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 of channel noise is represented by ion channel cluster size of the initiation region of neuron. It is shown that frequency synchronization only was achieved at arbitrary value of couple strength as long as two neurons' channel cluster sizes are the same. We also show that the relative phase of neurons can display profuse dynamic behavior under the combined action of coupling and channel noise. Both qualitative and quantitative descriptions are applied to describe the transitions between those behaviors. Relevance of our findings to controlling neural synchronization experimentally is discussed.