Gyrification



In the brain sciences, gyrification (or cortical folding or fissuration) refers to the folding of the cerebral cortex in mammals as a consequence of brain growth during embryonic and early postnatal development. In this process, gyri (ridges) and sulci (fissures) form on the external surface of the brain (i.e. at the boundary between the cerebrospinal fluid and the gray matter. The term gyrification is also sometimes used instead of the more common term foliation to describe the folding patterns of the cerebellum, which is highly convoluted in other taxa, too, e.g. in birds.

Phylogeny
As illustrated in the figure, gyrification occurs across mammals in a gradually different manner: It increases slowly with overall brain size, following a power law , and a range of theoretical models exist as to the degree to which it hints at the evolution of cognitive abilities in a given range of species.

Ontogeny
The folding process usually starts during embryonic development, in humans around mid-gestation . While the extent of cortical folding has been found to be partly determined by genetic factors  , the underlying biomechanical mechanisms are not yet well understood. The overall folding pattern, however, can be mechanistically explained in terms of the cerebral cortex behaving as a gel that buckles under the influence of non-isotropic forces. Possible causes of the non-isotropy include thermal noise, variations in the number and timing of cell divisions, cell migration, cortical connectivity, synaptic pruning, brain size and metabolism (phospholipids in particular), all of which may interact.

Medical relevance
This multitude of underlying processes has rendered the concept of gyrification increasingly important for clinical diagnostics in recent years, since gyrification in some areas of the human brain appears to reflect functional development and thus to correlate with measures of intelligence, and disturbances in the folding pattern &mdash; as determined by non-invasive neuroimaging &mdash; can be taken as indicators of neuropsychiatric diseases if differences due to gender and age are accounted for. Apart from being a marker of disorders like schizophrenia or Williams syndrome, a number of disorders exist of which abnormal gyrification is a dominant feature, e.g. polymicrogyria or lissencephaly ranging from agyria to pachygyria.

Quantification
Folding of a brain can be described in both local and global terms, once a suitable representation of a brain surface has been obtained from neuroimaging data by some surface extraction technique. The latter usually delivers a triangulated surface representing either the boundary between the cerebrospinal fluid and the gray matter or between the latter and the white matter but in principle, any surface in between would do as well (e.g. the central layer which is also sometimes used). Leaving the multiple issues of resolution and artifacts in these surface representations aside, the brain surface mesh, like any mesh of a closed three-dimensional manifold, can then be analyzed in terms of local curvature measures, from which global measures can be derived. Over the last decades, several such measures have been proposed. Following the developments in imaging techniques, they were initially focused on quantification in two-dimensional spaces, later in three-dimensional ones. Some examples that are commonly used include:
 * $ L^2 norms$:
 * $$LN_G = \tfrac{1}{4\pi} \textstyle \sqrt{\sum_A K^2}$$, with $$K = k_1 k_2$$ being the Gaussian curvature, computed from the two principal curvatures $$k_1$$ and $$k_2$$
 * $$LN_M =\tfrac{1}{4\pi} \textstyle \sum_A H^2$$, with $$H=\tfrac{1}{2}(k_1 + k_2)$$ being the Mean curvature
 * Folding index
 * $$FI =\tfrac{1}{4\pi} \textstyle \sum_A k^{\ddagger}$$, with $$k^{\ddagger}=|k_1|(|k_1|-|k_2|)$$
 * Intrinsic curvature index
 * $$ICI =\tfrac{1}{4\pi} \textstyle \sum_A K^+$$, with $$K^+$$ being the positive Gaussian curvature


 * Gyrification index
 * Cortical complexity
 * Fractal dimension
 * Global gyrification index
 * Local gyrification index
 * Shape index
 * Curvedness
 * Roundness