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. [1]
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) [2] who suggested the system in 1961 [3] and Jinichi Nagumo et al. who created the equivalent circuit the following year. [4]
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. [5]
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.
The type and stability of the index +1 can be numerically computed by computing the trace and determinant of its Jacobian:
The point is a spiral point iff . That is, .
The limit cycle is born when a stable spiral point becomes unstable by Hopf bifurcation. [1]
Only when the linear nullcline pierces the cubic nullcline at three points, the system has a separatrix, being the two branches of the stable manifold of the saddle point in the middle.
Gallery figures: FitzHugh-Nagumo model, with , and varying . (They are animated. Open them to see the animation.)
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. [1]
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) [2] who suggested the system in 1961 [3] and Jinichi Nagumo et al. who created the equivalent circuit the following year. [4]
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. [5]
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.
The type and stability of the index +1 can be numerically computed by computing the trace and determinant of its Jacobian:
The point is a spiral point iff . That is, .
The limit cycle is born when a stable spiral point becomes unstable by Hopf bifurcation. [1]
Only when the linear nullcline pierces the cubic nullcline at three points, the system has a separatrix, being the two branches of the stable manifold of the saddle point in the middle.
Gallery figures: FitzHugh-Nagumo model, with , and varying . (They are animated. Open them to see the animation.)