Why does cerebral cortex fissure and fold

The folding process might enable increases in cortical surface area while limiting accompanying increases in axonal wiring costs Klyachko and Stevens, This association between network functioning and cortical folding may mediate relationships between cortical folding and brain function.

For instance, it has been shown that the cortical ribbon is thicker in the gyral areas and thinner in the sulcal areas and neurons located in the deep layers of gyri are squeezed from the sides and appear elongated while neurons that reside in the deep layers of sulci are stretched and look flattened Hilgetag and Barbas, , In addition, gyral and sulcal architecture is intrinsically related to the functional organization of the brain.

For instance, the central sulcus separates the somatosensory cortex from the motor cortex, the calcarine sulcus separates the superior and inferior visual hemifields and even in the very complex and variable prefrontal cortices, the cortex sulcation influences the functional organization Li et al. In addition, functional differences time-frequency activity, connectivity exist between sulcal and gyral areas Jiang et al. Human cortex development is a complex and dynamic process that begins during the first weeks of pregnancy and lasts until early adulthood Figure 2.

In parallel to cellular changes, early cortex development is characterized by dramatic changes in its macroscopic morphology due to the cortical folding process that begins from 10 weeks of fetal life Feess-Higgins and Larroche, ; Nishikuni and Ribas, During the third trimester of pregnancy, the cerebral cortex changes from a relatively smooth, lissencephalic surface to a complex folded structure that closely resembles the morphology of the adult cortex. The development of folds is relatively conserved across individuals and species: primary sulci, which develop first, are the less variable and most heritable Lohmann et al.

Some folding patterns are preserved across species, complex patterns in larger brains likely emerging from simplified patterns in smaller brains Borrell and Reillo, Dedicated MRI acquisition and morphometric tools have recently allowed to map the developing cortical surface and growth patterns in fetuses as young as 20 weeks of gestational age Habas et al.

These in vivo studies confirm earlier post-mortem data Chi et al. Although some inter-individual variability is observed, the regional pattern is relatively stable over the brain surface: sulcation starts in the central region with a first wave towards the temporal, parietal and occipital lobes, and a second wave towards the frontal lobe Ruoss et al.

In contrast to the primary sulci that highly consistently express across nearly all humans, certain secondary sulci can vary considerably between individuals Figure 1. At birth, the area of the cortical surface is three times smaller than in adults, but the cortex is similarly folded and the most variable sulci are the same in newborns and adults Hill et al. Longitudinal studies after birth indicate that the sulcal patterns are stable during development Cachia et al.

Cortical folding is a very complex process, involving factors at different scales; see Zilles et al. Briefly, several intermingled factors contribute to the fetal processes that influence the shape of the cerebral cortex, including cortical growth Kuida et al. Different hypotheses have been proposed to integrate these different factors within a coherent general theory. The measure of the cortex sulcation is a difficult issue because the cortical folds are complex 3D structures that are very variable among individuals Ono et al.

We provide below a selection of methods that can provide, directly or indirectly, information related to the sulcal patterns. Several studies of the cortex sulcation focused on the amount of cortex buried into the folds using the gyrification index GI. Such quantitative measure of the gyrification is indirectly related to some sulcal patterns; for instance, the GI is modulated by the presence vs.

This ratio was initially based on contour lengths measured in 2D sections of the brain Zilles et al. The main limitation of this measure is due to its 2D approach that cannot capture all the details of the 3D sulcal morphology. Using automatic segmentation of the cortex Mangin et al. A 3D version of the GI has been proposed, based on the ratio between the area of the sulcal surface area and the area of the convex envelope of the cortex and completed with regional and local indexes, restricted to some specific lobes or sulci Cachia et al.

The global GI can also be enriched by quantifying the amplitude of different folding wavelengths in different frequency bands Germanaud et al. This approach is of particular interest for the study of development, as there is a correspondence between these bands and primary, secondary and tertiary folds Dubois et al. Local sulcation measures, such as local GI, local curvature or fractal measures, have also been proposed Im et al. For each sulcus, it is also possible to quantify simple morphometric parameters such as the depth, length or opening of the folds distance between the two walls of each fold; Mangin et al.

More sophisticated features quantifying the complex 3D shape have also been introduced Mangin et al. However, the main limitation of all these quantitative measures is that they cannot accurately assess the qualitative features of the sulcal patterns Figure 1. In addition, these quantitative measures are state, and not trait, markers of the cortex anatomy and can therefore capture neurodegenerative as well as neurodevelopmental processes.

