## Frequent Links

# FitzHugh–Nagumo model

The **FitzHugh–Nagumo model** (**FHN**), named after Richard FitzHugh (1922 – 2007) who suggested the system in 1961 and J. Nagumo *et al*. who created the equivalent circuit the following year, describes a prototype of an excitable system (e.g., a neuron).

The FHN Model is an example of a relaxation oscillator because, if the external stimulus <math>I_{\text{ext}}</math> exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase space, before the variables <math>v</math> and <math>w</math> relax back to their rest values.

This behaviour is typical for spike generations (a short, nonlinear elevation of membrane voltage <math>v</math>, diminished over time by a slower, linear recovery variable <math>w</math>) in a neuron after stimulation by an external input current.

The equations for this dynamical system read

- <math>

\dot{v}=v-\frac{v^3}{3} - w + I_{\rm ext} </math>

- <math>

\tau \dot{w} = v+a-b w. </math>

The dynamics of this system can be nicely described by zapping between the left and right branch of the cubic nullcline.

The FitzHugh–Nagumo model is a simplified version of the Hodgkin–Huxley model which models in a detailed manner activation and deactivation dynamics of a spiking neuron. In the original papers of FitzHugh, this model was called **Bonhoeffer–van der Pol oscillator** (named after Karl Friedrich Bonhoeffer and Balthasar van der Pol) because it contains the van der Pol oscillator as a special case for <math> a=b=0 </math>. The equivalent circuit was suggested by Jin-ichi Nagumo, Suguru Arimoto, and Shuji Yoshizawa.[1]

## See also

## Further reading

- FitzHugh R. (1955) Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophysics, 17:257—278
- FitzHugh R. (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophysical J. 1:445–466
- FitzHugh R. (1969) Mathematical models of excitation and propagation in nerve. Chapter 1 (pp. 1–85 in H.P. Schwan, ed. Biological Engineering, McGraw–Hill Book Co., N.Y.)
- Nagumo J., Arimoto S., and Yoshizawa S. (1962) An active pulse transmission line simulating nerve axon. Proc. IRE. 50:2061–2070.

## External links

- FitzHugh–Nagumo model on Scholarpedia
- Interactive FitzHugh-Nagumo. Java applet, includes phase space and parameters can be changed at any time.
- Interactive FitzHugh–Nagumo in 1D. Java applet to simulate 1D waves propagating in a ring. Parameters can also be changed at any time.
- Interactive FitzHugh–Nagumo in 2D. Java applet to simulate 2D waves including spiral waves. Parameters can also be changed at any time.
- Java applet for two coupled FHN systems Options include time delayed coupling, self-feedback, noise induced excursions, data export to file. Source code available (BY-NC-SA license).