Abstract
We model the elasticity of the cerebral cortex as a layered material with bending energy along the layers and elastic energy between them in both planar and polar geometries. The cortex is also subjected to axons pulling from the underlying white matter. Above a critical threshold force, a ‘flat’ cortex configuration becomes unstable and periodic undulations emerge, i.e. a buckling instability occurs. These undulations may indeed initiate folds in the cortex. We identify analytically the critical force and the critical wavelength of the undulations. Both quantities are physiologically relevant values. Our model is a revised version of the axonal tension model for cortex folding, with our version taking into account the layered structure of the cortex. Moreover, our model draws a connection with another competing model for cortex folding, namely the differential growth-induced buckling model. For the polar geometry, we study the relationship between brain size and the critical force and wavelength to understand why small mice brains exhibit no folds, while larger human brains do, for example. Finally, an estimate of the bending rigidity constant for the cortex can be made based on the critical wavelength.
Highlights
The cerebral cortex, or grey matter, is the outermost layer of nerve tissue covering the cerebrum and plays a key role in high-level cognitive functions, such as decision-making
This combination of ingredients may make it reasonable to model the cortex as a layered liquid crystal with the neurons representing the liquid crystal molecules. With this “cortex as a liquid crystal structure” in what will turn out to be the smectic phase, we can revisit the axonal tension model and investigate the effect of pulling forces on the cortex. We will do this in both a planar geometry and a polar geometry and demonstrate that “vertical pulling” of the axons in the underlying white matter can lead to buckling in a layered structure
We assume that all the layers are equivalent in thickness and in elastic properties. Given this extra spatial structure, we model the elasticity of the cortex as a smectic liquid crystal and ask the following: What are the consequences of axons from the underlying white matter pulling vertically on the cortex in this planar geometry? The pulling of axons has been well-established [14] and given the orientation of axon highways in the underlying white matter [15], vertical pulling is in keeping with observations
Summary
The cerebral cortex, or grey matter, is the outermost layer of nerve tissue covering the cerebrum and plays a key role in high-level cognitive functions, such as decision-making. For the axonal tension model, as originally formulated, neuronal pathways connecting gyri should be denser than those connecting sulci (inward folds) Some data supports this notion, though the results may be a matter of defining which surrounding regions belong to gyri and which belong to sulci [8]. With this “cortex as a liquid crystal structure” in what will turn out to be the smectic phase, we can revisit the axonal tension model and investigate the effect of pulling forces on the cortex We will do this in both a planar geometry and a polar geometry and demonstrate that “vertical pulling” of the axons in the underlying white matter can lead to buckling in a layered structure.
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