Illustration of 1D and 2D cellular automaton models for repeatedly triggered autowaves.

<p>The 1D system consists of a linear row (x-direction) of <i>N</i> = 100 cells. States are decoded by color (0: white, 1: light blue, 2: dark blue). Each line corresponds to a global wave event. The 2D system consists of a hexagonal lattice. With a probability of <i>q</i> = 0.02, a resting cell is spontaneously excited (state 0 → state 1). This corresponds to the tips of triangular shapes in the space-time diagram of the 1D system. Two autowaves merge and annihilate when the wavefronts touch, resulting in an inverse tip or valley.</p

Mean (top) and standard deviation (bottom) of the distributions <i>p</i>(<i>T</i>_eff) of the effective triggering intervals, for gamma distributed spontaneous triggering distributions <i>p</i>(<i>T</i>_spon) with a mean of 100 and standard deviations of 10 (blue), 5 (magenta) and 2.5 (cyan).

<p>For a large number of cells <i>N</i>, the mean and the standard deviation of <i>p</i>(<i>T</i>_eff) both saturate at a value below the mean and standard deviation of <i>p</i>(<i>T</i>_spon).</p

Distribution of effective triggering intervals in the linear medium model.

<p>There are two separate peaks visible in the histogram: one centered around 80, corresponding to that of the Greenberg-Hasting model, and one around zero. The inset shows two wavefronts passing through each other.</p

Dependence of the effective triggering rate <i>R</i>_eff on the system size <i>N</i> for a fixed spontaneous triggering probability <i>q</i> = 0.01.

<p>1D corresponds to a 1 × <i>N</i> array, 2D corresponds to a <math><mrow><msqrt><mi>N</mi></msqrt><mo>×</mo><msqrt><mi>N</mi></msqrt></mrow></math> array of hexagonal cells. The inset shows the dependence of <i>R</i>_eff on <i>q</i> for a fixed number of cells <i>N</i> = 100.</p

Voxelized representation of collagen fibers.

<p>The figure shows three adjacent original confocal images (left) and the corresponding reconstruction result (right). The broadened fiber is reduced to a wiggly, continuous line with a ‘diameter’ of one voxel.</p

Adaptive template generation.

<p>Random sub-sections (yellow window) are selected from the 2D slices of the input stack. They are weighted with the intensity of the central pixel (right side) and then averaged to obtain a representative template for fiber cross sections.</p

Test of scale invariance.

<p>The same collagen gel has been recorded with three different optical resolutions (relative voxel sizes: high/medium/low 1/2/4). After reconstructing the three image stacks, the distribution of nearest obstacle distances were computed. The low and medium resolutions give similar results. Only at the highest resolution, the pores appear slightly smaller on average, because under these conditions even fine details of the network can be resolved.</p

Statistical properties of a real and a surrogate image stacks.

<p>(A) Comparison of the voxel intensity distributions in the real and surrogate image stacks. Both distributions are similar. (B) and (C) show angular distributions of the fiber segments. (B) Typical distributions of azimuthal angles in a real and a surrogate data set. The distributions are almost indistinguishable. The peaks are a result of voxelization. The principal directions, corresponding to the x- and y-direction, as well as the principal diagonals are over-represented in short fiber segments and lead to maxima at (C) Typical distributions of polar angles in a real and a surrogate data set. Again, the distributions are similar. Compared to an ideal isotropic network with , polar angles smaller than are increasingly suppressed due to the blind spot effect of confocal reflection microscopy <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0036575#pone.0036575-Jawerth1" target="_blank">[15]</a>.</p

Flow chart of the reconstruction algorithm.

<p>The method involves three independent 2D scans through the 3D image stack, along the x-, y- and z-directions. Since these scans are analogous, the diagram focuses on the x-scan only.</p

Small 3D stack of a collagen gel

<p>Dimensions collagen concentration 1.2 mg/ml. (A) Raw data, as recorded with confocal reflection microscopy, without any image processing. The lateral (x-, y-direction) resolution of the fibers is considerably better than the vertical (z-direction) resolution, due to the anisotropic point spread function. In addition, only fiber segments that run in small angles to the imaging plane are visible, due to the so-called blind spot effect. Moreover, the speckled appearance of the collagen network is an optical artifact of reflection microscopy; confocal images of fluorescently labeled collagen networks reveal continuous line networks <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0036575#pone.0036575-Mickel1" target="_blank">[16]</a>. (B) Corresponding reconstruction result, using the algorithm described in this paper. Note that voxels that appear to be missing in the reconstruction are located outside of the selected sub volume.</p