The nervous system is composed of a large number of neurons, and the electrical activities of neurons can present multiple modes during the signal transmission between neurons by changing intrinsic bifurcation parameters or under appropriate external forcing. In this review, the dynamics for neuron, neuronal network is introduced, for example, the mode transition in electrical activity, functional role of autapse connection, bifurcation verification in biological experiments, interaction between neuron and astrocyte, noise effect, coherence resonance, pattern formation and selection in network of neurons. Finally, some open problems in this field such as electromagnetic radiation on electrical activities of neuron, energy consumption in neurons are presented.