  We propose a novel mechanism leading to spatiotemporal oscillations in
extended systems that does not rely on local bulk instabilities. Instead,
oscillations arise from the interaction of two subsystems of different spatial
dimensionality. Specifically, we show that coupling a passive diffusive bulk of
dimension d with an excitable membrane of dimension d-1 produces a
self-sustained oscillatory behavior. An analytical explanation of the
phenomenon is provided for d=1. Moreover, in-phase and anti-phase
synchronization of oscillations are found numerically in one and two
dimensions. This novel dynamic instability could be used by biological systems
such as cells, where the dynamics on the cellular membrane is necessarily
different from that of the cytoplasmic bulk.
