There is suggestive evidence that pyramidal cell axons in neocortex may be coupled by gap junctions into an "axonal plexus" capable of generating very fast oscillations (VFOs) with frequencies exceeding 80 Hz. It is not obvious, however, how a pyramidal cell in such a network could control its output when action potentials are free to propagate from the axons of other pyramidal cells into its own axon. We address this problem by means of simulations based on three-dimensional reconstructions of pyramidal cells from rat somatosensory cortex. We show that somatic depolarization enables propagation via gap junctions into the initial segment and main axon, while somatic hyperpolarization disables it. We show further that somatic voltage cannot effectively control action potential propagation through gap junctions on minor collaterals; action potentials may therefore propagate freely from such collaterals regardless of somatic voltage. In previous work, VFOs are all but abolished during the hyperpolarization phase of slow oscillations induced by anesthesia in vivo. This finding constrains the density of gap junctions on collaterals in our model and suggests that axonal sprouting due to cortical lesions may result in abnormally high gap junction density on collaterals, leading in turn to excessive VFO activity and hence to epilepsy via kindling.