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Conceived and designed the experiments: JL JFM. Performed the experiments: JL. Analyzed the data: JL JFM. Contributed reagents/materials/analysis tools: JL. Wrote the paper: JL JFM.

The authors have declared that no competing interests exist.

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.

The development of the human brain from the early gestational weeks to the buckling of the first folds at around 20 weeks follows a narrow pathway between determinism and pure randomness. On the one hand normal adult individuals offer quite similar - from a pure qualitative and descriptive point of view - folding structures: gyri and sulci. On the other hand we observe morphological variabilities between different brains

Folding or buckling are very general processes in nature and among living organisms. Especially one of the most studied step in the morphogenesis of metazoans is gastrulation which corresponds to a symmetry breaking of the spherical embryo and an invagination. The origin of this folding remains unknown even if mechanical factors are undoubtedly implied

In these conditions it raises the issue of realistic modeling of far more complex buckling processes such as the gyrification of mammal brains. In this regard it is important to inspect carrefully previous models of gyrification.

Le Gros Clark

In the same extra-cortical point of view a recent and very popular model considers that the folding of the brain takes its origin in the mechanical tensions produced by the white matter fibers

At the opposite there are numerous hypotheses arguing that the cortical folding has intrinsic origins. In

In the intrinsic origins of folding we encounter also purely morphogenetic hypotheses in which cortical convolutions are under genetic control

In the next part we will see in detail another hypothesis for the cortical folding which is based on Turing instabilities

In this article we investigate the origin of anatomical variability from the early development and we propose a phenomenological model of the folding which is based on the putative existence of Turing morphogens. After recalling briefly some mathematical aspects of the model, we present the numerical schemes used for implementing the equations on a surface and for the deformation of the surface. We show some qualitative and quantitative results of the model. In particular we link sulcal pits maps to the average folding patterns across several realizations of a same noisy initial condition. And we study the variability of our model and demonstrate that it can lead to different modes of variability of one sulcus.

Cartwright

Very recently the Global Intermediate Progenitor (GIP) model has been proposed to explain the appearance of transversal or sectorial sulci following the Intermediate Progenitor hypothesis

We aim at extending the analogy first formulated by Cartwright and the GIP model using a system of reaction-diffusion equations that will modify the surface on which the equations take place. Namely the reaction diffusion system models the non-linear interaction of two morphogens

In our model initial conditions of the reaction-diffusion equations have strong similarities with the sulcal roots described in

Our choice of the reaction-diffusion equations differs from the one of Cartwright and the GIP model since we have adopted the Gray-Scott model:

The mathematical analysis of the model has been previously conducted in

Following ideas of

Since the surface on which evolve the morphogens is modified with time, we have to adapt the equations (1) and (2) to take into account the geometric changes. The problem of reaction-diffusion on growing domains has been well-studied in the past years. It leads generally to add convective and dilution terms to

The model proposed in

Since we work on discrete meshes we have used a finite element method to discretize the linear terms in the equations 1 and 2.

First we derive a weak formulation of the system on

Next we work on a discrete tessellation

The weak formulation becomes:

It is possible to treat the non-linear term with the following approximation as in

At last we need to compute

To be exact we should take into account the fact that the finite elements vary along time or in other terms that the mass matrix

For the surface deformation step, we translate equation (3) by simply modifying each vertex

When the area of a triangle ABC exceeds two times the averaged area of the triangles of the original mesh we divide the original triangle in four triangles constructed from the three midpoints A′,B′,C′ of each side. Moreover we divide each of the three triangles ABD, ACF, BCE in two triangles.

In our implementation we do not prohibit self-intersection which would increase considerably the computation time. However we can say that we escape this issue by not solving for too long time but also taking a parameter

First we can model the growth of a normal brain with the value

The pictures correspond to time instants 1, 1000, 2000, 3000, 4000 of the iteration process. The last graph indicates the evolution of surface area in the successive meshes.

Note that the evolution specified by the coupled reaction-diffusion equations and the surface deformation leads to a progressive folding of the initial sphere on

The blue line represents the average number across 50 simulations and the blue area around represents the standard deviation.

We can observe on

Moreover we propose a simple way to characterize the spatial stability of the folds along time. In other terms we demonstrate that the position of folds formed at different time instants remains relatively stable. We extract a map of the curvature

Time-average map

We can see on this figure that the average map

In this part we investigate the influence of noise in the spatial position of the folds. In particular we aim at demonstrating that the reaction diffusion mechanism is able to produce reproducible folds at certain specific locations but can also engender variability at other locations. For this we simulate

Top: Sum of the thresholded maps

On

First column: Three different modes of variability for the main fold observed on

More generally

It is possible to represent directly the influence of one or several parameters of the model (

The two axes correspond to the two parameters

As suggested by

First line: Growth patterns for different values of (F,k), respectively (0.04,0.06), (0.03,0.06) and (0.05,0.05). Second line: the corresponding brain patterns: normal, polymicrogyria, lissencephaly.

So we can see that for

Our model extends the initial proposal of Cartwright in

In our model we investigate also the variability of folding along the development of one individual and across several individuals - that is several realizations of the model. First we can see that for an unique development the position of the sulci remains stable along time. This result may seem trivial but is required for our model to produce definite patterns of gyrification that can be compared between different realizations of the model. Secondly the study of folding variability among 50 random realizations of the model reveals two important characteristics. The folding does not organize randomly even if we add noise to the initial condition of the reaction diffusion process. We have shown on one example that a main structure emerges that is strongly reproducible among several simulations. We can find a direct analogy between this main fold and the primary folds described in the literature

Moreover we have shown that in spite of its strong reproducibility the main fold could be broken in two separate parts by a gyrus. This result echoes previous studies

On a more theoretical point of view, our results on the reproducibility of the folds seem to confirm the impact of growth domain on the robust selection of patterns as it has been previously shown in

In conclusion we have proposed an extended framework for modelling the cortical folding. It is based on a system of coupled reaction-diffusion equations defined on a surface that evolves through the action of morphogens. We show that for some parameters the model gives rise to geometric patterns that can be related to cortical sulci. We also demonstrate that under the effect of noise the system yields morphological variability in these cortical structures. Moreover changing slightly the values of the parameters of the model can have an important influence on the nature of the created patterns which suggest a link toward pathologies of the brain development such as lissencephaly or polymicrogyria. In future developments we plan to investigate the difficult issue of estimating good values of parameters with respect to a given sequence of cortical surfaces across development.

We would like to thank Oliver Lyttelton, Maxime Boucher, Steven Robbins, and Alan Evans for having given us the possibility to use the average surface of 222 hemispheres.

We also would like to thank Roberto Toro for his comments on the manuscript.