A cortical folding model incorporating stress-dependent growth explains gyral wavelengths and stress patterns in the developing brain

P V Bayly; R J Okamoto; G Xu; Y Shi; L A Taber

doi:10.1088/1478-3975/10/1/016005

Introduction

Cortical folding is a critical process in brain development. In the human fetus, cortical folding normally occurs between the 25th and 40th weeks of gestation. Many other mammals also have folded (gyrencephalic) brains. In the ferret, cortical folding occurs post-natally (figure 1), roughly between the 5th and 30th days after birth. Folding in the ferret has been thoroughly documented in seminal papers by Smart and McSherry [1, 2] and in more recent work based on MR imaging [3–5] and detailed histology [6]. Abnormal folding patterns in the human brain are associated with severe mental or emotional disorders [7–11]. Disturbances of cortical folding in humans include complete absence of folds (lissencephaly), folds that are abundant but small and shallow (polymicrogyria), and folds that are fewer and coarser (pachygyria). Despite decades of intense study [12–15] and speculation, the mechanical basis of folding remains controversial.

**Figure 1.** Summary of cortical folding studies in the ferret [4, 5, 21, 49]. (a) Sequence of cortical surfaces generated from longitudinal MR imaging studies in the neonatal ferret. PXX = XX days after birth. (b) Coronal slice of P18 ferret brain near the conclusion of the folding process. The illustration on the right summarizes the results of dissection studies of tissue stress [21]. Initial cuts (dotted lines) open when tension is normal to the cut. 1—radial cuts through gyri stay closed (showing lack of tension between gyral walls) except at outer surface, where circumferential tension exists. 2—radial cuts through the bases (fundi) of sulci open in subcortical layers, showing circumferential tension in these locations. 3—circumferential cuts through gyri open, showing radial tension along the axes of gyri.
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Van Essen [16] has proposed that axonal tension produces folding by drawing sides of gyri (outward folds) together. This hypothesis is attractive because it is consistent with efficient wiring—axons that connect related areas will draw them together, decreasing the total length of axonal connections. Heidemann and co-authors [17–19] have shown that axons in vitro maintain tension. Dissection experiments [20, 21] have confirmed that white matter in mammalian brains is under tension, including the white matter in the adult mouse brain [20] and in both mature and developing ferret brains [21]. However the hypothesis that the walls of gyri are drawn together by axonal tension is not consistent with observed patterns of stress in the ferret brain, which reflect tension along, not across, gyri [21] (figure 1).

Other authors [22, 23] have pointed to differential growth, in which outer layers grow tangentially at faster rates than inner layers of the brain, as the driving mechanism of cortical folding. Richman and co-authors [23] proposed a model for the formation of cortical folds based on buckling of elastic surface layers on an elastic foundation. In this scenario, tangential growth of the outermost layer produces compressive stress that leads to buckling of this layer, modulated by the stiffness of the foundation. In terms of developmental neuroanatomy (see [24]) the outer layers in this model comprise the cortical plate, and the foundation, or core, captures the aggregate mechanical behavior of subplate, intermediate zone, subventricular zone, ventricular zone, and all deeper internal brain structures. In the Richman model [23], increasing the material stiffness, or elastic modulus, of the foundation leads to shorter wavelengths of the buckled layer (cortex). However, this model and other elastic buckling models [22] rely on a large mismatch in the elastic moduli of the outer layer (stiffer) and core (softer) to produce buckling patterns that approximate observed folds. In fact, the elastic modulus of the outer layer of cortex is not significantly different than that of inner regions of the brain [25–27].

The question we address is: what mechanism can produce folded surfaces with wavelengths consistent with those observed in nature, and stress patterns consistent with those seen in dissection experiments, without requiring a difference in elastic modulus between layers? We propose a model of folding in which tangential growth of the cortex drives the folding process. Instead of a purely elastic foundation [22, 23], or a hard constraint [28], the core is allowed to grow in response to the resulting stresses. Axons, which are the major functional component of white matter, are known to grow in response to imposed tension [19]. The proposed model produces surface folding in which the wavelength depends on the rate of cortical growth relative to the rate constant of the stress–growth relationship in the core. Patterns analogous to polymicrogyria and pachygyria can be obtained by adjusting this ratio. The pattern of folding-induced stress is consistent with observations of stress in the developing ferret brain, in which tension exists along the radial axis of gyri.

