# FitzHugh-Nagumo equation

## 1 The History

The FitzHugh-Nagumo equation  is a simplification of the Hodgkin-Huxley model devised in 1952. The Hodgkin-Huxley has four variables and the FitzHugh-Nagumo equation is a reduction  of that model. The reduction is from four variables to two variables where phase plane techniques may be used for the analysis of the model. The variables kept in the reduction of the model are the excitable variable and the recovery variable which are characterized as being the fast and slow variables respectively. The FitzHugh-Nagumo model was called, by FitzHugh, the Bonhoeffer - van der Pol model (BVP). FitzHugh explains that the BVP was devised in the same way as the van der Pol equation  “Its solution does not, to be sure, give an accurate fit to curves obtained from many physical oscillators. The equation was intended rather to represent the qualitative properties of a wide class of such oscillators, its algebraic form being chosen to be as simple as possible”.[FR] The name of Nagumo is added to FitzHugh by being able to represent the BVP model as an electrical device [NAY]. The BVP model “consists of three components, a capacitor representing the membrane capacitance, a nonlinear current-voltage device for the fast current, and a resistor, inductor, and battery in series for the recovery current.”[KS]

## 2 The Application

The application of the FitzHugh-Nagumo equation is to model the same phenomenon as the Hodgkin-Huxley model. The phenomenon that is modelled is the control of the electrical potential across cell membrane. This control is done by the change of flow of the ionic channels of the cell membrane. This results in the change in potential which is used to send electrical signals between cells. This is readily observed in muscle and other excitable cells. For example the FitzHugh-Nagumo equation is used to model electrical waves of the heart.[KS]

## 3 The Model

The FitzHugh-Nagumo model is defined by the following system of differential equations:

 $\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle\epsilon(y+x-x^{3}+I)$ $\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle\frac{-1}{\epsilon}(x-\beta+\gamma y)$

where $1-2\gamma/3<\beta<1$, $0<\gamma<1$, and $\gamma<\epsilon^{2}$.[FR] There are many equivalent     forms of the system, the more popular ones can be found in [RGG] with conversion for the different forms. A more general FitzHugh-Nagumo model is defined by the following system of differential equations:

 $\displaystyle\epsilon\dot{x}$ $\displaystyle=$ $\displaystyle f(x,y)+I$ $\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle g(x,y)$

where $f(x,y)=0$ resembles a cubic shape and $g(x,y)=0$ only has one intersection  with $f(x,y)=0$.[KS] In the above two formulas   the $I$ is the external current applied to the system. The following system is known as the spatially distributed FitzHugh-Nagumo model where diffusion is added to the general FitzHugh-Nagumo model.

 $\displaystyle\epsilon\frac{\partial v}{\partial t}$ $\displaystyle=$ $\displaystyle\epsilon^{2}\frac{\partial^{2}v}{\partial x^{2}}+f(v,w)+I$ $\displaystyle\frac{\partial w}{\partial t}$ $\displaystyle=$ $\displaystyle g(v,w)$

The above model has different travelling waves depending on the choice of parameter of the system.

## References

• FR FitzHugh, Richard: Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophysical Journal, Volume 1, 1961.
• HK Hale, Jack H. & Ko cak, Hüseyin: Dynamics and Bifurcations. Springer-Verlag, New York, 1991.
• IF Izhikevich, Eugene M., FitzHugh, Richard: http://www.scholarpedia.org/article/FitzHugh-Nagumo_ModelFitzHugh-Nagumo Model. http://www.scholarpedia.org/Scholarpedia, 2006.
• KS Keener, James & Sneyd, James: Mathematical Physiology. Springer-Verlag, New York, 1998.
• NAY Nagumo, J., S. Arimoto, and S. Yoshizawa: An active pulse transmission line simulating nerve axon, Proc IRE. 50: 2061-2070, 1964.
• RGG Rocs oreanu, C., Georgescu, A. & Giurgit eanu, N.: The FitzHugh-Nagumo Model: Bifurcation and Dynamics. Kluwer Academic Publishers, Boston, 2000.
Title FitzHugh-Nagumo equation FitzHughNagumoEquation 2013-03-22 14:29:45 2013-03-22 14:29:45 Daume (40) Daume (40) 5 Daume (40) Definition msc 34-00 Bonhoeffer - van der Pol model BVP FitzHugh-Nagumo model spatially distributed FitzHugh-Nagumo model