In this paper, an asynchronous filter is proposed for Markov jump neural networks (NNs) with time delay and quantized measurements where a logarithmic quantizer is employed. The filter and quantizer are both mode-dependent and their modes are asynchronous with that of the NN, which is described by hidden Markov models. By the Lyapunov-Krasovskii functional approach, a sufficient condition is derived and a filter is then designed such that the filtering error dynamics are stochastically mean square stable and strictly <inline-formula> <tex-math notation="LaTeX">\boldsymbol {(\mathscr U,\mathscr S,\mathscr V)} </tex-math></inline-formula>-dissipative. Finally, the effectiveness and practicability of the theoretical results are verified by two examples, including a biological network.