One of the major difficulties of devising qualitative measures of the sulcal patterns is the huge variability of the sulcal patterns Ono et al. Large sulci are often interrupted, each sulcus piece being susceptible to connecting with the others in various ways.

This recombination process often leads to ambiguous configurations for the usual anatomical nomenclature, which creates difficulties for the morphometric study of sulci. These difficulties have led to propose a generic nomenclature of cortical folding defined at a lower scale level Mangin et al. A classical way to perform such qualitative analysis of the sulcal patterns is based on visual inspection, which requires anatomical training by an expert, is time consuming and may suffer from subjective bias.

An important perspective is the development of methods for the fully automated recognition of cortical folding patterns Snyder et al. Several longitudinal studies have reported that small variations of the intrauterine environment, assessed by birth weight, are associated with differences in cognitive abilities after birth Shenkin et al.

As presented below, inter-individual variation in the sulcal patterns can therefore be used to search for prenatal differences. As detailed below, several studies analyzed the sulcal patterns of typically developed participants to investigate the long-term influence of fetal development on cognition.

Cognitive control CC , also referred as executive control or self-regulation, including inhibitory control—i. It plays an important role in academic Borst et al. It is also involved in the pathophysiology of numerous psychiatric disorders Diamond, The PCS is a complex structure that lies dorsal to the cingulate sulcus, found only in humans and chimpanzees Amiez et al.

In adults, asymmetry in the ACC sulcal pattern i. Similar behavioral findings were found during development; an asymmetrical ACC sulcal pattern is associated with increased cognitive control in children at age 5 Cachia et al. These early neurodevelopmental constraints on later cognitive efficacy are not fixed nor deterministic.

These additive effects of ACC and IFC sulcal patterns suggest that distinct early neurodevelopmental mechanisms, involving different brain regions, may contribute to CC. In addition, similarly to epigenetics, different environmental backgrounds, either after birth such as bilingualism Cachia et al. Interactions between sulcal patterns, similar to epistasis in genetics, have not yet been reported. Analysis of ACC sulcal pattern revealed that absence of the PCS in both left and right hemispheres is associated with lower reality monitoring, i.

In addition, a leftward asymmetric pattern was found to be associated with increased temperamental effortful control and decreased negative affectivity than a rightward pattern Whittle et al. This effect was found only for males.

In both females and males, a symmetric pattern was associated with increased temperamental affiliation compared to rightward asymmetric ACC sulcal pattern. Sulcal studies also revealed that academic abilities requiring intensive learning and training, such as numeracy or literacy, can also be traced back to fetal life.

Indeed, the pattern interrupted or continuous sulcus of the posterior part of the left lateral occipito-temporal sulcus OTS , which hosts the visual word form area VWFA , predicts reading abilities in year-old children Borst et al. The position of the sulcal interruption of the OTS plays a critical role since only interruption located in the posterior part of the OTS, hosting the VWFA, but not its anterior part, affects reading fluency Cachia et al.

Comparison of adults who learned to read during adulthood ex-illiterates and adults who learned to read during childhood literates revealed that age of reading acquisition modulates the effect of OTS sulcal pattern on reading abilities: interruption of the posterior left lateral OTS affected reading abilities in literates but not in ex-illiterates Cachia et al. In children with developmental dyslexia, the sulcal pattern in left parieto-temporal and occipito-temporal regions is not typical more sulcal pits basins of smaller size and correlates with reduced reading performance Im et al.

Regarding numeracy, the absence or presence of branches sectioning the horizontal branch of the intra-parietal sulcus IPS , a key region for processing numbers, was found to be related to individual differences in math fluency abilities and symbolic number comparison in children and adults Roell et al.

A large number of studies analyzed the cortex sulcation in these two disorders in order to investigate the contribution of early neurodevelopment in different clinical features, like symptomatology, age at onset, treatment resistance… We will also report more recent, but less systematic, studies of sulcation in other psychiatric.

Around one century ago, Elmert Ernest Southard visually inspected photos of cerebral cortex in order to investigate the neuropathology of schizophrenia dementia praecox and found atypical sulcal patterns, in particular in the temporal cortex in patients with hallucinations Southard, From visual inspection of clinical anatomical MRI scans, first-episode schizophrenia patients were found to have a reduction in the asymmetry of the lateral sulcus LS length, which borders the planum temporal Hoff et al.

