Mechanical stress connects cortical folding to fiber organization in the developing brain

Abstract

During development of the gyrencephalic brain, both the formation of cortical folds and the establishment of axonal tracts require large, coordinated mechanical deformations. Cortical folding enables a high ratio of cortical surface area to brain volume, which is thought to enhance overall processing power. Meanwhile, a complex network of axonal connections facilitates communication between distant brain regions. The mechanisms underlying the formation of brain folds and axon tract organization remain widely debated. However, evidence emerging from measurements of mechanical stress, combined with physical and mathematical models, suggests that constrained cortical expansion generates folds via mechanical instability. In this article, we highlight recent models and experimental data suggesting that mechanical stress induced by cortical folding also mediates axonal growth. We propose a key role for mechanics in establishing brain morphology and the organization of white matter fascicles of the mature brain.

The role of mechanics in brain development

In humans and other gyrencephalic species, the brain’s cortical surface folds to accommodate its rapid expansion during development (Fig. 1A). While the cortex is folding, the transient subplate emerges and is then transformed into an axon- and myelin-rich network of white matter fascicles that reflect connectivity in the adult brain [1, 2]. Altered sulcal and gyral patterns, as well as altered subcortical connectivity, have been associated with increased risk of neuropsychiatric and neurodevelopmental conditions such as schizophrenia, bipolar disorder, autism, epilepsy, and developmental delay [3]. Considerable effort is currently directed at understanding molecular genetic determinants and cell biological processes that influence cortical folding [4–8]. However, macroscopic morphological changes also require tissue-scale physical forces. Understanding the biophysical mechanisms that govern cortical folding and associated white matter organization could reveal links between the shape of the mature brain and the developmental changes associated with the aforementioned disorders.

Figure 1. Bending mechanics and the Constrained Cortical Expansion (CCE) model of cortical folding.

(A) Cortical mid-thickness surface reconstruction of the left hemisphere (lateral view) from magnetic resonance imaging of a preterm human at 27, 33, and 36 weeks postmenstrual age and again at 10 years of age [53, 59]. (B-C) Bending of an initially flat plate induces tangential compression along the inner curvature (blue arrows) and tangential stretch, or tension, along the outer curvature (red arrows). Cutting experiments have confirmed the presence of similar stresses in the developing and adult ferret brain (bottom images from [20]). (D) During sulcation, the cortex transitions from convex (outward bending) to concave (inward bending). Straightening of an initially convex plate induces inward bending stresses before the emergence of a concavity, which are increased in the final transition from flat plate to concave structure. (E) Models of brain folding driven by constrained cortical expansion (CCE) produce fold morphologies and stress patterns similar to the gyrencephalic brain. Top: Folding can be produced by physical gel models where the outer layer of gel expands faster than the inner layer. In the example shown, initial geometry (prior to gel swelling) was based on the shape of a 22 week fetal human brain [21]. Bottom: Finite element simulation of the same process indicates directions of tension consistent with cutting experiments shown in panels B-C [24].

This Opinion article focuses on the biomechanical relationships between macroscopic characteristics of brain anatomy and the dynamic cellular processes that occur in the developing brain. Mechanical studies of gyral/sulcal formation typically focus on growth and mechanical properties on the tissue scale, which describe the aggregate effects of cell populations. Moreover, mechanical forces can affect neural mechanobiology on the cellular scale. For example, axons elongate in response to applied stretch or tension [9], and axon growth cones migrate toward mechanically stiffer substrates [10]. Recent work has postulated that growth of the cortical plate and consequent folding create patterns of tension that differ between gyri and sulci, thereby influencing the ultimate organization of white matter fascicles [11]. By implication, altered cortical growth can affect cortical folding and axonal architecture in ways that may contribute to developmental mechanisms underlying neuropsychiatric dysfunction.

