Biomedical Engineering Reference
In-Depth Information
40 CHAPTER 3. ACTIVEMEMBRANES
When
V
m
reaches some predefined
V
t
m
, however, all solving of differential equations is suspended. The
simulation then abruptly jumps
V
m
to some value,
V
peak
, to simulate the upstroke of the action potential.
m
In some implementations,
V
m
is clamped to
V
peak
m
for some short time and then
reset
back to the resting
value as in Fig. 3.14. In other implementations, the reset is not exactly back to rest but below
V
t
m
. This
simple yet elegant model is called the
integrate and fire
model. It is still used in neural modeling because
it requires no updating of gating variables.
20
10
0
−10
−20
−30
−40
−50
−60
−1
0
1
2
3
Time(msec)
Figure 3.14:
Integrate and fire action potential.
3.6 NUMERICAL METHODS: TEMPLATE FOR AN ACTIVE
MEMBRANE
In Sec. 2.5, a way of numerically solving a differential equation was outlined. In the active membrane
models, we need to keep track of several differential equations as well as compute rate constants, steady-
state values and currents. Below is a template for how to write a program to solve the active equations.
Define constants (e.g.,
G
L
,g
Na
,g
K
,C
m
,dt
)
Compute initial
α
s and
β
s
Compute initial conditions for state variables
(e.g.,
V
rest
,m,h,n
)
m
for (time=0 to time=end in increments of
dt
)
Compute Currents (e.g.,
I
L
,I
Na
,I
K
)
Compute
α
s and
β
s
Update Differential Equations (
V
m
,m,n,h
)
Save values of interest into an array (e.g.,
V
m
)