Mathematical model and simulation

Our mathematical model of cortical folding uses the theory of volumetric growth developed by Rodriguez et al [29] and applied in many studies since [30–33]. Using standard continuum mechanics terminology, we designate the location of a material element in the reference configuration as X and the corresponding location of the same element in the deformed configuration as x. The deformation gradient tensor is ${\bm F} = \frac{{\partial {\bm x}}}{{\partial {\bm X}}}$ . According to the theory of Rodriguez et al [29], F is expressed as the product of a growth tensor, G, and an elastic tensor, F*:

$\begin{equation} {\bm F} = {\bm F}^* \cdot {\bm G}.\end{equation} \tag{ 1 }$

Numerous material models have been suggested to describe the mechanical behavior of brain tissue. Recent studies have proposed isotropic, hyperelastic or hyper-viscoelastic models for gray matter [26, 34, 35], and anisotropic, hyperelastic or hyper-viscoelastic models for white matter [36, 37]. At relatively short time scales (shorter than those associated with growth) cortical brain tissue may be roughly approximated as an isotropic, hyperelastic material [26], so the Cauchy stress tensor, σ, depends directly on the elastic deformation according to the constitutive relationship

$\begin{equation} \sigma = J^{* - 1} {\bm F}^* \cdot \frac{{\partial W}}{{\partial {\bm F}^{*T} }}.\end{equation} \tag{ 2 }$

Here W is the strain energy density function for the material and J* = det F* is the volume ratio of the elastic deformation. A standard neo-Hookean material model [38] is used here:

$\begin{equation} W = \frac{\mu }{2}( {I_1^* J^{* - 2/3} - 3} ) + \frac{\kappa }{2}( {J^* - 1} )^2 .\end{equation} \tag{ 3 }$

The strain energy depends on J* and on the first invariant (trace) of the elastic right Cauchy–Green strain tensor, $I_1^* = tr{\bm C}^*$ , where ${\bm C}^* = {\bm F}^{*T} \cdot {\bm F}^*$ . The shear modulus, μ, and the bulk modulus, κ, are the parameters of the hyperelastic model. Typically tissue is assumed to be nearly incompressible: κ ≫ μ.

Radial and tangential normal growth are included in the current model. Simulations are performed in (i) a two-dimensional (2D) rectangular domain in plane strain, approximating a section of tissue near the cortical surface, (ii) a 2D semi-elliptical domain in plain strain (approximating a section of an elliptical cylinder) and (iii) an axisymmetric ellipsoidal domain. In the 2D rectangular and elliptical domains, growth is modeled using

$\begin{equation} {\bm G} = G_r {\bm e}_r {\bm e}_r + G_t {\bm e}_t {\bm e}_t ,\end{equation} \tag{ 4 }$

where e_ie_j denotes the tensor product of unit vectors e_i and e_j. In the rectangular domain the 'radial' direction (normal to the free surface) is vertical and the tangential direction is horizontal.

In the axisymmetric ellipsoid, growth is modeled using

$\begin{equation} {\bm G} = G_R {\bm e}_R {\bm e}_R + G_t {\bm e}_t {\bm e}_t + G_t {\bm e}_\phi {\bm e}_\phi ,\end{equation} \tag{ 5 }$

where azimuthal (ϕ) growth is assumed equal to the in-plane tangential growth. In equations (4) and (5) the radial and tangential directions are defined in a standard elliptical or ellipsoidal coordinate system. In both models, the thin upper layer (the cortex) grows tangentially only, at a constant rate, G₀. In deeper layers (the foundation, or core) growth ensues in response to stress:

$\begin{eqnarray} &&\fl {\rm Cortex{:}} \;\; G_t = 1 + G_0 t,\quad G_r = 1,\end{eqnarray} \tag{ 6 }$

$\begin{eqnarray} &&\fl {\rm Core{:}} \;\; \frac{{\partial G_r }}{{\partial t}} = a(\sigma _{rr} - \sigma _{r0} )G_r , \quad \frac{{\partial G_t }}{{\partial t}} = a(\sigma _{tt} - \sigma _{t0} )G_t . \end{eqnarray} \tag{ 7 }$

The parameters σ_r0 and σ_t0 are 'target' stresses for the core; growth will continue until these stress values are reached. The parameter a (units: 1/Pa s or 1/Pa h) determines how rapidly variations in growth occur in response to variations in radial or tangential normal stress (σ_rr or σ_tt). For simplicity we assume a is the same in both directions. A very rough estimate of a ∼ 3 × 10⁻³/Pa h may be obtained for white matter from the experiments of Chada et al [19] who determined neurite elongation rates of 200 µm h⁻¹ under tension of 2 nN (200 µdyne).

The stiffness of the material in the core will also affect how quickly it responds to cortical growth. For an incompressible material undergoing small, uniaxial, plane-strain deformations, the stress–strain relationship is σ = 4μ_fε, where ε is the strain and μ_f is the shear modulus of the core (the subscript f denoting 'foundation' is used to distinguish core from cortex). Defining dimensionless stress $\bar \sigma = \sigma /4\mu _f$ , we can rewrite equation (7) as

$\begin{eqnarray} &&\fl \frac{{\partial G_r }}{{\partial t}} = 4\mu _f a(\bar \sigma _{rr} - \bar \sigma _{r0} )G_r ,\quad \frac{{\partial G_t }}{{\partial t}} = 4\mu _f a(\bar \sigma _{tt} - \bar \sigma _{t0} )G_t .\end{eqnarray} \tag{ 8 }$

