Numerical evidence is presented for the occurence of on-off intermittency and attractor bubbling in the time series of synaptic potentials of analog neurons with time-dependent synaptic noise. The cases of continuous noise with uniform distribution of values and (physiologically motivated) quantal noise with binomial distribution are considered. The results were obtained for a single neuron with synaptic self-connection and a network of two neurons with various weights of synaptic connections. In the latter case coexistence of a neuron showing on-off intermittency with another one, showing attractor bubbling, is possible.