en>Magioladitis: Fix template prefix + general fixes using AWB (8232)

2012-08-07T20:37:41Z

Fix template prefix + general fixes using AWB (8232)

New page

[[File:Fitzhugh Nagumo Model Time Domain Graph.png|thumb|right|Graph of v with parameters I=.5, a=0.7, b=0.8, and τ=12.5]]
[[File:Fitzhugh Nagumo Phase Space Graph.png|thumb|right|The blue line is the trajectory of the FHN model in phase space. The pink line is the cubic nullcline and the yellow line is the linear nullcline.]]
The '''FitzHugh–Nagumo model''' ('''FHN'''), named after [[Richard FitzHugh]] (1922 – 2007) who suggested the system in 1961 and J. Nagumo ''et al''. who created the equivalent circuit the following year, describes a prototype of an excitable system (e.g., a [[neuron]]).

The FHN Model is an example of a [[relaxation oscillator]] because, if the external stimulus <math>I_{\text{ext}}</math> exceeds a certain threshold value, the system will exhibit a characteristic excursion in [[phase space]], before the variables <math>v</math> and <math>w</math> relax back to their rest values.

This behaviour is typical for spike generations (=short elevation of membrane voltage <math>v</math>) in a neuron after stimulation by an external input current.

The equations for this dynamical system read

: <math>
\dot{v}=v-\frac{v^3}{3} - w + I_{\rm ext}
</math>

: <math>
\tau \dot{w} = v+a-b w.
</math>

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''' (named after [[Karl Friedrich Bonhoeffer]] and [[Balthasar van der Pol]]) because it contains the [[van der Pol oscillator]] as a special case for <math> a=b=0 </math>. The equivalent circuit was suggested by Jin-ichi Nagumo, Suguru Arimoto, and Shuji Yoshizawa.[http://www.siam.org/news/news.php?id=647]

==See also==
{{Div col}}
*[[Autowave]]
*[[Biological neuron models]]
*[[Computational neuroscience]]
*[[Hodgkin–Huxley model]]
*[[Reaction-diffusion]]
*[[Theta model]]
{{Div col end}}

==Further reading==
{{refbegin}}
*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 H.P. Schwan, ed. Biological Engineering, McGraw–Hill Book Co., N.Y.)
*Nagumo J., Arimoto S., and Yoshizawa S. (1962) An active pulse transmission line simulating nerve axon. [[Proc. IRE]]. 50:2061–2070.
{{refend}}

==External links==
*[http://scholarpedia.org/article/FitzHugh-Nagumo_Model FitzHugh–Nagumo model on Scholarpedia]
*[http://thevirtualheart.org/FHNindex.html Interactive FitzHugh-Nagumo.] Java applet, includes phase space and parameters can be changed at any time.
*[http://thevirtualheart.org/FHN1dindex.html Interactive FitzHugh–Nagumo in 1D. ] Java applet to simulate 1D waves propagating in a ring. Parameters can also be changed at any time.
*[http://thevirtualheart.org/FHN2dindex.html Interactive FitzHugh–Nagumo in 2D.] Java applet to simulate 2D waves including spiral waves. Parameters can also be changed at any time.
*[http://www.itp.tu-berlin.de/menue/lehre/owl/nichtlineare_dynamik/fitzhugh_nagumo/parameter/en/ Java applet for two coupled FHN systems] Options include time delayed coupling, self-feedback, noise induced excursions, data export to file. Source code available (BY-NC-SA license).

{{DEFAULTSORT:FitzHugh-Nagumo model}}
[[Category:Nonlinear systems]]
[[Category:Electrophysiology]]
[[Category:Computational neuroscience]]
[[Category:Biophysics]]

Defense-Independent Component ERA - Revision history

en>Magioladitis: Fix template prefix + general fixes using AWB (8232)