Chemically Based Mathematical Model for Development of Cerebral Cortical Folding Patterns

The mechanism for cortical folding pattern formation is not fully understood. Current models represent scenarios that describe pattern formation through local interactions, and one recent model is the intermediate progenitor model. The intermediate progenitor (IP) model describes a local chemically driven scenario, where an increase in intermediate progenitor cells in the subventricular zone correlates to gyral formation. Here we present a mathematical model that uses features of the IP model and further captures global characteristics of cortical pattern formation. A prolate spheroidal surface is used to approximate the ventricular zone. Prolate spheroidal harmonics are applied to a Turing reaction-diffusion system, providing a chemically based framework for cortical folding. Our model reveals a direct correlation between pattern formation and the size and shape of the lateral ventricle. Additionally, placement and directionality of sulci and the relationship between domain scaling and cortical pattern elaboration are explained. The significance of this model is that it elucidates the consistency of cortical patterns among individuals within a species and addresses inter-species variability based on global characteristics and provides a critical piece to the puzzle of cortical pattern formation.Author Summary: The size and shape of the cerebral cortex varies across species. The cortical folding pattern also varies from a smooth surface where no pattern is visible, as observed in the common treeshrew (Tupaia glis) and Eastern mole (Scalopus aquaticus), to an intricate labyrinthine pattern, as observed in humans. One current model, the intermediate progenitor model, describes the creation of a fold through local interactions in the ventricular zone which surrounds the lateral ventricle. Here we extend the local scenario described in the intermediate progenitor model to include global characteristics that differ between species. We approximate the lateral ventricle with a prolate spheroid and examine how patterns on a spheroidal surface change based on size and eccentricity. Our model reveals a direct correlation between pattern formation and lateral ventricular size and shape. This model's significance is that it elucidates the consistency of cortical patterns among individuals within a species and addresses inter-species variability based on global characteristics, such as size and shape of the lateral ventricle, and provides a critical piece to the puzzle of cortical pattern formation.