A Reaction-Diffusion Model of Human Brain Development

Abstract

Cortical folding exhibits both reproducibility and variability in the geometry and topology of its patterns. These two properties are obviously the result of the brain development that goes through local cellular and molecular interactions which have important consequences on the global shape of the cortex. Hypotheses to explain the convoluted aspect of the brain are still intensively debated and do not focus necessarily on the variability of folds. Here we propose a phenomenological model based on reaction-diffusion mechanisms involving Turing morphogens that are responsible for the differential growth of two types of areas, sulci (bottom of folds) and gyri (top of folds). We use a finite element approach of our model that is able to compute the evolution of morphogens on any kind of surface and to deform it through an iterative process. Our model mimics the progressive folding of the cortical surface along foetal development. Moreover it reveals patterns of reproducibility when we look at several realizations of the model from a noisy initial condition. However this reproducibility must be tempered by the fact that a same fold engendered by the model can have different topological properties, in one or several parts. These two results on the reproducibility and variability of the model echo the sulcal roots theory that postulates the existence of anatomical entities around which the folding organizes itself. These sulcal roots would correspond to initial conditions in our model. Last but not least, the parameters of our model are able to produce different kinds of patterns that can be linked to developmental pathologies such as polymicrogyria and lissencephaly. The main significance of our model is that it proposes a first approach to the issue of reproducibility and variability of the cortical folding. The anatomical variability of the human brain folds remains an unclear and challenging issue. However it is clear that this variability is the product of the brain development. Several hypotheses coexist for explaining the rapid development of cortical sulci and it is of the highest interest that understanding their variability would improve the comparison of anatomical and functional data across cohorts of subjects. In this article we propose to extend a model of cortical folding based on interactions between growth factors that shape the cortical surface. First the originality of our approach lies in the fact that the surface on which these mechanisms take place is deformed iteratively and engenders geometric patterns that can be linked to cortical sulci. Secondly we show that some statistical properties of our model can reflect the reproducibility and the variability of sulcal structures. At the end we compare different patterns produced by the model to different pathologies of brain development.