Flat Mapping the Brain

Annual Meeting ofThe Organization for Human Brain Mapping:Oral Session on DevelopmentHonolulu, Hawaii, USA June 14-18, 2015A Dynamically Growing Domain Model of Cortical Folding Pattern FormationPoster #4054 Dr. Monica K. Hurdal Department of Mathematics Florida State University, Tallahassee, FL mhurdal@math.fsu.edu http://www.math.fsu.edu/~mhurdal

1Research QuestionsWhat influences the size, location, and formation of cortical folds during development?Aim: Improve our understanding of cortical folding pattern formation in the brain Aim: Understand what constitutes normal cortical folding variabilityModel shape of brain and how its folding patterns develop and growHow do folds change in development, aging, and disease?

Modified from S. Zeki, A Vision of the Brain, 1993.OutlineUse mathematical modeling to elucidate information about cortical folding development.Not much is known about what influences the size, location, and formation of folds during development.Cortical Development a chemically-driven model Turing Systems used to model pattern formation in biological systemsProlate Spheroid eccentricity of a shape to be captured in the focal distanceResults modeling human cortical patterning malformations

universe-review.ca/I10-80-ventricles.jpg3Development of CortexLower Cortical Layer: Radial Unit Hypothesis (Rakic, 1988):founding population of radial glial cells (RGCs) created in VZRGCs create replacement RGCs and neuroblasts that travel to cortex to create a cortical layerneurons stack outwardly creating column; column and its RGC create a 1-1 mapping between cortex and VZP. Rakic, Science 241, 170-176, (1988).

RRRCortical plateIntermediate zoneSubventricular zone (SVZ)Ventricular zone (VZ)NNNNNNLateral VentricleR- Radial glial cellN- Neuron 4Symmetric cellular division stage creates founding popln of RGCs in VZ. Each additional cellular divisions results in a doubling of RGC popln.Assymetric cellular division stage creates a replacement RGC and neuroblast. Neuroblasts travel up radial arm of RGC to cortex to create a cortical layer located in the cortical plate. Neurons from different layers travel past the previous layer and stack outwardly, creating a columnar structure called an ontogenetic column. Each ontogenetic column and its associated RGC create a 1-1 mapping between cortex and VZ.RRRLateral VentricleNNNNNNIINNNNNNNNDevelopment of CortexUpper Cortical Layers: Intermediate Progenitor Hypothesis (Noctor, 2004) radial glial cells (RGCs) form and are activated to form intermediate progenitor (IP) cells IP cell travels to SVZ and it can create two neuroblasts per cellular division result is amplification of number of neuroblasts that travel to cortexR- Radial glial cellN- Neuron -Cortical plateIIntermediate progenitor cellNNNNIS. Noctor, V. Martinez-Cerdeno, L. Ivic, A. Kriegstein, Nat Neuroscience, 7, 136-144 (2004).Intermediate zoneSubventricular zone (SVZ)Ventricular zone (VZ)5Hypothesis matches what is observed in animals where upper layers of cortex have an amplified number of neurons as compared to lower layers.

Development of Sulcal PatternLateral VentricleA. Kriegstein, S. Noctor, V. Martinez-Cerdeno, Nature Reviews Neuroscience, 7(11), 883-890 (2006).R- Radial glial cellN- Neuron IIntermediate progenitor cell-Upper Layers: Intermediate Progenitor Model (Kriegstein, 2006) only subsets of RGCs are activated to create IP cells result is non-uniform distribution of IP cells, which create local amplication of neuroblasts surrounded by areas of non-amplificationRRRIINNNNNNCortical plateIntermediate zoneSubventricular zone (SVZ)Ventricular zone (VZ)6Lateral VentricleA. Kriegstein, S. Noctor, V. Martinez-Cerdeno, Nature Reviews Neuroscience, 7(11), 883-890 (2006).Development of Sulcal PatternR- Radial glial cellN- Neuron I-Intermediate progenitor cell