Of note, such atypical LS asymmetry was associated with better cognitive function in patients. Using MR three-dimensional surface rendering and visual classification of the temporal lobe sulcal patterns, schizophrenia patients were found to have a more vertical orientation to the sulci in the temporal lobe in the left hemisphere Kikinis et al.

With the development, of computerized brain morphometry methods, several studies in schizophrenia then investigated the sulcation in the whole brain using 2D GI index. Several studies also investigated the sulcal patterns in the PFC, particularly for the orbitofrontal cortex OFC and the dorsal anterior cingulate cortex which presents patterns that can be reliably and easily classified from anatomical MRI. Unusual sulcal pattern distributions of OFC have been repeatedly reported in patients with established schizophrenia Nakamura et al.

OFC sulcal pattern is associated with socioeconomic status, cognitive function, symptom severity and impulsivity Nakamura et al. These abnormalities are not restricted to schizophrenia but were also reported in other psychiatric conditions e. Reduced ACC asymmetry is associated with lower executive function in patients with schizophrenia Fornito et al. In addition, a shorter PCS is associated with a predisposition to hallucinations in patients with schizophrenia Garrison et al.

In patients with adolescent onset schizophrenia, the collateral sulcus, between the parahippocampal gyrus and the anterior part of the fusiform gyrus, was found shorter compared to typically developing adolescents Penttila et al.

In patients with auditory verbal hallucinations AVH , abnormal sulcation have been found in the language-related cortex, including shorter STS Cachia et al. Finally, impaired 3D GI was also associated with visual hallucinations VH , suggesting that VH, and likely the sensory complexity of hallucinations, could be a proxy of the neurodevelopmental weight of schizophrenia Cachia et al.

In addition, an incomplete hippocampal inversion Cachia et al. Impaired sulcation have also been found in patients with bipolar disorder BD.

Of note, no difference was detected between schizophrenia patients and BD patients McIntosh et al. Abnormal GI was also investigated in regard with treatment resistance in BD.

Patients with treatment-resistant depression, either bipolar or unipolar, were found to have reduced global 3D GI, while euthymic BD patients did not differ from healthy controls or depressed patients Penttila et al. These findings support the hypothesis that depression that responds particularly poorly to treatment might involve fetal neurodevelopmental factors Monkul et al.

However, this GI reduction could also be seen as the consequence of a neurodegenerative process Monkul et al. Besides, BD patients with at least one methionine alleles of brain-derived neurotrophic factor BDNF showed more important losses in 2D GI, an effect that was associated with gray matter loss in the left hemisphere. GI in PFC may also provide intermediate phenotype to distinguish subgroup of BD patients, a first step to define genetically more homogenous subtypes of BD.

Intermediate-onset BD patients have lower global 3D GI in the left and right hemispheres and a lower regional 3D GI in the right dorsolateral PFC in comparison to both early-onset patients and healthy subjects Penttila et al. This study was replicated on a large multi-site sample of BD patients and healthy controls Sarrazin et al. Early-onset BD patients had an increased 3D regional GI in the right dorsolateral PFC and patients with a positive history of psychosis had a decreased 3D regional GI in the left superior parietal cortex.

There was no difference between the whole patient cohort and healthy subjects. These different studies suggest that BD is associated with localized, but not generalized, abnormalities of sulcation, in particular in patients with a heavy neurodevelopmental loading. More recently, deviations of the cortex sulcation have also been found in other disorders Sasabayashi et al. Impaired OFC has also been found in addiction Patti et al.

Patient with ASD also exhibit poly microgyria Piven et al. Analysis of the sulcal position revealed spatial shifting of the superior and inferior frontal sulci, the superior temporal and the Sylvian fissure Levitt et al. More recently, analysis of the perisylvian area revealed that the right anterior caudal ramus of the posterior part of the STS is longer in ASD patients and associated with social cognition deficit Hotier et al.

This spectrum of brain malformations, due to the failure of migrating neurons to reach optimal positions in the developing cortex, leads to severe cognitive deficits. The sulcal patterns offer a window on the potential fetal constraints of the brain on cognitive abilities and clinical symptoms that manifest later in life. Sulcal studies can therefore inform us as to whether individual cognitive or clinical difference is associated in part to preexisting factors related to the structure of the brain defined during the fetal period.