Our objective is to illuminate, from a biomechanical perspective, determinants of human brain folding that potentially influence brain function at maturity. We begin by describing current competing theories (and evidence) of potential links between brain folding and connectivity. We argue that computational models and experimental measurements support constrained cortical expansion (CCE, see Glossary) as the driver of cortical folding, but with a critical role played by tension-induced growth (TIG) in neural tissues. We outline the evidence for this mechanobiological model and its implications, supported by computational work, experimental studies in gyrencephalic mammals, and physical experiments using materials that resemble developing brain tissue.

Constrained cortical expansion (CCE) as the driving force for gyrification

The mechanisms that drive folding of the cerebral cortical surface are still intensely debated. The idea that external forces, such as those that could be exerted by the skull or blood vessels, drive folding has been largely disproven [12]. Instead, current prevailing views focus on forces generated internally by differences between tissue compartments. One popular concept is that axons actively pull specific cortical regions together to form the opposing walls of a gyrus [13, 14]. Alternatively, heterogeneities in subplate growth or mechanical properties have been proposed to determine formation of a gyrus or sulcus [1, 15–19]. However, mounting evidence [20, 21] supports the idea that constrained cortical expansion drives folding. This mismatch between fast-growing cortex and slower-growing subcortical tissue leads to high compression within the cortical plate, which eventually causes it to buckle, or fold [11, 22–26].

The mechanical stress (tension or compression) observed within brain tissue offers critical evidence to support or refute competing theories of brain folding. In one set of simple experiments [20], investigators made physical cuts in the developing gyrencephalic brains of ferrets to probe the stress state within both the cortex and subcortical tissue beneath developing gyri and sulci. These experiments revealed that subcortical tension (internal pulling force) is directed primarily in the radial direction (perpendicular to the pial surface) beneath gyri and in the tangential direction (parallel to the pial surface) beneath sulci (Fig. 1C–D). While findings supported the idea that axons are in tension, they ran contrary to the idea that more connected regions are “tethered” in a way that actively pulls together the walls of gyri [13]. Conversely, subcortical pushing, which some investigators have inferred from the localized increase in subplate cells, axons, and thickness beneath gyri [1, 17, 18], would predict compression rather than tension in the predominant axon direction.

In contrast, observed patterns of tension do align with passive stresses predicted by a basic mechanical model of bending (Fig. 1B–C). When bending a beam or plate, the outer curvature (outer layer of a gyrus, inner layer of a sulcus) experiences tangential tension, and the inner curvature (outer layer of a sulcus, inner layer of a gyrus) experiences tangential compression. In addition, subcortical tissue beneath cortex destined to form a gyrus will experience radial tension, while tissue beneath a developing sulcus will experience radial compression. These stresses are consistent with the CCE mechanism, in which heterogeneous bending stresses emerge as a consequence of cortical buckling.

Physical models of non-biological materials designed to mimic the developing brain, such as differentially expanding gels and elastomers, have qualitatively confirmed many predictions of CCE (Fig. 1E). In a 2D acrylamide gel disk, folding can be produced and controlled by varying the stiffness and thickness of a swelling outer ring [26]. In 3D, gel models exhibit folded patterns reminiscent of human brain morphology [21, 24]. These physical models confirm the plausibility of CCE to produce folds but do not (yet) accurately replicate the arrangement of specific gyri and sulci, nor do they address the tissue stress state or its relationship to fiber architecture.

Mathematical and computational models of CCE have provided additional insight. The earliest mathematical studies, which considered growth of a simple elastic plate attached to an elastic substrate, demonstrated how fold morphology (wavelength and location) can be altered by subtle changes in initial brain geometry, initial thickness of the cortex, stiffness differences between the cortex and subcortical tissue, and heterogeneities in growth of the cortical layer [22, 27]. More recent finite element simulations have demonstrated how CCE can produce complex folding patterns in 3D, such as 3-hinge gyral morphologies [19], and have attempted to reproduce folding on a realistic human brain geometry [21]. Other studies have demonstrated important ways in which local heterogeneities in cortical thickness, cortical growth, or stiffness likely influence the precise location of folds in complex 3D folding scenarios [28]. Both physical and computational models have illustrated how large-scale differences in gyrification across species can be predicted by fundamental physical principles of buckling: thicker cortices or smaller brains require greater expansion to reach the critical point of folding, and folds will preferentially emerge along the long axis of the brain [24, 25].

