  We present a generic theory for the dynamics of a stiff filament under
tension, in an active medium with orientational correlations, such as a
microtubule in contractile actin. In sharp contrast to the case of a passive
medium, we find the filament can stiffen, and possibly oscillate, \textit{or}
buckle, depending on the contractile or tensile nature of the activity
\textit{and} the filament-medium anchoring interaction. We present experiments
on the behaviour of microtubules in the growth cone of a neuron, which provide
evidence for these apparently opposing behaviours. We also demonstrate a strong
violation of the fluctuation-dissipation (FD) relation in the effective
dynamics of the filament, including a negative FD ratio. Our approach is also
of relevance to the dynamics of axons, and our model equations bear a
remarkable formal similarity to those in recent work [PNAS (2001) {\bf
98}:14380-14385] on auditory hair cells. Detailed tests of our predictions can
be made using a single filament in actomyosin extracts or bacterial
suspensions.
