Detecting Synchronisation of Biological Oscillators by Model Checking

We define a subclass of timed automata, called oscillator timed automata, suitable to model biological oscillators. Coupled biological oscillators may synchronise, as emerging behaviour, after a period of time in which they interact through physical or chemical means. We introduce a parametric semantics for their interaction that is general enough to capture the behaviour of different types of oscillators. We instantiate it both to the Kuramoto model, a model of synchronisation based on smooth interaction, and to the Peskin model of pacemaker cells in the heart, a model of synchronisation based on pulse interaction. We also introduce a logic, Biological Oscillators Synchronisation Logic (BOSL), that is able to describe collective synchronisation properties of a population of coupled oscillators. A model checking algorithm is proposed for the defined logic and it is implemented in a model checker. The model checker can be used to detect synchronisation properties of a given population of oscillators. This tool might be the basic step towards the generation of suitable techniques to control and regulate the behaviour of coupled oscillators in order to ensure the reachability of synchronisation.