Mechanical consequences of folding, driven by CCE, may also explain several anatomical and histological observations that have drawn attention as potential biological drivers of gyrus formation. For example, the cortex of gyri is thicker than the cortex of sulci, and the distribution of thickness attributed to each cortical layer differs between gyri and sulci [14, 29]. However, both expanding gel models and finite element simulations predict emergence of this difference as a fundamental consequence of bending mechanics [30–33]. Similarly, increased thickness of the subplate [17] and deeper proliferative layers [34] beneath gyri have been reported. Again, simulations that combine cell migration and CCE mechanics show that local thickening of the subplate and outer subventricular zone could be either a cause or mechanical consequence of gyrus formation [35]. Mechanical bending strains may also explain the tangentially stretched appearance of cell bodies in bottom layers of sulci and top layers of gyri [14, 29], and the “bunching up” of radial fibers within the subplate beneath gyri [36].

Mechanical signals mediate axon growth and organization

Predictions of the CCE-based models qualitatively agree with measured directions of tissue stress, as well as anatomical observations on the tissue and cellular level. However, quantitative comparisons to experimental measurements fall short on several fronts. First, simulations (and gel models) that treat developing brain tissue as purely elastic or hyperelastic predict tension values that are approximately 10 times greater than experimental measurements. Secondly, elastic or hyperelastic models often rely on a cortex that is substantially stiffer than developing white matter to produce buckling[11, 22, 24], despite evidence that the cortex and subcortical layers do not differ dramatically in stiffness [20, 37, 38]. While hyperelastic simulations can produce a form of folding by assuming similar stiffness in both layers, the resulting sulci begin as sharp, self-contacting creases (a fundamentally different mechanical instability) [11, 22, 24], rather than the smooth undulations predicted by traditional buckling [11, 22, 24], which are observed in early stages of brain gyrification [25].

Intriguingly, the resting tension within axons, a phenomenon noted in the original axon tension (pulling) hypothesis [13], has emerged as the potential mechanism to reconcile CCE with experimental observations. In the new theory, however, axons are not viewed as elastic tension cables drawing regions of the brain together. Rather, experimental and computational evidence suggests that axons should be modeled as viscoelastic, exhibiting extreme compliance via tension-induced growth (TIG) over long morphogenic time scales. Numerous in-vitro experiments [9, 39, 40] have characterized this towed growth behavior when neurites are subjected to sustained stretch along their length. Moreover, in the absence of tension along the length of an axon, neurons tend to form new neurite branches, preferentially maintaining synaptic connections in neurites where axial tension is maintained [41–43]. Other cellular elements relevant to the developing brain, such as glial cell processes [44] and neuroepithelial cells [15, 16, 45] also exhibit the property of TIG.

TIG can be characterized by two key parameters: (1) the rate at which tissue grows (elongates) in response to sustained tension; and (2) the target stress, the slightly tensile stress state of axons at equilibrium. To date, simulations of CCE-driven cortical folding have focused primarily on the relationship between axon elongation rate and folding. Simulations that focused on reproducing the stresses observed in experimental data found that TIG in the nascent white matter was sufficient to reconcile the relatively low magnitude of stresses observed in cutting experiments of both mouse [37] and ferret [20]. Subsequent studies demonstrated how TIG can impact the wavelength of cortical folding, with slower TIG inducing shorter gyral wavelengths and faster TIG inducing longer wavelengths or lissencephaly [46]. Incorporation of TIG generally leads to more accurate predictions of sulcal curvature and global alignment of folds, without the assumption of higher cortical stiffness [24, 25].

