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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  November 2018 Early access  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 and Continuous Dynamical Systems - B, 2018, 23 (9) : 3935-3947. doi: 10.3934/dcdsb.2018118
##### References:
 [1] R. G. Casten, H. Cohen and P. A. Lagerstrom, Perturbation analysis of an approximation to the Hodgkin-Huxley theory, Quarterly of Applied Mathematics, 32 (1974/75), 365-402. [2] G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience, vol. 35, Springer Science & Business Media, 2010. [3] R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bulletin of Mathematical Biology, 17 (1955), 257-278.  doi: 10.1007/BF02477753. [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. [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. [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. [7] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. [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. [9] A. J. Hudspeth, T. M. Jessell, E. R. Kandel, J. H. Schwartz and S. A. Siegelbaum, Principles of Neural Science, 2013. [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. [11] M. Krupa, B. 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. [12] T. Ma and S. Wang, Attractor bifurcation theory and its applications to Rayleigh-BÃ©nard convection, Commun. Pure Appl. Anal., 2 (2003), 591-599.  doi: 10.3934/cpaa.2003.2.591. [13] T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53, World Scientific, 2005. [14] T. Ma and S. Wang, Phase Transition Dynamics, Springer, 2014.

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##### References:
 [1] R. G. Casten, H. Cohen and P. A. Lagerstrom, Perturbation analysis of an approximation to the Hodgkin-Huxley theory, Quarterly of Applied Mathematics, 32 (1974/75), 365-402. [2] G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience, vol. 35, Springer Science & Business Media, 2010. [3] R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bulletin of Mathematical Biology, 17 (1955), 257-278.  doi: 10.1007/BF02477753. [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. [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. [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. [7] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. [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. [9] A. J. Hudspeth, T. M. Jessell, E. R. Kandel, J. H. Schwartz and S. A. Siegelbaum, Principles of Neural Science, 2013. [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. [11] M. Krupa, B. 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. [12] T. Ma and S. Wang, Attractor bifurcation theory and its applications to Rayleigh-BÃ©nard convection, Commun. Pure Appl. Anal., 2 (2003), 591-599.  doi: 10.3934/cpaa.2003.2.591. [13] T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53, World Scientific, 2005. [14] T. Ma and S. Wang, Phase Transition Dynamics, Springer, 2014.
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