The brain is one of the most fascinating organs in the human body. Its complex structure on both micro- and macroscopic scales closely correlates with the unique cognitive abilities of humans. Cortical folding is one of the most important features of the human brain. Still, compared to other mammals, the human brain is neither the largest nor the most folded brain. However, relative to its size, it has the largest number of cortical neurons that connect with billions of neuronal synapses (Herculano-Houzel, 2009). This fact attracted the attention of neuroscientists over the past few years to explore the source of these cells and how they develop in the early stages of brain development.

The number of brain cells is determined in utero through the proliferation process. Previous studies on different lissencephalic species, such as mice, have shown that cell division in the brain is confined to a small region near the cerebral ventricles (Hansen et al., 2010). However, in gyrencephalic species, this seems to be different. Recent findings show that the human brain, for example, is characterized by two proliferation zones with two different types of progenitor cells. Both zones produce neurons that later migrate towards the outer brain surface and form the cortex Lui et al., 2011; Pebworth et al., 2021.

In rodents, progenitor cells around the ventricular zone (VZ) generate intermediate progenitor cells as their daughters, which accumulate above the VZ and form a new layer called the subventricular zone (Noctor et al., 2002). In humans, there is an additional outer layer of the subventricular zone, often referred to as outer subventricular zone (OSVZ) (Hansen et al., 2010; Lui et al., 2011; Noctor et al., 2007). This zone was first discovered in the monkey brain by Colette Dehay and her colleagues (Smart et al., 2002), and confirmed in the human brain by several following studies (Huttner and Kosodo, 2005). The OSVZ seems to play a significant role in the proliferation process and affects the size and complexity of the human cortex. The evidence for this allegation is the wave of cortical neurogenesis that coincides with the cell division in the OSVZ (Lukaszewicz et al., 2005). At the macroscopic scale, the high proliferation in the OSVZ coincides with a significant tangential expansion of the cortical layers. The latter is an essential factor for the formation of cortical folds (Reillo et al., 2011). Still, it remains unknown, how exactly this proliferation process in the OSVZ affects gyrification of the forming cortex. Different approaches have been used to understand the relation between cellular mechanisms at the microscopic scale and cortical development at the macroscopic scale. Genetic analyses and experimental studies using cell culture models and brain organoids have given first valuable insights concerning the source of cells and their behavior (Hansen et al., 2010). Here, we intend to complement these studies by using a numerical approach to bridge the scales from the behavior of different progenitor cell types at the cell scale to the emergence of cortical folds at the tissue or organ scale.

From a mechanics point of view, forces that are generated due to cellular processes may act as a link to understand the underlying mechanisms behind cortical folding (Budday et al., 2015b). Many previous studies tried to explain normal and abnormal cortical folding either from a purely biological or mechanical perspective (Tallinen et al., 2014; Razavi et al., 2015). However, it will not be possible to capture the folding mechanism without considering both perspectives at the same time (de Rooij and Kuhl, 2018; Zarzor et al., 2021; Wang et al., 2022). In other words, to fully understand the physiological and pathological mechanisms underlying cortical folding in the developing human brain, we need to study the coupling between cellular processes and mechanical forces – to eventually assess how disruption of cellular processes affect the folding pattern and lead to malformations of cortical development (Guerrini et al., 2008; Blumcke et al., 2021; Llinares-Benadero and Borrell, 2019).

To fill this knowledge gap, we establish a two-field computational model that accounts for both proliferating zones in the human brain, the VZ and OSVZ. The first field in the model describes the growth and deformation of brain tissue based on the theory of finite growth (Rodriguez et al., 1994; Göktepe et al., 2010). The second field describes the cellular processes occurring during human brain development, where we use an advection-diffusion equation to mimic the proliferation and migration in the subcortex and neuronal connectivity in the cortex (de Rooij and Kuhl, 2018; Zarzor et al., 2021). We add two source terms to consider the division in different zones. We validate our model through a comparison of the simulation results with histologically stained sections of the human fetal brain and address unresolved questions regarding the role of the OSVZ for cortical folding.

