Florida State University Libraries
2017 Modeling Cortical Folding Patterns on a Growing Oblate Spheroid Domain Raymond Morie
Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY
COLLEGE OF ARTS AND SCIENCES
MODELING CORTICAL FOLDING PATTERNS ON A
GROWING OBLATE SPHEROID DOMAIN
By
RAYMOND MORIE
A Thesis submitted to the Department of Mathematics in partial fulfillment of the requirements for graduation with Honors in the Major
Degree Awarded: Spring 2017 The members of the Defense Committee approve the thesis of Raymond Morie defended on April 26th, 2017.
Monica K. Hurdal Thesis Director
Eric Klassen Committee Member
James Justus Outside Committee Member
Richard Bertram Committee Member
ii TABLE OF CONTENTS
Abstract...... iv
1 Introduction 1
2 Biology and Models of the Cerebral Cortex 3 2.1 Neuroanatomy and Development ...... 3 2.1.1 Three- and Five-Vesicle Stages of Development ...... 5 2.2 Biological Theories of Development ...... 6 2.2.1 The Radial Unit Hypothesis ...... 6 2.2.2 Intermediate Progenitor Hypothesis ...... 7 2.2.3 Intermediate Progenitor Model ...... 7 2.2.4 Axonal Tension Hypothesis ...... 8 2.3 Holoprosencephaly and Ventriculomegaly ...... 9 2.3.1 Characteristics and Development of Ventriculomegaly ...... 9 2.3.2 Characteristics and Development of Holoprosencephaly ...... 10 2.4 Mathematical Models of Cortical Folding ...... 11 2.4.1 StaticModels...... 11 2.4.2 Growing Prolate Spheroid ...... 11 2.5 Conclusions...... 12
3 Turing Systems 13 3.1 StaticDomains ...... 13 3.2 Kinetics ...... 14 3.3 Growing Oblate Spheroid Coordinates ...... 14 3.4 ExponentialGrowth ...... 15 3.5 LogisticGrowth...... 17 3.6 Conclusions...... 18
4 Results 19 4.1 Exponentially Growing Oblate Spheroid Simulations ...... 20 4.1.1 Comparison to Static and Growing Prolate Spheroid Domain ...... 20 4.2 Logistically Growing Oblate Spheroid Simulations ...... 21
5 Conclusions and Future Directions 33
Bibliography ...... 35
iii ABSTRACT
Many of the mechanisms involved in brain development are not definitively understood. This in- cludes the mechanism by which the folds of the cerebral cortex, called gyri and sulci, are formed. This research adopts the Intermediate Progenitor Model, which emphasizes genetic chemical fac- tor control. This model is based on a hypothesis that activation of radial glial cells produces intermediate progenitor cells that in turn correspond to gyral wall formation. In previous work, Toole and Hurdal developed two biomathematical models that use a Turing reaction-diffusion system on a prolate spheroid domain. These models examined the effect that different domain growth functions had on cortical folding pattern formation. In this research, we modify those models to utilize an oblate spheroid domain. By analyzing the role of various parameters, we seek to use this modified domain to model diseases such as holoprosencephaly and ventriculomegaly, which alter the shape of the lateral ventricle.
iv CHAPTER 1
INTRODUCTION
Fundamental to the human experience is our ability to discern and model patterns found in our environment. This includes modeling the various phenomena that occur in our body. Fascinated by the way a homogeneous group of cells could differentiate into the various structures of the embryo, Alan Turing developed a system of differential equations capable of producing a variety of patterns formed in nature in his 1952 paper The Chemical Basis of Morphogenesis[19]. In this thesis, we utilize a Turing system to model the folding patterns of the cerebral cortex [17]. Previous work done by Toole and Hurdal created a model of cortical folding on a growing domain. A prolate spheroid, or an ellipsoid created by rotating an ellipse about its major axis, was used as the domain to model the shape the lateral ventricle. The model investigated the role of growth on cortical pattern formation and this was applied to simulate various diseases related to cortical folding. For the purposes of this thesis, we want to further investigate the role of domain shape in cortical pattern formation by using an oblate spheroid, or an ellipse rotating about its minor axis, as our domain. This allows us to model diseases that distort the shape of the lateral ventricles and cerebral cortex. In Chapter 2, we discuss the various structures of neuroanatomy and what is known about the developmental process of the nervous system. We present a number of hypotheses that describe a plausible genetic chemically-driven mechanism for which cortical folds are formed, and a competing hypothesis that emphasizes physical tension. We also present three previous models of cortical folding to illustrate the necessity of this new model. In Chapter 3, Turing systems are discussed in more detail. We consider the effects of domain growth versus the static domain case. Oblate spheroidal coordinates were derived from previous equations utilizing prolate spheroid coordiates. Lastly, two domain growth functions, an exponential function and a logistic function, are presented. The Turing pattern generating capabilities are examined and the connections to biological systems are explained.
1 Chapter 4 presents the simulation results of the two models presented in Chapter 3 and discusses the findings with respect to the results of previous models. The parameters utilized and altered are given. Finally, Chapter 5 presents the conclusions to the investigation on domain shape on growing domains. Future directions for models utilizing Turing systems are discussed.
