Abstract
This work is about the state transition of the stochastic Morris-Lecar neuronal model driven by symmetric α-stable Lévy noise. The considered system is bistable: a stable equilibrium (resting state) and a stable limit cycle (oscillating state), and there is an unstable limit cycle (borderline state) between them. Small disturbances may cause a transition between the two stable states, thus a deterministic quantity, namely the maximal likely trajectory, is used to analyze the transition phenomena in a non-Gaussian stochastic environment. According to the numerical experiment, we find that smaller jumps of the Lévy motion and smaller noise intensities can promote such transition from the sustained oscillating state to the resting state. It also can be seen that larger jumps of the Lévy motion and higher noise intensities are conducive for the transition from the borderline state to the sustained oscillating state. As a comparison, Brownian motion is also taken into account. The results show that whether it is the oscillating state or the borderline state, the system disturbed by Brownian motion will be transferred to the resting state under the selected noise intensity with a high probability.
Graphical abstract
Similar content being viewed by others
References
H.C. Tuckwell,Introduction to theoretical neurobiology (Cambridge: Cambridge University Press), 1988
E.M. Izhikevich,Dynamical Systems in Neuroscience (Cambridge: MIT press), 2007
M.D. McDonnell, S. Ikeda, J.H. Manton, Biol. Cybern. 105, 55 (2011)
T. Trappenberg,Fundamentals of computational neuroscience (Oxford, OUP Oxford, 2009)
W.W. Lytton,From computer to brain: foundations of computational neuroscience (Springer Science & Business Media, Berlin, 2007)
P. Dayan, L.F. Abbott,Theoretical neuroscience (MIT Press, Cambridge, 2001)
N. Chakravarthy, K. Tsakalis, S. Sabesan, L. Iasemidis, Ann. Biomed. Eng. 37, 565 (2009)
E.T. Rolls, G. Deco,The noisy brain: stochastic dynamics as a principle of brain function (Oxford University Press, Oxford, 2010)
B.G. Ermentrout, D.H Terman,Mathematical Foundations of Neuroscience (Springer Science & Business Media, Berlin, 2010)
I. Franović, K. Todorović, M. Perc, N. Vasović, N. Burić, Phys. Rev. E 92, 062911 (2015)
I. Franović, M. Perc, K. Todorović, S. Kostić, N. Burić, Phys. Rev. E 92, 062912 (2015)
Y. Wang, J. Ma, Y. Xu, F. Wu, P. Zhou, Int. J. Bifurc. Chaos 27, 1750030 (2017)
A. Longtin, Scholarpedia 8, 1618 (2013)
B. Lindner, J. Garcıa-Ojalvo, A. Neiman, L. Schimansky-Geier, Phys. Rep. 392, 321 (2004)
A. Patel, B. Kosko, IEEE Trans. Neural Networks 19, 1993 (2008)
A. Patel, B. Kosko, Lévy noise benefits in neural signal detection, inIEEE International Conference on Acoustics, Speech and Signal Processing, 2007, Vol. 3, pp. III–1413
J.A. Roberts, T.W. Boonstra, M. Breakspear, Curr. Opin. Neurobiol. 31, 164 (2015)
Y. Xu, J. Li, J. Feng, H. Zhang, W. Xu, J. Duan, Eur. Phys. J. B 86, 198 (2013)
X. Sun, Q. Lu, Chin. Phys. Lett. 31, 020502 (2014)
