Perturbation analysis of spontaneous action potential initiation by stochastic ion channels

A stochastic interpretation of spontaneous action potential initiation is developed for the Morris-Lecar equations. Initiation of a spontaneous action potential can be interpreted as the escape from one of the wells of a double well potential, and we develop an asymptotic approximation of the mean exit time using a recently developed quasistationary perturbation method. Using the fact that the activating ionic channel’s random openings and closings are fast relative to other processes, we derive an accurate estimate for the mean time to fire an action potential (MFT), which is valid for a below-threshold applied current. Previous studies have found that for above-threshold applied current, where there is only a single stable fixed point, a diffusion approximation can be used. We also explore why different diffusion approximation techniques fail to estimate the MFT.

Figure 1

Deterministic fast dynamics of the Morris-Lecar model (2.6). The double-well potential obtained from integrating $a\left(v\right)f\left(v\right)-g\left(v\right)$ [the right-hand side of (2.6)] over $v$ for different values of stimulus current $I$.

Figure 2

Mean time to fire an action potential (MFT) $T$ (ms) as a function of the applied current $I$ (mA) for $N=10$. (a) The quasistationary (solid line) and diffusion (dashed line) approximations are compared to 300 averaged Monte Carlo simulations (symbols). The relative error of each approximation, compared to Monte Carlo simulations, is shown by gray lines. (b) Coefficient of variation (CV) from the Monte Carlo simulations.

Figure 3

(Color online) The stability landscape $\Phi \left(v\right)$ from the diffusion (dashed line) and quasistationary (solid line) approximations, with no applied current ($I=0$). To normalize each curve we have taken $N=1$ ion channels. For ease of comparison we have vertically shifted each curve. The inset shows the difference of the height of each stability well approximation as a function of the applied current $I$.

Figure 4

(Color online) Strength-duration curves, with the quasistationary approximation (solid lines) diffusion approximation (dashed lines) and ${10}^2$ averaged Monte Carlo simulation (symbols). (a) The stimulus amplitude $I$ (mA) as a function of the stimulus duration $T$ (ms) for different values of $N$, the number of channels, with $\beta =0.8{\mathrm{s}}^{-1}$. (b) Curves for different values of $\epsilon$ (which scales the channel switching rate) and $N=8$. In each, the deterministic strength-duration curve, defined by (6.4), is compared to the stochastic behavior.

Figure 5

(Color online) Firing probability $\Pi$ as a function of the stimulus amplitude (fraction of threshold) with a stimulus duration of $70\mathrm{ms}$. The quasistationary approximation (solid lines) and the diffusion approximation (dashed lines) are compared to ${10}^6$ averaged Monte Carlo simulations (symbols).