2 CHAPTER 2
BIOLOGY AND MODELS OF THE CEREBRAL CORTEX
In order to understand the cortical folding patterns of the cerebral cortex, it is imperative to understand the key structures that are associated with cortical development. In this section, the central nervous system, the ventricular system, the germinal matrix, brain cells and structures and different theories of cortical development will be defined and discussed. In addition, the diseases ventriculomegaly and holoprocencephaly will be considered with respect to their influence on brain size and shape.
2.1 Neuroanatomy and Development
The major features of the brain on a macroscopic level are the cerebrum, cerebellum, and brain stem. The cerebrum is separated into two hemispheres (left and right), which are connected by a structure called the corpus callosum [13]. Each hemisphere is made up of four lobes: Frontal, parietal, occipital, and temporal based on their different functions and locations associated to each lobe. The ventricular system is made up of four ventricles. The first two ventricles are called the lateral ventricles, and there is one in each hemisphere. During development the lateral ventricles are shaped approximately like a prolate spheroid, a three-dimensional shape of revolution formed by rotating an ellipse about its major axis. In maturity, these lateral ventricles are c-shaped with posterior horns that extend backwards towards the occipital lobe. These lateral ventricles are attached to a third ventricle by the interventricular formina. The third ventricle is connected to a fourth ventricle which is located in the brainstem by the cerebral aqueduct. The ventricular system produces an iron-rich fluid called cerebral spinal fluid, which brings in nutrients, removes toxins and creates a physical cushion between the brain and the skull. Lining the walls of the lateral ventricles are the ventricular zone and the subventricular zone. These are layers of proliferative cells that are involved in cortical development. Passing from the ventricular
3 Figure 2.1: Major structures of the brain, figure adapted from [20].
zone to the subventricular zone involves continuing radially outward from the inside of one of the lateral ventricles. At about seven weeks gestational age, the subventricular zone produces a structure called the germinal matrix. This term has been used in literature to refer to both the part of the subventricular zone located ventrolateral to the lateral ventricle and extending along its lateral wall [8], and as a synonym for both the ventricular zone and the subventricular zone, referring to the collection of germ cells that are precursors to neurons in the cortex [3]. For the purposes of this thesis, the former definition is used. Through magnetic resonance imaging (MRI) data and computational reconstruction of 3- dimensional images, the growth of the germinal matrix has been examined [8]. The volume of the germinal matrix is found to increase exponentially from 11 − 23 weeks gestational age, and rapidly decrease from 25-28 weeks gestational age. Though there is some discrepancy in the liter- ature on when exactly cortical folding is considered to begin, it is clear that it and other crucial events related to cortical folding development coincide with the period of exponential germinal matrix growth. The two types of cells in the central nervous system are neurons and glial cells, with glial cells accounting for 90% of the total amount. Glial cells support neuron structure and provide insulation
4 which helps messages to be transmitted more efficiently, whereas neurons encode information by electrical impulses called action potentials that are transmitted in between them. The major components of a neuron are the soma (cell body), dendrites, axon and the presynaptic terminal. The soma contains the nucleus and perform cell-sustaining operations. The dendrites act as receptors for action potentials generated by other neurons. The axon connects the soma to the dendrites of other neurons, sending action potentials which are initialized in the axon hillock. The axon hillock is where the axon and soma connect. The presynaptic terminals electronically and chemically transmit action potentials to the post-synaptic terminal in the dendrites of adjacent neurons, typically by chemicals called neurotransmitters. The type of glial cell particularly relevant to this thesis is the radial glial cell. The soma of radial glial cells is attached to the ventricular zone. Radial glial cells are named for their radial fiber shaft which extends out from the ventricular zone to the cortical plate, the developmental precursor to the cerebral cortex. The somas and radial fiber of the radial glial cells create a 1-1 correspondence between the ventricular zone and the cortical plate.
2.1.1 Three- and Five-Vesicle Stages of Development
The three- and five-vesicle stages describe the differentiation of the nervous system from a developing embryo. Early on, there are three distinct layers of embryonic cells - the endoderm, the mesoderm, and the ectoderm [6]. The cells of the endoderm develop into the guts, lungs and liver. The cells of the mesoderm are the source of the vascular system, muscles, and connective tissue. The ectoderm is the origin of the central and peripheral nervous systems. As gestation continues, the ectoderm folds in on itself forming the neural tube. This is the precursor to the spinal column. The three-vesicle stage occurs upon maturation of the neural tube. The rostral, or head, region differentiates into three vesicles: the prosencephalon, the mes- encephalon, and the rhombencephalon. The five-vesicle stage occurs when the procencephalon further differentiates into the telencephalon and the diencephalon, and the rhombencephalon dif- ferentiates into the myelencephalon and the metencephalon. It is during the five-vesicle stage that the cerebral cortex populated with neuroblasts. Our model uses the radial unit hypothesis, the intermediate progenitor hypothesis, and the intermediate progenitor model to offer an explanation for how neuroblasts populate the cortical layers.
5 Figure 2.2: Ectoderm folding in on itself to form neural tube. Figure from [13].
2.2 Biological Theories of Development
There are a number of different existing biological theories that endeavor to explain the processes of cortical folding. In this thesis, we examine four separate theories that give some of the major biological viewpoints involved in the process. These are the radial unit hypothesis, the intermediate progenitor hypothesis, the intermediate progenitor model, and the axonal tension hypothesis. The axonal tension hypothesis is in direct opposition with the intermediate progenitor model, so it is not used in the formation of our oblate spheroid model.