The central cellular unit that plays a critical role in essential processes of brain development is a type of progenitor cells called radial glial cells. In the early stage of brain development, when neurogenesis begins, neuroepithelial cells transform into radial glial cells (Noctor et al., 2007). Around gestational week 5, these cells locate near cerebral ventricles, in the VZ, where they undergo interkinetic nuclear migration. The associated symmetric division behavior leads to a significant increase in the number of radial glial cells and results in both increased thickness and surface area of the VZ (Blows, 2003; Fish et al., 2008; Bystron et al., 2008). Subsequently, the cells switch to an asymmetric division behavior and generate intermediate progenitor cells (Noctor et al., 2004). The latter migrate to the subventricular zone, where they proliferate and produce neurons, as illustrated in Figure 1 (Noctor et al., 2007; Pebworth et al., 2021). Ultimately, the majority of cortical neurons are produced by intermediate progenitor cells (Lui et al., 2011; Libé-Philippot and Vanderhaeghen, 2021).

Figure 1

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According to the radial unit hypothesis proposed by Pasko Rakic over 30 years ago, the radial glial cell fibers organize the migration process, which starts around gestational week 6 (Nonaka-Kinoshita et al., 2013). He postulated that these fibers form a scaffold to guide neurons during their migration from the proliferating zones to their final destination in the cortex, which forms the outer brain surface (Rakic, 1988; Lui et al., 2011). In gyrencephalic species, those fibers have a characteristic fan-like distribution (Borrell and Götz, 2014; Nonaka-Kinoshita et al., 2013). However, it is still under debate whether this unique distribution is the cause or rather the result of cortical folding. Our recent computational analyses support the latter, i.e., that it is the result of cortical folding (Zarzor et al., 2021). The migration process synchronizes with a radial expansion of all brain layers. Still, the VZ does not expand remarkably as the intermediate progenitor cells move outwards to the subventricular zone. The migrated neurons finally organize themselves in the six-layered cortex in an inside-out sequence, where the early-born neurons occupy the inner layers (Gilmore and Herrup, 1997). Until gestational week 23, the outer brain surface is still smooth, although the bottom four layers of the cortex are already filled with neurons (Shinmyo et al., 2017). The first folds appear between gestational weeks 20 and 28, as the cortical layer significantly expands tangentially (Budday et al., 2015b; Habas et al., 2012). Importantly, at around gestational week 25, the cortical neuronal connectivity emerges and comes along with the horizontal elongation of neuronal dendrites (Takahashi et al., 2012).

While the processes summarized above are common among mammals, the human brain has some specific features that play a significant role in increasing the number of cortical neurons, and enhancing the complexity of cortical folds (Libé-Philippot and Vanderhaeghen, 2021). At the beginning of the second trimester, around gestational week 11, the original radial glial cells switch from producing intermediate progenitor cells to producing a special kind of cells that is found in all gyrencephalic species but is enriched in the human brain. The newly generated cells are similar to the radial glial cells in terms of shape and function, but unlike the original radial glial cells, they migrate to the outer layer of the subventricular zone, referred to as OSVZ, after they are born. Therefore, they are called outer radial glial cells (ORGCs) (Lui et al., 2011; Fietz et al., 2010; Hansen et al., 2010; Reillo et al., 2011; Nonaka-Kinoshita et al., 2013). We would like to note that some literature refers to this type of cells as basal radial glial cells. While the original radial glial cells have a bipolar morphology with two processes – one extending to the cerebral ventricle and one to the outer cortical surface – the ORGCs have a distinct unipolar structure with only a single process extending to the outer cortical surface (Hansen et al., 2010; Betizeau et al., 2013; Reillo et al., 2011; Nonaka-Kinoshita et al., 2013).

The OSVZ shows a significantly more pronounced radial expansion compared to the inner subventricular zone and VZ between gestational weeks 11.5 and 32. The immediate reason causing this difference is the characteristic division behavior of ORGCs: they translocate rapidly in radial direction before they divide, which scientists have referred to as ‘mitotic small translocation (MST)’ (Fietz et al., 2010). Importantly, the MST behavior pushes the boundary of the OSVZ outward, which increases its capacity to produce new neurons. The intermediate progenitor cells have enough space to undergo multiple rounds of division before producing neurons, which increases the overall number of generated neurons (Kriegstein et al., 2006; Lui et al., 2011).

