What do convolutions of the cerebral cortex achieve

Additionally, the brains were now folded, and therefore had an even larger surface area. Although the surface area of these mouse brains increased, the thickness of the brain tissue remained about the same. Chenn and Walsh were interested in why the brains were bigger. Were cells dividing more quickly? Were fewer cells dying? The answer to both of these questions was "No. In other words, they have the potential to become many different types of cells.

The cells continue to divide until they "choose" to become a specific type of cell. In the engineered mice with more beta-catenin, many of the young cells chose to stay in the cell cycle and keep dividing longer than usual. Riviere, A. Cachia, I. Do, Y. Samson, unpublished manuscript , or differential growth of the inner and outer strata of the cortex Richman et al. The results that we present in the following sections suggest that the morphogenesis of convolutions is a natural consequence of cortical growth, and that the early anisotropies of the cortical tissue can induce, modulate and guide the development of the folding pattern.

We construct a finite-elements implementation of the morphogenetic model so as to study the dynamics of cortical folding. For simplicity, the model is implemented in two dimensions. In the initial configuration the cortical layer is a circular structure composed of quadrilateral elements of width q w and thickness q h. The quadrilateral elements are attached to the centre of the model by linear elements that represent the radial elements of the morphogenetic model Fig.

In this section we introduce first the equations that control the elastoplasticity of the radial elements, and then the equations for elastoplasticity and growth of the cortical layer.

In the next section we present the results of the simulation of the morphogenetic model. Finite-elements model of the developing cerebral hemisphere. A Initial shape of the finite-elements model.

The cortical layer is composed of continuous quadrilateral elements attached to the centre by linear radial elements. B Dimensions of the finite-elements model. The initial length of the radial elements is r f 0 , the initial size of the quadrilateral elements of the cortical layer is q h in the radial direction and q w in the tangential direction. C Calculation of the deformation forces in the quadrilateral elements. Mechanically, the behaviour of each individual element of the cortical layer and of the radial elements is described by two equations: an equation for its elasticity and another for its plasticity.

The elastic forces for each quadrilateral element of the cortical layer are obtained from the difference between the actual deformed configuration and its rest configuration. The elastic constant k c determines both the tangential elasticity of the cortical layer and its rigidity which defines the force necessary to bend it. In the finite-elements model, bending the cortical layer deforms its quadrilateral elements: their external side is in tension while their internal side is in compression.

The bending force necessary to produce the same curvature in different cortical layers depends on the thickness q h of their quadrilateral elements.

This produces a growth function with a typical sigmoidal shape. The logistic-growth function has been used to model many growth processes concerning brain development Armstrong et al. The displacement of each point on the cortical layer is given by the addition of the forces exerted by the corresponding radial element and the two quadrilateral elements which share that point. In the current implementation of the morphogenetic model no detection of mechanical contact is performed.

Convolutional development in the morphogenetic model is illustrated in Figure 2. These convolutions are produced only by the growth of the cortical layer. Figure 2 A shows snapshots of the model at intervals of iterations, and Figure 2 B shows the curves for the total force Fig.

Development of convolutions in the model. A Screenshots of the model at iteration steps radial elements are not drawn to avoid burden. Initially the model expands symmetrically without developing convolutions, then convolutions develop, and finally the convolutions are accommodated in the cortical layer.

In the accommodation stage, convolutions can eventually fuse, as can be seen at the right side of the model, where convolutions a and b become a single convolution ab. B Evolution of the force, total radius, total area and total perimeter of the model.

The curves represent iterations of the simulation. During the stage of symmetric growth the forces and the total area augment rapidly, with a slight augmentation of the perimeter and the radius. The development of convolutions is accompanied by a marked decrease in the forces and an augmentation of the perimeter. We distinguish three main stages in the development of convolutions in the model:At the end of the simulation, the position of sulcal fundi is almost the same that at the beginning Fig.

Furthermore, gyri are significantly thicker than sulci Fig. Symmetric growth. In this first stage iterations 0— , the growth of the cortical layer produces a rapid increase of the total force in the cortical layer and radial elements Fig. The growth of the area of the cortical layer, controlled by the logistic-growth function, is shown in the Figure 2 B 3 heavy line.

Development of convolutions. In this second stage iterations — , folds of a similar tangential size and depth develop.

This moment is characterized by the rapid displacement of future gyri to the outside Fig. In fact, the total force in the convoluted model is similar to that at the beginning of the simulation, but for almost twice the perimeter.