2.2.1 The Radial Unit Hypothesis
The radial unit hypothesis describes how the initial population of radial glial cells and the 1-1 correspondence between the ventricular zone and the cortical plate are formed [14]. It describes two stages of cellular division: a symmetric stage, where a mother radial glial cell divides into two identical daughter cells, and an asymmetric stage, where a mother radial glial cell divides into two different types of daughter cells.
6 During the symmetric stage, the surface of the ventricular zone expands. The number of radial glial cells doubles with each successive division. The number of divisions has been shown to be controlled by the expression of specific genes. In the asymmetric stage, the radial glial cell divides into an identical daughter cell and a neuroblast. The neuroblast then travels up the radial arm of the radial glial cell from the ventricular zone to the cortical plate, creating a cortical layer. With successive divisions, neuroblasts pass previous layers and travel outward to create new cortical layers. This is the process by which columnar structures, or ontogenic columns, are formed. These ontogenic columns form a 1-1 correspondence between the ventricular zone and the cortical plate.
2.2.2 Intermediate Progenitor Hypothesis
The intermediate progenitor hypothesis takes the ideas of the radial unit hypothesis one step further. It posits that the lower cortical layers and the intial population of radial glial cells are formed similiarly, but adds another stage to the formation of the outer cortical layers [12]. The hypothesis states that in the asymmetric stage of cellular divison, radial glial cells divide into an identical daughter cell and a new cell type called an intermediate progenitor (IP) cell. These IP cells then travel to the subventricular zone and either divide into two identical neuroblasts, or perform up to two symmetrical divisions into identical daughter IP cells which ultimately each divide into two neuroblasts. As in Section (2.2.1), the neuroblasts travel up the radial fibers to the cortical plate. The layers of the cortical plate stack outwardly by age, so if neuroblasts were formed at the same time, they will end up in the same cortical layer.
2.2.3 Intermediate Progenitor Model
The intermediate progenitor Model employs the intermediate progenitor hypothesis to explain the formation of gyri and sulci in the cortex [9]. The model hypothesizes that distinct regional subsections of radial glial cells are activated to produce IP cells. The distribution of these activated subsection corresponds with an uneven distribution of IP cells in the subventricular zone, which then leads to uneven distribution of neuroblasts in the outer cortical layers. The idea is that sections of activated radial glial cells adjacent to inhibited sections corresponds with gyri and sulci, respectively, at the surface of the cerebral cortex.
7 (a)
(b)
Figure 2.3: The key feature of the intermediate progenitor model is the inhomogeneous distribution of IP cells. (a) IP cells create neuroblasts which begin to populate the cortical layers. (b) Areas of activated radial glial cells adjacent to inactivated radial glial cells correlate with the hill/valley effect in the outer cortical layers. In this figure CP refers to the cortical plate, and IZ, SVZ, and VZ refer to the intermediate zone, the subventricular zone, and the ventricular zone respectively.
2.2.4 Axonal Tension Hypothesis
The axonal tension hypothesis is a contending biological explanation at odds with the interme- diate progenitor model. The hypothesis states that tension from coroticocortical connections drives cortical folding. A corticocortical connection is the where the axon of one neuron connects to the dendrite of another in the cortex. The strength of these connection determines the folding pattern. Strongly connected areas bring gyral walls together, and sulci are formed in weakly connected ar- eas. Experimental data [5] shows that axons in highly interconnected areas follow straight paths, whereas axons in weakly connected areas curved around sulci. This lends support to the axonal
8 Figure 2.4: Figure showing axial MRI of normal brain (left) vs. brain with ventriculomegaly (right). Figure adapted from [11].
tension hypothesis.
2.3 Holoprosencephaly and Ventriculomegaly
The aim of our model is to simulate diseases which alter the shape of the lateral ventricle. In par- ticular, we will discuss the diseases holoprosencephaly and ventriculomegaly. Holoprosencephaly is a spectrum of disorders characterized by the failure of the prosencephalon to differentiate, partially or completely. Ventriculomegaly in fetuses is a condtion where the lateral ventricles are larger than normal, defined when the width of the lateral ventricle atrium exceeds 10 mm during gestation. In the absence of these diseases, the shape of the lateral ventricle can be approximated with a prolate spheroid. In the presence of these diseases, it is better approximated with an oblate spheroid.
2.3.1 Characteristics and Development of Ventriculomegaly
The severity of fetal ventriculomegaly is directly correlated with how large the lateral ventricles are dilated. Mild cases are within 10-12 mm, moderate cases are between 12 mm and 15 mm, and severe cases are 15+mm. It is associated with numerous other central nervous system disorders depending on the origins of the disease. The origins can be split into three categories: obstruction, dysgenesis, and destructive [15]. Obstructive cases are characterized by excess cerebral spinal fluid in the lateral ventricles, causing them to swell. Some examples of this are a Chiari II malformation and a Dandy-Walker malformation. Ventriculomegaly stemming from dysgenesis is associated with maldevelopment of cortical structures, irregular shape and orientation of the lateral ventricle, and a ”spectrum of
9 (a) (b)
Figure 2.5: Axial (a) and Sagittal (b) views of a brain with semilobar holoprosencephaly. Figure from [16].
absent midline structures.” In this case, ventriculomegaly may occur simultaneously with holopros- encephaly, absence of the corpus callosum, and/or septo-optic dysplasia. Destructive cases occur when an otherwise healthy brain is damaged in utero- so generally all structures are present [15].
