Intrinsic Geometry of Biological Surface Growth

by Philip H. Todd

Overview

1.1 General Introduction The work which comprises this essay formed part of a multidiscip­ linary project investigating the folding of the developing cerebral cortex in the ferret. The project as a whole combined a study, at the histological level, of the cytoarchitectural development concom­ itant with folding and a mathematical study of folding viewed from the perspective of differential geometry. We here concentrate on the differential geometry of brain folding. Histological results which have some significance to the geometry of the cortex are re­ ferred to, but are not discussed in any depth. As with any truly multidisciplinary work, this essay has objectives which lie in each of its constituent disciplines. From a neuroana­ tomical point of view, the work explores the use of the surface geo­ metry of the developing cortex as a parameter for the underlying growth process. Geometrical parameters of particular interest and theoretical importance are surface curvatures. Our experimental portion reports the measurement of the surface curvature of the ferret brain during the early stages of folding. The use of sur­ face curvatures and other parameters of differential geometry in the formulation of theoretical models of cortical folding is discussed.

Product Details

ISBN-13:	9783540164821
Publisher:	Springer Berlin Heidelberg
Publication date:	06/19/1986
Series:	Lecture Notes in Biomathematics , #67
Pages:	132
Product dimensions:	6.69(w) x 9.61(h) x 0.01(d)

Table of Contents

1: Introduction.- 1.1 General Introduction.- 1.2 Introduction from Mathematical Biology.- 1.3 Neuroanatomical Introduction.- 2: Some Geometrical Models in Biology.- 2.1 Introduction.- 2.2 Hemispherical Tip Growth.- 2.3 The Mouse Cerebral Vesicle.- 2.4 The Shape of Birds1 Eggs.- 2.5 The Folding Pattern of the Cerebral Cortex.- 2.6 Surface Curvatures of the Cerebral Cortex.- 2.7 Coda.- 3: Minimum Dirichlet Integral of Growth Rate as a Metric for Intrinsic Shape Difference.- 3.1 Introduction.- 3.2 Isotropic and Anistropic Biological Growth.- 3.3 Some Properties of the Minimum Dirichlet Integral.- 3.4 Minimum Dirichlet Integral as a Metric for Shape.- 3.5 Comparison with other Dirichlet Problems.- 4: Curvature of the Ferret Brain.- 4.1 Material & Methods.- 4.2 Results and Interpretation.- 4.3 Discussion.- 4.4 The nearest plane region to a given surface.- 4.5 Conclusions.- References.- Appendix A: Numerical Surface Curvature.