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.

Figure 1

(a) The joint density function $P(\mathcal{C},z)$ according to different values of $k^{+}$. The bifurcation behavior can be observed at the threshold $k^{+}=0.3$. (b) The sample paths according to the $P(\mathcal{C},z)$ plotted in the left panel. As $k^{+}$ becomes smaller than the threshold, there emerge forms of pulse in the curve of $V\left(t\right)$, meanwhile $V\left(t\right)$ becomes more regular. Here $k^{-}=2.55$.

Figure 2

(a) The joint density function $P(\mathcal{O},z)$ according to different values of $k^{-}$. Note the bifurcation near the threshold $k^{+}=(g_L/C_m)=2.3$. (b) The sample paths of $V\left(t\right)$. Here $k^{+}=0.06$.

Figure 3

The steady-state density of voltage $P\left(z\right)$ under four situations.

Figure 4

(a) The dependence of the mean voltage $E\left(z\right)$ on $k^{+}$ and $k^{-}$. (b) The dependence of the voltage fluctuation $D\left(z\right)$ on $k^{+}$ and $k^{-}$.

Figure 5

The schematic illustration of the gating currents.

Figure 6

(a) The dependence of the EPR on $k^{+}$ and $k^{-}$, which is calculated from the steady-state density. (b) The dependence of the number of transitions on $k^{+}$ and $k^{-}$, which is calculated from the sample paths.