The role of mechanics during brain development

Abstract

Convolutions are a classical hallmark of most mammalian brains. Brain surface morphology is often associated with intelligence and closely correlated to neurological dysfunction. Yet, we know surprisingly little about the underlying mechanisms of cortical folding. Here we identify the role of the key anatomic players during the folding process: cortical thickness, stiffness, and growth. To establish estimates for the critical time, pressure, and the wavelength at the onset of folding, we derive an analytical model using the Föppl-von-Kármán theory. Analytical modeling provides a quick first insight into the critical conditions at the onset of folding, yet it fails to predict the evolution of complex instability patterns in the post-critical regime. To predict realistic surface morphologies, we establish a computational model using the continuum theory of finite growth. Computational modeling not only confirms our analytical estimates, but is also capable of predicting the formation of complex surface morphologies with asymmetric patterns and secondary folds. Taken together, our analytical and computational models explain why larger mammalian brains tend to be more convoluted than smaller brains. Both models provide mechanistic interpretations of the classical malformations of lissencephaly and polymicrogyria. Understanding the process of cortical folding in the mammalian brain has direct implications on the diagnostics of neurological disorders including severe retardation, epilepsy, schizophrenia, and autism.

Keywords: Brain development; cortical folding; growth; instabilities; thin films.

Figure 1

Surface to volume ratio of the mammalian brain. Larger mammals have larger brains [65]. The dashed line with a slope of 2/3 indicates isometric scaling for which brain surface area would scale proportionally with brain volume. The solid line with a slope of 0.9 indicates that the mammalian brain surface area increases disproportionally faster than brain volume [32]. The degree of gyrification increases with brain size.

Figure 2

Surface morphology of the mammalian brain. Larger mammals have larger brains: The cow brain, right, is larger than the pig brain, middle, which is larger than the sheep brain, left. Photographs of the entire brain, upper row, and a frontal coronal brain section, lower row, illustrate the characteristic folding pattern: The surface of the cow brain, right, is more folded than the surface of the pig brain, middle, which is more folded than the sheep brain, left. The degree of gyrification increases with brain size.

Figure 3

Analytical model of confined, layered medium subjected to growth-induced compression. Growing layer on an elastic foundation, left, and on a growing foundation, right. We model cortical folding using the classical fourth order Föppl-von-Kármán plate theory and adopt a sinusoidal ansatz for the deflection w, which generates a sinusoidal transverse force q. This provides analytical estimates for the critical cortical pressure P^crit and for the wavelength λ^crit parameterized in terms of the cortical thickness t_c, the cortical and subcortical Young’s moduli E_c and E_s, and the cortical and subcortical growth rates G_c and G_s.

Figure 4

Growth-induced cortical pressure P as a function of the wavenumber n for varying cortical stiffnesses E_c, left, and varying subcortical stiffnesses E_s, right. The dotted line characterizes the critical pressure P^crit at which folding occurs. The corresponding wavenumber n characterizes the critical folding pattern.

Figure 5

Critical folding pressure P^crit, left, and critical folding time t^crit, right, for varying growth ratio G_c/G_s and varying stiffness ratio E_c/E_s. The folding pressure P^crit increases with increasing cortical growth G_c and increasing cortical stiffness E_c. The folding time t^crit decreases with increasing cortical growth G_c and increasing cortical stiffness E_c.

Figure 6

Critical wavelength λ^crit for varying stiffness ratios E_c/E_s and varying cortical thicknesses t_c, left, and for varying growth ratios G_c/G_s and varying cortical thicknesses t_c, right. The wavelength λ^crit increases with increasing cortical stiffness E_c, with increasing subcortical growth rate G_s, and with increasing cortical thickness t_c.

Figure 7

Multiscale continuum model for cortical and subcortical growth. The cortex, the gray matter, grows morphogenetically at a constant rate G_c. Cortical growth induces subcortical deformation, which triggers subcortical growth. The subcortex, the white matter, grows at a stretchdependent rate as G_s 〈 J^e − J⁰ 〉, where G_s mimics the axon elongation rate and 〈 J^e − J⁰ 〉 activates growth only, if the elastic volume stretch J^e exceeds its baseline value J⁰.

Figure 8

Sensitivity of surface morphology with respect to initial cortical thickness for constrained growth in a rectangular domain. The dots illustrate the computationally predicted wavelengths λ^crit for varying cortical thicknesses t_c. The solid line shows the averaged computational wavelength-thickness relation for a morphogenetically growing cortex on a stretch-driven growing subcortex. The dashed line shows the analytical wavelength-thickness relation for a growing cortical layer on a growing subcortical foundation.

Figure 9

Sensitivity of surface morphology with respect to initial cortical thickness and stiffness ratio for elliptic geometry. The wavelength λ^crit increases with increasing cortical thickness t_c, from top to bottom, and with increasing stiffness ratio, E_c/E_s, from left to right. Larger wavelengths induce larger subcortical stretch resulting in larger subcortical growth.

Figure 10

Sensitivity of surface morphology with respect to initial cortical thickness and stiffness ratio for elliptic geometry. The dots illustrate the computationally predicted average wavelengths λ^crit for varying cortical thicknesses t_c and varying stiffness ratios E_c/E_s. The average wavelength increases with increasing cortical thickness t_c, from left to right, and with increasing stiffness ratio, E_c/E_s, from lower blue dots to upper red dots.

Figure 11

Sensitivity of surface morphology with respect to initial cortical thickness and growth ratio for elliptic geometry. The wavelength of primary folding λ^crit increases with increasing cortical thickness t_c, from top to bottom. The overall wavelength increases with increasing growth ratios G_c/G_s, from left to right.