Because sulcal patterns are mainly determined before birth and stable across the lifespan Chi et al. However, a direct causal link has yet to be provided. For instance, for reading abilities Cachia et al. Regarding the interpretation of sulcal studies, it is important to stress that early neurodevelopmental factors assessed with the cortex sulcation only explain a part of the inter-individual variability.

Indeed, other factors, including environmental factors like socio-economic status SES , schooling, culture, physical activity, stress… also contribute to the cognition and clinical symptoms. Furthermore, beside cumulative effects of sulcation and environmental factors, some evidence suggest possible interactions between sulcation and environmental factors Gay et al. Such interaction between experiential diversity and early neurodevelopment could explain why trauma is critical in some hallucinations, but plays a minor or no role in others Luhrmann et al.

It also has to be discovered whether cortical sulcation, in addition to its effect on the cognitive efficiency and clinical symptoms, can also modulate the pedagogical and clinical interventions. In the clinical domain, if therapeutic intervention is found to have different effects in patients with different sulcal patterns, it would open new perspectives toward individualized and precision medicine.

Sulcal studies have been performed at the group level to investigate general mechanisms of early neurodevelopment on cognition and clinical symptoms. Even though statistically significant findings have been reported, it is important to emphasize the very important variability at individual level.

The translation of sulcal findings towards pedagogical or clinical applications, which requires moving from group-level to individual-level, will therefore raise complex methodological issues; see for instance Duchesnay et al. Another critical methodological issue regards the classification of the sulcal pattern that may raise some difficulties due to the very high inter-individual variability Figure 3 ; Ono et al.

Manual Garrison, or semi-automated Snyder et al. A way to overcome sulcal ambiguities is the establishment of a dictionary of the frequent local folding patterns Sun et al. Another challenging methodological perspective is the development of fully automated techniques for sulcal pattern labeling. The development of such techniques is very complex because of the possible sulcal ambiguities. Figure 3. Inter-individual variability of the sulcal patterns. A Example of individual ACC sulcal patterns in 12 healthy subjects.

B Superimposition of individual sulcal patterns in a common reference space MNI after linear spatial normalization. AC wrote the first draft of the manuscript and created the figures. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Thiery, eds. Pandya, D. Papez, J. Passingham, R. Patten, B. Paul, R. Payne, B. There are, however, a number of other evidences against a GM driven mechanism of cortical folding. Partial removal of the skull during development does not have a dramatic effect on the fissure pattern, and lesion experiments suggest that cortical folding is not primarily dependent on a disproportionate growth between cortical and subcortical structures reviewed in Kaas, Thus, the primary source of fissure formation must be sought in factors within the cortex itself — or underneath it.

Based on our findings on the scaling of cortical connectivity and WM volume in primates Herculano-Houzel et al. According to our model, rather than driving the folding of the WM surface, the folding of the external surface of the GM results from folding of the WM surface, which, in turn, results from increased tension within the WM due to increased numbers of axons composing the WM depending on their physical properties of caliber and tension.

Our model is quantitative; acknowledges that the cortex scales differently in size across mammalian orders as different power functions of its number of neurons; is therefore applicable, in principle, to all mammalian species; and makes easily testable predictions for all of them. The following is a description of the model, its assumptions, and a discussion of its implications and predictions, and how they can be tested. So far, we have found the size of the different brain structures, the numbers of cells that compose them, and their average densities, and therefore average cell size, to be parameters related to one another by power functions Herculano-Houzel et al.

Generically, one should expect brain and cortical allometric scaling rules that are valid over several orders of magnitude across species within a particular mammalian order to take the form of power laws i. This is because relations which are expected to remain valid over many orders of magnitude should not be given in terms of parameters that specify a particular size scale, that is, they should be scale-independent.

As can be easily demonstrated, a one-variable function is scale-independent if and only if it is a power law. We therefore expect to find the scaling rules that determine cortical folding to also be scale-invariant, and therefore power laws. Even more importantly, we assume that the number of neuronal cells is the main free parameter that coordinates the scaling of every other quantity of interest, measurable, or estimated.

