  We extend the theory of weakly coupled oscillators to incorporate slowly
varying inputs and parameters. We employ a combination of regular perturbation
and an adiabatic approximation to derive equations for the phase-difference
between a pair of oscillators. We apply this to the simple Hopf oscillator and
then to a biophysical model. The latter represents the behavior of a neuron
that is subject to slow modulation of a muscarinic current such as would occur
during transient attention through cholinergic activation. Our method extends
and simplifies the recent work of Kurebayashi (Physical Review Letters, 111,
214101, 2013) to include coupling. We apply the method to an all-to-all network
and show that there is a waxing and waning of synchrony of modulated neurons.

Dynamical Systems;Neurons and Cognition