Upper Layers: Intermediate Progenitor Model (Kriegstein, 2006) only subsets of RGCs are activated to create IPCsresult is non-uniform distribution of IP cells, which create local amplication of neuroblasts surrounded by areas of non-amplificationRRRNNNNNNIINNNNNNNNNNIntermediate zoneSubventricular zone (SVZ)Ventricular zone (VZ)Corticalplate7Macaque Neocortex Figure: The subventricular zone (SVZ) is thicker in areas underlying gyrus formation and thinner in areas underlying sulcus formation. a | Parasagittal sections of the macaque occipital lobe. In embryonic day (E) 78 macaque cortex, a thickened SVZ (indicated by black arrows under 1) presages the gyral formation that can be seen just over 2 weeks later at E94 (1*). By contrast, a much thinner SVZ (indicated by black arrows under 2) is located under a region of sulcal formation (2, arrow). b | Similar features are observed in coronal sections of the developing human cortex. At gestational week (GW) 20, areas of thickened SVZ (indicated by brackets under 1, 3 and 5) presage gyral formation that can be seen in the same region of the cortex four weeks later at GW24. By contrast, areas of thinner SVZ (indicated by brackets under 2 and 4) are located under regions of sulcal formation that are observed four weeks later at GW24. CP, cortical plate; IZ, intermediate zone; LV, lateral ventricle; SVZ*, encompasses stratified transitional fields 16; VZ, ventricular zone. Panel a modified, with permission, from Ref. 56 (2002) Oxford Univ. Press. Panel b modified, with permission, from Ref. 63 (2005) Taylor & Francis.Model the development and growth of brain folding with a Turing reaction-diffusion system using dynamic growthTuring System: stability in the absence of diffusion and diffusion driven instabilityG. Toole, M.K. Hurdal, J Dynamics and Diff Equations, 26, 315-332, (2014), A.M. Turing, Phil. Trans. Roy. Soc. of Lon. B, 237, 37-72 (1952),Growing Domain Turing SystemR.A. Barrio, C. Varea, J. L. Aragon, P. K. Maini, Bull Math Bio, 61(3), 483-505 (1999).Derive Growing Domain Turing System with Exponential Growthu = activator, v = inhibitorF(u,v) and G(u,v) = kinetics functionsD 0 implies diffusion-driven instability.Asterisks correspond to values of k2 when n = 1, 2, 3.

x[u]Solvingon a 1D domain , 0 < x < P, with periodic boundary conditions we obtain solutions of the form:where and n is an integer.

10N=1:9.8596N=2:39.4384N=3:88.73641D Turing System - ExampleSolvingon a 1D domain , 0 < x < P, with periodic boundary conditions we obtain solutions of the form:where and n is an integer. 0, 2 0, 2

x[u] = 102.0408 = 416.6667

x[u]An increase in domain scale will increase k2.11Activator and inhibitor reactants in the Turing system regulate IP cell productionModel shape of lateral ventricle (LV) with a prolate spheroid with major axis of spheroid corresponding to major axis of LV; surface represents VZGerminal matrix growth initially modeled using exponential, logistic growth estimated from Kinoshita et al.Examine how growth rate R and overall level of genetic expression affect pattern formation

Proposed Model

Images: http://embryology.med.unsw.edu.au/Notes/neuron5.htm54-56 embryonic daysY. Kinoshita, et al., Am J Neuroradiol, 22, 382-388 (2001).12Since the LV and VZ are critical components in the development of cortical patterning and their shape in early development is representative of a prolate spheroid, we model the LV with a prolate spheroid and the VZ with a prolate spheroid surface.Prolate Spheroid CoordinatesProlate spheroid created by rotating an ellipse about its major axisCoordinates expressed as (, , ); overall shape encapsulated by focal distance f

radial term: angular term: rotational term:f focal distance: a semi-major axis, b semi-minor axis

Solve (Helmholtzs equation) in terms of prolate spheroidal coordinates. Copper regions: areas where activator concentration u controls IP cell production and exceeds inhibitor v, resulting in gyrus formation.

13Copper regions represent areas where activator concentration u controls IP cell production and exceeds inhibitor v, resulting in gyrus formation; dark-colored regions represent areas where inhibitor concentration v exceeds activator u, resulting in sulcus formation

Growing Domain: Exponentially Growing Prolate Spheroidexponential growth rate function = (t) = eRt growth rate R allows domain size to be controlled: increasing R leads to more complex patterns at a given elapsed timedomain scale parameter represents overall level of genetic expression of activator and inhibitor morphogens: increasing leads to more complex patternsGrowing Domain: Logistically Growing Prolate Spheroid

Snapshots of logistical domain growth over time; R = 0.015, = 145Pattern evolves and changes rapidly during domain growth; converges when domain growth stops.

