A study of fixed points and Hopf bifurcation of Fitzhugh-Nagumo model

Phan Van Long Em *

* Corresponding author (pvlem@agu.edu.vn)

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Received: 21 Feb 2017
Revised: 21 Jun 2017
Accepted: 09 Mar 2018
Published: 30 Mar 2018
DOI: 10.22144/ctu.jen.2018.015
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Phan, V. L. E. (2018). A study of fixed points and Hopf bifurcation of Fitzhugh-Nagumo model. CTU Journal of Innovation and Sustainable Development , (08), 112-121. https://doi.org/10.22144/ctu.jen.2018.015
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No. 08 (2018)

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Abstract

In this article, a class of FitzHugh-Nagumo model is studied. First we are finding all necessary conditions for the parameters of system such that we have just one stable fixed point which presents the resting state for this famous model. After that, by using the Hopf’s theorem, we proof analytically the existence of a Hopf bifurcation that is a critical point where a system’s stability switches and a periodic solution arises. More precisely, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues cross the complex plane imaginary axis. With  reasonably generic assumptions for the dynamical system, a small-amplitude limit cycle branches from the fixed point.
Keywords: FitzHugh-Nagumo model, fixed point, Hopf bifurcation, limit cycle

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References

Ambrosio, B., and Aziz-Alaoui, M.A., 2012. Synchronization and control of coupled reaction diffusion systems of the FitzHugh-Nagumo-type. Computers and Mathematics with Applications. 64: 934-943.

Ambrosio, B., 2009. Propagation of waves in an excitable zone: numerical simulations and analytical approach. PhD thesis. The university of Pierre and Marie Curie-Paris 6. Paris, France (in french).

Ambrosio, B., and Aziz-Alaoui, M.A., 2012. Synchronization in a network of coupled reaction diffusion systems of the FitzHugh-Nagumo-type. Actes du colloque EDP-Normandie, Le Havre, 119-131 (in French).

Ambrosio, B., and Aziz-Alaoui, M.A., 2013. Synchronization and control of a network of coupled reaction-diffusion systems of generalized FitzHugh-Nagumo type. ESAIM: Proceedings. 39: 15-24.

Aziz-Alaoui, M.A., 2006. Synchronization of Chaos. Encyclopedia of Mathematical Physics. Elsevier. 5: 213-226.

Corson, N., 2009. Dynamics of a neural model, synchronization and complexity. PhD thesis. The university of Le Havre. Le Havre, France (in french).

Dang-Vu Huyen, and Delcarte, C., 2000. Bifurcations and Chaos, an introduction to dynamics contemporary with programs in Pascal, Fortan et Mathematica. Eds Elipses, Université – Mécanique (in french).

Ermentrout, G. B., Terman, D. H., 2009. Mathematical Foundations of Neurosciences. Springer.

Hodgkin, A.L., and Huxley, A. F., 1952. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117: 500-544.

Izhikevich, E. M., 2007. Dynamical Systems in Neuroscience. The MIT Press.

Izhikevich, E. M., and FitzHugh, R., 2006. FitzHugh-Nagumo model. Scholarpedia, 1(9):13-49.

Keener, J. P., and Sneyd, J., 2009. Mathematical Physiology. Springer.

Murray, J. D., 2010. Mathematical Biology. Springer.

Nagumo, J., Arimoto, S., and Yoshizawa, S., 1962. An active pulse transmission line simulating nerve axon. Proc. IRE. 50: 2061-2070.