  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.