It is apparent that the rate constant governing the response of the core to imposed deformation is R_f = 4aμ_f (units: 1/s). Fully dimensionless versions of equations (6)–(7) can be obtained using τ = G₀t:

$\begin{eqnarray} &&\fl {\rm Cortex:}\;\; G_t = 1 + \tau ,\quad G_r = 1,\end{eqnarray} \tag{ 9 }$

$\begin{eqnarray} &&\fl {\rm Core:}\;\; \frac{{\partial G_r }}{{\partial \tau }} = \frac{{R_f }}{{G_0 }}(\bar \sigma _{rr} - \bar \sigma _{r0} )G_r , \quad \frac{{\partial G_t }}{{\partial \tau }} = \frac{{R_f }}{{G_0 }}(\bar \sigma _{tt} - \bar \sigma _{t0} )G_t .\nonumber\\ \end{eqnarray} \tag{ 10 }$

Finite element simulations of the rectangular, elliptical and ellipsoidal models were performed using COMSOL Multiphysics software (V.4.3, COMSOL Inc., Burlington, MA). Routine simulations in the rectangular domain were performed with 2412 rectangular elements; in the 2D ellipse and axisymmetric ellipsoid the domain was discretized into 2498 triangular elements. Quasi-static, time-dependent simulations were performed with the COMSOL backward differentiation formula (BDF) algorithm with maximum absolute error tolerance <10⁻⁷ and relative error tolerance <10⁻⁶. Since the locations of spatial instabilities are extremely sensitive to imperfections in numerical models, small imperfections were deliberately introduced by simulating low levels of radial growth in the core in the early stages of the simulation. Folding was typically initiated by these imperfections but the folding pattern was not solely determined by the imperfection field. The robustness of solutions with respect to discretization (mesh size and time step) was confirmed by reproducing results with finer mesh resolution and smaller error tolerances.

Analytical prediction of critical stress and wavelength

The time scale of folding is long enough for the brain to grow significantly in response to stress; such growth in the core would lead to relaxation of the stresses induced by cortical growth. The behavior of the core thus approximates the response of a viscoelastic material, specifically a Maxwell fluid. Biot [39] developed a theory for the folding instability of a thin, laterally compressed, viscoelastic plate embedded in a viscoelastic continuum. The theory of Biot [39] may be extended to the situation in which compressive stress is due to tangential growth of a thin elastic plate with thickness h (the cortex) and the embedding medium grows according to the laws in equation (7). According to this analysis (shown in the appendix) we predict that the wavelength of folds, λ, will depend on the ratio Γ_G = G₀/R_f, the ratio of cortical growth to the rate constant of stress-induced growth, and β = μ/μ_f, the ratio of the short-term elastic moduli of cortex (μ) and core (μ_f). Specifically we predict that

$\begin{equation} \frac{\lambda }{h} = 2\pi \left( {\frac{{\beta (\Gamma + 1)}}{{3\Gamma }}} \right)^{1/3} ,\end{equation} \tag{ 11 }$

where Γ is the largest positive real root of the polynomial equation,

$\begin{equation} \Gamma ^5 - \frac{{64}}{9}\beta ^2 \Gamma _G ^3 (\Gamma ^2 + 2\Gamma + 1) = 0.\end{equation} \tag{ 12 }$

In this analysis, existence of at least one positive real root of equation (12) indicates that folding will occur at the given values of growth rate and material parameters. In this study, for each parameter combination only one real root of equation (12) was found and thus only one wavelength is predicted.

Results

Simulations

Growth was simulated using the mathematical description given by equations (1)–(10). The material models for both cortex and subcortical foundation were hyperelastic (neo-Hookean). In the cortex, the constant growth rate in the direction tangent to the surface was specified to be G₀, and there was no growth in the radial direction (normal to the cortical surface). In the foundation, tangential and radial growth were stimulated by corresponding stress components, as described by the first-order relationships in equation (7).

In all simulations the modulus ratio β between cortex and core was set to 1 unless noted otherwise. Using the rough estimates a ∼ 3 × 10⁻³/Pa h and shear modulus μ ∼ 300 Pa leads to an estimate of the growth rate constant R_f ∼ 1/h for axons in vitro, which is probably an upper bound for heterogeneous tissue in vivo. In the ferret the surface area doubles roughly every 4 days during the period of folding; this corresponds to a circumferential growth rate G₀ ∼ 0.01/h–0.04/h. Results are described at specific values of scaled time: τ = G₀t.

