The proton conductivity of a Nafion 112 membrane is measured with a high spatial resolution using electrochemical atomic force microscopy. Image analysis reveals an inhomogeneous conductivity distribution which is attributed to the limited connectivity of hydrophilic domains. This information relates to the micro-morphology which is due to phase separation of the hydrophobic polymer backbone and the hydrophilic pendant groups. The direct images relate to a different length scale and are complementary to the x-ray diffraction investigations which provide only average information. Furthermore, the measured current values reveal an interesting correlation with the size of the conductive areas. A bimodal conductivity distribution suggests that there are different mechanisms which contribute to the proton current in Nafion. Additionally, time dependence in local conductivity is found and interpreted in terms of redistribution of water in the membrane. A statistical analysis of the current distribution is performed and compared with theoretical simulations. Evidence is found for the existence of a critical current density. On a timescale of seconds the response of the conductive network is probed by applying voltage steps to the atomic force microscope tip.