1. MATHEMATICAL MODEL FOR ACTION POTENTIAL
Amirkabir University of Technology
MATHEMATICAL MODEL FOR ACTION POTENTIAL
Supervisor: Dr Gharibzadeh
Designed by Yashar Sarbaz

4. Action Potential

7. In the real world, neurons have a variety of additional channels that shape their action potentials

8. Attention

9. A.L. HODGKIN and A. F. HUXLEY
The Nobel Prize in Physiology or Medicine 1963 (with Eccles): "for their discoveries concerning the ionic mechanisms involved in excitation and inhibition in the peripheral and central portions of the nerve cell membrane"

10. Separation of Current into its Na and K Components

11. HH Experiments in two Case
Normal Seawater
Low Na Seawater:
Replace 90% sodium chloride by choline chloride while K and remaining chloride ions are unchanged

12. Three Assumption of HH 1. T: Time of peak inward current 2.
Same voltage clamp but different 3.

13. Calculating Current of Na and K

14. HH Equations

15. Conductance Changes with Time

16. The Hodgkin-Huxley Model
Central concept of model: Define three state variables that represent (or “control”) the opening and closing of ion channels
m controls Na channel opening
h controls Na channel closing
n controls K channel opening

17. The Potassium Channel The potassium has 4 similar sub units
Each subunit can be either “open” or “closed”
(Protein 3D Configurations)
The channel is open if and only if all 4 subunits are open

18. The Potassium Channel
The probability of a subunit being open:
The probability of the channel being open:
The conductance of a patch of membrane to K+ when all channels are open: (Constant obtained by experiments)
The conductance of a patch of membrane to K+ when the probability of a subunit being open is n:

19. The Kinetics of Potassium Channel Subunits

20. The potassium channel is closed in the resting membrane potential
Dependence of the Potassium Channel Parameters to the Membrane Potential
The potassium channel is closed in the
resting membrane potential

21. Mathematical Model for K
:Fraction of Open Channels
:Conductance When all
Channels are Open

22. Calculating n
Assuming n to Obey First Order Kinetics:

23. Solving n Equation

24. Curve Fitting for Rate Constants

25. Na+ Channels Have Two Gates
F8-15

26. The Sodium Channel
The potassium has 3 similar fast subunits and a single slow subunit
Each subunit can be either “open” or “closed”
(Protein 3D Configurations)
The channel is open if and only if all 4 subunits are open

27. The Sodium Channel The probability of a slow subunit being open:
The probability of a fast subunit being open:
The probability of a slow subunit being open:
The probability of the channel being open:
The conductance of a patch of membrane to Na+ when all channels are open: (Constant obtained by experiments)
The conductance of a patch of membrane to Na+ :

28. The Kinetics of Sodium Channel Subunits

29. Dependence of the Sodium Channel Parameters to the Membrane Potential
The slow subunit is open in the resting potential
The fast subunit is closed in the resting potential
The Sodium Channel is closed in the resting potential

30. Comparison of Voltage Dependence of channel kinetics

31. Mathematical Model for Na

32. Mathematical Model for Na

33. Solving m, h Equations

34. Solving m, h Equations

35. Border Condition For Na Channels in
In the Steady State Conductance of Na is Near the Zero and Since m is increasing Function, then:
At the Rest Conductance of Na is relatively Slow, So:

36. Main Relation for

37. Curve Fitting for Rate Constants

38. Curve Fitting for Rate Constants

39. Obtaining H for all V

40. H As Membrane Potential

41. Total Current of Membrane

42. Simulation of Action Potential

43. Calculation Changes in Membrane Potential

44. THE END