Evolutionary Kuramoto dynamics

Abstract

Biological systems have a variety of time-keeping mechanisms ranging from molecular clocks within cells to a complex interconnected unit across an entire organism. The suprachiasmatic nucleus, comprising interconnected oscillatory neurons, serves as a master-clock in mammals. The ubiquity of such systems indicates an evolutionary benefit that outweighs the cost of establishing and maintaining them, but little is known about the process of evolutionary development. To begin to address this shortfall, we introduce and analyse a new evolutionary game theoretic framework modelling the behaviour and evolution of systems of coupled oscillators. Each oscillator is characterized by a pair of dynamic behavioural dimensions, a phase and a communication strategy, along which evolution occurs. We measure success of mutations by comparing the benefit of synchronization balanced against the cost of connections between the oscillators. Despite the simple set-up, this model exhibits non-trivial behaviours mimicking several different classical games - the Prisoner's Dilemma, snowdrift games, coordination games - as the landscape of the oscillators changes over time. Across many situations, we find a surprisingly simple characterization of synchronization through connectivity and communication: if the benefit of synchronization is greater than twice the cost, the system will evolve towards complete communication and phase synchronization.