FitzHugh-Nagumo model

The FitzHugh-Nagumo Model (named after Richard FitzHugh, 1922-2007) describes a prototype of an excitable system, i.e., a neuron.

If the external stimulus $$ I_{\rm ext} $$ 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-v^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, since it contains the van der Pol oscillator as a special case for $$ a=b=0 $$.