The ORGCs, like radial glial cells, play an important role in the proliferation process: they divide symmetrically and asymmetrically to produce further ORGCs and intermediate progenitor cells (Libé-Philippot and Vanderhaeghen, 2021). Intermediate progenitor cells divide to generate a pair of neurons (Lui et al., 2011). According to previous studies, 40% of produced neurons are generated by ORGCs at gestational week 13, but this ratio increases to 60% by gestational week 14, and exceeds 75% by gestational week 15.5. Then, after gestational week 17, the ORGCs become the only source of cortical neurons in the upper cortical layers (Hansen et al., 2010). Besides their role in increasing the number of neurons, the ORGCs generate additional scaffolds that elongate to the outer brain surface and serve as paths for neuronal migration (Llinares-Benadero and Borrell, 2019; Nonaka-Kinoshita et al., 2013). Compared to other mammals, the neurogenesis of the human cortex can thus be divided into two main stages. The first stage is characterized by migration along a continuous scaffold consisting of radial glial cell fibers, which run from the ventricular surface to the outer cortical layer around gestational week 15. During the second stage, the migration path switches to a discontinuous form. After gestational week 17, the radial glial cell fibers run from the ventricular surface to the inner subventricular zone, while ORGC fibers run from the OSVZ to the outer brain surface, as indicated in Figure 1 (Nowakowski et al., 2016). Consequently, migrating neurons follow a sinuous path through numerous radial fibers before they reach its final location in the cortex (Lui et al., 2011). The additional scaffolds formed by ORGC fibers are not only important for neuronal migration, but also for the tangential expansion of the cortex. Previous studies show that a reduced number of ORGCs lead to reduced tangential expansion. In these cases, the cortex is less folded or even lissencephalic (Poluch and Juliano, 2015). In contrast, increasing the number of ORGCs leads to more excessive folding (Florio et al., 2017; Borrell, 2018). Still, it remains unknown whether these effects are a result of the specific proliferation behavior of ORGCs or the associated scaffold of ORGC fibers. What is known, though, is that the existence of ORGCs is a necessary but not sufficient condition for cortical folding (Llinares-Benadero and Borrell, 2019).

Around gestational week 28, the migrating neurons occupy the first (top) cortical layer, while the migration and proliferation processes come to an end. After finishing their role during the neurogenesis stage, radial glial cells and ORGCs switch to produce different types of glial cells, e.g., astrocytes and oligodendrocytes (Schmechel and Rakic, 1979). Also, radial glial cells may later convert into ependymal cells that locate around the cerebral ventricles. The oligodendrocytes form myelin sheaths around neuronal axons, wherefore the subcortical layer gains its characteristic white color.

To numerically study the effect of the VZ and the OSVZ on the resulting folding pattern, we simulate human brain development by using the finite element method. The influence of various factors on the emergence of cortical folds can be best shown on a simple two-dimensional (2D) quarter-circular geometry (Darayi et al., 2022), as illustrated in Figure 2A. In addition, we also investigate the folding evolution on a simplified half-sphere three-dimensional (3D) geometry. In the following, we introduce the main equations describing the coupling between cellular mechanisms in different proliferating zones and cortical folding, which we solve numerically.

Figure 2

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We validate our computational model by comparing the simulation results with histologically stained sections of the human fetal brain. For details on the corresponding preparation and staining, we refer to Zarzor et al., 2021. The sections belong to human fetuses aborted at gestational weeks 17, 24, 30, and 34. After defining the areas representative of the simulation domain introduced in the previous section (see Figure 2A) for each gestational week, we detect and assess the cell density using the software Qupath. This study was approved by the ethics review board of the Friedrich-Alexander-Universität Erlangen-Nürnberg with the reference number 22-209-Bp, and all procedures were conducted in accordance with the Declaration of Helsinki. In the following, we summarize the steps we followed to determine the cell density in human fetal brain sections. Since this analysis is intended to mainly serve as a means of comparison between different gestational weeks, a more detailed analysis using individual markers was not necessary at this stage.

1. Identify regions of interest to which the analysis should be applied, as shown in Figure 3A.

2. Pre-process the annotated area to ensure better cell detection through image color transfer, contrast ratio modification, and stain vector settings.

3. Automatically detect the cells in the relevant area by using the ‘positive cell detection’ command in Qupath, which distinguishes between cell types according to the staining. Here, the negative cells (blue) are mostly glial cells or extracellular matrix components, while positive cells (red) are mostly neurons or progenitor cells, as shown in Figure 3B. While this distinction between cell types might not be fully accurate, the results are satisfactory for our specific application.

