Gyrification/Addendum

Quantitative measures of gyrification

Commonly used measures of the exent of cortical folding 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 and $A$ the area of the surface in question
• 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
• Curvedness
  • $C={\sqrt {\tfrac {k_{1}^{2}+k_{2}^{2}}{2}}}$
• Sharpness of folding
  • $S=(k_{1}-k_{2})^{2}$
• Bending energy
  • $E_{b}=\int _{A}{(k_{1}+k_{2})}^{2}dA$
• Willmore energy
  • $E_{W}=\int _{A}{(k_{1}-k_{2})}^{2}dA=\int _{A}{H}^{2}dA-\int _{A}{K}dA$
• Gyrification index
  • $GI(n)={\tfrac {A(n)_{pial}}{A(n)_{gc}}}$, with $n$ indicating the number of the slice, and $A(n)_{pial}$ and $A(n)_{gc}$ being the pial surface area in that slice and the surface area of the boundary between gray matter and cerebrospinal fluid, respectively
• Gyrification-White index
  • $GWI={\tfrac {A_{gw}}{A_{gc}}}$, with $A_{gw}$ being the surface area of the boundary between gray matter and white matter
• White matter folding
  • $WMF={\tfrac {A_{gw}}{{V_{w}}^{2/3}}}$, with $V_{w}$ being the volume of the white matter
• Cortical complexity
• Fractal dimension
• Global gyrification index
• Local gyrification index
• Shape index
• Roundness
  • $RN={\tfrac {A}{^{3}{\sqrt {36\pi V^{2}}}}}$