K. Ýr Jónsdóttir, A. Rønn-Nielsen, K. Mouridsen, E.B. Vedel Jensen, Scand. J. Stat. 40, 511 (2013)
M. Vinaya, R.P. Ignatius, Nonlinear Dyn. 94, 1133 (2018)
Z. Wang, Y. Xu, H. Yang, Sci. China Technol. Sci. 59, 371 (2016)
J. Wu, Y. Xu, J. Ma, PLoS One 12, e0174330 (2017)
B. Dybiec, E. Gudowska-Nowak, J. Stat. Mech. 2009, P05004 (2009)
Y. Li, Y. Xu, J. Kurths, Phys. Rev. E 96, 052121 (2017)
Y. Li, Y. Xu, J. Kurths, X. Yue, Phys. Rev. E 94, 042222 (2016)
Y. Xu, J. Feng, J.J. Li, H. Zhang, Chaos 23, 013110 (2013)
Y. Xu, J. Feng, J.J. Li, H. Zhang, Physica A 392, 4739 (2013)
Y. Xu, Y. Li, J. Li, J. Feng, H. Zhang, J. Stat. Phys. 158, 120 (2015)
M. Perc, Phys. Rev. E 76, 066203 (2007)
K. Ishimura, A. Schmid, T. Asai, M. Motomura, Nonlinear Theory Appl. IEICE 7, 164 (2016)
S.F. Duki, M.A. Taye, J. Stat. Phys. 171, 878 (2018)
I. Bashkirtseva, L. Ryashko, P. Stikhin, Int. J. Bifurc. Chaos 23, 1350092 (2013)
M. Frey, E. Simiu, Physica D 63, 321 (1993)
I. Bashkirtseva, S. Fedotov, L. Ryashko, E. Slepukhina, Int. J. Bifurc. Chaos 26, 1630032 (2016)
Y. Zheng, L. Serdukova, J. Duan, J. Kurths, Sci. Rep. 6 (2016)
F. Wu, X. Chen, Y. Zheng, J. Duan, X. Li, Chaos 28, 075510 (2018)
S. Lim, J. Rinzel, J. Comput. Neurosci. 28, 1 (2010)
S. Tanabe, K. Pakdaman, Biol. Cybern. 85, 269 (2001)
J. Touboul, Physica D 241, 1223 (2012)
C. Morris, H. Lecar, Biophys. J. 35, 193 (1981)
D. Terman, J. Nonlinear Sci. 2, 135 (1992)
J.M. Newby, P.C. Bressloff, J.P. Keener, Phys. Rev. Lett. 111, 128101 (2013)
T. Tateno, K. Pakdaman, Chaos 14, 511 (2004)
J.P. Keener, J.M. Newby, Phys. Rev. E 84, 011918 (2011)
Y. Liu, R. Cai, J. Duan, Physica A 531, 121785 (2019)
C. Liu, X. Liu, S. Liu, Biol. Cybern. 108, 75 (2014)
K. Tsumoto, H. Kitajima, T. Yoshinaga, K. Aihara, H. Kawakami, Neurocomputing 69, 293 (2006)
O. Zeitouni, A. Dembo, Stochastics 20, 221 (1987)
K. Sato,Lévy Processes and Infinitely Divisible Distributions (Cambridge University Press, Cambridge, 1999)
J. Bertoin,Lévy Processes (Cambridge University Press, Cambridge, 1998)
D. Applebaum,Lévy Processes and Stochastic Calculus (Cambridge University Press, Cambridge, 2009)
J. Duan,An Introduction to Stochastic Dynamics (Cambridge University Press, Cabridge, 2015)
H. Wang, X. Chen, J. Duan, Int. J. Bifurc. Chaos 28, 1850017 (2018)
X. Chen, F. Wu, J. Duan, J. Kurths, X. Li, Appl. Math. Comput. 348, 425 (2019)
T. Gao, J. Duan, X. Li, Appl. Math. Comput. 278, 1 (2016)
J. Wei, R. Tian, J. Math. Phys. 56, 031502 (2015)
X. Yang, J. Cao, Appl. Math. Modell. 34, 3631 (2010)
M. Perc, M. Gosak, New J. Phys. 10, 053008 (2008)
M. Gosak, D. Korošak, M. Marhl, New J. Phys. 13, 013012 (2011)
Q. Zhu, J. Cao, R. Rakkiyappan, Nonlinear Dyn. 79, 1085 (2015)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cai, R., Liu, Y., Duan, J. et al. State transitions in the Morris-Lecar model under stable Lévy noise. Eur. Phys. J. B 93, 38 (2020). https://doi.org/10.1140/epjb/e2020-100422-2
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2020-100422-2