Effect of growth rate on folding wavelength

Simulations of a growing cortical plate of thickness h = 0.05 on a rectangular foundation of length L = 2 and H = 1 were performed while varying the ratio (Γ_G) of the cortical growth rate, G₀, to the rate constant, R_f, governing the stress-induced growth of subcortical layers. To ensure that folding was initiated in the center of the domain, a small initial imperfection was introduced by specifying small, positive radial growth in a narrow central band of width δ in the core, using the following modified version of equation (10). The final pattern was insensitive to δ for δ ⩽ 0.1, so δ = 0.05 was used:

$\begin{eqnarray} &&\frac{{\partial G_r }}{{\partial \tau }} = \frac{1}{{\Gamma _G }}\left( {\bar \sigma _{rr} - \bar \sigma _{r0} } \right)G_r + F(x,\tau );\nonumber\\ F\left( {x,\tau } \right) = \left\{ {\begin{array}{@{}ll@{}} \dsty \frac{1}{{\Gamma _G }}\tau ,& {\rm for}\;\left| x \right| < \delta \quad {\rm and}\quad \tau < 0.01 \\\ms 0,&{\rm otherwise} \\ \end{array}} \right.\!.\end{eqnarray} \tag{ 13 }$

The results of these simulations are shown in figure 2, which depicts the growth ratios G_r and G_t, and the growth-induced stresses σ_rr, and σ_tt for Γ_G = G₀/R_f = 2.5 × 10⁻², 2.5 × 10⁻³, 2.5 × 10⁻⁴ and 2.5 × 10⁻⁵. The wavelength clearly increases as the cortical growth ratio Γ_G decreases. Also, as expected, the wavelength in simulations scales with the thickness of the cortical layer (figure 3). The wavelengths observed in simulations are close to the wavelengths predicted using equations (11) and (12) (figure 4) and clearly follow the analytically-predicted trend.

**Figure 2.** Effects of cortical growth rate on wavelength, subcortical growth, and stress in the basic model of cortical folding. Target stresses $\bar \sigma _{r0} = \bar \sigma _{t0} = 0$ . Columns: radial growth *G_r*; tangential growth *G_t*; radial stress σ_r; tangential stress σ_t. (Row 1) Γ_G = 2.5 × 10⁻², scaled time τ = 0.060; (Row 2) Γ_G = 2.5 × 10⁻³, τ = 0.035; (Row 3) Γ_G = 2.5 × 10⁻⁴, τ = 0.014; (Row 4) Γ_G = 2.5 × 10⁻⁵, τ = 0.008. The modulus ratio β = 1 in all cases.
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**Figure 3.** Folding wavelength is proportional to cortical thickness for a given value of cortical tangential growth rate. (a) Cortical thickness h = 0.03. (b) h = 0.07. Parameters: Γ_G = 2.5 × 10⁻³; β = 1.
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**Figure 4.** (a) Effect of cortical growth rate on the gyral wavelength predicted by equations (11)–(12) for three values of the stiffness ratio β. (b) Solid line: wavelengths predicted from equations (11) and (12) for β = 1. Dotted line and open circles: wavelengths observed in simulations.
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Patterns of differential growth and stress

Some features of the patterns of growth-induced stress observed in figure 2 are consistent with previous experimental observations of tissue stress [21] summarized in figure 1. Radial tension is evident within the core of gyri, and tangential tension in the core is highest below fundi. However, in the model tangential tension is also seen within the core of gyri. We hypothesized that the tangential tension in the outermost layer of the core arises because in the basic model growth is imposed only in the cortex, which is modeled as a completely separate outer layer. Tangential growth thus occurs in the outer core only in response to stress, leading to a sharp discontinuity in tangential growth at the interface. In reality, there is a fairly smooth gradient in tangential growth [6] from the outer cortical layer. We model this by having the target tangential stress in the outer core track the imposed compressive tangential stress in the cortex and decay smoothly and rapidly with distance from the interface (r = H):

$\begin{equation} \sigma _{t0} = - 4\mu \tau \,{\rm e}^{20(r - H)} .\end{equation} \tag{ 14 }$

With this modification the dependence of wavelength on cortical growth rate is maintained, but the tangential stress within gyri becomes compressive (figure 5), consistent with observations (figure 1) [21].

**Figure 5.** Effects of cortical growth rate on wavelength, subcortical growth, and stress in the cortical folding model with a compressive target stress in the outer core. Columns: radial growth *G_r*; tangential growth *G_t*; radial stress σ_r; tangential stress σ_t. (Row 1) Γ_G = 2.5 × 10⁻², τ = 0.059; (Row 2) Γ_G = 2.5 × 10⁻³, τ = 0.034; (Row 3) Γ_G = 2.5 × 10⁻⁴, τ = 0.013; (Row 4) Γ_G = 2.5 × 10⁻⁵, τ = 0.0048.
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Effects of shape, initial conditions and cortical growth rate

To consider the effects of the geometry of the entire brain while retaining computational simplicity, we simulated cortical growth in (i) a cylinder with elliptical cross section, and (ii) an axisymmetric ellipsoid. A 2D elliptical region undergoing plane strain was used to model the cylindrical case. In both cases the major and minor semi-axes of the elliptical cross-section were A = 1.2 and B = 1.0 and the thickness of the cortical layer was h = 0.05. In the 2D ellipse, the radial coordinate $r = \ssty \sqrt {\frac{{x^2 }}{{A^2 }} + \frac{{y^2 }}{{B^2 }}}$ and in the axisymmetric ellipsoid $R = \ssty\sqrt {\frac{{z^2 }}{{A^2 }} + \frac{{r^2 }}{{B^2 }}}$ .

