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Answer and explanation the assignment 2 pdf in Modelling Organs, Tissues, Cells and Devices
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Physiological Modelling BME 623
Assignment 2
1) In the Hodgkin-Huxley formulation of neural activation, three gating variables n, m, h
are employed, satisfying the ODE:
dx
x (V )(1 x) x (V ) x
dt
where x n, m, h and x(V) and x(V) are known functions of membrane voltage.
Assuming a voltage clamp experiment is performed, whereby the membrane voltage is
stepped suddenly from a value Vhold to a new value Vclamp and held at this value via a
feedback mechanism. x and x are now constant.
a) Solve this equation analytically for x, with initial value x (0) = x0. What are the
homogeneous, particular and general solutions?
b) What is the steady-state value of x? Hence, what is a reasonable estimate for x0?
2) Consider the system below of two coupled masses of mass M, connected to each other
and to fixed supports via three springs with spring constants k.
k
k
M
x1
M
k
x2
a) Write down a pair of ODEs describing the motion of this system.
R
b) Solve this system analytically for the displacements x1 and x2, assuming the masses
are initially at rest and displaced by amounts u1 and u2.
[Hint: Use the variable substitutions y1 = x1+x2, y2 = x1-x2].
3) The modified Hodgkin-Huxley equations for a space-clamped neural impulse (action
potential) are given below:
iNa
iK
iL
dV
1
3
4
g Na m h V VNa g K n V VK g L V VL istim
dt
Cm
dn
n (1 n ) n n
dt
dm
m (1 m ) m m
dt
dh
h (1 h ) h h
dt
with
10(V 50)
s 1
(V 50)
1 exp
10
(V 60) 1
n 125 exp
s
80
n
100(V 35)
s 1
(V 35)
1 exp
10
(V 60) 1
m 4000 exp
s
18
m
(V 60) 1
s
20
1000
h
s 1
(V 30)
1 exp
10
h 70 exp
Model parameters and initial values are:
Parameter
g Na
Value
120000
Units
S/cm2
gK
36000
S/cm2
gL
300
V Na
55
S/cm2
mV
VK
-72
mV
VL
-49.387
mV
Cm
1
F/cm2
Variable
V
n
m
h
Initial Value
-60 mV
0.3177
0.0529
0.5961
Assume that a brief stimulus current is applied as a square-pulse waveform according to
20,000 nA/cm 2
i stim (t )
0
0.001 t 0.002 s
all other times
Using MATLAB, solve for and plot the action potential as well as the ion currents iNa, iK,
and iL for 0 < t
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