L Measure functions

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About L Measures

L Measures is a research tool for making quantitative morphological measurements from neuronal reconstructions. There are 43 measurements including area measurements of soma, length measurements of processes, angular twists and turns of processes, calculation of the number of terminals and bifurcations, and many more.


L-Measure website

Quick Table of L Measures and Additional Functions(*)

Features Formula or Equation Image Description
Dimensions
Width X w Neuronwidth.png Measured along the x-axis of a neuron.
Height Y h Neuronheight.png Measured along the y-axis of a neuron.
Depth Z d Neurondepth.png Measured along the z-axis of a neuron.
Soma Features
Soma Position (*) Somasurface.png XYZ coordinate of the soma centroid.
Soma Radii (*) Somasurface.png Average radius of the soma.
Soma Surface Area (*) Number of surface voxels of the soma Somasurface.png Surface area of the soma.
Soma Volume (*) Number of voxels of the soma Somasurface.png Volume of the soma.
Skewness X, Y, Z (*) Xskew.png Yskew.png Zskew.png Center of the soma deviation from the center of the cell (see Dimensions).
Euclidean Skewness (*) Failed to parse (Cannot write to or create math temp directory): \sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2} Distance from the center of the soma to the center of the cell.
Counts
Stems n Nstems.png Number of stems attached to the soma.
Branch Pt n Nbif.png Number of bifurcations of a neuron.
Segments (or Branch) n Nbranch.png Number of branches of a neuron. (A branch is between two critical points or between a critical point and the tip.)
Tips n Ntip.png Number of terminal tips of a neuron.
Compartment features
Type t = 1, 2, 3, or 4 Compartments are assigned four different types: 1=soma, 2=axon, 3=dendrites, and 4=apical dendrites (are dendrites that emerge from the apex of a pyramidal cell).
Diameter D Lmdiameter.png Diameter of each compartment of the neuron.
Diameter Power D1.5 Diameter of each compartment of the neuron raised to the power of 1.5.
Length l Lmlength.png Measured length of each compartment.
Surface Area 2\pi\,\!rh

(area of cylinder w/o top and bottom)

Lmsurface.png Surface area of each compartment.
Section Area \pi\,\!r^2 Lmsectionarea.png Cross-sectional area of each compartment.
Volume \pi\,\!r^2h Lmvolume.png Volume of the compartments.
Burke Taper \dfrac{D_a - D_b}{L} Burketaper.png The change in diameter over pathlength between two critical points (the current and previous one).
Hillman Taper \dfrac{D_a - D_b}{D_a} Hillmantaper.png The ratio of the change in diameter to the initial diameter of two critical points. The initial diameter is usually larger.
Branch (Segment) features
Branch Order n = 0,1,2,3... Branchorder.png Order of branch with respect to the soma starting at 0 for branches attached to soma.
Euclidean Distance \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}

Straight line distance

EuclideanDistance.png Straight line distance between the two endpoints of a branch.
Path Length A + B + C + D + E + ... PathLength.png Summation of the individual compartment lengths that form a branch.
Contraction EucDistance\over Pathlength Contraction.png Ratio between the Euclidean distance of a branch and its path length.
Fragmentation n Fragmentation.png Total number of compartments that form a branch between two bifurcation points, or between a bifurcation point and a terminal tip.
Helix {( \overrightarrow{A} \times \overrightarrow{B} ) \times \overrightarrow{C}} \over 3(|\overrightarrow{A}| \cdot |\overrightarrow{B}| \cdot |\overrightarrow{C}|) Helicity of the branches of the neuronal tree. Needs to be at least 3 compartments long to compute the helicity.
Fractal_Dim \log EucDistance \over \log PathDistance or

(n-1) \sum e_ip_i - \sum e_i \sum p_i \over {(n-1) \sum e_i^2 - (\sum e_i)^2}

(ei is EucDistance and pi is PathDistance)

Fractal dimension metric of the branches in the dendrite trees.
Bifurcation features
Daughter Ratio Da / Db

