Minimum Action Path Theory Reveals the Details of Stochastic Transitions Out of Oscillatory States

Cell state determination is the outcome of intrinsically stochastic biochemical reactions. Transitions between such states are studied as noise-driven escape problems in the chemical species space. Escape can occur via multiple possible multidimensional paths, with probabilities depending nonlocally on the noise. Here we characterize the escape from an oscillatory biochemical state by minimizing the Freidlin-Wentzell action, deriving from it the stochastic spiral exit path from the limit cycle. We also use the minimized action to infer the escape time probability density function.

Figure 1

Comparison of escape trajectories from the Hopf model limit cycle to the stable fixed point. Results show 5 trajectories of the CLE (green) compared with the MAP (blue) for different values of $\omega$ and $\mathrm{\Omega }$. For $\mathrm{\Omega }=150$, results are also compared with 5 Gillespie trajectories (red). The unstable limit cycle separating the basins of attraction (shaded area) is found by temporal inversion of Eq. (1). For the sake of clarity, only the last turn of each trajectory prior to escape is shown. The rest of the parameters are $x_c=100$, $y_c=100$, $x_0=5$, $a=8$, $b=5$, $c=15$.

Figure 2

Comparison of escape trajectories of the Morris-Lecar model limit cycle. (Left) MAP prediction (blue) compared with 30 CLE trajectories (red) $\mathrm{\Omega }={10}^4$. For the sake of clarity, only the last fraction of each trajectory prior to escape is shown. (Right) Comparison of the escape voltage distribution from the limit cycle for $\mathrm{\Omega }=20000$ (green) with MAP prediction (cyan line). The results are also compared with the distribution of escape values of $V$ from a distance $\mathrm{\Delta }w=0.0005$ from the limit cycle (red), and the occupation distribution along the limit cycle, i.e., before escape, (dark blue).

Figure 3

Comparison of the escape angle distribution from the limit cycle (green) with the escape angle predicted by the MAP (cyan line) for $\omega =75$ and $\mathrm{\Omega }=150$ in the Hopf model. The results are also compared with the distribution of escape angles from an annulus of radius 0.001 around the limit cycle (red), and the angular distribution along the limit cycle, i.e., before escape, (dark blue). Inset: Action density (Lagrangian) along MAP normalized to the maximum density. Other parameters are the same as in Fig. 1.

Figure 4

Comparison of predictions of the MFPT for the Hopf model. (Top) Comparison of MFPTs calculated from CLE simulations (circles) with the exponential dependence of the MFPT on $\mathrm{\Omega }$ given by $\mathcal{S}$ (lines). Each line is computed by minimizing the action $\mathcal{S}$ for different $\omega$ and fitting the prefactor $C$. (Bottom) Following the same procedure, the values of the action $\mathcal{S}$ and $C$ are compared for different values of $\omega$. Parameters values are the same as those of Fig. 1, error bars are standard error of the mean from the CLE.

Figure 5

(Top) Comparison of the cumulative distribution function (CDF) of the number of turns $t/(2\pi \omega )$ for 10 000 realizations of the Hopf model with $\omega =75$. For each distribution the Cramer–von Mises (CvM) criterion is computed to compare the resulting distribution with the geometric and exponential distribution determined by the value of the action. (Inset) Detail of the CDF for small revolution numbers. (Bottom) CDF of the number of spikes per train for the Morris-Lecar model comparing a CLE simulation of a total time 2 100 000, with the geometric distribution predicted by the MAP method. (Inset) Trajectory exhibiting spike trains.