  We use linear stability analysis and hybrid lattice Boltzmann simulations to
study the dynamical behaviour of an active nematic confined in a channel made
of viscoelastic material. We find that the quiescent, ordered active nematic is
unstable above a critical activity. The transition is to a steady flow state
for high elasticity of the channel surroundings. However, below a threshold
elastic modulus, the system produces spontaneous oscillations with periodic
flow reversals. We provide a phase diagram that highlights the region where
time-periodic oscillations are observed and explain how they are produced by
the interplay of activity and viscoelasticity. Our results suggest new
experiments to study the role of viscoelastic confinement in the
spatio-temporal organization and control of active matter.