Da = larger daughter diameter
Db = smaller daughter diameter

Daughterratio.png Ratio between the diameter of the bigger daughter and the smaller daughter of the current bifurcation. The diameters are measured at the initial part of the branches right when they exit the bifurcation.
Parent Daughter Ratio Da / Dp,Db / Dp Parentdaughterratio.png Ratio between the diameter of a daughter and its father for each critical point. The diameters are measured immediately before and after the bifurcation point.
Partition Asymmetry {|n1 - n2|} \over {n1+n2-2} Compares the symmetry of the processes in terms of the number of tips on the left daughter to the number of tips on the right daughter of a bifurcation.
Rall Power D_p^n = D_a^n + D_b^n Parentdaughterratio.png Best value that fits the equation with extreme 0-5, resolution 1/1000 over the range.
Pk (Rall's Ratio) {D_a^n+D_b^n} \over {D_p^n} Parentdaughterratio.png n = [0, 5]
Pk_classic (Rall's Ratio) {D_a^{1.5}+D_b^{1.5}} \over {D_p^{1.5}} Parentdaughterratio.png Rall power is set to 1.5
Pk_2 (Rall's Ratio) {D_a^2+D_b^2} \over {D_p^2} Parentdaughterratio.png Rall power is set to 2
Azimuth (*) ϴ (in degrees) Azimuth of the daughter segment at a bifurcation point on the x-y plane.
Elevation (*) ϴ Elevation of the daughter segment at a bifurcation point extending into the z plane from the azimuth.
Bif_ampl_local ϴ Bifamplocal.png The local angle formed right at the vertex of the branches; initial angle formed as the two daughter branches exit the bifurcation.
Bif_ampl_remote ϴ Bifampremote.png From the bifurcation, it is the angle to the tips of the two daughter branches. Angle between two branches; angle between the ends of two growing segments (in degrees) at the current bifurcation; angle between the two daughter bifurcations and/or terminal ends (whichever comes first) at the current bifurcation. (Daughter bifurcation means the next bifurcation encountered after the current bifurcation.)
Bif_tilt_local ϴ Biftiltlocal.png The angle between the end of the parent branch and the initial part of the daughter branch at the bifurcation; angle between the last compartment of the parent branch and the first compartment of each daughter branch at the bifurcation.
  • There are two daughter branches for each bifurcation, which gives two angles.
Bif_tilt_remote ϴ Biftiltremote.png The angle between the first daughter compartment of the current bifurcating father and the previous father compartment of the previous father bifurcation; angle between the last part of the previous branch and the initial part of the next branches (skipping the branch in-between).
Bif_torque_local ϴ Angle between the current plane of bifurcation and the previous plane of bifurcation; the planes are defined by the initial part of the daughter branches as they exit the point of bifurcation.
Bif_torque_remote ϴ Angle between the current plane of bifurcation and the previous plane of bifurcation; the plane of bifurcation is defined by a bifurcation and its two daughter bifurcations or terminal ends (whichever comes first).
Terminal bifurcation features
Terminal degree n Terminaldegree.png Total number of tips each segment will terminate into.
Terminal Compartments 0 (if not terminal), 1 (if terminal) Terminalsegment.png Number of compartments that comprise the terminal branch.
Terminal Parent Diameter D Lastparentdiameter.png Diameter of last bifurcation before the terminal tips.
Diameter Threshold D Diameterthreshold.png Diameter of first compartment after the terminal bifurcation leading to a terminal tip; initial diameter of the branch exiting the terminal bifurcation.
Hillman Threshold 0.5P + 0.25D1 + 0.25D2

P: parent diameter,
D1 = 1st daughter diameter,
D2 = 2nd daughter diameter

Parentdaughterratio.png Computes the weighted average diameter between 50% of father and 25% of daughter diameters of the terminal bifurcation.
Other features
Tip to Soma Euclidean Distance (*) D TipToSoma.png Straight line distance from the soma point to the terminal tip.
Branch Pt to Soma Euclidean Distance (*) D TipToSoma.png Straight line distance from the soma point to the branch point.

Note: Branches are modeled as a series of compartments. Critical points refer to bifurcation points and terminal tips.