Most studies, including those described above, have applied TIG behavior uniformly throughout the nascent white matter. However, in more recent work, axons and their accompanying TIG behavior have been modeled explicitly, by incorporating discrete fibers throughout the simulation [47] or by definining the volume fraction of axon fibers directly within the tissue [23]. In both approaches, regional variations due to heterogeneities in underlying axon density or alignment can alter the precise location or pattern of folding.

Recently, models of TIG have been extended to explore not only macroscopic folding behavior, but also changes at the microscopic level [11]. By explicitly attributing TIG to axons within the developing subplate, CCE-driven models of folding predict preferential growth of axons in the direction of folding-induced tension: tangential beneath sulci and radial beneath gyri (Fig. 2A), consistent with predominant fiber orientations observed in adulthood [48]. Furthermore, by recognizing that axons make up only a small percentage of early subplate tissue, these models predict increasing axon density over the course of development, driven by CCE-induced tension (Fig. 2B). Ultimately, TIG results in remodeling of subcortical tissue consistent with observations in real brains, including radially-oriented water diffusion anisotropy (reflecting radial cell processes) and increased overall fiber content beneath developing gyri [1, 2, 11], decreased diffusion anisotropy beneath developing sulci [1, 11], radial axon fascicles beneath gyri of the adult brain [48], and tangential axon fascicles beneath sulci of the adult brain [48]. Models also predict alignment of deeper fiber tracts along the ventricles and anterior-posterior axis, consistent with experimental observations [11] and adult brain organization [48]. More recent finite element models suggest that mechanical cues at the axonal tip [10] may also contribute to radial axon organization beneath gyri [47, 49].

Figure 2. Tension Induced Growth (TIG) and the Constrained Cortical Expansion (CCE) model of cortical folding.

(A) In CCE-driven models of cortical folding, the tension introduced by folding induces elongation of radial fibers beneath gyri and tangential fibers beneath sulci, as well as higher overall fiber volume fractions in gyri. Top row illustrates the direction (black tick marks) and magnitude (color) of maximum tension over the course of fold progression. Note that the emergence of stress heterogeneities within the subplate occurs prior to visible folding. Bottom row illustrates the volume fraction of fibers perpendicular f₁ (red) and parallel f₂ (blue). Fiber reorganization (bottom row) occurs slowly and is most pronounced after folds have emerged, in the direction of maximum tension induced by folding. (B) Theory underlying axonal fiber reorganization observed in A, shown for a block of tissue (similar to the block outlined in the bottom row of panel A). When fibers are allowed to grow in response to tension along their length, stretch oriented in the direction of the red fibers leads to growth of fibers in that direction. As tension is sustained, the tissue slowly remodels and becomes dominated by fibers in the direction of applied tension. Fiber volume fractions in radial, tangential, and out-of-plane directions are denoted by f₁ (red), f₂ (blue), and f₃ (green), respectively. All images modified from [11].

The explicit representation of axons in models of TIG has introduced new possibilities for how observed heterogeneities in subplate organization may influence the location or morphology of folds. Just as a local reduction in subcortical stiffness can provide a mechanical bias to induce formation of a gyrus [27, 50], a localized increase in radial axon density has a similar effect in TIG models [11, 23]. Conversely, a local sparsity of axons, or predominance of tangential axons, may encourage formation of a sulcus [11, 23]. However, many other factors can introduce bias for gyrus or sulcus formation, including local variations in cortical expansion, stiffness or geometry. Models suggest that these alternative perturbations can override the bias from subplate organization, and that TIG-induced remodeling can override an initial opposing predominance of fibers [11]. Notably, while TIG predicts that the dominant axon orientation will follow the direction of maximum tension, these models include fibers in perpendicular directions (e.g., crossing fibers) [11], which may emerge in response to other signaling mechanisms and persist (in smaller numbers) after folding [13, 15, 16].