2.3.2 Characteristics and Development of Holoprosencephaly
Holoprosencephaly is a disorder also has three defined levels of severity [7]. They are called lo- bar, semilobar, and alobar. Lobar holoprosencephaly, unlike the other two forms, has separation of the brain into hemispheres. The corpus callosum is partially to completely developed, and thalami are separated. In semilobar (moderate) holoprosencephaly, there is incomplete separation of the brain into left and right hemispheres, partially fused thalami, a monoventricle (an underdeveloped and fused lateral ventricle), and an absent or underdeveloped corpus callosum. Alobar holoprosen- cephaly is the most severe form, it is characterized by a monoventricle, completely fused thalami, absent corpus callosum and third ventricles, no olfactory tract and a potentially missing or fused optic tract. There are also severe facial malformations absent in milder forms of the disease. Direct causes of holoprosencephaly are currently unknown. However, there are a number of associations including four known mutations in genes that have been associated with the disease. SHH, SIX3, TGIF, and ZIC2. Other associations are trisomy 13, trisomy 18, and deletion of chromosomes 2q and 7q. Figure 2.5 shows MR images of a semilobar case of holoprosencephaly.
10 2.4 Mathematical Models of Cortical Folding
To illustrate the necessity of our new model, this section describes three previous models of cortical folding. All of these models assume the intermediate progenitor Model, previously discussed in section 2.2.3. The models can be differentiated from one another by either domain shape or the presence of growth in the model.
2.4.1 Static Models
A static prolate spheroid model [17] and a static oblate spheroid model [4] were developed using a two equation Turing System given by 2 2 ut = D∇ u + ω(u + av − Cuv − uv ), (2.1) 2 2 vt = ∇ v + ω(bv + hu − Cuv + uv ) with u,v as concentrations of the activator and inhibitor, D = Du/Dv is the ratio of the morphogen diffusion coefficients, a, b, C, h representing Barrio-Varrea-Maini kinetics parameters which have been nondimensionalized to match System 3.3 [18] [1], and ω > 0 is the domain scale parameter. The patterns generated by the Turing system represented areas of activation and inhibition on the spheroidal domain. By the intermediate progenitor model, these areas of activation and inhibition correspond with gyri and sulci formation, respectively, and therefore represent the cortical folding pattern. The key result from these two models with respect to this thesis is that patterns generated on an oblate spheroid domain were shown to have no similarity to the patterns on a prolate spheroid domain [4].
2.4.2 Growing Prolate Spheroid
Toole and Hurdal developed a framework to implement a domain growth in a Turing system on a prolate spheroidal domain to better reflect the nature of cortex development [18]. The equations and conditions used will be discussed in more detail in Chapter 3. In his dissertaion, Toole demonstrated that domain size was the most important factor in determining the complexity of cortical pattern formation. In addition, it was found that transiency of the pattern (i.e. the pattern not converging to a final state) was driven by rapidly increasing the growth rate. Using this model, many diseases related to cortical folding were able to be modeled, including polymicrogyria and lissencephaly. However, this model cannot be used to model diseases that alter the shape of a normal lateral ventricle. This is the goal of our new model on an oblate spheroid domain.
11 2.5 Conclusions
In this chapter, basic neuroanatomy and development relevant to the model was presented. Holoprosencephaly and ventriculomegaly were discussed with respect to their effect on domain shape. The radial unit hypothesis, intermediate progenitor hypothesis, intermediate progenitor model and their proposed mechanism of cortical folding were outlined. The axonal tension hypoth- esis was discussed as a competing mechanism. In addition, three previous mathematical models of cortical folding and their differences with respect to domain shape and growth were discussed. These demonstrate the need of a model on a growing oblate spheroid domain, which is the subject of Chapter 3.
12 CHAPTER 3
TURING SYSTEMS
A Turing System is a reaction-diffusion system with a distinct ability to produce spatially in- homogenous patterns from a uniform steady state. These patterns are represented by chemical concentration gradients of two morphogens, an activator and an inhibitor. The equations of the system describe their interaction as they diffuse across the domain. Turing systems have been used to model a number of different biological phenomena, such the density of hair follicles developing of mice [10] and patterns on fish skin [1]. In this chapter, following the results in Toole’s dissertation, we show the differences between Turing systems on static and growing domains. We then determine the conditions when growing domains exhibit Turing behavior, the types of reaction kinetics used, and incorporate these equations on a new oblate spheroidal domain. Two growth functions and the effect they have on the Turing system are discussed.
3.1 Static Domains
A generalized Turing reaction diffusing system is given by ∂u = D∇2u + ωf(u,v), ∂t (3.1) ∂v = ∇2v + ωg(u,v), ∂t Where u, v are concentrations of the activator and inhibitor, respectively. We let D = Du ∈ (0, 1) Dv denote the ratio of diffusion coefficients where 0
13 3.2 Kinetics
The functions f,g must be nonlinear to properly reflect the biological systems. The type of reaction kinetics used in this model are Barrio-Varea-Maini (BVM) kinetics [1]. This particular type of reaction kinetics is referred to as phenomenological kinetics, which are utilized to recreate patterns when the true reaction kinetics remain to be elucidated. Let a,b,C,h denote the kinetics parameters. The kinetic functions in this thesis are then given by
f(u,v)= u + av − Cuv − uv2,
g(u,v)= bv + hu − Cuv + uv2,
These kinetics are designed to produce striped patterns with C = 0 and spotted patterns with C > 0.
