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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 A neuron, neurone, or nerve cell is an electrically excitable cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous tissue in all animals except sponges and placozoa. N ...
). The FHN Model is an example of a
relaxation oscillator In electronics a relaxation oscillator is a nonlinear electronic oscillator circuit that produces a nonsinusoidal repetitive output signal, such as a triangle wave or square wave. on Peter Millet'Tubebookswebsite The circuit consists of a feedba ...
because, if the external stimulus I_ exceeds a certain threshold value, the system will exhibit a characteristic excursion in
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
, before the variables v and w relax back to their rest values. This behaviour is typical for spike generations (a short, nonlinear elevation of membrane voltage v, diminished over time by a slower, linear recovery variable w) in a neuron after stimulation by an external input current. The equations for this
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
read : \dot=v-\frac - w + RI_ : \tau \dot = v+a-b w. 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 2D version of the
Hodgkin–Huxley model The Hodgkin–Huxley model, or conductance-based model, is a mathematical model that describes how action potentials in neurons are initiated and propagated. It is a set of nonlinear differential equations that approximates the electrical charact ...
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 Karl-Friedrich Bonhoeffer (13 January 1899 – 15 May 1957) was a German chemist. Education and career Born in Breslau, he was an older brother of martyred theologian Dietrich Bonhoeffer. His father was neurologist Karl Bonhoeffer and his moth ...
and
Balthasar van der Pol Balthasar van der Pol (27 January 1889 – 6 October 1959) was a Dutch physicist. Life and work Van der Pol began his studies of physics in Utrecht in 1911. J. A. Fleming offered van der Pol the use of the Pender Electrical Laboratory at ...
) because it contains the
Van der Pol oscillator In dynamical system, dynamics, the Van der Pol oscillator is a Conservative force, non-conservative oscillator with nonlinearity, non-linear Damping ratio, damping. It evolves in time according to the second-order differential equation: :-\mu(1-x ...
as a special case for a=b=0 . The equivalent circuit was suggested by Jin-ichi Nagumo, Suguru Arimoto, and Shuji Yoshizawa.


See also

*
Autowave Autowaves are self-supporting non-linear waves in active media (i.e. those that provide distributed energy sources). The term is generally used in processes where the waves carry relatively low energy, which is necessary for synchronization or ...
*
Biological neuron model Biological neuron models, also known as a spiking neuron models, are mathematical descriptions of the properties of certain cells in the nervous system that generate sharp electrical potentials across their cell membrane, roughly one millisecon ...
*
Computational neuroscience Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is a branch of neuroscience which employs mathematical models, computer simulations, theoretical analysis and abstractions of the brain to u ...
*
Hodgkin–Huxley model The Hodgkin–Huxley model, or conductance-based model, is a mathematical model that describes how action potentials in neurons are initiated and propagated. It is a set of nonlinear differential equations that approximates the electrical charact ...
*
Morris–Lecar model The Morris–Lecar model is a biological neuron model developed by Catherine Morris and Harold Lecar to reproduce the variety of oscillatory behavior in relation to Ca++ and K+ conductance in the muscle fiber of the giant barnacle . Morris–Le ...
* Reaction–diffusion *
Theta model The theta model, or Ermentrout–Kopell canonical model, is a biological neuron model originally developed to model neurons in the animal Aplysia, and later used in various fields of computational neuroscience. The model is particularly well suit ...


References


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

Java applet, includes phase space and parameters can be changed at any time.

Java applet to simulate 1D waves propagating in a ring. Parameters can also be changed at any time.

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). {{DEFAULTSORT:FitzHugh-Nagumo model Nonlinear systems Electrophysiology Computational neuroscience Biophysics