Analytical solution of the steady membrane voltage fluctuation caused by a single ion channel

An analytical steady-state solution of the stochastic model describing the integrated dynamics of the membrane voltage and the gating of a single channel is presented. The voltage density function experiences bifurcation in the parameter space, and the threshold is determined by the comparison between the rates of the voltage evolution and that of the channel gating. It is singular at the reversal potential if the voltage evolves faster than the channel gating. As an application, the entropy production rate associated with the gating currents is calculated; its variation tendency in parameter space is consistent to that of the number of transitions counted from the sample paths. The analysis in this paper identifies the values of parameters that justify the formation of the voltage pulse and the efficient energy cost, which is associated with the singularity of the voltage density functions.