Dynamics of learning in coupled oscillators tutored with delayed reinforcements

In this work we analyze the solutions of a simple system of coupled phase oscillators in which the connectivity is learned dynamically. The model is inspired in the process of learning of birdsong by oscine birds. An oscillator acts as the generator of a basic rhythm, and drives slave oscillators which are responsible for different motor actions. The driving signal arrives to each driven oscillator through two different pathways. One of them is a "direct" pathway. The other one is a "reinforcement" pathway, through which the signal arrives delayed. The coupling coefficients between the driving oscillator and the slave ones evolve in time following a Hebbian-like rule. We discuss the conditions under which a driven oscillator is capable of learning to lock to the driver. The resulting phase difference and connectivity is a function of the delay of the reinforcement. Around some specific delays, the system is capable to generate dramatic changes in the phase difference between the driver and the driven systems. We discuss the dynamical mechanism responsible for this effect, and possible applications of this learning scheme.