4. Count the nearby detections in a small circle with an arbitrarily chosen radius $ϵ=100\mathrm{\mu }\mathrm{m}$ around each detected positive cell (i) to determine the corresponding cell density by dividing the number of nearby detections by the circular area, as demonstrated in Figure 3C.

5. Visualize the cell density, as illustrated in Figure 3A.

Figure 3

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Figure 3 shows the cell density distribution around gestational week 17 and demonstrates the densely packed VZ due to the high proliferation rate of radial glial cells, with about ${\displaystyle 13,600\text{cells}/\mathrm{m}\mathrm{m}{}^2}$. In addition, we can locate the other zones introduced in Figure 1. The OSVZ shows a higher cell density (approximately ${\displaystyle 9550\text{cell}/\mathrm{m}\mathrm{m}{}^2}$) than both the inner subventricular and the intermediate zone. This zone is composed of several types of cells, the original intermediate progenitor cells that migrated from the VZ, migrated neurons, ORGCs that produce more intermediate progenitor cells, and newborn neurons that are produced through the intermediate progenitor cells’ asymmetric division. Our analysis of the histologically stained sections of the human fetal brain (see Figure 3) also shows that the OSVZ is 1.5 times thicker than the inner subventricular zone, even though it did not emerge at an earlier stage of development (the inner subventricular zone emerges around gestational week 7 and the OSVZ around gestational week 11). This implies that the thickness of the OSVZ increases with time. The intermediate zone is characterized by a low cell density with about ${\displaystyle 1600\text{cells}/\mathrm{m}\mathrm{m}{}^2}$. Still, it is a transit area for the migrating neurons. The higher cell density in the inner subventricular zone ${\displaystyle 4800\text{cells}/\mathrm{m}\mathrm{m}{}^2}$ corroborates the presence of another type of cell besides migrating neurons, i.e., intermediate progenitor cells. The migration process in gestational week 17 is still ongoing – the cortex is not yet fully developed and filled with neurons with only about ${\displaystyle 12,000\text{cells}/\mathrm{m}\mathrm{m}{}^2}$. Therefore, at this stage of development, the VZ still has the highest cell density in the brain. Figure 4 compares the cell density distribution along a line from the ventricular surface to the outer cortical surface in the human fetal brain at gestational week 17 (indicated by line L in Figure 3) with different simulation results. For better comparability and to avoid differences in the dimensions between the histologically stained sections of the human fetal brain and the simulation domain, we normalize the domain’s radius according to the extension from the ventricular to the outer cortical surface in the stained sections. In addition, we normalize the cell density with respect to its maximum value in the cortex. According to our previous work (Zarzor et al., 2021), we include a varying stiffness in the cortex during human brain development in our simulations, adopt a stiffness ratio of 3, and a division rate in the VZ of 120.

Figure 4

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The simulation results for an initial division rate in the OSVZ of $G_{\text{osvz}}=10$ well capture the trends observed in the histologically stained sections of the human fetal brain. The cell density shows a first local peak representing the VZ for a normalized radius of approximately 0.05. It then gradually decreases to reach its first local minimum in the inner subventricular zone for a normalized radius of 0.3. This effect is less pronounced in the simulations than in the actual human fetal brain. The curves start to rise again in the OSVZ to reach the second peak for a normalized radius of 0.4. Again, the simulation results capture this peak quite accurately, with a difference of only 3%. The second local minimum represents the intermediate zone for a normalized radius of 0.7, while the third peak represents the cortex. The simulation results for $G_{\text{osvz}}=20$ and 30 result in a higher cell density in the OSVZ than in the actual human fetal brain. In contrast, the curve for $G_{\text{osvz}}=0$ (i.e., without including the effect of the OSVZ) shows a significantly decreased cell density in the OSVZ and intermediate zone.

