Patch of membrane with voltage-gated sodium
Consider a patch of membrane with voltage-gated sodium and potassium conductances described by:
( ) ( ) ( )
v v
3
Na Na g t m = g t h t
and
( ) ( )
v v
4
K K g t n t = g ,
where
v
2
Na g = 240 mS cm ,
v
2
K
g =120 mS cm
, and:
d
1
d
m m
m
m m
t
= − − ,
d
1
d
h h
h
h h
t
= − −
, and
d
1
d
n n
n
n n
t
= − − ,
where:
( )
(25.41 6.06 )
0.374 25.41
1 e m
m
m v
v
−
−
=
−
&
( )
( 21) 9.41
0.795 21
1 e m
m
m v
v
−
−
−
= ,
( )
( 27.74 9.06 )
0.110 27.74
1 e m
m
h v
v
+
−
=
+
−
&
(56 12.5 )
1 e
4.514
m
h v
−
=
+
,
and
( )
(35 10 )
0.0516 3
1 e
5
m
m
n v
v
−
−
−
=
&
( )
( 35) 10
0.1 35
e
29
1
m
m
n v
v
−
−
−
= ,
where the αs and βs have units of
1 ms−
and
m
v
is the relative transmembrane potential in mV.
1. In MATLAB, create a script gatingparticle_dynamics.m that calculates
m
, h
and
n
as
a function of
m
v
from the equations for the αs and βs given above and plots them on the same axis
over the range
−30 mV,140 mV
, and that calculates
m
,
h
and
n
as a function of
m
v
from the
equations for the αs and βs given above and plot them together on a different axis over the range
−30 mV,140 mV .
2. Consider the case where the membrane is at a holding potential
( ) 0mV for 0 m h v t v t = =
and then
at time
t = 0
the membrane steps to a clamp potential of
( ) 90mV for 0 m c
v t v t = =
. Derive
expressions for
( ) Nav
g t t for 0
and
( ) Kv
g t t for 0 .
3. In MATLAB, create a function gatingparticlesODE.m that has the input variables t and Y and
the output variable dYdt. The input variable Y is a 3×1 array where Y(1)= m , Y(2)= h, and
Y(3)= n. The output variable is a 3×1 array where dYdt(1)= dmdt, dYdt(2)= dhdt, and
dYdt(3)= dndt, and the values of the derivatives dmdt, dhdt, and dndt are obtained from the
equations given above.
4. Create a script gatingparticle_demo.m that uses the function gatingparticlesODE.m
with ode15s to simulate the time course of the conductances for the case described in Part 2 above
and compare the simulation results to the theoretical curves from Part 2. Find values of the ODE
solver parameters RelTol and MaxStep that provide good accuracy for the simulation results.