grooves on the smooth model cortical surface become deeper and steeper as Cfm increases.

122 n0 = 10, RSD = 0.052095% (a) (b) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.25 , AVG = 1.5767 , GI = 1.0024 C = 0.5 , AVG = 3.1535 , GI = 1.0185 fm fm

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 0.75 , AVG = 4.7302 , GI = 1.0495 C = 1 , AVG = 6.3069 , GI = 1.0919 fm fm

(e) (f) 40 40

20 20

0 0 −40 −20 0 20 40 −40 −20 0 20 40 C = 1.25 , AVG = 7.8836 , GI = 1.1406 C = 1.5 , AVG = 9.4604 , GI = 1.1904 fm fm

Figure B.13: Changing the magnitudes of the applied forces when n0 = 10. The magni- 2 tudes of the applied forces are determined by the scaling constant Cu = 25 n0 Cfm × × where n0 = 10 and (a) Cfm = 0.25, (b) Cfm = 0.5, (c) Cfm = 0.75, (d) Cfm = 1.0, (e) Cfm = 1.25, and (f) Cfm = 1.5. The average (AVG) of the magnitudes of the applied forces increases Cfm times as much as in Figure 4.1c. The other parameters are as in Fig- ure 4.1c. The rudimentary shallow grooves on the smooth model cortical surface become deeper and steeper as Cfm increases.

123 B.3.2 Distance between the Applied Forces

The magnitude of each force = 4 (a) 60 (b) 60 (c) 60

40 40 40

20 20 20

0 0 0 −50 0 50 −50 0 50 −50 0 50 n0 = 3, GI = 1.3782 n0 = 5, GI = 1.151 n0 = 10, GI = 1.0334

Figure B.14: Changing distance between the applied pulling forces when f = 4.0 N. On | | the initial semi-circular domain, the arc lengths between each pair of the pulling forces are (a) 19.6805 mm when n0 = 3, (b) 11.3323 mm when n0 = 5, and (c) 5.3681 mm when n0 = 10. The magnitude of each force is 4.0 N. All parameters except the strengths of the forces are as in Figure 4.1a–c.

The magnitude of each force = 5 (a) (b) (c) 60 60 60

40 40 40

20 20 20

0 0 0 −50 0 50 −50 0 50 −50 0 50 n0 = 3, GI = 1.4855 n0 = 5, GI = 1.2339 n0 = 10, GI = 1.056

Figure B.15: Changing distance between the applied pulling forces when f = 5.0 N. On | | the initial semi-circular domain, the arc lengths between each pair of the pulling forces are (a) 19.6805 mm when n0 = 3, (b) 11.3323 mm when n0 = 5, and (c) 5.3681 mm when n0 = 10. The magnitude of each force is 5.0 N. All parameters except the strengths of the forces are as in Figure 4.1a–c.

124 B.4 Preliminary numerical results of a three-dimensional model

r = 18.351 mm, r −r = 2.5 mm, |f| = 0.04 N 2 2 1

20 20 15 15

z 10 z 10 5 5 0 0 40 40 20 20 20 20 0 0 0 −20 0 −20 y x y x

20 10 0 −10 −20 0 20 z 10 0 10

20 y 20 15

z 30 10 40

5 20 y 40 0 0 x −20 −10 0 10 20 x

Figure B.16: Diﬀerent views and skeletons of Figure 6.3a.

r = 51.088 mm, r −r = 2.5 mm, |f| = 0.04 N 2 2 1

60 60

40 40 z z 20 20

0 0 100 100 50 50 50 0 50 0 0 −50 0 −50 y x y x

0 60 40 20 0 −20 −40 −60 50 z 20 0

40 60 y 60 40 80 z

20 100 100 y 50 0 0 x −60 −40 −20 0 20 40 60 x

Figure B.17: Diﬀerent views and skeletons of Figure 6.3b.

125 r = 40.464 mm, r −r = 4.5 mm, |f| = 0.04 N 2 2 1

40 40 z z 20 20

0 0 80 50 50 50 60 0 40 0 20 0 −50 0 −50 y x y x

0 50 0 −50 50 z 0 20 50

40 y 40

30 60 z 20 80 80

60 y 10 40 20 0 0 x −50 0 50 x

Figure B.18: Diﬀerent views and skeletons of Figure 6.3c.

r = 40.464 mm, r −r = 2.5 mm, |f| = 0.024 N 2 2 1

40 40 z z 20 20

0 0 80 50 50 50 60 0 40 0 20 0 −50 0 −50 y x y x

0 50 0 −50 50 z 0 20 50

40 y 40

30 60 z 20 80 80

60 y 10 40 20 0 0 x −50 0 50 x

Figure B.19: Diﬀerent views and skeletons of Figure 6.3d.

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136 BIOGRAPHICAL SKETCH

Sarah Kim grew up in Ansan, Gyeonggi-do, South Korea. After graduating from Ansan Dongsan Christian High School in 2004, she attended Hanyang University with the highest entrance ex- amination score of the College of Science and Technology and received full tuition scholarships. Sarah graduated summa cum laude with the President’s Honor in 2008 with a Bachelor’s degree in Applied Mathematics. She continued her study at Hanyang University and earned her Master’s degree in Applied Mathematics in 2010 under the advisement of Dr. Do Wan Kim. She received the Graduate Research Fellowship from LG Corporation Yonam Foundation and full conference travel grants from National Research Foundation of Korea. While volunteering with a medical service team in 2006, Sarah pondered how she could con- tribute to improving people’s health through her area of expertise, applied mathematics. She thus decided to study abroad for biomedical mathematics at Florida State University. She also worked as a teaching assistant and an instructor with full classroom responsibility at Florida State University. She received the Pi Mu Epsilon Honor and scholarships from Southern Scholarship Foundation. She earned her Master’s degree in Biomedical Mathematics in 2012 and her doctorate in Biomedical Mathematics in 2015, both under the advisement of Dr. Monica K. Hurdal.

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