As in the rectangular domain, folding was facilitated by specifying initial radial growth in the core, either in a small band (as in equation (13), but for |θ| < δ), or as a harmonic or random field equation

$\begin{eqnarray} &&\frac{{\partial G_r }}{{\partial \tau }} = \frac{1}{{\Gamma _G }}\left( {\bar \sigma _{rr} - \bar \sigma _{r0} } \right)G_r + F(r,\theta ,\tau )\nonumber\\ &&\left\{ \begin{array}{@{}ll@{}} F\left( {r,\theta ,\tau } \right) = F_0 \tau \,\cos k\theta ;\;\tau < 0.10\;{\rm or} \\ F\left( {r,\theta ,\tau } \right) = F_0 \tau \;{\rm random}(r,\theta );\;\tau < 0.10 \\ \end{array} \right.\!\!\!. \end{eqnarray} \tag{ 15 }$

Here F₀ defines the magnitude of the initial shape perturbation. Both the elliptical geometry and the initial conditions influenced folding patterns. At similar parameter values, longer wavelengths were seen in the ellipse and ellipsoid than in the rectangular domain (figure 6), but the general dependence of wavelength on cortical growth rate remained. Converged folded solutions were found in a smaller range of Γ_G; at small Γ_G the growth of the core prevented critical compressive stress from being reached and at large Γ_G the core seemed too stiff to allow significant deflection.

**Figure 6.** Folding in (a) the elliptical cylinder (plane strain) model and (b) the axisymmetric ellipsoid model. Color encodes tangential growth *G_t*.
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Cortical growth rate affects wavelength in the 2D elliptical domain and axisymmetric ellipsoid, as in the rectangular domain. In the ellipse and ellipsoid, simulations typically show that the instability occurs later, allowing the domain to expand significantly (figure 6). We observe that, as in the rectangular domain, the wavelength increases as Γ_G decreases; however the number of waves per angular increment remains fairly constant. For a given value of Γ_G the wavelength λ_rectangle < λ_ellipse < λ_ellipsoid, which is consistent with the structural stiffening effect of curvature.

Folding patterns exhibited by the elliptical cylinder in response to cortical growth (equations (1)–(10), (15)) are shown in figure 7 for three sets of harmonic and random initial conditions. Note that the final shape appears to be affected by even slight variations in the initial shape. However, fine-grained random initial conditions lead to a relatively coarse folding pattern, confirming that material and structural properties play an important role in determining wavelength.

Heterogeneous initial conditions do not completely determine the final shape; the cortical growth rate remains important. In figure 9 (row 1) we see that the initial field with spatial frequency four time less than the initial field in figure 8 leads to a final field with the same short wavelength; the growth rate Γ_G = 0.50 appears to be the critical factor. For the identical initial conditions but slower cortical growth (Γ_G = 0.20) almost no folding occurs (figure 9, row 2). Amplifying the initial radial growth of the core to the point that initial shape is clearly deformed (figure 10) leads to hybrid shapes with both wavelengths represented. One interpretation of figure 10 is that the initial shape determines the locations of primary folds, and the cortical growth rate mediates the formation of secondary gyri and sulci.

**Figure 8.** Effects of cortical growth on wavelength, subcortical growth, and stress in the axisymmetric ellipsoid model. Initial radial growth in the core was attained with equation (15) with F(R, θ, τ) = 10τcos 32θ for τ < 0.1. Columns: initial radial growth *G_R* at τ = 0.10; radial growth *G_R*; tangential growth *G_t*; radial stress σ_r; tangential stress σ_t; 3D view of radial stress σ_r. Row 1: Γ_G = 0.50, τ = 0.90. Row 2: Γ_G = 0.20, τ = 1.25.
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**Figure 10.** The effects of cortical growth rate and larger initial perturbations. Initial radial growth in the core was imposed according to equation (15) with F(R, θ, τ) = 50τcos 8θ for τ < 0.10. Columns: initial radial growth *G_R* at τ = 0.10; radial growth *G_R*; tangential growth *G_t*; 3D view of radial stress σ_r. Row 1: Γ_G = 0.50, τ = 0.90. Row 2: Γ_G = 0.20, τ = 1.25.
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Effects of spatial variations in tangential cortical growth

The consistent locations of primary cortical folds might, alternatively, be due to intrinsic spatial variations in cortical growth rate. The qualitative effects of such variations can be captured by replacing the dimensionless equation for uniform cortical growth (equation (9)) with

$\begin{equation} {\rm Cortex:}\;\; G_t = 1 + \tau (1 + 0.1\cos 8\theta ).\end{equation} \tag{ 16 }$

No initial imperfection or growth in the core was imposed. These spatial variations in tangential growth appear to drive the ellipsoid toward a final shape with the same number of lobes as the growth pattern, with gyri at regions of greatest expansion and sulci at regions of least expansion (figure 11). However, short wavelength instabilities may be superimposed on this pattern (figure 11, row 1) if the cortical growth rate is high enough. If intrinsic variations in tangential growth rate do indeed lead to consistently located primary folds, growth-induced instability remains the likely cause of secondary gyri.