Accommodation of convolutions. In this third stage beginning at iteration , the newly developed convolutions are accommodated in the available perimeter and eventually fuse.

Figure 2 A shows an example of such an accommodation. As the size of convolutions depends mainly on the thickness and mechanical properties of the cortical layer and not on its total perimeter, convolutions can develop that do not have enough space to be completely deployed in the available cortical perimeter.

In subsequent iterations they fuse to form a composed convolution at the right side arrow ab, iteration , which decreases the total number of convolutions from 13 to Without the plastic component of the model, this stage of accommodation could last indeterminately.

The adaptation of the radial elements and the cortical layer to deformations of long duration ensures the stability of the model.

Gyral and sulcal thickness. Mean percentual thickness of the cortical layer for gyri and sulci. Gyral and sulcal thickness have been evaluated in the model after iterations. Thickness is reported as a percent of the mean radius of the cortical layer. Error bars represent SD. One striking characteristic of the mammalian brain is the relationship observed between cortical surface and the total brain volume through evolution.

If the brains of different species were just scaled versions of one another, the cortical surface should vary to two-thirds the power of the brain volume. However, allometric studies of this relationship show that in gyrencephalic brains the cortical surface increases almost linearly with brain volume Jerison, , ; Prothero and Sundsten, ; Hofman, ; Prothero, ; Changizi, Hence, during brain growth, gyrification allows for an increasingly large cortical surface.

We have studied this relationship by analysing the variation of the perimeter of the cortical layer as a function of the total area of the model.

In our analysis the cortical layer's perimeter represents the cortical surface area, and the total brain volume is represented by the area of the smallest disk containing the model. The different models are obtained by setting different values for the carrying capacity parameter K in the logistic-growth function equation 5 , which controls the total growth of the cortical layer.

If different models were simply scaled versions of one another, the perimeter of the cortical layer should vary as the square root of the area of the model.

This is the case when the values of K produce non-convoluted models. However, when the models develop convolutions the growth of the cortical layer's perimeter dissociates from the growth of the total area, and the perimeter of the cortical layer is larger than expected by homothetic scaling.

In Figure 4 the coefficient of the relation becomes closer to 1, and the perimeter increases almost linearly with the area. Figure 4 shows also that thinner cortical layers have more convolutions, and that these convolutions begin to develop earlier than for the thicker layers.

Cortical layer perimeter versus total area. Before convolutions develop, the perimeter grows proportionally to the square root of the area slope 0. This relation becomes close to linear when convolutions develop. Thinner cortices have more convolutions which begin to develop earlier q h , thickness of the cortical layer; n , average number of convolutions. The graphic shows the perimeter versus area data points for 20 different growths K between 0 and and three different thicknesses of the cortical layer.

Grey zones indicate data points belonging to a convoluted model. Our model suggests that without any additional hypotheses the simple mechanical properties of the cortex are sufficient to produce cortical folding. Yet, the structural asymmetries induced by the early regionalization of the cerebral cortex can trigger the development of convolutions and influence their distribution. We have studied three types of asymmetries, illustrated in Figures 5—11 :. It has been observed that convolutions in fact appear to follow the global geometry of the brain, i.

Mechanic asymmetries, which can be associated with the early differences in cortical cytoarchitecture. Regulatory genes have been found to express in discrete zones and gradients Levitt et al.

In the adult brain many cytoarchitectonic areas with well-defined borders can be identified; however, the existence of gradual variations between cytoarchitectonic fields is controversial Horton, Growth asymmetries, which can be associated with the different growth dynamics of different cortical regions. For example, the growth of the striate cortex in the macaque is up to 2. Step differences in growth can also represent the decreased growth of a cortical region caused by the disruption of its normal afferents Goldman-Rakic, ; Rakic, ; Dehay et al.

Growth asymmetries can also be gradual, and antero-posterior and latero-medial gradients in the cortical proliferative kinetics have been identified at early stages in development Sanderson and Weller, ; Levitt et al. Geometric asymmetries: perturbation. Development of convolutions in the model where the point indicated by the arrowhead has been displaced to the centre by 0. The letters a, b, … signal the same convolution at different iteration steps.

Geometric asymmetries: order of development. Evolution of convolutional development in the perturbed model of Figure 5. First convolutions to develop are deeper in the final configuration, and induce a cascade of smaller secondary and tertiary convolutions in the order a, b, c, d, e. Geometric asymmetries: elliptical shape. Development of convolutions in an elliptical model.