3.3 Growing Oblate Spheroid Coordinates
The main goals of this thesis are to investigate the role of domain shape and growth on cortical pattern formation. To do this, we alter a Turing system previously used to simulate patterns on a prolate spheroid domain to use an oblate spheroid domain. An oblate spheroid is a surface of revolution created by rotating an ellipse about its minor axis. f − 2 2 f − 2 2 The coordinate system is given by x = 2 (1 η )(ξ + 1) cos φ, y = 2 (1 η )(ξ + 1) sin φ, z = f p ∈ ∞ p 2 ηξ, where f is the interfocal distance, ξ = sinh µ, µ (0, ) is the radial term, η = cos θ where η ∈ [−1, 1] and φ ∈ [0, 2π) is the azimuthal angle. To aid the computational implementation, we ∈ φ replace φ with ζ [0, 1) where ζ = 2π . Overall, a position vector X on a growing oblate spheroid is defined as f0 − 2 2 2 (1 η )(ξ + 1) cos 2πζ X f0 − 2 2 (ζ,η,t)= ρ(t) 2 p(1 η )(ξ + 1) sin 2πζ , (3.2) p f0 ηξ 2 where ρ(t) is the function that defines the growth rate and f0 is the initial focal distance at t = 0. The implementation of a growth function on a Turing System was previously done by Toole [18] on a prolate spheroid domain. This thesis modifies the equations used to create an oblate spheroid domain using the transformation ξ → ıξ and f →−ıf. The equations used were
∂u D ρ˙ 2 = ∆†u − 2 u + ω(u + av − Cuv − uv ), ∂t ρ2 ρ (3.3) ∂v 1 ρ˙ = ∆ v − 2 v + ω(bv + hu − Cuv + uv2), ∂t ρ2 † ρ 14 ρ˙ The main difference from the static case is the presence of a dilution term, −2 ρ φ for φ ∈ u,v, ρ˙ which alters the chemical concentration gradients with respect to the growth function ρ(t). ρ is defined as the ratio of the first derivative of the growth function ρ at t = 0 with itself. On an oblate spheroid domain,
1 4(1 − η2) 4η(2ξ2 + η2 + 1) − ∆†φ = 2 2 2 2 φζζ + 2 2 2 φηη 2 2 2 2 φη. (3.4) π f0 (1 − η )(ξ + 1) f0 (ξ + η ) f0 (ξ + η )
Figure 3.1: Oblate Spheroid Coordinates. An ellipse cross-section parametrized by (ξ,η) with constant φ. Figure from [2].
3.4 Exponential Growth
The first domain growth function incorporated into the Turing System is the exponential growth function given by
ρ(t)= eRt where R> 0 represents the growth rate of the domain and t ≥ 0 represents time. The use of the exponential growth function has a variety of supporting evidence [18]. The subventricular zone, as previously stated in Chapter 2, is the place from which IP cells self-amplify, which corresponds with gyral wall formation. This zone produces the germinal matrix, which grows
15 exponentially during the time of IP cell self-amplification [8]. In addition, the cerebral hemispheres are shown to grow exponentially from weeks 8-13 of gestation. With exponential growth, the dilution term is given by ρ˙ −2 φ = −2Rφ (3.5) ρ so system (3.3) becomes
∂u D 2 = ∆†u − 2Ru + ω(u + av − Cuv − uv ), ∂t ρ2 (3.6) ∂v 1 = ∆ v − 2Rv + ω(bv + hu − Cuv + uv2). ∂t ρ2 † This is identical to the static case when R = 0. Linear stability analysis was performed by Toole [18] on these equations to determine when the parameters meet the Turing Criteria. These conditions were determined to be the following:
ω(fu + gv) − 4R< 0, 2 − 2 ω (fugv fvgu) + 4R > 0, 2R(1 + D) − ω(fu + Dgv) < 0, (3.7) 2 2 (1 + D) 2 R 4 − + ω (fugv − fvgu) D 1 ω2 − 2 +Rω (1 + D)(fu + Dgv) 2(fu + gv) < (fu + Dgv) . D 4D The first two conditions satisfy the Turing criteria of stability in the absence of diffusion, and the latter two satisfy the conditions of instability in the presence of diffusion. Our nondimensionalized
BVM kinetics have the following partial derivatives evaluated at the steady state (u0,v0) = (0, 0):
2R fu = 1, fv = a, gu = ω − 1, and gv = b,
By substituting these values into System (3.7) we obtain ω(1 + b) − 4R< 0, ω2(a + b) − 2Rω(1 + a + b) + 4R2 > 0, − 2R(1 + D) ω(1 + Db) < 0, (3.8) 1 ω2(a + b)+ R2 2 − − D D 2 1 ω 2 +Rω − b − 1+ Db − 2a < (1 + Db) . D 4D 16 These conditions can be applied to our oblate spheroid domain because the Helmholtz equation, which gives the spatial components of the Turing system, is separable in oblate spheroid coordinates [18].
