FitzHugh–Nagumo model
Encyclopedia
The FitzHugh–Nagumo model, 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
).
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 typical for spike generations (=short elevation of membrane voltage
) in a neuron after stimulation by an external input current.
The equations for this dynamical system read
Neuron
A neuron is an electrically excitable cell that processes and transmits information by electrical and chemical signaling. Chemical signaling occurs via synapses, specialized connections with other cells. Neurons connect to each other to form networks. Neurons are the core components of the nervous...
).
If the external stimulus
exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase spacePhase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
, before the variables
and relax back to their rest values.This behaviour is typical for spike generations (=short elevation of membrane voltage
) in a neuron after stimulation by an external input current.The equations for this dynamical system read
-
-
The dynamics of this system can be nicely described by zapping between the left and right branch of the cubic nullclineNullclineNullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equationsx_1'=f_1x_2'=f_2In a two-dimensional linear system, the nullclines can be represented by two lines on a two-dimensional plot.-External links:* from MIT OpenCourseWare*...
.
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 BonhoefferKarl Friedrich Bonhoeffer-Life:Born in Breslau, he was an older brother of martyred theologian Dietrich Bonhoeffer.Bonhoeffer studied from 1918 in Tübingen and Berlin, finishing his PhD in 1922 in Berlin with Walther Nernst. From 1923 to 1930 he was an assistant with Fritz Haber at Kaiser Wilhelm Institute for Physical...
and Balthasar van der PolBalthasar van der PolBalthasar van der Pol was a Dutch physicist.Van der Pol studied physics in Utrecht, and in 1920 he was awarded his doctorate . He studied experimental physics with John Ambrose Fleming and Sir J. J. Thomson in England...
) 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.http://www.siam.org/news/news.php?id=647
See also
- Biological neuron models
- Computational neuroscienceComputational neuroscienceComputational neuroscience is the study of brain function in terms of the information processing properties of the structures that make up the nervous system...
- Hodgkin–Huxley model
Further reading
- 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.
External links
- FitzHugh-Nagumo model on Scholarpedia
- Interactive FitzHugh-Nagumo. Java applet, includes phase space and parameters can be changed at any time.
- Interactive FitzHugh-Nagumo in 1D. Java applet to simulate 1D waves propagating in a ring. Parameters can also be changed at any time.
- Interactive FitzHugh-Nagumo in 2D. Java applet to simulate 2D waves including spiral waves. Parameters can also be changed at any time.
- 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).
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