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.