FitzHugh-Nagumo equation

FitzHugh-Nagumo equation

1

The History

The

FitzHugh-Nagumo equation

is a simplification of the Hodgkin-Huxley model devised in 1952. The Hodgkin-Huxley has four

variables

and the FitzHugh-Nagumo equation is a

reduction

of that model. The reduction is from four variables to two variables where

phase plane

techniques may be used for the

analysis

of the model. The variables kept in the reduction of the model are the

excitable variable

and the

recovery variable

which are characterized as being the fast and slow variables respectively. The FitzHugh-Nagumo model was called, by FitzHugh, the

Bonhoeffer - van der Pol model (BVP)

. FitzHugh explains that the BVP was devised in the same way as the

van der Pol equation

“Its solution does not, to be sure, give an accurate fit to curves obtained from many physical oscillators. The equation was intended rather to represent the qualitative

properties

of a wide class of such oscillators, its algebraic form being chosen to be as simple as possible”.

[

FR

]

The name of Nagumo is added to FitzHugh by being able to represent the BVP model as an electrical device

[

NAY

]

. The BVP model “consists of three components, a capacitor representing the membrane capacitance, a nonlinear current-voltage device for the fast current, and a resistor, inductor, and battery in series for the recovery current.”

[

KS

]

2

The Application

The application of the FitzHugh-Nagumo equation is to model the same phenomenon as the Hodgkin-Huxley model. The phenomenon that is modelled is the control of the electrical potential across cell membrane. This control is done by the change of flow of the ionic channels of the cell membrane. This results in the change in potential which is used to send electrical signals between cells. This is readily observed in muscle and other excitable cells. For example the FitzHugh-Nagumo equation is used to model electrical waves of the heart.

[

KS

]

3

The Model

The

FitzHugh-Nagumo model

is defined by the following system of differential equations:

x

˙

=

ϵ

⁢

(

y

+

x

-

x

3

+

I

)

y

˙

=

-

1

ϵ

⁢

(

x

-

β

+

γ

⁢

y

)

where

1

-

2

⁢

γ

/

3

<

β

<

1

,

0

<

γ

<

1

, and

γ

<

ϵ

2

.

[

FR

]

There are many

equivalent

forms of the system, the more popular ones can be found in

[

RGG

]

with conversion for the different forms. A more general FitzHugh-Nagumo model is defined by the following system of differential equations:

ϵ

⁢

x

˙

=

f

⁢

(

x

,

y

)

+

I

y

˙

=

g

⁢

(

x

,

y

)

where

f

⁢

(

x

,

y

)

=

0

resembles a cubic shape and

g

⁢

(

x

,

y

)

=

0

only has one

intersection

with

f

⁢

(

x

,

y

)

=

0

.

[

KS

]

In the above two

formulas

the

I

is the external current applied to the system. The following system is known as the

spatially distributed FitzHugh-Nagumo model

where diffusion is added to the general FitzHugh-Nagumo model.

ϵ

⁢

∂

⁡

v

∂

⁡

t

=

ϵ

2

⁢

∂

2

⁡

v

∂

⁡

x

2

+

f

⁢

(

v

,

w

)

+

I

∂

⁡

w

∂

⁡

t

=

g

⁢

(

v

,

w

)

The above model has different travelling waves depending on the choice of

parameter

of the system.

References

FR

FitzHugh, Richard: Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophysical Journal, Volume 1, 1961.

HK

Hale, Jack H. & Ko cak, Hüseyin: Dynamics and

Bifurcations

. Springer-Verlag, New York, 1991.

IF

Izhikevich, Eugene M., FitzHugh, Richard:

http://www.scholarpedia.org/article/FitzHugh-Nagumo_Model

FitzHugh-Nagumo Model.

http://www.scholarpedia.org/

Scholarpedia, 2006.

KS

Keener, James & Sneyd, James: Mathematical Physiology. Springer-Verlag, New York, 1998.

NAY

Nagumo, J., S. Arimoto, and S. Yoshizawa: An active pulse transmission line simulating nerve axon, Proc IRE. 50: 2061-2070, 1964.

RGG

Rocs oreanu, C., Georgescu, A. & Giurgit eanu, N.: The FitzHugh-Nagumo Model: Bifurcation and Dynamics. Kluwer Academic Publishers, Boston, 2000.

Title

FitzHugh-Nagumo equation

Canonical

name

FitzHughNagumoEquation

Date of creation

2013-03-22 14:29:45

Last modified on

2013-03-22 14:29:45

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

5

Author

Daume (40)

Entry type

Definition

Classification

msc 34-00

Synonym

Bonhoeffer - van der Pol model

Synonym

BVP

Synonym

FitzHugh-Nagumo model

Defines

spatially distributed FitzHugh-Nagumo model

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