The system (15) contains a singular small parameter, the
ratio of the characteristic times of the intracellular conductivity
, 
, and of the membrane excitability, 
.
Biophysical data [Weidmann, 1952] give the intracellular resistivity of
the order of 250 Ohm 
cm, which for the cell size of 10-80 
m
gives of the order of several S, so 
, which is much less than 
.
Numerical calculations of the simplified two-compartment model
(15) would require time steps not greater than , while if we can get rid of the small parameter, times steps
of around would be adequate.
We exclude the small parameter by ``quasi-stationary'' arguments. Rewrite
the first two equations of (15) as
where
and
Now, in (17) the term 
dominates over
, and omitting the nonlinear term ,
(17) becomes a linear equation. If the characteristic time
of 
change is bigger than , then this equation
describes the fast approach of 
to its quasi-stationary value,
and we can substitute this value into (16), which yields
Equations (18) and the last pair of (15) form a
closed system, which depends only on the ratio 
. If we
assume that the duration of pulses is shorter than the
characteristic time 
of the slow variables then 
. This final simplification gives:
This model is almost as simple as the original (1)
ordinary differential equation (e.g., it has three equations more than
the 17 variable Noble et al. [1990] ordinary differential system we use
for ventricular excitation), but describes the effect of external
current. This has been obtained from (15) assuming the
characteristic time of the external curent pulses, 
, is
In practice , and 
are all of the order of
1 ms.
The general model (14) can be simplified by quasi-stationary arguments to
with separate ordinary differential equations for v at each point of the
membrane. If the external field 
is fixed in direction and varies
only in magnitude, then the surface integral in (20)
can be reduced, in a Lebesgue style, to an ordinary integral
where the kernel K(s) is determined by the cell geometry, the conductivities, and the direction of the external field, and because of electroneutrality of the cell
The simple model (19) corresponds to evaluation of the integral in
(21) at two points 
.
Below we explore numerically some properties of the model (19) and its generalisation to spatially extended media. To validate the two-point approximation of the integral (21), we compare the two-point results with those of a five-point evaluation of (21), in the form