In this section, we apply our computational model to answer some of the major questions regarding the role of the OSVZ for cortical folding during human brain development. As mentioned above, we consider both cases, constant and varying cortical stiffness. Before addressing the individual questions, we would like to first highlight some general features of the mechanical instability problem as the underlying mechanism of cortical folding. Figure 5 shows an exemplary temporal course of both sulcus depth and folding evolution – defined as the ratio between the outer perimeter at time step $t$ and the initial perimeter (see Figure 6) – for the varying stiffness case with $G_{\text{vz}}=120$ and $G_{\text{osvz}}=20$. The outer brain maintains its smooth surface until the first instability point, where compressive stresses in the cortex induced by cell-density-driven cortical growth reach a critical value, the cortex starts to fold and sulci deepen rapidly and uniformly. At the instability point, the curve of the folding evolution shows a kink, as the outer brain surface now expands more rapidly. After a second instability point, where a secondary instability occurs and we see a pitchfork-like bifurcation in Figure 5, only every second sulcus continues to deepen, while those in between become shallower again. We refer to the resulting folding pattern as period doubling pattern. For a detailed discussion of secondary instabilities, period doubling, and even period tripling patterns and their role in cortical folding, we refer to our previous works in Budday et al., 2015a and Budday et al., 2015c.

Figure 5

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Figure 6

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How does cell proliferation in the OSVZ affect cortical folding patterns?

Many previous studies have tried to address this point (Hansen et al., 2010). However, through purely experimental approaches, it is difficult to answer this question. In our model, we can apply different values of the division rate in the OSVZ ($G_{\text{osvz}}$) to show how ORGC proliferation affects cortical folding. Figure 7 shows the folding patterns emerging at gestational week 36 for both varying and constant stiffness cases and different values of the initial division rate in the OSVZ $G_{\text{osvz}}$. In the case of constant stiffness, the effect is marginal. In the case of varying stiffness, there is a more noticeable change in the folding patterns. In general, the distance between neighboring sulci decreases with increasing $G_{\text{osvz}}$, as marked in Figure 7. For the displayed cases, the distance decreases from d = $8.796\mathrm{mm}$ for $G_{\text{osvz}}=0$ to d = $8.67\mathrm{mm}$ for $G_{\text{osvz}}=10$ and d = $8.2\mathrm{mm}$ for $G_{\text{osvz}}=20$. Interestingly, the cortical thickness and effective stiffness ratio at the first instability point (denoted by w in Figure 5) are the same for all these cases. Therefore, we attribute the observed differences to the faster increase in the cell density and thus cortical growth, cortical stiffness, and the effective stiffness after the instability has been initiated. In addition, we observe period doubling patterns emerge (Budday et al., 2015a; Budday et al., 2015c), which are most pronounced at a value of $G_{\text{osvz}}=20$. This indicates that the proliferation in the OSVZ enhances secondary mechanical instabilities and leads to more complex folding patterns earlier.

Figure 7

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Figure 8 demonstrates that the observed trends also hold true when extending the model to 3D. For the case of varying stiffness with a stiffness ratio of 3, a growth ratio of 3, and an initial division rate in the VZ $G_{\text{vz}}=600$, the folding complexity increases with increasing initial division rate in the OSVZ $G_{\text{osvz}}$.

Figure 8

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Besides the direct relation between the proliferation in the OSVZ and the folding morphology, there are indirect effects and other aspects concerning ORGC proliferation, which will be discussed in more detail in the following sections.

How does cell proliferation in the OSVZ affect the cell density and folding evolution?

After we have assessed the effect of cell proliferation in the OSVZ on the final folding pattern, we investigate its effect on the evolution of both cell density and folding morphology. Figure 9 shows the temporal evolution of the maximum cell density in the domain and the folding evolution between time steps 260 and 380 for different initial division rates in the OSVZ $G_{\text{osvz}}$ and a constant division rate in the VZ $G_{\text{vz}}=120$. Again, we consider both cases, constant and varying cortical stiffness. Increasing the initial division rate in the OSVZ leads to a significant increase in both the cell density and folding evolution. Consequently, for the case of $G_{\text{osvz}}=30$, the cell density reaches the highest value of ${\displaystyle 1100\mathrm{m}{\mathrm{m}}^{-2}}$, corresponding to a folding evolution of 1.25. For the case of $G_{\text{osvz}}=0$, in contrast, the cell density does not even exceed the migration threshold. Comparing the cases $G_{\text{osvz}}=20$ and $G_{\text{osvz}}=30$ shows that the instability occurs earlier with increasing initial division rate in the OSVZ $G_{\text{osvz}}$. Thus, the ORGC proliferation decreases the time required to reach the final folding pattern. In general, the differences between the results for the constant and varying cortical stiffness case are minor. However, the curves for the varying cortical stiffness case rise faster than for the constant case. We note that we could not generate results for the initial division rate in the OSVZ $G_{\text{osvz}}=30$ and the constant cortical stiffness case due to numerical issues.

