  This work introduces a methodology for studying synchronization in adaptive
networks with heterogeneous plasticity (adaptation) rules. As a paradigmatic
model, we consider a network of adaptively coupled phase oscillators with
distance-dependent adaptations. For this system, we extend the master stability
function approach to adaptive networks with heterogeneous adaptation. Our
method allows for separating the contributions of network structure, local node
dynamics, and heterogeneous adaptation in determining synchronization.
Utilizing our proposed methodology, we explain mechanisms leading to
synchronization or desynchronization by enhanced long-range connections in
nonlocally coupled ring networks and networks with Gaussian distance-dependent
coupling weights equipped with a biologically motivated plasticity rule.
