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This addendum is a continuation of the article Gyrification.

## Quantitative measures of gyrification

Commonly used measures of the exent of cortical folding include[1][2]:

• ${\displaystyle L^{2}norms}$:
• ${\displaystyle LN_{G}={\tfrac {1}{4\pi }}\textstyle {\sqrt {\sum _{A}K^{2}}}}$, with ${\displaystyle K=k_{1}k_{2}}$ being the Gaussian curvature, computed from the two principal curvatures ${\displaystyle k_{1}}$ and ${\displaystyle k_{2}}$
• ${\displaystyle LN_{M}={\tfrac {1}{4\pi }}\textstyle \sum _{A}H^{2}}$, with ${\displaystyle H={\tfrac {1}{2}}(k_{1}+k_{2})}$ being the Mean curvature and ${\displaystyle A}$ the area of the surface in question
• Folding index
• ${\displaystyle FI={\tfrac {1}{4\pi }}\textstyle \sum _{A}k^{\ddagger }}$, with ${\displaystyle k^{\ddagger }=|k_{1}|(|k_{1}|-|k_{2}|)}$
• Intrinsic curvature index
• ${\displaystyle ICI={\tfrac {1}{4\pi }}\textstyle \sum _{A}K^{+}}$, with ${\displaystyle K^{+}}$ being the positive Gaussian curvature
• Curvedness
• ${\displaystyle C={\sqrt {\tfrac {k_{1}^{2}+k_{2}^{2}}{2}}}}$
• Sharpness of folding
• ${\displaystyle S=(k_{1}-k_{2})^{2}}$
• Bending energy
• ${\displaystyle E_{b}=\int _{A}{(k_{1}+k_{2})}^{2}dA}$
• Willmore energy
• ${\displaystyle E_{W}=\int _{A}{(k_{1}-k_{2})}^{2}dA=\int _{A}{H}^{2}dA-\int _{A}{K}dA}$
• Gyrification index
• ${\displaystyle GI_{slice}(n)={\tfrac {A(n)_{outer}}{A(n)_{inner}}}}$, with ${\displaystyle n}$ indicating the number of the slice, and ${\displaystyle A(n)_{outer}}$ and ${\displaystyle A(n)_{inner}}$ being the outer and inner cortical contour in that slice. Anatomically, the inner contour can be thought of as representing the pia mater, the outer one the arachnoid mater. The latter correspondence is rough, since the arachnoid also encloses venous sinuses.
• ${\displaystyle GI_{mesh}(n)={\tfrac {A(n)_{outer}}{A(n)_{inner}}}}$, with ${\displaystyle n}$ indicating the number of the region, and ${\displaystyle A(n)_{outer}}$ and ${\displaystyle A(n)_{inner}}$ being the outer and inner cortical surface area in that region. The anatomical correspondences apply equally to the slice-based and regional definitions.
• Gyrification-White index
• ${\displaystyle GWI={\tfrac {A_{gw}}{A_{gc}}}}$, with ${\displaystyle A_{gw}}$ being the surface area of the boundary between gray matter and white matter
• White matter folding
• ${\displaystyle WMF={\tfrac {A_{gw}}{{V_{w}}^{2/3}}}}$, with ${\displaystyle V_{w}}$ being the volume of the white matter
• Cortical complexity
• Fractal dimension
• Shape index
• Roundness
• ${\displaystyle RN={\tfrac {A}{^{3}{\sqrt {36\pi V^{2}}}}}}$

## References

1. Rodriguez-Carranza, C.E.; P. Mukherjee & D. Vigneron et al. (2008), "A framework for in vivo quantification of regional brain folding in premature neonates", Neuroimage 41: 462, DOI:10.1016/j.neuroimage.2008.01.008
2. Pienaar, R.; B. Fischl & V. Caviness et al. (2008), "A methodology for analyzing curvature in the developing brain from preterm to adult", International Journal of Imaging Systems and Technology 18 (1): 42–68, DOI:10.1002/ima.20138