FitzHugh model - Nagumo

Graph v with values I = 0.5, a = 0.7, b = 0.8 and τ = 12.5

The blue line is the trajectory of the PCN model in the phase space. The pink line is a cubic nullcline. The yellow line is a linear nullcline.

The FitzHugh-Nagumo model describes a prototype of an excitable system (for example, a neuron). It is named after Richard FitzHugh (1922–2007), who proposed the system in 1961, and J. Nagumo, who proposed a similar scheme the following year.

This model is an example of a generator of relaxation oscillations , due to the fact that if an external stimulus $I_{\text{ext}}$ ${\ displaystyle I _ {\ text {ext}}}$ ${\ displaystyle I _ {\ text {ext}}}$ exceeds a certain threshold value, the system will exhibit a characteristic reciprocating motion (excursion) in the phase space, until such time as the variables $v$ ${\ displaystyle v}$ $v$ and $w$ ${\ displaystyle w}$ $w$ Do not "relax" to the previous state.

This behavior is characteristic of spikes excited in the neuron by stimulation with an external input signal.

The following equations correspond to this dynamic system:

{\dot {v}}=v-{\frac {v^{3}}{3}}-w+I_{\rm {ext}}

{\ displaystyle {\ dot {v}} = v - {\ frac {v ^ {3}} {3}} - w + I _ {\ rm {ext}}}

{\ displaystyle {\ dot {v}} = v - {\ frac {v ^ {3}} {3}} - w + I _ {\ rm {ext}}}

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

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

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

The dynamics of this system can be beautifully described as switching between the left and right branches of a cubic nullcline .

The Fitzhugh – Nagumo model is a simplified version of the Hodgkin – Huxley model , which explains in some detail the dynamics of activation and deactivation of a pulsating neuron.

This model includes the use of a van der Pol oscillator for a particular case. $a=b=0$ ${\ displaystyle a = b = 0}$ ${\ displaystyle a = b = 0}$ . A similar solution was proposed by Jin-iti Nagumo, Suguru Arimoto and Shuji Yoshizawa. [one]

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Sources

FitzHugh R. (1955) Mathematical models of the membrane membrane. Bull. Math Biophysics 17: 257-278
FitzHugh R. (1961) Impulses and physiological states of the membrane membrane. Biophysical J. 1: 445-466
FitzHugh R. (1969) Mathematical models of nerve. Chapter 1 (pp. 1–85 in HP Schwan, ed. Biological Engineering, McGraw – Hill Book Co., NY)
Nagumo J., Arimoto S., and Yoshizawa S. (1962) An active pulse line simulating nerve axon. Proc. Ire . 50: 2061–2070.

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Links

FitzHugh – Nagumo model on Scholarpedia
Interactive FitzHugh-Nagumo. A java application that includes phase space and parameters that can be changed at any time.
Interactive FitzHugh – Nagumo in 1D. Java application for modeling one-dimensional waves propagating in a ring. Parameters can also be changed at any time.
Interactive FitzHugh – Nagumo in 2D. Java application for modeling 2D waves, including spiral waves. Parameters can also be changed at any time.
Java applet for two coupled FHN systems Parameters include connection delay, self-response caused by excursion noise, data export to a file. Source code is available (BY-NK-SA license).

FitzHugh model - Nagumo

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