Structure of Membranes - Membrane Biophysics, Biological and Medical Physics, Biomedical Engineering - page 28

I

V

R

C m

W

L

Tunnel diode

E

Fig. 2.6 An equivalent electric circuit diagram for the FiztHugh-Nagumo equations

The model equations have been transformed into dimensionless form, where rep-

resents the fast variable which is the electrostatic potential, u represents the slow
variable, the sodium gating variables, and β , δ and γ are constants which satisfy the
conditions 0

v

< β <

1 and δ 1, accounting for the slow kinetics of sodium channel.

Later, Nagumo constructed a circuit using tunnel diodes for the nonlinear channel
element, whose model equations are those of FitzHugh. Hence, the previous dimen-
sionless equations are now generally accepted as the so-called FiztHugh-Nagumo
model. The previous equations representing the FiztHugh-Nagumo model are often
expressed in more generalized forms as:

d d t =

f (v, u ) + I (2.29)

and

d u
d t = g(v,

u ) (2.30)

where, as described previously, the nullcline f 0 represents a cubic shape.

That means that for a finite range of values of u , there are three solutions

(v, u ) =

v = v( u )

of the equation f (v, u ) = 0. The nullcline g(v, u ) = 0 is assumed to have one

intersection with the curve f (v, u ) = 0.