The quasi-threshold phenomenon in Higgins model with mono-stability driven by Gaussian white noise is investigated. The initial excitation phase is identified as escaping event with a specific trajectory defined as the quasi-threshold. In the limit of weak noise, a group of differential equations governing the optimal exit path, quasi-potential and exponential prefactor are deduced via WKB approximation. Results show that the optimal path approaches the quasi-threshold with a nearly tangent way and almost follows the deterministic flow subsequently in the quasi-potential plateau, making it analogous to the bistable system. Numerical experiments verify not only the results of the optimal path but also the ones of the optimal fluctuational forces. Then the difference between the practical exit location and the position with minimal quasi-potential is also revealed by including the exponential prefactor into the expression of exit location distribution. Finally, the mean first passage time (MFPT) is evaluated theoretically with the secondary-order-approximation taken into consideration. These findings and computations shed light on exploring underlying qualitative mechanism and quantitative feature of excitation behaviors in biological systems.

研究了由高斯白噪声驱动的具有单稳定性的希金斯模型的准阈值现象。初始激励阶段被标识为逃逸事件，其特定轨迹定义为准阈值。在微弱噪声的限制下，通过WKB近似推导了一组控制最优出口路径，准电势和指数前因数的微分方程。结果表明，最优路径以近似切线的方式接近准阈值，并且随后在准电位平台上几乎遵循确定性流，使其类似于双稳态系统。数值实验不仅验证了最优路径的结果，而且还验证了最优波动力的结果。然后，通过在出口位置分布的表达式中包括指数前因，也揭示了实际出口位置与具有最小准电位的位置之间的差异。最后，在理论上考虑了二次近似，对平均第一次通过时间（MFPT）进行了评估。这些发现和计算为探索生物系统中激发行为的基本定性机理和定量特征提供了启示。