An important implication of the CCE+TIG hypothesis is that the rate of cortical expansion relative to the rate of TIG (growth of nascent white matter) must be relatively fast for gyrification [46] and TIG-induced remodeling [11] to occur (Fig. 3). The prediction of increased short-range connectivity (u-fibers) in gyrencephalic cases, at the expense of long-range radial connections, is consistent with connectivity differences observed across primate species [51]. Within humans, this dynamic could also contribute to clinically observed abnormalities in cortical folding. For example, preterm infants experience altered trajectories of cortical expansion, with slower expansion during the period of brain folding and increased expansion over later developmental periods [52–54]. Reductions in folding [55, 56] and connectivity [57] have been observed in these cases, ultimately increasing risk of cognitive and mental health disorders [56]. Other deviations from the normal cortical growth trajectory could similarly contribute to abnormal folding and connectivity, which have been reported across a range of neurodevelopmental or psychiatric conditions [3, 58].

Figure 3. Biological observations consistent with Tension Induced Growth (TIG) and the Constrained Cortical Expansion (CCE) model of cortical folding.

Schematic summary of cell- and tissue-level observations explained by mechanical stresses. Across all brain layers, thickness differences between gyri and sulci emerge naturally from mechanical bending stresses. In the cortex, differences in thickness between layers, as well as cell morphology, may be explained in part by mechanical stress. The subplate contains outgoing axon projections from nearby cortical neurons (pink) and incoming axons from other brain regions (blue), among other components (not shown). Elastic deformation, TIG-induced elongation, and the influence of mechanical signals on axon tip guidance all likely contribute to the emergence of dense, predominantly radial fibers beneath gyri and relatively sparse, predominantly tangential fibers beneath sulci. TIG-induced proliferation may also contribute to the increase in basal radial glial cells observed in the proliferative outer subventricular zone of prospective gyri. As illustrated in the upper right-hand corner, the CCE+TIG mechanism predicts that these changes will only emerge if the rate of cortical expansion exceeds the rate at which subcortical layers can increase volumetric expansion, dictated in part by the rate of axon elongation (CCE>TIG). Conversely, if CCE is sufficiently slow, no new folds will form, and mechanical factors will encourage more uniform, radial organization.

Concluding remarks

There is an intrinsic relationship between gyral-sulcal morphology and tissue organization (Fig. 3). Although it is not yet certain which of these patterns emerges first, or whether they emerge simultaneously, they are inextricably linked by mechanics. Mechanical measurements and computational models support important roles for CCE and TIG. In principle, concentrated radial fibers could induce formation of a gyrus, and the reduction in radial organization beneath budding sulci has been interpreted as evidence that the subplate drives folding. However, formation of a gyrus (by CCE) will stimulate growth of radial fibers beneath gyri and tangential fibers beneath sulci (by TIG) even prior to the visible emergence of a sulcus (Figs. 1D, 2A). Similarly, investigators have pointed to gyral-sulcal differences in the thickness or microstructure of the cortex (and deep proliferative zones) as evidence of patterning, but early emerging patterns of stress accompanied by TIG may explain these differences (see Outstanding Questions).

Outstanding questions.

• What are realistic parameters and limits for tension-induced growth (TIG)? TIG, which reflects tension-induced elongation of developing axons, is expected to influence cortical morphology and axonal connectivity in a rate-sensitive manner. Future work should relate explicit measures of each axon’s response to tension to the implicit, tissue-level response of nascent white matter.

• How do mechanical stresses affect connectivity and microstructure in the cortex? Measures of cortical microstructure suggest differences in cell morphology, including greater dendritic and axonal arborization in the cortex of gyri compared to sulci. While current TIG models focus on the subplate, tension may also influence neural and glial cell processes in cortical grey matter.

• How do mechanical stresses impact deep subcortical zones? Recent studies have pointed to subventricular proliferative zones in determining the ultimate location of primary gyri. Although current TIG models focus on axon elongation, tension-induced growth may also influence non-neuronal cell processes, cell body size, or even proliferation of neural progenitors.

