ece602 bme i ordinary differential equations in biomedical engineering (cont’d)
DESCRIPTION
ECE602 BME I Ordinary Differential Equations in Biomedical Engineering (Cont’d). BME Model Examples. Hodgkin – Huxley Model. An empirical model of an action potential in a squid giant axon. BME Model Examples. Hodgkin – Huxley Model. Separation of charges across a membrane - PowerPoint PPT PresentationTRANSCRIPT
ECE602 BME I
Ordinary Differential Equations in Biomedical Engineering (Cont’d)
BME Model Examples
Hodgkin – Huxley Model
• An empirical model of an action potential in a squid giant axon
BME Model Examples
Hodgkin – Huxley Model
Separation of charges across a membrane
• Negative ions along the inside of the cell membrane
• Positive ions along the outside of the cell membrane
• Elsewhere the negative and positive ions are approximately evenly distributed
BME Model Examples
Hodgkin – Huxley Model
Channels allow ions to pass through the membrane
• Selective
• Either passive or active
BME Model Examples
Hodgkin – Huxley Model
Passive Channels
• Always open
• Ion specific
• Responsible for resting potential (Nernst Equation)
BME Model Examples
Hodgkin – Huxley Model
Active Channels
• Open or close
• Ion specific
• Responsible for action potential
BME Model Examples
Hodgkin – Huxley Model
Action Potential (V)
• V Reaches threshold->Na+ channel open->inward flow of Na+->further depolarization->increases Na+ conductance->more Na+ current->driving V -> concludes with the closure of the Na+ channel
• A similar, but slower change in K+ conductance drives V back to the resting potential.
BME Model Examples
Hodgkin – Huxley Model
Electrical circuit model of the cell membrane
0 ionm Idt
dVC
V: the internal minus the external potential
BME Model Examples
Hodgkin – Huxley Model
The principle ionic currents:
• The sodium current (INa)
• The potassium current (IK)
• The leakage current (IL)
)(
)(
)(
LLL
KKK
NaNaNa
VVgI
VVgI
VVgI
gx: the membrane conductance for X
VX: the membrane resting potential for X
BME Model Examples
Hodgkin – Huxley Model
The current flow through a population of channels:
)(),( VctVsI
• s: the proportion of open channels in a population
• c: the I-V curve of a single channel
BME Model Examples
Hodgkin – Huxley Model
(1-s)Closed
sOpen(V)
(V)
ssdt
ds )1(
)(Vs (Steady state value)
1)(Vs (The time constant)
BME Model Examples
Hodgkin – Huxley Model
• The sodium conductance
hmgg NaNa3
mmdt
dmmm )1(
hhdt
dhhh )1(
mm
mm
mmm
1
hh
hh
hhh
1
BME Model Examples
Hodgkin – Huxley Model
• The potassium conductance
4ngg KK
nndt
dnnn )1(
nn
nn
nnn
1
BME Model Examples
Hodgkin – Huxley Model
• The leakage conductance
LL gg
BME Model Examples
Hodgkin – Huxley Model
The complete model:
)()()( 34LLNaNakKm VVgVVhmgVVng
dt
dVC
nndt
dnnn )1(
mmdt
dmmm )1(
hhdt
dhhh )1(
BME Model Examples
Hodgkin – Huxley Model
1
1 07.0
4
1
251.0
125.0
1
1001.0
10
3020
18
10
25
80
10
10
Vh
V
h
V
mVm
V
nVn
e
e
e
e
V
e
e
V
BME Model Examples
Hodgkin – Huxley Model
2
222
/1
5961.0)0( 0529.0)0( 3177.0)0( 8)0(
6.10 115 12
/3.0 /120 /36
cmFC
hmnmVV
mVVmVVmVV
cmmSgcmmSgcmmSg
m
LNaK
LNaK
BME Model Examples
Hodgkin – Huxley Model
Reference:
J. Keener and J. Sneyd, “Mathematical Physiology”, Springer: 1998.