BrainCellMembraneDeformableModel

To acheive automatic segmentation of 3D electron microscopic neuropil images, two major challenges must be met. (1) Accuracy must be high enough to eliminate human correction from the process. Massive datasets are needed to examine brain circuitry and the reconstruction is not feasible by hand. (2) Automatic segmentation techniques must recognize enough detail to create models of neural activity. Certainly this requires identifying all synapses, and it will require identifying other subcellular components also (because subcellular components tell us where the synapses are and they can affect the flow of ions).
Accuracy of human traces (compared to current automatic techniques) is probably due to the fact that humans have and intuitive understanding of the typical morphology of cells and membranes. Current approaches to circuit reconstruction (from electron microscopy) do not include a model of the cell morphology. A deformable model of the cell would be useful because then standard model matching techniques could be used to match the deformable neuron to the 3D volumetric data. However, a deformable model of a brain cell needs to be developed.
Ideally a deformable model would be developed that models an entire brain cell. It would need to model complex morphologies with branches, oddly shaped soma, and cross-sections that change along the length of the axon. Suppose that the fitness for the deformable model is that the model matches with the 3D data. The cell membrane is stained (black); a virtual force tends to push the deformable model surface into alignment with the membrain in the 3D image. (See Ahlberg)
The attractor image could be the raw image with membranes stained. (See Ahlberg for definition of attractor image.) However a more accurate attractor image would likely be the output of the techniques mentioned in Jain or Andres, where a filter serves to reduce the noise of the image and emphasize the membranes of cells. If the internal components of the cell are stained, it is essential to remove items that would confuse the system and cause attraction of the surface to noise. Two components that tend to cause confusion are mitochondria and vesicles, and there are others. Mishcenko detected vesicles and removed them and suggested the usefulness of a mitochondria detector but did not implement one. Building vesicle and mitochondria detectors and detectors for any other structures inside the cell would be useful. The distracting components could then be removed selectively from the attractor image.
[warning: the rest of the writing on this page is in an extremely rough state]
As a way of working up to the more complex cell model, here is a model of just an axon:
As with an active contour, the deformable surface that is fit to the neuron membrane should have certain aspects --- has certain configurations more probable than others. For the active contour -- an energy is defined. For the active surface model of a neuron provided here, the energy is defined in terms of the curvature of the skeleton, in the same way that curvature is modeled for an active contour. The values for the spoke lengths are dependent on each other so that the energy of the system is lower if the spoke lenghts at a give node are more equal - to model the effect that the cross-section of an axon is typically somewhat circular.
SWC is limited to describing branch structure of neurons. The cell membrane of neurons (typically visible in electron microscopy) is not a simple branch structure but rather a branch structure with a skin-like surface.
This article describes an extension of SWC Format to handle the surfaces morphology of cell membranes.
===Method I. Subdivision surfaces for a tube-like representation===
Currently this method is well defined only for a path of nodes, not a structure with branches.
List of 3D points connected with segments. At the center of each segment a set of segments radiate outward perpendicular to the segment (like spokes of a tire, roughly speaking). Each spoke has a specified length. N spokes radiate out of the center of each segment. The angle between each spoke is (2 Pi) / N. Each segment is labeled seg1, seg2, seg3, etc. Each spoke on a is labeled seg1spoke1, seg1spoke2, seg1spoke3, etc (using seg1 as an example). A mesh is created using quads of the form seg1spoke1, seg1,spoke2, seg2,spoke2, seg2,spoke1.
In general seg(i)spoke(j), seg(i)spoke(j+1), seg(i+1)spoke(j+1), seg(i+1),spoke(j) for all valid i and j.
The butterfly subdivision method is then applied to the quads to create a mesh that has more triangles and describes a "smoother" surface than the raw quads described above.
PreliminaryCodeAxonRepresentation
===Method II. Integral of Super Ellipsoid===
This method handles branched structures.
http://farsight-toolkit.org/wiki/SuperEllipseTrace3D
Use the SWC format to define a tree structure. Integrate a super ellipsoid along the tree structure. The parameters of the super ellipsoid are allowed to vary as you move along the branches of the tree. (An in the NEURON simulator, values can be defined to vary with respect to position of a branch of the tree and these values are parameters of the super ellipsoid.)
==References==
Jain V, Murray JF, Roth F, Turaga S, Zhigulin V, Biggman KL, Helmstaedter M, Denk W, Seung HS, ICCV 2007
Jörgen Ahlberg Active Contours in Three Dimensions