Fig 1.
Tessellating domains and sub-domain structure in biological systems.
At markedly different length scales, the skin of giraffes and the stained neocortices of rodents display similar arrangements of polygonal domains, many of which appear further divided into sub-structures. A Image of the skin of a giraffe (Giraffa camelopardalis reticulata), credited to O. Berger, and described by Koch & Meinhardt (1994; [56]) as a Voronoi tessellation. The dark panels overlap with a vascular structure that is important for thermoregulation. B Image of a tangential section of the cytochrome oxidase stained primary somatosensory cortex of an adult laboratory rat (Sprague-Dawley), revealing a pattern of large cortical columns known as ‘barrels’, which have also been described formally as a Voronoi tessellation. C Sub-structures apparent in larger barrel columns have been described in terms of the four categories depicted below, which correspond to the stable patterns generated by a reaction-diffusion model parameterised to amplify modes of increasing spatial frequency. Images in B and C are from Land & Erickson (2005; [57]) and are shown at a common scale. Photograph in A reprinted with permission from Koch A.J. & Meinhardt H., Reviews of Modern Physics, 66, 1481 (1994). Copyright (1994) by the American Physical Society (http://dx.doi.org/10.1103/RevModPhys.66.1481).
Fig 2.
The distribution of pattern correlations along common edges of tessellating triangles should be bimodal.
Colour images show typical patterns generated by a reaction-diffusion model with a large diffusivity term, using a colour map in which red and blue mark extreme high and low concentration values, and green marks zero concentration. A Solved within the boundary of an equilateral triangle, two basic patterns emerge, with extreme concentrations in one corner and along the opposite edge (left) or at two corners (right). Along the edges, three pattern types are apparent. Type 1 varies between the two extremes, type 2 varies between one extreme and zero, and type 3 does not vary. The probability of type i is given below as p(i). The table gives the probability that the absolute correlation between patterns sampled along two randomly chosen edges will be high (pa), medium (pb), or low (pc). As pc < pb < pa the distribution of correlations should be bimodal. B Patterns that emerge within the boundary of an isosceles triangle will be of type 1 or 3 only, changing the distribution of correlations across random edge pairs while retaining an overall bimodal distribution (pc < pa). However, if pairs of edges are restricted to those which may be adjacent in a tessellation then only pairs of type 1 and pairs of type 3 are possible, and pa = 1. Hence, in more general terms, the distribution of correlations between patterns measured along the edges of adjacent tessellation domains should be even more strongly bimodal.
Fig 3.
Correlated pattern formation in adjacent tessellation domains without communication.
A system of reaction-diffusion equations (Eq 1; Dn = χ = 36) was solved using boundary shapes that tessellate in different ways (left column), with blue and red corresponding to extreme positive and negative values, and black lines delineating the domains. Values were sampled along the individual vertices of each domain and samples were correlated between edges of different domains, either amongst pairs of edges that are adjacent in the tessellation (center column) or randomly selected (right column). Histograms show the distributions (f) of correlation coefficients (r) obtained in either case, which were fit by the beta-distribution (dotted line) parameterized by α (see text for details; q is the sum of squared differences between the data and the fit). Rows A-E show data obtained from tessellations comprising domains with different shapes: A equilateral triangles; B isosceles triangles; C scalene triangles; D a Voronoi tessellation; E a Voronoi tessellation with rounded vertices. Peaks at ±1 in the histograms indicate that while pattern formation occurs entirely independently within each domain, patterns may become correlated between (adjacent) domains due to common constraints that derive from the fact that their boundary shapes tessellate.
Fig 4.
Analysis of control patterns formed without shaped boundary constraints registers only very weak correlations.
Combinations of eight values of the diffusion constant Dn and eight values of the constant χ that weights the non-linear coupling term were evaluated on domains of a Voronoi tessellation generated from random seed points. The remaining free parameter, Dc was set to 0.3Dn, and increments in χ were expressed as proportions of Dn to cover a large parameter space. In the ‘constrained’ condition, the shapes of the domain boundaries were a constraint on pattern formation. In the ‘control’ condition, solutions were obtained in circular domains, and the tessellation boundaries were imposed only after pattern formation, to allow a corresponding set of correlations to be measured for comparison. A Values of α were obtained in each condition and for each parameter combination by fitting the resulting distribution of adjacent-edge correlations. Only weak bimodality (high α) was observed in the control condition. Following a log transformation to each axis, α values were clearly linearly separable, as confirmed by the success of a perceptron in discriminating the two conditions (perceptron decision boundary shown in green). Example solutions in the constrained and control conditions are shown in B and C, respectively, for the combination of parameters (Dn = χ = 36) that yielded the lowest α values in the control condition (α = 0.64).
Fig 5.
Correlated pattern formation in tessellated domains is predicted to emerge robustly across a wide range of pattern-forming systems.
Correlations between self-organized patterns in adjacent domains of randomly seeded Voronoi tessellations were measured across a wide range of parameters. Panel A shows values of α estimated from the distribution of 1000 pairwise correlation coefficients obtained from each of sixty four combinations of parameter values (as in Fig 4A; ‘constrained’). The overlaid contour corresponds to the threshold, α = 0.5, at or below which the hypothesis that the domain boundaries constrained pattern formation is very strongly supported. Based on this threshold, patterns are expected to be correlated by the tessellation boundary constraints across a large portion of the parameter space. Panel B shows example patterns for four extreme cases.
Fig 6.
Emergence of correlated patterns in adjacent domains of the developing neocortex.
We analysed images of immunohistochemical stains for serotonin transporter (5-HTT) expression on the surface of the rat barrel cortex, obtained at postnatal days 5 (P5), 8 (P8), and 10 (P10). This stain reveals the shapes of the barrel columns, each corresponding to a whisker on the animal’s snout, as large dark polygonal patches forming a Voronoi tessellation. From P8, sub-barrel structures become apparent and by P10 they clearly identify several regions of high synaptic density within many barrels. Panel A shows the details of the analysis method for the P10 image. Overlaid pairs of parallel red and blue lines show the extents along which image intensity was sampled for each pairwise comparison. Each line marks a vertex of the barrel boundary, and samples were constructed by averaging the grayscale intensity of pixels in one of 50 regularly spaced rectangular bins extending a short distance in from the line towards the corresponding barrel center. The correlation coefficient for each pair of samples is shown in black text, and the plots above show sampled data used for three example pairwise comparisons. Distributions of correlation coefficients obtained from pairs of edges from adjacent barrels are shown for each postnatal day in B, showing a clear progression from a unimodal shape at P5 to a bimodal shape at P10, and supporting the hypothesis that pattern formation within the barrels occurs postnatally and is constrained by the barrel boundary shapes. Original images from [11].