Hierarchy of Feature Computation

OntologyLevel1.jpg
* indicates important feature?
ID Features Level Type Description Derived from ID L-Measure?
0 Radius Node length
1 X,Y,Z Node point
2 Average Radius Compartment length 0
3 Average Diameter Compartment length 2 Y
4 Diameter Power Compartment length 3 Y
5 Section Area Compartment size 2 Y
6 Surface Area Compartment size 2,8 Y
7 Volume Compartment size 5,8 Y
8 Length Compartment length 1 Y
9 Stem Distance Segment Soma length 1
10 Branch Order * Segment Branch label Y
11 Average Diameter Segment Branch length 3
12 Average Diameter Power Segment Branch length 4
13 Section Area Segment Branch size 5
14 Surface Area Segment Branch size 6
15 Volume Segment Branch size 7
16 Euclidean Length Segment Branch length 1 Y
17 Path Length Segment Branch length 8 Y
18 Contraction Segment Branch ratio 16,17 Y
19 Fragmentation Segment Branch count Y
20 Burke Taper Segment Branch ratio 3,17 Y
21 Hillman Taper Segment Branch ratio 3 Y
22 Terminal Degree Segment Branch count Y
23 Daughter Ratio Branch Point Branch ratio 3 Y
24 Parent Daughter Ratio Branch Point Branch ratio 3 Y
25 Rall Power Branch Point Branch ratio 3 Y
26 Rall's Ratio Branch Point Branch ratio 3 Y
27 Rall's Ratio (ralls = 1.5) Branch Point Branch ratio 3 Y
28 Rall's Ratio (ralls = 2) Branch Point Branch ratio 3 Y
29 Partition Asymmetry Branch Point Branch ratio Y
30 Azimuth Branch Point Branch angle 1
31 Elevation Branch Point Branch angle 1
32 Local Bifurcation Amplitude Branch Point Branch angle 1 Y
33 Local Bifurcation Tilt Smaller Branch Point Branch angle 1 Y
34 Local Bifurcation Tilt Larger Branch Point Branch angle 1
35 Local Bifurcation Torque Smaller Branch Point Branch angle 1 Y
36 Local Bifurcation Torque Larger Branch Point Branch angle 1
37 Remote Bifurcation Amplitude Branch Point Branch angle 1 Y
38 Remote Bifurcation Tilt Smaller Branch Point Branch angle 1 Y
39 Remote Bifurcation Tilt Larger Branch Point Branch angle 1
40 Remote Bifurcation Torque Smaller Branch Point Branch angle 1 Y
41 Remote Bifurcation Torque Larger Branch Point Branch angle 1
42 Width Cell length Y
43 Height Cell length Y
44 Depth Cell length Y
45 Stems * Cell Soma count Y
46 Branch Pt * Cell Branch count Y
47 Segments * Cell Branch count Y
48 Tips * Cell Branch count Y
49 Soma X Pos * Cell Soma point 1
50 Soma Y Pos * Cell Soma point 1
51 Soma Z Pos * Cell Soma point 1
52 Skewness X * Cell Soma length 49
53 Skewness Y * Cell Soma length 50
54 Skewness Z * Cell Soma length 51
55 Euclidean Skewness * Cell Soma length 52,53,54
56 Soma Radii Cell Soma length 1
57 Soma Surface Area * Cell Soma size
58 Soma Volume * Cell Soma size
59 Average Stem Distance Cell Soma length 9
60 Average Diameter * Cell Branch length 11
61 Average Diameter Power Cell Branch length 12
62 Average Section Area Cell Branch size 13
63 Average Surface Area Cell Branch size 14
64 Average Volume Cell Branch size 15
65 Average Euclidean Length Cell Branch length 16 Y
66 Average Path Length * Cell Branch length 17 Y
67 Average Contraction * Cell Branch ratio 18 Y
68 Average Fragmentation Cell Branch count 19 Y
69 Average Burke Taper * Cell Branch ratio 20 Y
70 Average Hillman Taper Cell Branch ratio 21 Y
71 Average Daughter Ratio Cell Branch ratio 23 Y
72 Average Parent Daughter Ratio * Cell Branch ratio 24 Y
73 Average Rall Power Cell Branch ratio 25 Y
74 Average Rall's Ratio Cell Branch ratio 26 Y
75 Average Rall's Ratio (ralls = 1.5) Cell Branch ratio 27 Y
76 Average Rall's Ratio (ralls = 2) Cell Branch ratio 28 Y
77 Average Partition Asymmetry Cell Branch ratio 29 Y
78 Average Azimuth Cell Branch angle 30
79 Average Elevation Cell Branch angle 31
80 Average Local Bifurcation Amplitude * Cell Branch angle 32 Y
81 Average Local Bifurcation Tilt Cell Branch angle 33,34
82 Average Local Bifurcation Torque Cell Branch angle 35,36
83 Average Remote Bifurcation Amplitude Cell Branch angle 37 Y
84 Average Remote Bifurcation Tilt Cell Branch angle 38,39
85 Average Remote Bifurcation Torque Cell Branch angle 40,41
86 Average Leaf Level * Cell Branch label 10
87 Average Leaf Path Length * Cell Branch length 3
88 Average Terminal Compartments Cell Branch count 19 Y
89 Average Diameter Threshold * Cell Branch length 3 Y
90 Average Terminal Parent Diameter Cell Branch length 3 Y
91 Average Terminal Hillman Threshold Cell Branch ratio 3 Y
92 Average Tip to Soma Euclidean Distance Cell Branch length 1
93 Average Branch Pt to Soma Euclidean Distance Cell Branch length 1
Retrieved from "http://farsight-toolkit.ee.uh.edu/wiki/L_Measure_functions"
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