3.5 Logistic Growth
The second growth function implemented into our Turing system is the logistic growth function. Logisitic growth is more biologically accurate in that growth stops asymptotically, as the lateral ventricle and the rest of the brain eventually stop growing. The function was constructed to match the starting and ending sizes of the exponential growth function and to reflect biological data. A typical logistic growth function is given by
K ρ(t)= , (3.9) 1+ e−r(t−t0) with t ≥ 0, K represents the asymptotic value of the function as t → ∞, r is the logistic growth rate (distinct from the exponential growth rate R) and t0 lets us shift the graph of the function horizontally. Parameter selection was based off of germinal matrix volume data by Kinoshita et al [8]. To faciliate comparison with the exponential growth function, let K∗ = eRt − 1, and vertically shift the graph of the logistic function by 1. This scales the equation such that ρlog(0) ≈ ρexp(0) and ρlog(tfinal) ≈ ρexp(tfinal). Then 3.9 becomes
K∗ ρlog(t)= + 1. (3.10) 1+ e−r(t−t0)
As with the exponential growth function, implementing ρlog(t) into our Turing System intro- duces a dilution term. For logistic growth,
ρ˙ r(ρ − 1) ρ − 1 −2 φ = −2 1 − φ = −2L(t)φ. (3.11) ρ ρ K∗
So System (3.3) becomes
∂u D 2 = ∆†u − 2L(t)u + ω(u + av − Cuv − uv ), ∂t ρ2 (3.12) ∂v 1 = ∆ v − 2L(t)v + ω(bv + hu − Cuv + uv2). ∂t ρ2 †
17 The dilution term L(t) being time dependent creates some drawbacks for this model. A notable effect is that the steady state of the system is also time dependent. This makes it impossible to perform linear stability analysis, because it would be necessary to re-linearize the system for every point of t. Since we cannot perform linear stability analysis, we cannot derive the Turing conditions for a logistically growing domain. However, this does not preclude System (3.12) from producing Turing behavior, as we will show in Chapter 4.
3.6 Conclusions
In this chapter, generalized Turing systems and BVM kinetics were outlined. The equations from a growing prolate spheroid framework were altered to utilize an oblate spheroid domain. Motivations for including exponential and logistic growth were discussed and the domain growth functions were implemented into the Turing system.
18 CHAPTER 4
RESULTS
In this chapter, simulation results for the two domain growth models are presented. The results generated by the growing oblate spheroid domain are compared to static domains with similar parameters to illustrate the role of growth and domain size in pattern formatiom. In addition, the results are compared with the growing prolate spheroid domain of [18] to see if the role of domain shape is similar to the static case found in [4]. Simulations were done using a forward time central-space difference scheme in FORTRAN and visualized in MATLAB. Despite Turing systems having sensitivity to initial conditions, the overall pattern generating behavior of a Turing system have been shown to be the same despite different initial conditions [18]. Therefore, all simulations used an identical seed of initial conditions to facilitate comparison. The areas of higher activator concentration correspond to copper colored regions and the areas of higher inhibitor concentrations are colored black. The diffusion coefficient D, and kinetics parameters a, b, and C were constant for all of the following simulations with the following values:
D = 0.516, a = 1.112, b = −1.01 C = 0
All simulations on the growing domain models utilized an initial interfocal distance of f = 2 and a constant radial term ξ = 0.7227. Thus the initial surface area of our oblate spheroid is 4π, the same surface area of the unit sphere. To facilitate comparison between a growing and static domain, simulations on a static domain used a scaled focal distance that was the same as the final focal distance of the growing oblate spheroids. The effects of increasing focal distance with constant ξ isotropically increases the size of the oblate spheroid. The values of ξ are inversely proportional to the eccentricity of the oblate spheroid, therefore a higher value of ξ will cause the ellipsoid to more so resemble a sphere, and a lower ξ value will cause the oblate spheroid to look flatter. All simulations were run from t = 0 to t = 35 to allow the pattern to converge on logistically growing and static domains.
19 (a) (b) (c)
Figure 4.1: Simulations showing varying values of R and ω. R = 0.010 and ω = 115 in (a), R = 0.010 and ω = 70 in (b), R = 0.025 and ω = 70 in (c). For increasing values of R or omega, the complexity of the pattern also increases.
4.1 Exponentially Growing Oblate Spheroid Simulations
The Turing patterns generated by the exponentially growing model on an oblate spheroid have a continually transient quality rather than converging to a final pattern. This is consistent with results on a prolate spheroid domain, providing evidence that the effects of domain shape and domain growth are independent. To preserve the Turing conditions delineated in System (3.7), the two parameters altered across the simulations are the growth rate R and the domain scale ω. Figure 4.1 shows how the resulting Turing Patterns increase in complexity, or number of stripes/spots, with higher levels of R and ω. Figures 4.5, 4.6, 4.7, 4.8, 4.9, and 4.10 show the evolution of the Turing pattern for various values of R and ω.
4.1.1 Comparison to Static and Growing Prolate Spheroid Domain
Simulations involving the static domain case were done by scaling the focal distance to the final size of the oblate spheroids with exponential growth. The cortical folding patterns are shown to converge to a final pattern, similar to the logistic case discussed in Section (4.2). The final patterns on a static growing domain of the same size are different, but similar in complexity to the exponentially growing case. This is consistent with the findings on a prolate spheroid domain [18]. The oblate spheroid domain responded to domain growth parameters in similarly predictable ways to simulations run on a prolate spheroid domain. Figures 4.2, 4.3 show that patterns formed on a prolate spheroid domain are qualitatively similar to the oblate spheroid domain. It was previously shown on there were cases of the final Turing pattern showing no similarity between a
20 static oblate spheroid domain and a static prolate spheroid domain [4]. Those results were unable to be replicated with the new computational implementation used in this thesis.
