Emergent Patterns from Probabilistic Generalisations of Lateral Activation and Inhibition

The combination of laterally activating and inhibiting feedbacks is well known to spontaneously generate spatial organisation. It was introduced by Gierer and Meinhardt as an extension of Turing’s great insight, that two reacting and diffusing chemicals can spontaneously drive spatial morphogenesis per se. In this study, we develop an accessible nonlinear and discrete probabilistic model to study simple generalisations of lateral activation and inhibition. By doing so, we identify novel modes of morphogenesis beyond the familiar Turing-type modes; notably, beyond stripes, hexagonal nets, pores, and labyrinths, we identify labyrinthine highways, Kagome lattices, gyrating labyrinths, and multi-colour travelling waves and spirals. The results are discussed within the context of Turing’s original motivating interest: the mechanisms which underpin the morphogenesis of living organisms.