**Figure 11.** The effect of spatial variation of cortical growth rate. Spatial variations of tangential growth in the cortex were imposed according to equation (16) with *G_t* = 1 + τ(1 + 0.1cos 8θ). Columns: tangential growth *G_t* at τ = 0.10; radial growth *G_R*; tangential growth *G_t*; 3D view of radial stress σ_r. Row 1: Γ_G = 0.50, τ = 1.00. Row 2: Γ_G = 0.20, τ = 1.25.
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Discussion

Evaluation of cortical folding hypotheses

The analysis and simulations presented here support the hypothesis that differential tangential growth of the cortex, rather than axonal tension, drives the folding process. The current model is distinct from previous models of differential growth that treat the brain as a layered elastic body [22, 23]. The essential feature of the current model is that deeper layers grow in response to stresses developed by cortical expansion. While axonal tension-driven models [16] and models in which the cortex is pushed outward by growth [40] may produce realistic shapes similar to the folded brain, geometric similarity is not strong evidence of the underlying hypotheses. The stress states predicted by such models are not consistent with experimental observations described in [21]. The predictions of the current model, using reasonable estimates of parameters from experimental studies [5, 19] are consistent with both observed folding patterns and observed stress distributions [21].

We previously introduced a model of cortical folding [21] which also included stress-dependent growth in deeper layers. The prior model included more layers with different specified growth rates in each layer, but relied on limiting cortical growth to a specific region to induce a single local fold in that region. In contrast, the current model relies on instability to produce folding patterns in which wavelength depends on the relative rates of growth.

While growth-induced instability is suggested to underlie the more random, shorter-wavelength secondary folds, two candidate mechanisms are identified for the formation of primary gyri. (i) Heterogeneous radial growth before the period of folding may set up small variations in initial conditions (shape, stiffness, or stress) that are amplified by tangential expansion of the cortex. We note that axonal tension, while not actively producing folds, may be an important feature of such an initial state. (ii) Tangential cortical expansion itself may be heterogeneous, as in [21]. Current experimental data do not disprove either of these possibilities. It is clear that the shape of the brain before folding is not perfectly smooth [1–4]; we have also observed that on a coarse scale, cortical expansion is non-uniform [5]. More precise and better resolved measurements are needed to determine the relative importance of these mechanisms in determining gyral location.

Some investigators have made important predictions regarding cortical folding without directly addressing the causal mechanism. Todd [41] suggested that initial curvature determines the folding pattern, with sulci evolving from lines of minimal curvature. This is consistent with the effects of initial shape exhibited by the current model. Prothero and Sundstren [42] use scaling arguments to arrive at plausible shapes, but do not address mechanical forces.

Elastic and hyperelastic models of wrinkling and creasing of soft tissue

A number of recent theoretical studies [28, 43–47] have demonstrated that instabilities of surfaces or layers in elastic and hyperelastic materials arise in response to compression or constrained growth. These models of soft hyperelastic materials can produce folds or creases like those seen in the brain. The current model shares many of the features of these elastic models, but emphasizes the role of stress-induced growth, rather than elastic deformation, in predicting the geometry, wavelengths and stress fields associated with folding. The current model not only produces reasonable predictions of folding patterns and stress distributions, but more accurately reflects the behavior of the living brain tissue on these time scales.

Limitations

Both analytical predictions of gyral wavelength and numerical simulations were performed. There is a consistent discrepancy of approximately 25% between the wavelengths observed in simulations and the predictions of the stability analysis (figure 4, equations (11)–(12), and the appendix). The analysis is based on the simplest theoretical model developed by Biot [39] for folding of a viscoelastic layer embedded in a viscoelastic medium. In this model, the adhesion (tangential traction) between core and cortex was neglected, and the dimensions of the foundation (L, H) were assumed to be infinite. Furthermore, the replacement of the operator R by the inverse of the time to reach critical stress, $R = \frac{1}{{T_P }} = 4\mu G_0 /\sigma _P$ , is an ad hoc assumption that is physically reasonable but not mathematically precise. Despite the relatively small quantitative discrepancy between equations (11) and (12) and simulations, the correct prediction of the inverse relationship between growth rate and wavelength is a valuable insight from the analysis.

