FitzHugh-Nagumo model

The FitzHugh-Nagumo model (after Richard FitzHugh (* 1922) and J. Nagumo, who developed the model independently of one another) describes a prototype of a stimulable system, for example a neuron or axon . If the external stimulus exceeds a threshold value, the system makes a characteristic excursion in the phase space before the variables and return to their rest values . This behavior is a model for the generation of spikes (= brief increase in membrane tension ) in a neuron after stimulation by an external current . ${\displaystyle I_{\rm {ext}}}$ ${\displaystyle (v,w)}$ ${\displaystyle v}$ ${\displaystyle w}$ ${\displaystyle (v_{0},w_{0})}$ ${\displaystyle v}$ ${\displaystyle I_{\rm {ext}}}$

Spike dynamics of the FitzHugh-Nagumo model after a short stimulation ${\displaystyle I_{\rm {ext}}\neq 0}$

Zero lines of the FitzHugh-Nagumo model (blue) and example trajectory (red)

Here v is a membrane potential and w is a variable that describes a negative feedback with linear dynamics that is necessary for recovery after the electrical excitation.

The equations of this dynamic system are

${\displaystyle {\dot {v}}=v-{\frac {1}{3}}v^{3}-w+I_{\rm {ext}}}$

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

The excitation dynamics can be clearly shown with the help of the zero lines. The stationary point (rest values) is the intersection of the - and - zero lines. If the system is excited for a short time ( ), it describes an excursion in phase space that can be divided into four stages: first, the trajectory describes an almost horizontal trajectory, because due to apply . As soon as the trajectory reaches the cubic -zero line, it sinks rapidly and the trajectory follows the -zero line. At the upper vertex of the zero lines, there is another horizontal passage to the left branch of the zero line, and then another phase in which the trajectory of these zero lines follows. ${\displaystyle (v_{0},w_{0})}$ ${\displaystyle {\dot {v}}}$ ${\displaystyle {\dot {w}}}$ ${\displaystyle I_{\rm {ext}}\neq 0}$ ${\displaystyle \tau \gg 1}$ ${\displaystyle {\dot {v}}\gg {\dot {w}}}$ ${\displaystyle {\dot {v}}}$ ${\displaystyle {\dot {v}}}$ ${\displaystyle {\dot {v}}}$ ${\displaystyle {\dot {v}}}$ ${\displaystyle {\dot {v}}}$

The FitzHugh-Nagumo model, which maps the activation and deactivation dynamics in detail in a spiking neuron, is a simplified version of the Hodgkin-Huxley model . In the original FitzHugh articles, this model is also referred to as the Bonhoeffer van der Pol oscillator because it contains the van der Pol oscillator as a special case for . ${\displaystyle a=b=0}$

literature

• FitzHugh R. (1955) Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophysics, 17: 257-278
• FitzHugh R. (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophysical J. 1: 445-466
• FitzHugh R. (1969) Mathematical models of excitation and propagation in 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 transmission line simulating nerve axon. Proc IRE. 50: 2061-2070.

Web links

• Article . In: Scholarpedia . (English, including references) by Eugene Izhikevich, Richard Fitzhugh 2006
• Java applet for two coupled FHN systems