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

Abstract

In humans and many other mammals, the cortex (the outer layer of the brain) folds during development. The mechanics of folding are not well understood; leading explanations are either incomplete or at odds with physical measurements. We propose a mathematical model in which (i) folding is driven by tangential expansion of the cortex and (ii) deeper layers grow in response to the resulting stress. In this model the wavelength of cortical folds depends predictably on the rate of cortical growth relative to the rate of stress-induced growth. We show analytically and in simulations that faster cortical expansion leads to shorter gyral wavelengths; slower cortical expansion leads to long wavelengths or even smooth (lissencephalic) surfaces. No inner or outer (skull) constraint is needed to produce folding, but initial shape and mechanical heterogeneity influence the final shape. The proposed model predicts patterns of stress in the tissue that are consistent with experimental observations.

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.

Figure 2

Effects of cortical growth rate on wavelength, subcortical growth, and stress in the basic model of cortical folding. Target stresses σ̄_r0 = σ̄_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 = 0.08. The modulus ratio β = 1 in all cases.

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.

Figure 4

(a) Effect of cortical growth rate on the gyral wavelength predicted by Eqs. 11–12 for three values of the stiffness ratio β. (b) Solid line: wavelengths predicted from Eqs. 11–12 for β = 1; Dotted line and open circles: wavelengths observed in simulations.

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 =0.048.

Figure 6

Folding in (a)the elliptical cylinder (plane strain) model and (b) the axisymmetric ellipsoid model. Color encodes tangential growth G_t.

Figure 7

Effects of initial shape perturbation on wavelength, subcortical growth, and stress in the 2D elliptical (cylinder) model with a compressive target stress in the outer core. Initial radial growth in the core was attained with Eq. 15 with F(R, θ, τ) = F₀τ cos kθ 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. Row 1: F(r, θ, τ) = 10τ cos 32θ for τ < 0.10, shown at τ =0.90. Row 2: F(r, θ, τ) = 10τ cos 8θ for τ < 0.10, shown at τ =0.60; Row 3: F(r, θ, τ) = 50τ random (r, θ) for τ < 0.1, shown at τ =1.2. In all rows Γ_G = 0.20.

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 Eq. 15 with F(R, θ, τ) = 10τ cos 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.

Figure 9

The effects of cortical growth rate and small variations in initial conditions. Initial radial growth in the core was imposed according to Eq. 15 with F(R, θ, τ) = 10τ cos 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.

Figure 10

The effects of cortical growth rate and larger initial perturbations. Initial radial growth in the core was imposed according to Eq. 15 with F(R, θ, τ) = 50τ cos 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.

Figure 11

The effect of spatial variation of cortical growth rate. Spatial variations of tangential growth in the cortex were imposed according to Eq. 16 with G_t = 1 + τ(1 + 0.1 cos 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.

Figure A.1

A thin elastic beam on an elastic foundation, under compressive loading.

Figure A.2

(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.