Although cortical folding is truly a 3D process, simulations were performed in 2D, assuming either plane strain conditions or axisymmetric deformation. While 2D mechanical models have proven useful for understanding 3D behavior, it is clear that ultimately 3D models will be required to study characteristic folding patterns. Similarly, the boundary conditions in the current model, particularly the assumption of zero force on the cortical surface, are idealized. Our results show that boundary forces are not necessary to produce folding, but do not exclude the possibility that boundary conditions play an important role. Also, other mechanisms (such as axonal connectivity) could modulate shapes produced by differential cortical expansion and stress-induced growth.

Future work

The current study is focused on the generic mechanisms that govern the initial formation and wavelengths of gyri. Further numerical studies should explore the large-deformation, 'post-buckled' behavior of these models. Such studies will need to exploit more advanced simulation techniques to achieve convergence and accuracy under these conditions. The domains of the current simulations are extremely simple; the extension of modeling and simulation to the full 3D case with realistic initial shapes (as in [43]) is clearly warranted.

Conclusions

A model of cortical folding based on differential growth, in which the stiffnesses of the cortex and interior regions of the brain are similar, can explain both (i) variations in wavelength of folds and (ii) the stress fields observed in the developing brain. Cortical growth rate, relative to how quickly the core grows in response to stress, affects the wavelength of cortical folds; more rapid cortical expansion generally leads to shorter wavelengths. Wavelengths also scale directly with cortical thickness. Finally, the initial shape before tangential cortical expansion, and spatial variations in tangential expansion itself, also affect the final shape.

Acknowledgments

Funding from NIH grant R01 NS070918 (Taber), R21 EB005834 (Bayly) and NSF grant DMS0540701 (Taber) is gratefully acknowledged. C D Kroenke provided valuable suggestions.

Appendix:

A closed-form prediction for cortical wavelength is derived, in terms of the cortical growth rate and the mechanical properties of the cortical and subcortical regions. We follow the analysis by Biot [39] of folding of a viscoelastic plate on a viscoelastic foundation. We begin by modeling the cortex as a thin elastic plate (thickness h, width b, length L, Young's modulus E, Poisson's ratio ν) on a continuous elastic foundation, or core (thickness H, width b, length L, Young's modulus E_f, Poisson's ratio ν_f) undergoing plane strain deformation while under a compressive axial load, P (figure A1). The equation for quasi-static deflections of the plate, neglecting tangential traction forces between the plate and foundation, is

$\begin{equation} \frac{E}{{1 - \nu ^2 }}\frac{{bh^3 }}{{12}}\frac{{\partial ^4 w}}{{\partial x^4 }} + P\frac{{\partial ^2 w}}{{\partial x^2 }} = q.\end{equation} \tag{ A.1 }$

The vertical force per unit length q is related to the deflection w by the stiffness of the foundation. If w is sinusoidal, then q is also sinusoidal with amplitude dependent on the amplitude, w₀, and spatial frequency, γ, of w [39]:

$\begin{equation} w = w_0 \,{\rm cos}\,\gamma x,\end{equation} \tag{ A.2 }$

$\begin{equation} q = q_0 \,{\rm cos}\,\gamma x,\end{equation} \tag{ A.3 }$

$\begin{equation} q_0 = \frac{{ - \gamma bE_f }}{{2(1 - \nu _f ^2 )}}w_0 .\end{equation} \tag{ A.4 }$

The wavelength of the sinusoidal solution is λ = 2π/γ. (As an aside, we note that in the classic Winkler foundation model, represented by an array of linear springs, the foundation stiffness k = q₀/w₀ is independent of spatial frequency.) The compressive load, P, on the thin plate can be expressed in terms of the compressive stress, σ_P:

$\begin{equation} P = \sigma _P bh.\end{equation} \tag{ A.5 }$

**Figure A1.** A thin elastic beam on an elastic foundation, under compressive loading.
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Substituting the expressions for w, q and P above into the equation of equilibrium, and limiting our discussion to the case where both plate and foundation are incompressible ( $\frac{E}{{1 - \nu ^2 }} = 4\mu$ and $\frac{{E_f }}{{1 - \nu _f ^2 }} = 4\mu _f$ ) the solution becomes unstable when

$\begin{equation} \frac{{\mu h^3 }}{3}\gamma ^4 - \sigma _P h\gamma ^2 = - 2\mu _f \gamma ,\end{equation} \tag{ A.6 }$

or

$\begin{equation} \sigma _P = \frac{\mu }{3}h^2 \gamma ^2 + \frac{{2\mu _f }}{{h\gamma }}.\end{equation} \tag{ A.7 }$

The critical stress σ_Pcr at which a specific harmonic solution becomes unstable depends on the scaled wavenumber hγ = 2πh/λ. The solution with minimal critical stress is found by setting $\frac{{\partial \sigma _P }}{{\partial (h\gamma )}} = 0$ , leading to the following expressions for critical stress and wavelength:

$\begin{equation} \sigma _{Pcr} = 3\mu \left( {\frac{\mu }{{3\mu _f }}} \right)^{1/3} ,\end{equation} \tag{ A.8 }$

