Dorian Luis Linero Segrera

A model of cerebral cortex formation during fetal development using reaction-diffusion-convection equations with Turing space parameters

The cerebral cortex is a gray lamina formed by bodies of neurons covering the cerebral hemispheres, varying in thickness from 1.25 mm in the occipital lobe to 4mm in the anterior lobe. The brains surface is about 30 times greater that of the skull because of its many folds; such folds form the gyri, sulci and fissures and mark out areas having specific functions, divided into five lobes. Convolution formation may vary between individuals and is an important feature of brain formation; such patterns can be mathematically represented as Turing patterns. This article describes how a phenomenological model was developed by describing the formation pattern for the gyri occurring in the cerebral cortex by reaction diffusion equations with Turing space parameters. Numerical examples for simplified geometries of a brain were solved to study pattern formation. The finite element method was used for the numerical solution, in conjunction with the Newton-Raphson method. The numerical examples showed that the model can represent cerebral cortex fold formation and reproduce pathologies related to gyri formation, such as polymicrogyria and lissencephaly.

Appearance and formation of seed and pericarp may be explained by a reaction–diffusion mechanism? A mathematical modeling

Abstract This paper proposes a phenomenological model describing plant seed cortex (testa) and pericarp pattern formation by reaction–diffusion equations with Turing space parameters. Numerical examples for the seed’s simplified geometries in order to study pattern formation were done. The finite element method in conjunction with the Newton–Raphson method was used for the numerical solution to approximate nonlinear partial differential equations. Numerical examples showed that the model can represent the formation of different types of seed cortex and pericarp.