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).

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. In this case ${\displaystyle I_{\rm {ext}}=0.5,a=0.8,b=0.7}$.

The FHN Model[1] is an example of a relaxation oscillator because, if the external stimulus ${\displaystyle I_{\text{ext}}}$ exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase space, before the variables ${\displaystyle v}$ and ${\displaystyle w}$ relax back to their rest values.

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

The equations for this dynamical system read

${\displaystyle {\dot {v}}=v-{\frac {v^{3}}{3}}-w+RI_{\rm {ext}}}$

${\displaystyle \tau {\dot {w}}=v+a-bw.}$

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 2D 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 ${\displaystyle a=b=0}$. The equivalent circuit was suggested by Jin-ichi Nagumo, Suguru Arimoto, and Shuji Yoshizawa.[2]