Frequency and phase synchronization of two coupled neurons with channel noise

We study the frequency and phase synchronization in two coupled identical and nonidentical neurons with channel noise. The occupation number method is used to model the neurons in the context of stochastic Hodgkin-Huxley model in which the strength of channel noise is represented by ion channel cluster size of neurons. It is shown that channel noise allows the two neurons to achieve both frequency and phase synchronization in the regime where the deterministic Hodgkin-Huxley neuron is unable to be excited. In particular, the identical channel noises lead to frequency synchronization in weak-coupling regime. However, if the coupling is strong, the two neurons could be frequency locked even though the channel noises are not identical. We also show that the relative phase of neurons displays profuse dynamical regimes under the combined action of coupling and channel noise. Those regimes are characterized by the distribution of the cyclic relative phase corresponding to antiphase locking, random switching between two or more states. Both qualitative and quantitative descriptions are applied to describe the transitions to perfect phase locking from no synchronization states.