R and have similar effects as in exponential growth.

Copper regions: areas where activator concentration u controls IP cell production and exceeds inhibitor v, resulting in gyrus formation.Application: Human Malformations:Polymicrogyria (many small gyri)

Neural migration disorder characterized by an excessive number of small prominent convolutions spaced out be shallow and enlarged sulciFocal polymicrogyrias are often associated with malformations in GPR56 (regulates regional cortical patterning). (Rakic, 2004)Many different types of polymicrogyriaFocal / DiffuseBilateral / UnilateralAlone / Associated with other diseasesPMG with LV enlargement: model by increasing RSome cases of bilateral frontoparietal PMG present with microcephaly and LV enlargement: model by increasing

P. Rakic, Science, 303, 1983-84 (2004).

http://www.neuropathologyweb.org/chapter11/chapter11dNMD.htmlNRS: Norman-Roberts Syndrome

Characterized by type 1 lissencephaly: below-normal number of cortical folds which are broader in widthVery rare congenital diseaseSevere mental retardation, epilepsyMicrocephaly smaller than normal brain, LVs, head, reduced head growth rateSome cases present with non-enlarged LVs: model reduced head growth rate by decreasing RSome cases present with enlarged LVs: model by decreasing Results also generated for type 1 lissencephaly in normocephalic brains with enlarged LVs

Sagittal MRI SlicesAxial MRI SlicesImages A & B: Norman Roberts SyndromeImages C & D: Normal PatientFetal MRI at 23 weeks gestational ageImages adapted from F. Natacci et al., Prenat Diagn, 27, 567-572 (2007)

Pattern FormationModeling and understanding cortical folding pattern formation is important for quantifying cortical developmentTogether R and control the complexity of the labyrinthine pattern that is produced

Healthy Simulation: R = 0.015, = 115PMG Simulation: R = 0.021, = 115Animations are for logistic growth.18Modeling Disease: Growth Rate Effects

NRS: R=0.005, =115NRS with microcephaly, non-enlarged LVs => decrease RNormal: R=0.015, =115PMG: R=0.021, =115PMG with enlarged LVs => increase RChanging growth rate R leads to changes in pattern formationFigures are for exponential growth.NRS with microcephaly, non-enlarged LVs: assume LVs in smaller brain are smaller than normal LVs => decrease RPMG with enlarged LVs: assume LVs are larger than normal LVs => increase R

19Modeling Disease: Domain ScaleNRS: R=0.015, =30NRS with microcephaly, enlarged LVs => decrease Normal: R=0.015, =115PMG: R=0.015, =150PMG with microcephaly, enlarged LVs => increase Changing domain scale parameter affects overall level of genetic expression of activator and inhibitor morphogens

Figures are for exponential growth.NRS with microcephaly, enlarged LVs: assume larger LVs in smaller brain are equivalent to normal LVs => keep R constant; generate pattern by decreasing omegaPMG with microcephaly, enlarged LVs: assume larger LVs in smaller brain are equivalent to normal LVs => keep R constant; generate pattern by increasing omega20Discussion and ConclusionsOur growing domain Turing models enabled us to investigate the role of chemical morphogens in cortical folding Our model is able to elucidate parameters which can lead to excessive or below-normal cortical folding by altering parameters controlling domain growth R, genetic expression levels These models represent an important step in improving our understanding of cortical folding pattern formation in the brain and the influence that domain growth and genetic expression can have on cortical foldingData is needed to help test and modify the hypotheses generated by our model (collaborators wanted!)Lateral ventricular shape information and growth over timeRegulation of the production of IP cells Estimates of model kinetic parameters

21More InformationPoster #4054

AcknowledgementsFormer PhD students Deborah Smith and Gregory Toole

More information on Mathematical Modeling and Brain Mapping:URL: http://www.math.fsu.edu/~mhurdalEmail: mhurdal@math.fsu.edu

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