• Previous Article
    Improved extensibility criteria and global well-posedness of a coupled chemotaxis-fluid model on bounded domains
  • DCDS-B Home
  • This Issue
  • Next Article
    Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives
November  2018, 23(9): 3935-3947. doi: 10.3934/dcdsb.2018118

Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain

Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

Received  May 2017 Revised  October 2017 Published  April 2018

Fund Project: The author is grateful for Professor Shouhong Wang for his advice and suggestions.This research is supported in part by the National Science Foundation (NSF) grant DMS-1515024, and by the Office of Naval Research (ONR) grant N00014-15-1-2662

The main objective of this article is to study the dynamic transitions of the FitzHugh-Nagumo equations on a finite domain with the Neumann boundary conditions and with uniformly injected current. We show that when certain parameter conditions are satisfied, the system undergoes a continuous dynamic transition to a limit cycle. A mixed type transition is also found when other conditions are imposed on the parameters. The main method used here is Ma & Wang's dynamic transition theory, which can be used generally on different set-ups for the FitzHugh-Nagumo equations.

Citation: Yiqiu Mao. Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3935-3947. doi: 10.3934/dcdsb.2018118
References:
[1]

R. G. CastenH. Cohen and P. A. Lagerstrom, Perturbation analysis of an approximation to the Hodgkin-Huxley theory, Quarterly of Applied Mathematics, 32 (1974/75), 365-402. Google Scholar

[2]

G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience, vol. 35, Springer Science & Business Media, 2010. Google Scholar

[3]

R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bulletin of Mathematical Biology, 17 (1955), 257-278. doi: 10.1007/BF02477753. Google Scholar

[4]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6. Google Scholar

[5]

S. Hagiwara and Y. Oomura, The critical depolarization for the spike in the squid giant axon, The Japanese Journal of Physiology, 8 (1958), 234-245. doi: 10.2170/jjphysiol.8.234. Google Scholar

[6]

S. Hastings, On the existence of homoclinic and periodic orbits for the Fitzhugh-Nagumo equations, Quart. J. Math. (Oxford), 27 (1976), 123-134. doi: 10.1093/qmath/27.1.123. Google Scholar

[7]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. Google Scholar

[8]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117 (1952), p500. Google Scholar

[9]

A. J. Hudspeth, T. M. Jessell, E. R. Kandel, J. H. Schwartz and S. A. Siegelbaum, Principles of Neural Science, 2013.Google Scholar

[10]

C. K. Jones, Stability of the travelling wave solution of the Fitzhugh-Nagumo system, Transactions of the American Mathematical Society, 286 (1984), 431-469. doi: 10.1090/S0002-9947-1984-0760971-6. Google Scholar

[11]

M. KrupaB. Sandstede and P. Szmolyan, Fast and slow waves in the Fitzhugh-Nagumo equation, Journal of Differential Equations, 133 (1997), 49-97. doi: 10.1006/jdeq.1996.3198. Google Scholar

[12]

T. Ma and S. Wang, Attractor bifurcation theory and its applications to Rayleigh-Bénard convection, Commun. Pure Appl. Anal., 2 (2003), 591-599. doi: 10.3934/cpaa.2003.2.591. Google Scholar

[13]

T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53, World Scientific, 2005. Google Scholar

[14]

T. Ma and S. Wang, Phase Transition Dynamics, Springer, 2014. Google Scholar

show all references

References:
[1]

R. G. CastenH. Cohen and P. A. Lagerstrom, Perturbation analysis of an approximation to the Hodgkin-Huxley theory, Quarterly of Applied Mathematics, 32 (1974/75), 365-402. Google Scholar

[2]

G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience, vol. 35, Springer Science & Business Media, 2010. Google Scholar

[3]

R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bulletin of Mathematical Biology, 17 (1955), 257-278. doi: 10.1007/BF02477753. Google Scholar

[4]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6. Google Scholar

[5]

S. Hagiwara and Y. Oomura, The critical depolarization for the spike in the squid giant axon, The Japanese Journal of Physiology, 8 (1958), 234-245. doi: 10.2170/jjphysiol.8.234. Google Scholar

[6]

S. Hastings, On the existence of homoclinic and periodic orbits for the Fitzhugh-Nagumo equations, Quart. J. Math. (Oxford), 27 (1976), 123-134. doi: 10.1093/qmath/27.1.123. Google Scholar

[7]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. Google Scholar

[8]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117 (1952), p500. Google Scholar

[9]

A. J. Hudspeth, T. M. Jessell, E. R. Kandel, J. H. Schwartz and S. A. Siegelbaum, Principles of Neural Science, 2013.Google Scholar

[10]

C. K. Jones, Stability of the travelling wave solution of the Fitzhugh-Nagumo system, Transactions of the American Mathematical Society, 286 (1984), 431-469. doi: 10.1090/S0002-9947-1984-0760971-6. Google Scholar

[11]

M. KrupaB. Sandstede and P. Szmolyan, Fast and slow waves in the Fitzhugh-Nagumo equation, Journal of Differential Equations, 133 (1997), 49-97. doi: 10.1006/jdeq.1996.3198. Google Scholar

[12]

T. Ma and S. Wang, Attractor bifurcation theory and its applications to Rayleigh-Bénard convection, Commun. Pure Appl. Anal., 2 (2003), 591-599. doi: 10.3934/cpaa.2003.2.591. Google Scholar

[13]

T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53, World Scientific, 2005. Google Scholar

[14]

T. Ma and S. Wang, Phase Transition Dynamics, Springer, 2014. Google Scholar

[1]

Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457

[2]

Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. Evolution Equations & Control Theory, 2018, 7 (4) : 571-585. doi: 10.3934/eect.2018027

[3]

Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216

[4]

Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203

[5]

Wenqiang Zhao. Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3453-3474. doi: 10.3934/dcdsb.2018251

[6]

Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems. Evolution Equations & Control Theory, 2017, 6 (4) : 559-586. doi: 10.3934/eect.2017028

[7]

John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851

[8]

Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072

[9]

Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150

[10]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839

[11]

Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157

[12]

Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891

[13]

Jyoti Mishra. Analysis of the Fitzhugh Nagumo model with a new numerical scheme. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 781-795. doi: 10.3934/dcdss.2020044

[14]

Pierluigi Colli, Antonio Segatti. Uniform attractors for a phase transition model coupling momentum balance and phase dynamics. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 909-932. doi: 10.3934/dcds.2008.22.909

[15]

Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101

[16]

Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106

[17]

Yangrong Li, Jinyan Yin. A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1203-1223. doi: 10.3934/dcdsb.2016.21.1203

[18]

Abiti Adili, Bixiang Wang. Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 643-666. doi: 10.3934/dcdsb.2013.18.643

[19]

Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441

[20]

Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (53)
  • HTML views (377)
  • Cited by (0)

Other articles
by authors

[Back to Top]