There are several reasons for this assumption. First, neurons, rather than glia, are the first cells to populate the developing brain in large numbers, and their connectivity begins to be established at the same time as convolutions begin to form, even before the final neuronal complement is in place Goldman-Rakic, ; Goldman-Rakic and Rakic, Therefore, and in contrast to most earlier studies on brain allometry that implicitly or explicitly regarded the number of neurons as a consequence of brain size, we believe instead that any biologically plausible model of brain allometry must consider brain size, in all its aspects, to be a consequence of its number of neurons, according to scaling laws that may vary across different phyla.

Second, we have found that, in contrast to the order- and structure-specific neuronal scaling rules, the scaling of different brain structures seems to occur as a universally shared function of their numbers of glial cells, both across orders and structures Herculano-Houzel, This result, combined with the late onset of gliogenesis in post-natal development Sauvageot and Stiles, , suggests that we can assume that cortical composition is determined essentially by the number of neurons and their average mass which is itself closely related to the number of neurons by an order-specific power law ; after neurogenesis, nearly invariant glia will then infiltrate the intraneuronal space in proportion to the total neuronal mass Herculano-Houzel, The development of an adult mammalian cortex can then be viewed as a process whereby total numbers of neurons, numbers of neurons connected through the WM, their size which includes the soma, all dendrites, and an axon of a particular caliber , and cortical folding vary in lockstep, over an invariant background of glia.

Our model thus assumes that all parameters related to cortical scaling and folding can be described as power functions of the total number of cortical neurons. Note that the fact that general allometric rules exist for cortical morphology in each order, expressible as power laws of their number of neurons, does not mean that the latter is the only significant degree of freedom in brain or cortical evolution.

Rather, the power laws tell us that any other significant degrees of freedom must be present either at a substructure level, thus being erased by measurements that average over the entire structure, or at the microscopic level of detailed connectivity, which is not accessible to our methods, but also not relevant to the model at hand.

Our model, as presented for cross-species comparisons, considers total cortical volumes and areas, and average values of neuronal density and cortical thickness for the whole cortex, in line with the empirical studies that generated the numerical data used here Herculano-Houzel et al. Neuronal density is now known to vary across the cortical surface within primate species Collins et al. However, comparative studies on the scaling of cortical gyrification traditionally analyze whole-brain patterns Hofman, ; Pillay and Manger, , and attempts to understand the scaling of cortical gyrification have similarly been directed toward whole-brain comparisons.

We therefore developed our model based on average values for whole cortex that can be compared across orders, but predict that the scaling rules proposed here to govern gyrification at the level of the whole cortex might also be applicable at the local level across cortical areas.

White matter is largely composed of axons connecting neurons in the GM, mostly with each other but also with subcortical structures, along with the glial cells that support their function.

The volume of each axon is simply its cross-sectional area multiplied by its length. If there is no significant correlation between these two latter quantities which can be proven mathematically that will be the case when axon bundle volume is constrained and average signal propagation time is minimized , then the total axonal volume is the product of average axon cross-section area a , the average axonal length l , and the total number of axons present in the WM.

We can assume further that the volume of the intra-axonal space, including in particular the myelin sheath and the myelinating oligodendrocytes, is proportional to axonal volume, given the experimental support for a linear relation between axon diameter and myelin sheath diameter Sadahiro et al.

Using the common assumption of a linear relation between total number of oligodendrocytes and the total axon length Barres and Raff, , , then the total volume of the WM can be written to scale with the product of the total axon length or total number of oligodendrocytes and average axonal cross-sectional area. This is a simplifying assumption, since it can observed from direct imagery that multiple fiber orientations can be present very close to each other even at the WM—GM interface.

However, we believe on theoretical grounds that this is a reasonable if imperfect approximation of a somewhat more complex anatomical reality. Indeed, axons typically cross this interface in parallel bundles Mori et al. We must make this assumption because we are unfortunately aware of no systematic studies of the distribution axonal incidence angles in the literature, although published diffusion tensor imaging tracing studies show a clear but unquantified preference for perpendicular angles of incidence for instance, Mori et al.

A systematic variation in the average incidence angle across species would alter our results somewhat, but not appreciably except for a very large range of values 1. Finally, our model assumes that connections through the WM are formed early in development, at the same time as the GM becomes folded, an assumption that is supported by experimental evidence reviewed in Welker, ; that most axons in mammalian cortical WM are myelinated Olivares et al.

We consider that the surface of the WM—GM interface, with total area A W , is crossed nearly perpendicularly by most axons leaving or entering the WM, of an average cross-sectional area a , and which, together with the ensheathing glial cells, comprise the entirety of the WM surface. A W can thus be quantified as the product of the number of cortical neurons, N ; the fraction n of these neurons that are connected through the WM; and their cross-sectional area, a Figure 1.

