# FitzHugh–Nagumo model

Graph of v with parameters I=0.5, a=0.7, b=0.8, and τ=12.5
The blue line is the trajectory of the FHN model in phase space. The pink line is the cubic nullcline and the yellow line is the linear nullcline.

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 $I_{\text{ext}}$ exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase space, before the variables $v$ and $w$ relax back to their rest values.

This behaviour is typical for spike generations (=short elevation of membrane voltage $v$) in a neuron after stimulation by an external input current.

The equations for this dynamical system read

$\dot{v}=v-\frac{v^3}{3} - w + I_{\rm ext}$
$\tau \dot{w} = v+a-b w.$

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 $a=b=0$. The equivalent circuit was suggested by Jin-ichi Nagumo, Suguru Arimoto, and Shuji Yoshizawa.[1]

• 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.