A generalized version of sequential logic circuit based neuron models is presented, where the dynamics of the model is modeled by an asynchronous cellular automaton. Thanks to the generalizations in this paper, the model can exhibit various neuron-like waveforms of the membrane potential in response to excitatory and inhibitory stimulus. Also, the model can reproduce four groups of biological and model neurons, which are classified based on existence of bistability and subthreshold oscillations, as well as their underlying bifurcations mechanisms.