The WM volume V W is the sum total of the volumes of all fibers and is thus equal to one half of the product of A W and the average axonal length in the WM, l , such that. Figure 1. Schematic of the cortical layout used in the model. The two volumes on the right illustrate the cortical gray matter top and white matter bottom. The gray matter is composed of an N number of neurons, a fraction n of which are connected through the white matter darker gray , either sending or receiving axons of an average cross-sectional area a through it.

Glial cells, which have been found to be distributed at a fairly constant density across species Herculano-Houzel, , are not shown. The surface area of the interface between the gray and white matter, A W , is given as the product nNa , and the volume of the white matter, V W , is proportional to the product of A W and the average axonal length in the white matter, l.

If the WM scales under tension, the cubic root of its volume should increase more slowly than the square root of its surface area, leading to deformation of the latter, that is, to folding of the GM—WM surface.

Thus, a F W value of exactly one implies a spherical WM, and larger values imply more convoluted forms. In this case of isometric growth, which would ensue if the WM did not scale under tension, then we would expect F W to be invariant as function of N.

Note that if we took into account a systematic variation of the incidence angle of fibers at the GM—WM interface as a power law of N , we would have to introduce a non-zero new coefficient at the expression for A w.

There is unfortunately currently no experimental way of estimating the value of such coefficient. We have assumed throughout that it is small enough to be disregarded, but should it prove to be otherwise we will have to recalculate the other coefficients accordingly, and revisit the conclusion obtained.

Given that V G scales as a power function of N , with N v , then. Average cortical thickness, therefore, scales a function of a GM-related variable the scaling of neuronal density with N , and two WM-related variables that, together with N , determine WM folding the scaling of connectivity and of axonal cross-sectional area with N.

Note that although this equation is not an exact power law, it can in practice be well approximated by one since V W and V G scale in fairly similar ways with N. In this case, F G becomes. Further, the thickness of the GM is thus a consequence of some of the same parameters that determine how the cortex folds, and not a determinant of it.

A schematic of the model is depicted in Figure 2. Figure 2. Schematic of our connectivity-driven model of the scaling of cortical folding with increasing numbers of cortical neurons N.

To the left are shown what we propose to be the fundamental parameters determining cortical folding, probably determined genetically, and which we postulate to vary alometrically with N : the fraction of cortical neurons connected through the gray matter n , the average cross-sectional area of the axons in the white matter a , the average neuronal density in the gray matter D , which is approximately proportional to the inverse of average neuronal cell volume in the gray matter , and the average axonal length in the white matter l.

Next, white matter surface A W and volume V W are organized as shown, depending on N and the scaling exponents, and thus determine the folding of the white matter surface F W. On top of A W , the gray matter becomes organized depending on the average size of its neurons, which, combined to a and n , determine cortical thickness, T.

The degree of folding of the gray matter, F G , is thus a consequence of the folding of the white matter, which is in turn dependent on how the parameters determining cortical connectivity c , a , and l scale with N.

Useful mathematical models are those that lead to a number of testable predictions. This is one major advantage of our model: it allows us to derive not only testable qualitative insights on the scaling of cortical folding, but also quantitative predictions that can be tested experimentally. Our model predicts that the folding of the GM is related to the folding of the WM, and the scaling of the former across species depends on the scaling of the latter.

We predict that WM folding scales across mammalian species with the number of cortical neurons; the fraction of these neurons that are interconnected through the WM; the average length of the myelinated fibers in the WM; and their average cross-sectional area.

GM folding then scales depending, additionally, on the scaling of the GM thickness, which in turn is determined by the scaling of neuronal density in the GM besides the scaling of connectivity and average axonal cross-sectional area in the WM. Figure 3 illustrates how the interplay across the scaling of these parameters determines cortical morphology and folding. Figure 3. Schematics of various manners of cortical scaling and folding, not shown depending on the interplay between the scaling parameters and how they vary with the number of cortical neurons as it increases left to right.

One remarkable characteristic of our model is that, in principle, it applies universally across mammalian orders and therefore describes the scaling of cortical folding universally , even though relationships such as those among folding index, cortical thickness, and cortical size are different across orders Pillay and Manger, In fact, we can predict that these relationships will be different across orders depending on the particular defining exponents that apply to each order; order-specific characteristics of the scaling of cortical folding will result from combinations of these exponents.

