FitzHugh–Nagumo model

The FitzHugh–Nagumo model (FHN) describes a prototype of an excitable system (e.g., a neuron). It is an example of a relaxation oscillator because, if the external stimulus exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase space, before the variables and relax back to their rest values. This behaviour is a sketch for neural spike generations, with a short, nonlinear elevation of membrane voltage , diminished over time by a slower, linear recovery variable representing sodium channel reactivation and potassium channel deactivation, after stimulation by an external input current. The equations for this dynamical system read 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 turn, the Van der Pol oscillator is a special case of the FitzHugh–Nagumo model, with . It was named after Richard FitzHugh (1922–2007) who suggested the system in 1961 and Jinichi Nagumo et al. who created the equivalent circuit the following year. 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 . The equivalent circuit was suggested by Jin-ichi Nagumo, Suguru Arimoto, and Shuji Yoshizawa. Qualitatively, the dynamics of this system is determined by the relation between the three branches of the cubic nullcline and the linear nullcline. The cubic nullcline is defined by . The linear nullcline is defined by . In general, the two nullclines intersect at one or three points, each of which is an equilibrium point. At large values of , far from origin, the flow is a clockwise circular flow, consequently the sum of the index for the entire vector field is +1. This means that when there is one equilibrium point, it must be a clockwise spiral point or a node. When there are three equilibrium points, they must be two clockwise spiral points and one saddle point.