Advertisement

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
  • 296 Downloads

Abstract

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.

Keywords

Large deviation theory Quasi-potential Discontinuity Sliding set Switching line 

Notes

Acknowledgements

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.

Funding

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.

References

  1. 1.
    Kloeden, P., Potzsche, C.: Nonautonomous Dynamical Systems in the Life Sciences. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  2. 2.
    Hodgkin, A.: The local electric changes associated with repetitive action in a non-medullated axon. J. Physiol. 107, 165–181 (1948)CrossRefGoogle Scholar
  3. 3.
    Izhikevich, E.: Dynamical Systems in Neuroscience: the Geometry of Excitability and Bursting. MIT Press, Cambridge (2007)Google Scholar
  4. 4.
    Sekerli, M., Del Negro, C., Lee, R., Butera, R.: Estimating action potential thresholds from neuronal time-series: new metrics and evaluation of methodologies. IEEE Trans. Biomed. Eng. 51, 1665–1672 (2004)CrossRefGoogle Scholar
  5. 5.
    Prescott, S., Koninck, Y., Sejnowski, T.: Biophysical basis for three distinct dynamical mechanisms of action potential initiation. Plos Comput. Biol. 4, e1000198 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Mitry, J., Mccarthy, M., Kopell, N., Wechselberger, M.: Excitable neurons, firing threshold manifolds and canards. J. Math. Neurosci. 3, 12 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Wieczorek, S., Ashwin, P., Luke, C., Cox, P.: Excitability in ramped systems: the compost-bomb instability. Proc. R. Soc. A Math. Phys. Eng. Sci. 467, 1243–1269 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fitzhugh, R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biol. 17, 257–278 (1955)Google Scholar
  9. 9.
    Freidlin, M., Wentzell, A.: Random Perturbations of Dynamical Systems. Springer, Berlin (2002)zbMATHGoogle Scholar
  10. 10.
    Grassberger, P.: Noise-induced escape from attractors. J. Phys. A Math. Gen. 22, 3283–3290 (1989)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Khovanov, I., Polovinkin, A., Luchinsky, D., McClintock, P.: Noise-induced escape in an excitable system. Phys. Rev. E 87, 032116 (2013)CrossRefGoogle Scholar
  12. 12.
    Chen, Z., Zhu, J., Liu, X.: Crossing the quasi-threshold manifold of a noise-driven excitable system. Proc. R. Soc. A Math. Phys. Eng. Sci. 473, 20170058 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hilborn, R., Erwin, R.: Coherence resonance in models of an excitable neuron with noise in both the fast and slow dynamics. Phys. Lett. A 322(1–2), 19–24 (2004)CrossRefzbMATHGoogle Scholar
  14. 14.
    DeVille, R., Vanden-Eijnden, E., Muratov, C.: Two distinct mechanisms of coherence in randomly perturbed dynamical systems. Phys. Rev. E 72(3), 031105 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Franovic, I., Perc, M., Todorovic, K., Kostic, S., Buric, N.: Activation process in excitable systems with multiple noise sources: Large number of units. Phys. Rev. E 92(6), 062912 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Newby, J., Bressloff, P., Keener, J.: Breakdown of fast-slow analysis in an excitable system with channel noise. Phys. Rev. Lett. 111, 128101 (2013)CrossRefGoogle Scholar
  17. 17.
    Zhu, Q., Wang, H.: Output feedback stabilization of stochastic feedforward systems with unknown control coefficients and unknown output function. Automatica 87, 166–175 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Newby, J.: Spontaneous excitability in the Morris–Lecar model with ion channel noise. SIAM J. Appl. Dyn. Syst. 13(4), 1756C1791 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35, 193–213 (1981)CrossRefGoogle Scholar
  20. 20.
    Chen, Z., Liu, X.: Subtle escaping modes and subset of patterns from a nonhyperbolic chaotic attractor. Phys. Rev. E 95(1), 012208 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Crandall, M., Lions, P.: Vicosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277, 1–42 (1983)CrossRefzbMATHGoogle Scholar
  22. 22.
    Crandall, M., Evans, L., Lions, P.: Some properties of viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 282, 487–502 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ishii, H.: A simple, direct proof of uniqueness for solutions of the Hamilton–Jacobi equations of eikonal type. Proc. Am. Math. Soc. 100, 247–251 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sethian, A., Vladimirsky, A.: Ordered upwind methods for static Hamilton–Jacobi equations: theory and algorithms. SIAM J. Numer. Anal. 41, 325–363 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Cameron, M.: Finding the quasipotential for nongradient SDEs. Phys. D Nonlinear Phenom. 241, 1532–1550 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Heymann, M., Vanden-Eijnden, E.: The geometric minimum action method: a least action principle on the space of curves. Commun. Pure Appl. Math. 61, 1052–1117 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Whitney, H.: On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane. Ann. Math. 62, 374–410 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Dykman, M., Luchinsky, D., Mcclintock, P., Smelyanskiy, V.: Corrals and critical behavior of the distribution of fluctuational paths. Phys. Rev.Lett. 77, 5229–5232 (1996)CrossRefGoogle Scholar
  29. 29.
    Chen, Z., Liu, X.: Patterns and singular features of extreme fluctuational paths of a periodically driven system. Phys. Lett. A 380, 1953–1958 (2016)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Chen, Z., Liu, X.: Noise induced transitions and topological study of a periodically driven system. Commun. Nonlinear Sci. Numer. Simul. 48, 454–461 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Dykman, M.: Observable and hidden singular features of large fluctuations in nonequilibrium systems. Phys. Lett. A 195, 53–58 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Filippov, A.: Differential equations with discontinuous right-hand side. Am. Math. Soc. Transl. 2, 199–231 (1964)zbMATHGoogle Scholar
  33. 33.
    Kuznetsov, Y., Rinaldi, S., Gragnani, A.: One-parameter bifurcations in planar Filippov systems. Int. J. Bifurc Chaos 13, 2157–2188 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Guardia, M., Seara, T., Teixeira, M.: Generic bifurcations of low codimension of planar Filippov systems. Int. J. Bifurc. Chaos 250, 1967–2023 (2011)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174, 312–368 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Fenichela, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 53–98 (1979)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Smelyanskiy, V., Dykman, M., Maier, R.: Topological features of large fluctuations to the interior of a limit cycle. Phys. Rev. E 55, 2369–2391 (1997)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Maier, R., Stein, D.: Limiting exit location distributions in the stochastic exit problem. SIAM J. Appl. Math. 57(3), 752C790 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Luchinsky, D., Maier, R., Mannella, R., Mcclintock, P., Stein, D.: Observation of saddle-point avoidance in noise-induced escape. Phys. Rev. Lett. 82, 1806–1809 (1999)CrossRefGoogle Scholar
  40. 40.
    Ermentrout, B.: Type I membranes, phase resetting curves, and synchrony. Neural Comput. 8, 979–1001 (1996)CrossRefGoogle Scholar
  41. 41.
    Lucken, L., Yanchuk, S., Popovych, O., Tass, P.: Desynchronization boost by non-uniform coordinated reset stimulation in ensembles of pulse-coupled neurons. Front. Comput. Neurosci. 7, 63 (2013)CrossRefGoogle Scholar

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

Personalised recommendations