Nonlinear Dynamics

, Volume 96, Issue 4, pp 2293–2305 | Cite as

Non-differentiability of quasi-potential and non-smooth dynamics of optimal paths in the stochastic Morris–Lecar model: Type I and II excitability

  • Zhen Chen
  • Jinjie Zhu
  • Xianbin LiuEmail author
Original Paper


We studied the stochastic Morris–Lecar model of both Type I and II excitability using the large deviation theory. A mechanical interpretation for the discontinuity of the optimal path that has been found for decades in nearly all systems driven by weak white noise was provided. By doing that, we connected some long-separated concepts in mechanics, mathematics and physics: sliding set of non-smooth dynamical systems, non-differentiable set of the viscosity solution to the Hamilton–Jacobi equation and switch line, the singular structure which occurs in the Lagrangian manifold to characterize the sudden switch of optimal paths. We found that the existence of the sliding set engendered some novel dynamics in the case of Type I excitability. As for Type II, we found that the canard separatrix forms a boundary threshold of different dynamical behaviors and a crossing point in weak noise limit is located. Besides, the validity of this separatrix as the threshold is verified by a comparison with a level curve of quasi-potential at the cusp of the switching line.


Large deviation theory Quasi-potential Discontinuity Sliding set Switching line 



We thank gratefully Prof. Hebai Chen at the Fuzhou University for informative discussions in topics like non-smooth dynamics and sliding sets. Z.Chen would also like to thank the support and care from M.J.Wang over the passed years

Author Contributions

ZC, JZ and XL designed the study and organized the entire research program. ZC wrote the manuscript, MATLAB code and performed the simulations. JZ offered considerable advice during the research. XL corrected and revised the manuscript, supervised the modeling and simulation. All authors gave final approval for publication.


This work was supported by the National Natural Science Foundation of China (Nos. 11472126, 11232007, 1177020530), the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) (Grant No. MCMS-0116G01).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Data accessibility

The algorithm to calculate the quasi-potential can be found in [22] and necessary parameters for all the numerical results are shown in the captions of figures. Readers can perform the method to reproduce the present numerical results by any program language such as C, Fortran or MATLAB.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.State Key Lab of Mechanics and Control for Mechanical Structures, College of Aerospace EngineeringNanjing University of Aeronautics and AstronauticsNanjingChina
  2. 2.Department of Brain and Cognitive SciencesUniversity of RochesterRochesterUSA

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