Learning spatio-temporal patterns with Neural Cellular Automata

Many complex natural phenomena--such as organ growth, the structure of materials or the patterns of neural activity in our brains--are emergent [1]. These are typically characterised by many simple interacting components that collectively exhibit behaviour that is far richer than that of the individual parts, and cannot easily be predicted from them. Emergence is especially prevalent in complex systems of biological nature across a wide range of scales - from gene expression dictating cell fates, interacting cells forming structures during morphogenesis, synaptic connections in the brain, or the interactions of organisms in ecology. Cellular Automata (CA) provide simple models of spatio-temporal emergent behaviour, where a discrete lattice of'cells' are equipped with an internal state and a rule that updates each cell state depending on itself and its local neighbours. The classic Game of Life [2] is a famous example, where cell states and the update rule utilise simple Boolean logic, but the emergent complexity has fascinated and inspired much research [3, 4]. CA are a natural modelling framework of a wide range of biological processes such as: skin patterning [5, 6], limb polydactyly [7], chimerism [8], cancer [9] and landscape ecology [10]. In these cases the CA rules are constructed with expert knowledge of likely mechanisms, however in general the space of possible CA rules is vast, and there is a non-uniqueness by which several rules can result in qualitatively similar emergent behaviours. As such the inverse problem of inferring mechanistic interactions (CA rules) that might generate a given observed emergent behaviour is much more challenging than the forward problem.