4.2 Logistically Growing Oblate Spheroid Simulations
Simulations on a logistically growing domain were implemented using System (3.12). We use identical kinetics parameters, approximately equivalent final domain size, and identical ini- tial conditions to the exponentially growing case to facilitate comparison. The parameters spe- cific to the logistic growth function in equation 3.10 were defined as were r = 0 : 6603; ∗ Rt t0 = 19.9258;K = e final − 1 where R = 0.01 and tfinal = 35. Simulations were done by varying values of ω. Figure 4.4 shows that the increasing the domain scale parameter results in an increase in pattern complexity, consistent with the exponential model. Pattern transiency corre- sponded directly with the rapid growth of the logistic model before converging to a final pattern. Figures 4.11 and 4.12 show the change in pattern at eight different time steps to illustrate this.
21 (a) ω = 70 R =0 (b) ω = 70 R =0.010 (c) ω = 70 R =0.010
(d) ω = 115 R =0 (e) ω = 115 R =0.010 (f) ω = 115 R =0.010
(g) ω = 70 R =0 (h) ω = 70 R =0.025 (i) ω = 70 R =0.025
Figure 4.2: Figures (a),(d), and (g) show final generated patterns on a static oblate spheroid domain, Figures (b), (e), and (h) show final patterns on an exponentially growing oblate spheroid domain, and the evolution of their Turing patterns are shown in Figures 4.5, 4.6, and 4.7 respectively. Figures (c), (f), and (i) are the final generated patterns on an exponentially growing prolate spheroid domain. The static oblate spheroids were generated with focal distance scaled by expR∗tfinal with R = 0.010 in (a),(d) and R = 0.025 in (g). This illustrates that domain size is an incredibly strong predictor of final pattern complexity. Comparison of domain shape shows that prolate spheroids and oblate spheroids generate patterns with similar width of the stripes and qualitatively similar patterns with identical growth parameters.
22 (a) ω = 115 R =0 (b) ω = 115 R =0.025 (c) ω = 115 R =0.025
(d) ω = 140 R =0 (e) ω = 140 R =0.025 (f) ω = 140 R =0.025
(g) ω = 140 R =0 (h) ω = 140 R =0.010 (i) ω = 140 R =0.010
Figure 4.3: Figures (a),(d), and (g) show final generated patterns on a static oblate spheroid domain, Figures (b), (e), and (h) show final patterns on an exponentially growing oblate spheroid domain, and Figures (c), (f), and (i) are final generated patterns on an exponentially growing prolate spheroid domain. The evolution of the Turing patterns in (b), (e), and (h) are shown in figures 4.8, 4.9 and 4.10 respectively. The focal distance was scaled by expR∗tfinal with R = 0.025 in (a),(d) and R = 0.010 in (g) to give the static oblate spheroids the same final size as the growing domains. At higher values of ω and R pattern complexity continues to increase.
23 (a) ω = 70 (b) ω = 70
(c) ω = 115 (d) ω = 115
Figure 4.4: Comparison of the final generated patterns on a logistically growing oblate spheroid domain in Figures (a), (c) and logistically growing prolate spheroid domain (b), (d). The effect of increasing values of ω leads to more complex patterns across domains
24 Figure 4.5: Progression of pattern formation on an exponentially growing Turing System with ω = 70 and R = 0.01. The snapshots correspond to the various time values indicated by the red dots on the graph of the growth function.
25 Figure 4.6: Progression of pattern formation on an exponentially growing Turing System with ω = 115 and R = 0.01. The snapshots correspond to the various time values indicated by the red dots on the graph of the growth function.
26 Figure 4.7: Progression of pattern formation on an exponentially growing Turing System with ω = 70 and R = 0.025. The snapshots correspond to the various time values indicated by the red dots on the graph of the growth function.
27 Figure 4.8: Progression of pattern formation on an exponentially growing Turing System with ω = 115 and R = 0.025. The snapshots correspond to the various time values indicated by the red dots on the graph of the growth function.
28 Figure 4.9: Progression of pattern formation on an exponentially growing Turing System with ω = 140 and R = 0.025.
29 Figure 4.10: Progression of pattern formation on an exponentially growing Turing System with ω = 140 and R = 0.01. The snapshots correspond to the various time values indicated by the red dots on the graph of the growth function.
30 Figure 4.11: Progression of pattern formation on an logistically growing Turing System with ω = 70. The snapshots correspond to the various time values indicated by the red dots on the graph of the growth function.
31 Figure 4.12: Progression of pattern formation on an logistically growing Turing System with ω = 115. The snapshots correspond to the various time values indicated by the red dots on the graph of the growth function.