Figure 9

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Which proliferation zone is more influential for fetal human brain development?

To answer the question which one of the proliferation zones (VZ or OSVZ) is more important for cellular brain development and cortical folding, we have implemented four sets of division rates. The first set assumes a high initial division rate in the VZ $G_{\text{vz}}$ and a low initial division rate in the OSVZ $G_{\text{osvz}}$. For the following parameter sets, we gradually decrease $G_{\text{vz}}$, while we increase $G_{\text{osvz}}$. Figure 10 shows the corresponding results for the maximum cell density in the domain and the folding evolution between time steps 250 and 450 for both cases, constant and varying cortical stiffness. Unexpectedly, the cell density for the set ($G_{\text{vz}}=30,G_{\text{osvz}}=30$) rises faster than the one for the set ($G_{\text{vz}}=120,G_{\text{osvz}}=0$). Indeed, not only the cell density is affected by increasing the division rate in the OSVZ (at the expense of decreasing it in the VZ), but also convolutions (cortical folds) appear earlier, as illustrated by the curve for the folding evolution (Figure 10, right). Our results thus indicate an unproportionally strong effect of ORGC proliferation in the OSVZ on cellular brain development and cortical folding. We attribute this observation to the larger volume occupied by the OSVZ compared to the VZ. Concerning the difference between the results for the constant and varying cortical stiffness case, the curves for a varying stiffness rise faster than for a constant stiffness.

Figure 10

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How does the MST behavior of ORGCs affect brain development?

We have introduced a specific simulation parameter to capture the MST behavior of ORGCs, which now allows us to study the effect of the MST factor on cortical folding. For this parameter study, we limit ourselves to the varying cortical stiffness case with an initial division rate in the VZ $G_{\text{vz}}=30$ and an initial division rate in the OSVZ $G_{\text{osvz}}=30$. Figure 11 shows the temporal evolution of the maximum cell density and the folding evolution between time steps 250 and 430 for different values of the MST factor. Our simulations indicate that with increasing MST factor, the value of the maximum cell density in the domain increases exponentially. Since the MST factor incorporates the expansion of the OSVZ with time, these results are consistent with the observation in the previous section, showing an increasing spatial expansion of the OSVZ.

Figure 11

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Is the OSVZ affected by cortical folding?

After we have discussed the effect of the OSVZ on the formation of cortical folds, we will now discuss the opposite – the effect of cortical folding on the OSVZ. Although the OSVZ has a constant thickness in our model throughout the entire domain in the intermediate stage of human brain development (before cortical folds emerge), as demonstrated in red in Figure 12, left, the simulations show that this quickly changes after the first folds start to emerge. The thickness of the OSVZ starts to show spatial variations and becomes thicker beneath gyri and thinner beneath sulci, as shown in Figure 12, right. This result is consistent with what was previously observed experimentally in the human brain (Kostović et al., 2002). While it is to date not clear whether this phenomenon is rather the cause or the result of cortical folding, our study clearly indicates that the (mechanics-driven) process of cortical folding is sufficient to induce OSVZ variations. The forces generated by the mechanical growth problem not only fold the cortical layer but also lead to undulations in the deeper zones. Still, the deeper proliferating zones (i.e., inner subventricular zone and VZ) remain equally smooth as the ventricular surface.

Figure 12

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Are cortical folds affected by regional proliferation variations in the OSVZ?