Thus, it is conceivable that cortical folding increases in larger brains with no change in connectivity and no change in the average cross-sectional area of the axons in the WM; with decreasing connectivity and increasing average cross-sectional area of the WM; and so forth. Indeed, one of the strengths of the model is that one can predict how the different scaling exponents will be related. In these conditions, it can be predicted that the thickness of the GM will scale depending on the scaling of neuronal density alone, and thus occur indeed with a constant number of neurons beneath the cortical surface, as intended in some models Rockel et al.

Thinner mammalian cortices are usually found to be more folded than thicker cortices of a similar size Hofman, ; Pillay and Manger, This finding has been attributed to thinner cortices being supposedly more pliable than thicker cortices, which would render the former less resistant to being folded Pillay and Manger, In contrast, our model predicts that cortical thickness is actually determined by two WM-related factors that also determine the degree of cortical folding connectivity and average axonal cross-sectional area in the WM , and a third, GM-related variable neuronal density.

Remarkably, the relationship between cortical thickness, connectivity, neuronal density, and axonal cross-sectional area predicts that a uniform number of neurons underneath a cortical surface area will only be found across species Rockel et al. In all other cases, the number of neurons underneath a cortical surface will scale with a non-zero combination of d and t.

Notice that this prediction is valid both for the scaling of the entire cerebral cortex and for different cortical areas. Ratios larger than 0 mean that axons shorten relative to isometry more slowly than the cortex thickens as it gains neurons.

Another way of thinking about F W is to express it in terms of the average axon length l and the WM characteristic length R W defined as the radius of a sphere with volume V W. Thus, the more axonal tension curtails the growth of l , the greater the F W and the more convoluted the WM becomes. Compared to a smooth WM surface, folding the WM results in axons having to travel shorter distances to connect GM neurons.

A more folded WM will have shorter axons, as a fraction of its characteristic size. Since the whole purpose of the axons in WM is to transmit signals, it makes sense to quantify how well and quickly they do it, in terms of the scaling rules obtained above. It is well known that an action potential impulse propagates along a myelinated axon in a time proportional to the axon length and inversely proportional to the square root of axon cross-sectional area.

From the equation above, it is clear that, with all else being equal, increasing axonal thickness would result in smaller propagation times. However, if all axons in the WM were to grow thicker by the same factor, then a tightly packed WM would also have to expand to accommodate the extra volume.

But a larger WM would mean that GM neurons would be further apart, and axons would have to be longer to connect them. For the sake of argument, let us considering an isometric doubling of all the lengths and diameters i. According to the formula given above, the average axon impulse propagation time in this isometrically scaled up WM would also be unchanged.

Signals have to propagate further due to the doubling of l , but the propagation speed is proportionally faster due to the quadrupling of a. Thus, as far as average propagation times in the WM are concerned, there is no difference between scaled up or scaled down isometric versions of the WM.

In contrast, the hypometric scaling of axonal length in the WM has the obvious consequence of decreasing propagation time: the smaller the value of l , the smaller the increase in axonal cross-sectional area required to maintain a constant average propagation time as the cortex gains neurons interconnected through the WM.

This means that cortices in which connectivity through the WM and average propagation time scale similarly will have folding indices that also scale similarly.

This also means that, for a given value of c , a faster upscaling of WM folding will be accompanied by a slower increase in propagation times. Increasing cortical folding in larger brains is thus associated with the advantage of a diminished increase in the average propagation time that would otherwise be expected if the WM grew isometrically.

Signal propagation times tell us how fast a cortex computes information, but not how effectively. To quantify computational capacity in a simple way, consider a simple neuronal circuit composed of a few neurons connected by axons passing through the WM. A typical such operation is memory retrieval: The circuit receives as input an incomplete pattern that is a partial match to a stored pattern. Clearly, each such computational cycle which can be as simple as two neurons with reciprocal connections is completed in the time it takes for a signal to propagate along the axons of its interconnected, constituting neurons.

In this case, C , the number of operations involving WM fibers i. Note that C ef is highly dependent on the scaling of axonal length, but only weakly so on the scaling of axonal cross-sectional area. If on the contrary the WM scales under tension, with a smaller increase in V W than expected, that implies that l is scaling more slowly than expected.