Measurements from the JET neutron profile monitor are analysed tomographically to deduce the two-dimensional spatial distribution of neutron emissivity and its temporal evolution. The most dramatic change in the emissivity profile is produced by a sawtooth crash. The profiles before and after a sawtooth crash are determined and changes in their characteristics, such as width and amplitude, are compared to those deduced from other diagnostics, including soft X-rays, electron density interferometers and electron cyclotron emission. The drop in the central neutron emissivity can be much larger than that obtained from the integrated global neutron emission, and hollow emissivity profiles are produced. The observed inversion radius is the same on neutron, soft X-ray, and electron temperature profiles. This hollow profile can be understood by observations of the soft X -ray emissivity on the fastest time-scale. A major theme of this paper is that much more information and understanding can be gained if several diagnostics are used together, with their different resolutions, time-scales and physical properties measured.