Tessellations emerge in many natural systems, and the constituent domains often contain regular patterns, raising the intriguing possibility that pattern formation within adjacent domains might be correlated by the geometry, without the direct exchange of information between parts comprising either domain. We confirm this paradoxical effect, by simulating pattern formation via reaction-diffusion in domains whose boundary shapes tessellate, and showing that correlations between adjacent patterns are strong compared to controls that self-organize in domains with equivalent sizes but unrelated shapes. The effect holds in systems with linear and non-linear diffusive terms, and for boundary shapes derived from regular and irregular tessellations. Based on the prediction that correlations between adjacent patterns should be bimodally distributed, we develop methods for testing whether a given set of domain boundaries constrained pattern formation within those domains. We then confirm such a prediction by analysing the development of 'subbarrel' patterns, which are thought to emerge via reaction-diffusion, and whose enclosing borders form a Voronoi tessellation on the surface of the rodent somatosensory cortex. In more general terms, this result demonstrates how causal links can be established between the dynamical processes through which biological patterns emerge and the constraints that shape them.

In many sensory systems, information transmission is constrained by a bottleneck, where the number of output neurons is vastly smaller than the number of input neurons. Efficient coding theory predicts that in these scenarios the brain should allocate its limited resources by removing redundant information. Previous work has typically assumed that receptors are uniformly distributed across the sensory sheet, when in reality these vary in density, often by an order of magnitude. How, then, should the brain efficiently allocate output neurons when the density of input neurons is nonuniform? Here, we show analytically and numerically that resource allocation scales nonlinearly in efficient coding models that maximize information transfer, when inputs arise from separate regions with different receptor densities. Importantly, the proportion of output neurons allocated to a given input region changes depending on the width of the bottleneck, and thus cannot be predicted from input density or region size alone. Narrow bottlenecks favor magnification of high density input regions, while wider bottlenecks often cause contraction. Our results demonstrate that both expansion and contraction of sensory input regions can arise in efficient coding models and that the final allocation crucially depends on the neural resources made available.