Growth in a Turing Model of Cortical Folding

Abstract

The brain's cerebral cortex is folded into many gyri (hills) and sulci (valleys). Little is known about how the cortex folds or why the folds are located where they are. We have developed a spatio-temporal mathematical model of cortical folding to address this question. Our model utilizes a Turing reaction-diffusion system on an exponentially growing prolate spheroidal domain. This domain approximates the shape of the lateral ventricle (LV) during cortical development. The Intermediate Progenitor Model (IPM) of cortical folding states that regional patterning of self-amplication of intermediate progenitor cells (IPCs) in the subventricular zone of the LV corresponds with the formation of cortical folding. As self-amplication of IPCs is genetically controlled via chemical gradients, a Turing system is a logical choice to create a mathematical representation of the IPM. A growing domain model of cortical folding may be more realistic than previous static domain models of cortical folding since it incorporates the growth that naturally occurs as the brain develops. By comparing patterns generated by our growing prolate spheroid Turing system with those generated by a static prolate spheroid Turing system, we show that the addition of growth causes a significant change in system behavior; the system produces transient patterns instead of converging to one final pattern. Our model illustrates the importance of including growth in a model of cortical folding and can be utilized to explain certain human diseases of cortical folding.