Previous studies have emphasized the existence of variations in the ORGC proliferation rate in different brain regions (Hansen et al., 2010). Some of those have suggested that the proliferation rate is higher beneath gyri than beneath sulci (Borrell, 2018). Similarly, in the ferret brain, where a region close in structure to the primate’s OSVZ was found, this region shows a unique mosaic-like structure (Fietz et al., 2010; Reillo and Borrell, 2012). In this section, we aim to assess the effect of regional proliferation variations in the OSVZ on the emerging cortical folding pattern. We discuss two different heterogeneous patterns here, but have included more variations online through our user interface on GitHub, as described in the Data availability section. In the first case, the OSVZ division rate gradually decreases along the circumferential direction. In the second case, the division rate varies in a more random pattern. Figures 13 and 14 show how cortical folds develop in both cases for the varying cortical stiffness case, an initial division rate in the VZ $G_{\text{vz}}=120$, and an initial division rate in the OSVZ $G_{\text{osvz}}=20$. As expected, the evolving folding patterns slightly differ. In both cases, the first folds appear where the cell proliferation rate is highest. As expected, those regions also show a higher cell density in the cortex than regions nearby. However, both cases lead to final patterns with similar distances between sulci and folding complexity (one period doubling pattern). In addition, gyri and sulci are distributed equally – regardless of the division rate. Therefore, we may conclude that inhomogeneous cell proliferation in the OSVZ controls the location of first gyri and sulci but does not necessarily affect the distance between sulci (also referred to as folding wavelength) and the overall complexity of the emerging folding pattern. This agrees well with our previous finding that the characteristic wavelength of folding remains relatively stable for inhomogeneous cortical growth patterns (Budday and Steinmann, 2018). The simulation results are also consistent with the previously found remarkable surface expansion above the regions with higher proliferation in the OSVZ (Llinares-Benadero and Borrell, 2019).

Figure 13

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Figure 14

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In this work, we have made use of a computational model for brain growth to provide insights into the role of the OSVZ – a unique additional proliferating zone in humans – during cortical folding in the developing brain. By using computational tools, we have addressed different open questions with the aim of valuably supplementing classical experimental approaches. The advantage of the computational model is that we can, on the one hand, clearly isolate purely physical mechanisms. On the other hand, it is possible to systematically study and understand the effect of individual parameters, which is much more difficult through in vitro and in vivo experiments. Our simulations have demonstrated how the proliferation in the OSVZ enhances the complexity of cortical folding patterns in 2D and 3D. Our results show that the existence of the OSVZ particularly triggers the emergence of secondary mechanical instabilities leading to more complex folding patterns. Furthermore, the proliferation of ORGCs reduces the time required to induce the mechanical instability and thus cortical folding. Interestingly, our simulation results suggest that the generated mechanical forces not only ‘fold’ the cortex but also deeper subcortical zones including the OSVZ, which becomes thicker beneath gyri and thinner beneath sulci as a result of cortical folding. Consequently, our analyses suggest that the purely mechanics-driven process of cortical folding is sufficient to induce regional differences in the thickness of the OSVZ. Finally, our simulations reveal that inhomogeneous cell proliferation patterns in the OSVZ can control the location of first gyri and sulci but do not necessarily affect the distance between sulci and the overall complexity of the emerging folding pattern.

In conclusion, our physics-based computational modeling approach has allowed us to systematically assess the role of the OSVZ during human brain development and its effect on cortical folding – the classical hallmark of the human cortex at the organ scale. In the future, the computational framework can be used to not only better understand physiological brain development but also pathological processes – especially those involving abnormal cortical folding patterns. The computational model is able to shed new light on the interplay between the multiple processes at different scales and can help identify the main controlling parameters. However, it can only complement, not substitute, sophisticated experimental approaches that are still needed to answer questions, e.g., regarding the functional difference between progenitor cell types and corresponding lineage decisions.

The datasets and code generated and/or analyzed during the current study are available on GitHub with a user interface to change model parameters and run simulations: https://github.com/SaeedZarzor/BFSimulator (copy archived at swh:1:rev:5fd9298c69a446b5ff21fff2ecaff5ef8a0ac085).

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Author details

1. Mohammad Saeed Zarzor

   Friedrich-Alexander-Universität Erlangen-Nürnberg, Institute of Applied Mechanics, Erlangen, Germany

   Contribution

   Conceptualization, Software, Validation, Investigation, Visualization, Methodology, Writing - original draft

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-3005-6115

2. Ingmar Blumcke

   University Hospitals Erlangen, Institute of Neuropathology, Erlangen, Germany

   Contribution

   Resources, Validation, Writing - review and editing

   Competing interests

   No competing interests declared

3. Silvia Budday

   Friedrich-Alexander-Universität Erlangen-Nürnberg, Institute of Applied Mechanics, Erlangen, Germany

   Contribution

   Conceptualization, Supervision, Funding acquisition, Methodology, Writing - original draft, Project administration, Writing - review and editing

   For correspondence

   silvia.budday@fau.de

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-7072-8174

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