• What are the clinical implications of TIG? Prior work has found correlations between cortical expansion, folding, connectivity and function. Future work should explore whether CCE+TIG models involving altered trajectories of cortical expansion can explain observed differences in brain morphology and connectivity and relate these differences to clinical outcomes.

The biological drivers of cortical folding and connectivity may involve multiple redundant mechanisms, or vary by fold and developmental timing. Comprehensive characterization of the mechanobiology leading to the development of a specific fold may require precise measurement of growth, morphology, and mechanical stress over the period of fold formation. However, taken together, existing data strongly indicate a role for mechanical signals in brain growth and tension-induced remodeling. Computational work to clarify the implications of CCE and TIG mechanisms, as well as further measurements of brain shape, microstructure and mechanics, are warranted to explore the associations and potential causal relationships between stress, folding and function in the developing mammalian brain.

Highlights.

• The idea that subcortical axons drive cortical folding is attractive due to the strong correlation between fold morphology and the underlying axon organization, but it is not consistent with the available physical evidence of mechanical stress in brain tissue.

• The idea that constrained cortical expansion (CCE) drives folding is consistent with measurements of tensile stress in brain tissue, but it does not explain the correlation between folding and axon organization.

• Tension-induced growth (TIG), mediated by the stress from CCE, provides an explanation for the relationship between folding and white matter organization that is consistent with available evidence.

• Future experiments are needed to elucidate the implications of tension-induced growth in short- and long-range brain connectivity, and why these anatomical characteristics are associated with neurodevelopmental disorders.

Glossary

Stress (tension or compression)

Mechanical stress is the internal force per unit area in a material. Normal stress (with ‘normal’ in the geometrical sense, as the direction penpendicular to the surface) is positive (tensile) if it describes an outward force on a face of a material element, which tends to lengthen the element. Normal stress is negative (compressive) if it describes an inward force on an element face that tends to shorten the element.

Target stress

In stress-induced tissue growth, target stress refers to the value of stress at which no growth occurs. If stress differs from this value, the tissue will grow until the target stress is reached.

Strain

Mechanical strain refers to the change in size and shape (deformation) of a material element. In the context of tissue deformation, strain may be due to biological growth, or a passive response to applied loads. Positive strain represents lengthening of an element, and negative strain describes shortening of an element. Strain is related to stress by stiffness or more complex material models (see other Glossary definitions).

Elastic

In mechanics, this term is used to describe deformation that is reversible when forces are removed, or a material that possesses this behavior. Deformations that are not elastic include plastic (permanent) deformation, viscous (rate-dependent) deformation, and growth.

Stiffness

Mechanical stiffness defines the resistance to deformation of a structure, which in general depends on both the material and geometry of the structure. Elastic modulus is the intrinsic resistance of a material to elastic deformation. The inverse of stiffness is compliance.

Hyperelastic

A material exhibiting elastic, nonlinear behavior under large deformations can be described as hyperelastic. Rubber is an example of a hyperelastic material, whereas steel exhibits elastic behavior only under small deformations (it fails or deforms permanently under large deformations).

Viscoelastic

A viscoelastic material contains both an elastic (or hyperelastic) resistance to deformation over some time scales and a viscous resistance proportional to rate of deformation over other time scales. Silly Putty^™ is an example of a viscoelastic material that is elastic at short time scales and viscous over longer times.

Rates and time constants

Rates describe changes in a parameter per unit time (i.e., rate of area growth in m²/s). Time constants or rate constants typically characterize the relationship between a parameter and its rate of change.

Constrained cortical expansion (CCE)

The cortex expands, or grows, faster than subcortical tissue, to which it is constrained to remain in contact. In gyrencephalic species, this difference in growth is hypothesized to induce cortical folding, thereby accommodating a large cortical surface area in a relatively small volume.

Tension-induced growth (TIG)

Mechanical stress, specifically tension, can induce growth of biological tissues. If tissue is stretched to a specific length, tension-induced growth provides a mechanism by which the tissue can grow and accommodate this stress, ultimately reducing tension to a “target” stress value.