Convolutions begin to develop at the minor axis of the model and propagate to the sides. The letters indicate the same convolutions throughout iteration steps. Mechanic asymmetries: step. Development of convolutions in the model with a step in the elastic constant of the cortical layer.

The elastic constant of the cortical layer k c is smaller at the top of the model more elastic. A convolution develops at the border and induces decreasing convolutions to the sides. A model with the same shape, growth, and mean elasticity does not develop detectable convolutions. Letters indicate the same convolution throughout iteration steps. Mechanic asymmetries: gradient. Development of convolutions in the model with a gradient in the elastic constant of the cortical layer more elastic at top, i.

Convolutions develop simultaneously and are more pronounced as the elasticity decreases. A model with the same shape, growth, and mean elasticity does not develop convolutions. Growth asymmetries: step. Development of convolutions in the model with a step in the growth of the cortical layer value of the parameter K. The growth is bigger at the top of the model.

Convolutions develop at the side with more growth. A model with the same shape, mechanical properties, and mean growth does not develop convolutions. Growth asymmetries: gradient. Development of convolutions in the model with a gradient in the growth of the cortical layer smaller growth at bottom. Convolutions develop simultaneously and are less pronounced as the growth decreases.

Figure 5 illustrates the effect of the displacement of a single point of the cortical layer in the initial configuration of the model. At the position of the perturbation arrowhead in Fig. Figure 6 shows the evolution of the size of convolutions labelled a—d in Figure 5. At the end of the simulation the first convolutions to develop are the deepest ones, while those developing afterwards are progressively shallower. Convolutions are termed primary, secondary or tertiary, according to their time of development and final depth.

Figure 7 shows the effect of a global change in the initial geometry of the model. Instead of a circular initial shape, we have used an elliptical model. We observe the development of a cascade of convolutions beginning at the minor axis of the ellipse followed by convolutions of decreasing depth toward both sides. This kind of geometric asymmetry may induce the development of convolutions in an otherwise smooth model convolutions are detected through the zero-crossings of the first derivative of the external contour of the cortical layer.

Figure 8 illustrates the effect of mechanic asymmetries on the model by introducing steps and gradients in the elastic constant of the cortical layer k c. For the mechanic step, the elastic constants of the top and bottom halves of the cortical layer have been set to different values Fig. The simulation shows the development of a convolution at the border between both halves. This triggers a cascade of convolutions which become progressively smaller towards the sides being however deeper in the more elastic half.

The exact location of the elasticity step is not in a sulcal fundus nor in a gyral crown, but at the inflexion. The behaviour of the model with a top-down gradient in the elastic constant of the cortical layer k c is shown in Figure 9. We observe that convolutions develop simultaneously approximately at iteration and that they are deeper in the less rigid half. Figure 10 illustrates the effect of asymmetries in the growth of the cortical layer by introducing steps and gradients in the carrying capacity parameter K of the logistic-growth function.

When a growth step is introduced between the top and bottom halves of the cortical layer, the first convolutions develop at the border between them, and then propagate to the sides developing progressively shallower convolutions. Convolutions are deeper in the region with more growth Fig. The effect of a top-down gradient in the carrying capacity of the cortical layer is shown in Figure To the bottom, the zone of less growth develops less pronounced convolutions, which increase in depth as we move to the zone of more growth.

The convolutions in the model develop simultaneously approximately at iteration The ontogeny of cortical gyrification in the mammalian brain follows three main stages: i a period of growth where the hemispheres remain smooth; ii a stage of rapid convolution development; and iii a long period of accommodation without changes in the degree of convolution Connolly, ; Chi et al.

These three stages are reproduced by our model simply by the action of cortical growth Fig. Our model suggests that cortical growth can induce the development of convolutions by itself, and does not require the specific folding mechanisms that have been proposed, such as resistance opposed by the cranium Le Gros Clark, , differential growth of the inner and outer cortical layers Richman et al.

Our results see Fig. Furthermore, the apparent immobility of the sulcal fundi, reported by Smart and McSherry a , and the differences in thickness between gyri and sulci, which represent a major justification for the gyrogenetic theory of Welker , may also be an effect of growth-driven cortical folding.