$\begin{equation} \lambda = 2\pi h\left( {\frac{\mu }{{3\mu _f }}} \right)^{1/3} .\end{equation} \tag{ A.9 }$

Because stress induces growth over time scale of interest [19] the core does not act like a purely elastic system. Rather it relaxes after the application of stress, like the spring-dashpot foundation in figure A.2. This behavior is approximated by the classic Maxwell model of a viscoelastic fluid. For a Maxwell fluid, the force–displacement relation (equation (A.4)) for the foundation is replaced by

$\begin{equation} \dot q_0 + \frac{1}{{\tau _f }}q_0 = - 2\mu _f \gamma b\dot w_0 .\end{equation} \tag{ A.10 }$

Defining the operator $R = \frac{{\rm d}}{{{\rm d}t}}$ as in [39], we rewrite the equation above as

$\begin{equation} q_0 = - 2\mu _f \gamma b\frac{R}{{R + R_f }}w_0 ,\end{equation} \tag{ A.11 }$

where $R_f = \frac{1}{{\tau _f }}$ is the rate constant of the foundation. When R ≪ R_f the foundation behaves like a fluid; when R ≫ R_f it acts like a solid.

**Figure A2.** (a) If no subcortical growth occurs in response to stress, the cortex is like the beam on an elastic foundation. (b) If growth occurs in response to stress, the core acts like a viscoelastic (Maxwell) foundation, responding like a solid for fast deformations and like a fluid at slow strain rates.
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Applying the correspondence principle [48], we substitute the new expression into the equation of motion and evaluate the conditions for stability. The critical load and wavelength now depend on R and R_f [39]:

$\begin{equation} \sigma _{Pcr} = 3\mu _f \frac{R}{{R + R_f }}\left( {\frac{{\mu (R + R_f )}}{{3\mu _f R}}} \right)^{1/3}\! ,\end{equation} \tag{ A.12 }$

$\begin{equation} \lambda = 2\pi h\left( {\frac{{\mu (R + R_f )}}{{3\mu _f R}}} \right)^{1/3} \!.\end{equation} \tag{ A.13 }$

In these expressions it is clear that choosing R to be small (slow deformations) leads to small critical load and long wavelength—the foundation appears soft. When R is large (relatively rapid, but still quasi-static deformation), the critical load increases and the wavelength decreases, approximating the response of a beam on a stiff foundation.

A.1. Extension to growth: an elastic layer growing on a viscoelastic foundation

The 'viscoelastic rate constant' of the growing foundation is R_f = 4aμ_f (equation (8)). To choose the value of the operator R, we consider how long it will take to achieve the critical stress in the growing cortex. If the growth rate in the cortex is G₀, the critical compressive stress σ_Pcr ≈ 4μG₀T_P, where T_P is the 'critical time' when buckling occurs. If we set the value of R = 1/T_P, and thus set σ_P = 4μG₀/R in the equation for the critical stress (A.12) we obtain a polynomial in R:

$\begin{equation} R^5 - \frac{{64\mu ^2 G_0 ^3 }}{{9\mu _f^2 }}\big(R^2 + 2R_f R + R_f ^2 \big) = 0.\end{equation} \tag{ A.14 }$

This equation can be rewritten in terms of the scaled operator Γ = R/R_f and the dimensionless parameters Γ_G = G₀/R_f and β = μ/μ_f. We obtain

$\begin{equation} \Gamma ^5 - \frac{{64}}{9}\beta ^2 \Gamma _G ^3 (\Gamma ^2 + 2\Gamma + 1) = 0.\end{equation} \tag{ A.15 }$

The existence of at least one positive real root of equations (A.15) and (12 in the main text) indicates that the critical load (equation (A.12)) will be reached at the given ratios of growth rate and material parameters. The corresponding wavelength is given by equations (A.17) and (11 in the main text):

$\begin{equation} \frac{{\sigma _{Pcr} }}{\mu } = \frac{{3\Gamma }}{{\beta (\Gamma + 1)}}\left( {\frac{{\beta \left( {\Gamma + 1} \right)}}{{3\Gamma }}} \right)^{\frac{1}{3}} ,\end{equation} \tag{ A.16 }$

$\begin{equation} \frac{\lambda }{h} = 2\pi \left( {\frac{{\beta (\Gamma + 1)}}{{3\Gamma }}} \right)^{1/3} .\end{equation} \tag{ A.17 }$

Equations (A.15–A.17) show that the wavelength and critical stress depend on these two dimensionless parameters: β, the ratio of the short-term elastic moduli of cortex and core, and Γ_G, the ratio of cortical expansion rate to the rate constant of stress-induced growth.

A cortical folding model incorporating stress-dependent growth explains gyral wavelengths and stress patterns in the developing brain

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Abstract

Introduction

Mathematical model and simulation

Analytical prediction of critical stress and wavelength