480
8 Morphogenesis
Models for Cortical Folding The models discussed in the following paragraphs
are based on differential growth. 17 Unless stated otherwise, growth in all models is
simulated using RHM theory.
The first quantitative biomechanical model for cortical folding was proposed by
Richman et al. (1975), who treated the brain as a bilayered spherical shell attached
to a core of elastic tissue. Both layers are assumed to grow at different rates and
considerably faster than the core. With curvature effects neglected, compressive
stresses in the cortical layers cause them to buckle into a pattern roughly consistent
with observations. Like a beam on an elastic foundation, the buckling wavelength
decreases as the foundation stiffness increases [see Eq. (8.75)].
About 35 years later, Xu et al. (2010b) proposed a 2D model for the ferret brain,
which conveniently folds during the first few weeks after birth (Fig. 8.62a). In the
initial configuration, the brain is modeled as a solid circular cylinder containing four
layers (Fig. 8.62b): cortex, subplate, white matter, and deep gray matter. Consistent
with experimental measurements, all layers have the same material properties.
The simulation includes the following steps:
1. With G z = 1, increasing values of G r = G θ are specified uniformly in all
layers except the white matter. This pattern of differential growth generates
circumferential tension in the white matter (Fig. 8.62c, left).
2. Additional circumferential growth (G θ ) is specified in region 1 of the cortex
(defined in Fig. 8.62b), causing it to buckle locally outward, creating a gyrus
(Fig. 8.62c, center). Simultaneously, since the subplate contains axons, which
grow in response to tension (Chada 1997), this layer grows (and relaxes)
according to the evolution equations
˙
G r = aσ r G r ,
˙
G θ = aσ θ G θ ,
˙
G rθ = ˙
G θr = aσ rθ ,
where a is a constant and the shear terms are included because of the complex
deformation in the subplate. The initial conditions are G r (0) = G θ (0) = 1 and
G rθ (0) = G θr (0) = 0.
3. After region 1 stops growing, the same procedure is applied to cortical region 2,
creating a second gyrus (Fig. 8.62c, right).
The successive gyri created in this model are consistent with findings that the
developing brain folds progressively from lateral to medial locations (Kroenke
et al. 2009). The resulting morphology resembles that of an 18-day-old ferret brain
(compare Figs. 8.62c and 8.63a).
To test the model further, residual stresses were probed by making linear cuts in
slices of ferret brains (Xu et al. 2010b). At day 18, for example, radial cuts through
gyri and sulci revealed significant circumferential tension in the white matter tract,
as well as near the outer surface in gyri. Less tension or even compression was found
17 Other computational models for cortical folding are reviewed elsewhere (Wyczalkowski et al.
2012; Garcia et al. 2018).
8 Morphogenesis
Models for Cortical Folding The models discussed in the following paragraphs
are based on differential growth. 17 Unless stated otherwise, growth in all models is
simulated using RHM theory.
The first quantitative biomechanical model for cortical folding was proposed by
Richman et al. (1975), who treated the brain as a bilayered spherical shell attached
to a core of elastic tissue. Both layers are assumed to grow at different rates and
considerably faster than the core. With curvature effects neglected, compressive
stresses in the cortical layers cause them to buckle into a pattern roughly consistent
with observations. Like a beam on an elastic foundation, the buckling wavelength
decreases as the foundation stiffness increases [see Eq. (8.75)].
About 35 years later, Xu et al. (2010b) proposed a 2D model for the ferret brain,
which conveniently folds during the first few weeks after birth (Fig. 8.62a). In the
initial configuration, the brain is modeled as a solid circular cylinder containing four
layers (Fig. 8.62b): cortex, subplate, white matter, and deep gray matter. Consistent
with experimental measurements, all layers have the same material properties.
The simulation includes the following steps:
1. With G z = 1, increasing values of G r = G θ are specified uniformly in all
layers except the white matter. This pattern of differential growth generates
circumferential tension in the white matter (Fig. 8.62c, left).
2. Additional circumferential growth (G θ ) is specified in region 1 of the cortex
(defined in Fig. 8.62b), causing it to buckle locally outward, creating a gyrus
(Fig. 8.62c, center). Simultaneously, since the subplate contains axons, which
grow in response to tension (Chada 1997), this layer grows (and relaxes)
according to the evolution equations
˙
G r = aσ r G r ,
˙
G θ = aσ θ G θ ,
˙
G rθ = ˙
G θr = aσ rθ ,
where a is a constant and the shear terms are included because of the complex
deformation in the subplate. The initial conditions are G r (0) = G θ (0) = 1 and
G rθ (0) = G θr (0) = 0.
3. After region 1 stops growing, the same procedure is applied to cortical region 2,
creating a second gyrus (Fig. 8.62c, right).
The successive gyri created in this model are consistent with findings that the
developing brain folds progressively from lateral to medial locations (Kroenke
et al. 2009). The resulting morphology resembles that of an 18-day-old ferret brain
(compare Figs. 8.62c and 8.63a).
To test the model further, residual stresses were probed by making linear cuts in
slices of ferret brains (Xu et al. 2010b). At day 18, for example, radial cuts through
gyri and sulci revealed significant circumferential tension in the white matter tract,
as well as near the outer surface in gyri. Less tension or even compression was found
17 Other computational models for cortical folding are reviewed elsewhere (Wyczalkowski et al.
2012; Garcia et al. 2018).