In Figure 2 , the position of the sulcal fundi at the beginning of the simulation for the smooth model is almost the same as at the end, when convolutions have developed see Fig. Figure 3 shows that gyri can be significantly thicker than sulci for convolutions developed by cortical growth alone.

Many recent experimental studies have shown the importance of cortical growth for cortical folding and show, for example, that a global increase in cortical growth induces convolutions in the normally lissencephalic mouse brain Haydar et al.

In our model, decreasing the growth of the cortical layer prevents the development of folds, while increasing the growth can induce folding in an otherwise smooth model. The number and size of cortical convolutions vary in different gyrencephalic brains. The degree of convolution of a brain is frequently estimated according to the gyrification index introduced by Zilles et al.

This index is defined as the ratio between the length of the pial contour and the length of the external contour in a coronal section. In our model, cortical growth plays a fundamental role in the development and deepening of convolutions, yet their number and size are determined by the thickness of the cortical layer. As illustrated in Figure 4 , thin cortical layers develop more convolutions than thicker ones which also begin to develop earlier.

If the degree of convolution of our model were to be studied according to the gyrification index, this parameter would not discern between the effect of cortical growth and thickness. In fact, the gyrification index does not allow us to distinguish between a cortex with a small number of deep convolutions and one with a large number of superficial convolutions. Based on our model, we suggest that a measure of the relative thickness, such as the ratio between the mean cortical thickness and the external contour length, should effectively complement the gyrification index.

The allometric study of different parameters such as body size, brain size and white matter volume among different mammalian species is a major tool for investigating brain evolution Jerison, In particular, allometric studies show that in gyrencephalic brains the cortical surface area increases almost proportionally to the brain volume.

This relationship reveals a trend towards the expansion of the cerebral cortex in larger mammals Rakic, ; Caviness et al. The increased degree of convolution of larger brains results in an increased cortical surface without a comparable increase in the total cerebral volume.

Our model is able to reproduce this relationship, and the increase in the cortical layer's perimeter of a convoluted model is larger than for a smooth model of equal total area see Cortical Surface and Brain Volume. There are cases, however, where the linear relationship between cortical surface area and brain volume does not applies. A well-known exception is the manatee, a large marine mammal with an almost lissencephalic cortex. The lack of convolutions in the manatee brain can be explained by the dependence of the degree of convolution on cortical thickness, as suggested by our model.

The manatee's cortex is in fact exceptionally thick, with a mean cortical thickness of 4 mm Reep and O'Shea, — compared, for example, with the human brain, whose mean cortical thickness is 2.

On the other hand, cetaceans such as whales and dolphins have thin and profusely convoluted cortices, in agreement with our results.

Experimental support for the idea of the dependence of the degree of convolution on cortical thickness can be found in the results of allometric studies of Hofman and Prothero The relationship between cortical surface area and brain volume presents a strong correlation.

Cortical volume associated with cortical growth is the parameter which most strongly correlates with brain volume, while the weakest correlation is observed for the mean cortical thickness. This may indicate that the main linear trend in the evolution of surface area depends on cortical growth, while the most plausible source of variability is the mean cortical thickness. The steady growth of the cortical layer in the morphogenetic model produces convolutions of the same size and with almost the same time of development.

This type of folding is different from that observed in gyrencephalic brains, which show gradients in the degree of convolution and a hierarchical organization of sulci with differences in their sizes and onset time.

Our results suggest that this organization can be produced by early asymmetries gradual or abrupt changes in the geometry, mechanical properties and growth dynamic of the cerebral cortex, which can induce and guide the development of convolutions. After being exposed to the solvent for sixteen minutes, the brain model developed convolutions and folds that greatly resembled those of real brains. In addition, the stages in the development of these shapes were similar to those observed via MRI imaging.

Although the results already supported the hypothesis presented in , the researchers still wanted to improve on their simulations. This process has certain limitations.

First of all, the observations reveal a notable asymmetry between the two hemispheres of the imitation brain. Why is this? Secondly, due to the need to ensure the feasibility of the experiment, the researchers considered the growth of the cortex to be uniform, although they knew this was not the case. Finally, the mechanical model is not yet able to attain a thirtyfold increase in its volume by absorbing the solvent, as the human brain does during its growth.

The researchers will seek to correct all of these details in the next stages of their work. And their exploration of this subject does not stop there.

Beyond improvements to the model, the scientists want to take the simulations a step further, seeking to make the folding process take place in reverse order.