  We study the dynamics of a simple adaptive system in the presence of noise
and periodic damping. The system is composed by two paths connecting a source
and a sink, the dynamics is governed by equations that usually describe food
search of the paradigmatic Physarum polycephalum. In this work we assume that
the two paths undergo damping whose relative strength is periodically modulated
in time and analyse the dynamics in the presence of stochastic forces
simulating Gaussian noise. We identify different responses depending on the
modulation frequency and on the noise amplitude. At frequencies smaller than
the mean dissipation rate, the system tends to switch to the path which
minimizes dissipation. Synchronous switching occurs at an optimal noise
amplitude which depends on the modulation frequency. This behaviour disappears
at larger frequencies, where the dynamics can be described by the time-averaged
equations. Here, we find metastable patterns that exhibit the features of
noise-induced resonances.