32 CHAPTER 5
CONCLUSIONS AND FUTURE DIRECTIONS
The purpose of this thesis was to examine the effects of domain shape and domain growth in cortical pattern development in order to more accurately model diseases that alter the shape of the lateral ventricles. We ultilized an oblate spheroid domain to model the shape of the lateral ventricle and introduced two growth functions– an exponential growth function, and a logistic growth function. The effects of domain growth on the oblate spheroid domain were consistent with the effect of growth on prolate spheroid domain. Furthermore, the size of the stripes and spots of the generated patterns were consistent regardless of domain shape, although final patterns generated were different. Domain size is the most powerful indicator of pattern complexity, as shown when compared to the static case with focal distance scaled to the final size of the exponential models. Because of the demonstrated consistent effects of the final generated patterns regardless of domain shape, future work includes utilizing this knowledge of parameters with MRI data of holo- prosencephaly and ventriculomegaly in order to model their effect on cortical folding patterns. All of the simulations used for this thesis utilized the same eccentricity of the oblate spheroid with radial term ξ = .7227; however in vivo the cerebral cortex would have varying levels of eccentricity, which has been shown to affect pattern formation [17]. The addition of isotropic exponential and logistic growth functions is able to mimic the growth of the lateral ventricles during gestation, however it does not account for the formation of the posterior horns of the mature lateral ventricles. As the role of domain shape has been shown to affect pattern development, a growth function that allows for the domain shape to deform into the c-shaped mature appearence would generate more accurate results. The model of cortical folding presented in this thesis is based on the intermediate progenitor Model, which emphasizes genetic, chemical factor control of development. However, the existence of the morphogens required to activate or inhibit the formation of intermediate progenitor cells is still theoretical. Experimental evidence for the axonal tension hypothesis suggests the true mechanism for cortical folding is at least partially physical [5]. Thus, an updated model taking into account
33 both genetic chemical factors of the intermediate progenitor model and the physical effects of the axonal tension hypothesis could be more biologically relevant. A particularly useful area of future work would be creation of a quantitative metric for domain complexity. This would give a concrete measure and more insight into the the characteristics of the generated patterns. It would also streamline comparison of Turing patterns on three-dimensional models. In conclusion, the role of domain growth is consistent across various domain shapes. The role of domain shape in pattern development has a demonstrable effect on the final pattern but can be predicted to a certain complexity given domain growth rate and domain scaling parameters.
34 BIBLIOGRAPHY
[1] R. A. Barrio, C. Varea, J.L. Aragon, and P.K. Maini. A two-dimensional numerical study of spatial pattern formation in interacting Turing systems. B Math Biol, 61(3):483–505, 1999.
[2] K. Flammer. Spheroidal Wave Functions. Stanford University Press, Palo Alto, 1957.
[3] O. Glenn. Normal development of the fetal brain by MRI. Semin Perinatol, 33:208–219, 2009.
[4] C. Grazzini. An oblate spheroid model of cortical folding. Florida State University Honors Thesis, April 2015.
[5] C.C. Hildgetag and H Barbas. Role of mechanical factors in the morphology of the primate cerebral cortex. PLoS Comput Biol, 2(3):146–159, 2006.
[6] E.R. Kandel, J.H. Schwartz, and T.M. Jessel. Principles of Neural Science. McGraw-Hill, New York, 4 edition, 2000.
[7] O. Kang and R. Agrawal. Holoprosencephaly. https://radiopaedia.org/articles/holoprosencephaly. Accessed: 2017-03-30.
[8] Y. Kinoshita, T. Okudera, E. Tsuru, and A. Yokota. Volumetric analysis of the germinal matrix and lateral ventricles performed using MR images of postmortem fetuses. Am J Neuroradiol, 22:382–388, 2001.
[9] A. Kriegstein, S. Noctor, and V. Martinez-Cerdeno. Patterns of neural stem and progenitor cell division may underlie cortical expansion. Nat Rev Neurosci, 7(11):883–890, 2006.
[10] P.K. Maini, T.E. Woolley, R.E. Baker, E.A. Gaffney, and S.S. Lee. The Turing model comes of molecular age. Science, 314:1397–1398, 2006.
[11] Ja. Neumiller, Jo. Neumiller, B. Gates, S. Setter, and D. Greely. Normal pressure hydro- cephalus. US Pharm, 1:56–61, 2007.
[12] S.C. Noctor, V. Martinez-Cerdeno, L. Ivic, and A. Kriegstein. Cortical neurons arise in sym- metric and asymmetric division zones and migrate through specific phases. Nat Neurosci, 7(2):136–144, 2004.
[13] J. Nolte. The Human Brain: An Introduction to Its Functional Anatomy. Mosby, Philadelphia, PA, 6 edition, 2009.
[14] P. Rakic. Specification of cererbral cortical areas. Science, 241(4862):170–176, 1988.
35 [15] J. Ritner and M. Frates. Fetal CNS: A systematic approach. Radiol Clin N Am, 52:1253–1264, 2014.
[16] M. Rutherford, P. Tortori-Donati, and A. Rossi. MRI of the Neonatal Brain. Saunders Ltd, Philadelphia, PA, 4 edition, 12 2001.
[17] D. Striegel. Modeling the Folding Pattern of the Cerebral Cortex. PhD thesis, Florida State University, Tallahassee, FL, October 2009.
[18] G. Toole. Modeling Cortical Folding Patterns on a Growing Domain. PhD thesis, Florida State University, Tallahassee, FL, March 2013.
[19] A.M. Turing. The chemical basis of morphogenesis. Philos T R Soc Lond, 237(641):37–72, August 1952.
[20] S. Zeki. Visions of the Brain. Wiley, Hoboken, NJ, 1993.
36