Biological action at a distance: Correlated pattern formation in adjacent tessellation domains without communication
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).