1. Lecture 4: more ion channels and their functions Na + channels: persistent K + channels: A current, slowly inactivating current, Ca-dependent K currents I C, I AHP Ca 2+ channels: low-threshold I T and high- threshold I L, non-ohmic currents Refs: Dayan and Abbott, Ch 6; Gerstner and Kistler, Sect.2.3, T F Weiss. Cellular Biophysics (MIT Press) Ch 7.

2. General formalism: ohmic channels General equation

3. General formalism: ohmic channels General equation Currents have form

4. General formalism: ohmic channels General equation Currents have form m : activating variables h : inactivating variables

5. General formalism: ohmic channels General equation Currents have form m : activating variables h : inactivating variables HH Na channel:

6. Persistent (noninactivating) Na channel

7. No h !

8. Persistent (noninactivating) Na channel No h !

9. Persistent (noninactivating) Na channel No h ! Increases neuronal excitability

10. K channels: “A currents” (same form as HH Na channel)

11. K channels: “A currents” (same form as HH Na channel) fast slow-inactivating current

12. K channels: “A currents” (same form as HH Na channel) fast slow-inactivating current 2 kinds of each

13. Effect of A currents  h ~ 10-20 ms

14. Effect of A currents  h ~ 10-20 ms Opposite direction from Na current: hyperpolarizes membrane

15. Effect of A currents  h ~ 10-20 ms Opposite direction from Na current: hyperpolarizes membrane Slows spike initiation: have to wait for I A to inactivate:

16. Effect of A currents  h ~ 10-20 ms Opposite direction from Na current: hyperpolarizes membrane Slows spike initiation: have to wait for I A to inactivate:

17. Type I and Type II neurons Type I: arbitrarily slow rate possible (fx with A current) Type II: minimum firing rate >0 (fx Standard HH)

18. Ca 2+ -dependent K conductances (1): I C

19. (persistent)

20. Ca 2+ -dependent K conductances (1): I C (persistent)

21. Ca 2+ -dependent K conductances (1): I C (persistent)

22. Ca 2+ -dependent K conductances (1): I C (persistent) Activation is [Ca 2+ ] -dependent

23. Ca 2+ -dependent K conductances (1): I C [Ca 2+ ] = 0.1, 0,2, 0.5, 1.0, 2.0, 5.0  mol/l (persistent) Activation is [Ca 2+ ] -dependent

24. Ca 2+ -dependent K conductances (1): I C [Ca 2+ ] = 0.1, 0,2, 0.5, 1.0, 2.0, 5.0  mol/l Contributes to repolarization after spikes (persistent) Activation is [Ca 2+ ] -dependent

25. Ca 2+ -dependent K conductances (2): I AHP After-hyperpolarization current

26. Ca 2+ -dependent K conductances (2): I AHP After-hyperpolarization current

27. Ca 2+ -dependent K conductances (2): I AHP After-hyperpolarization current

28. Ca 2+ -dependent K conductances (2): I AHP Slow, no voltage dependence! After-hyperpolarization current

29. Ca 2+ -dependent K conductances (2): I AHP Ca 2+ enters (through other channels) during action potentials Slow, no voltage dependence! After-hyperpolarization current

30. Ca 2+ -dependent K conductances (2): I AHP Ca 2+ enters (through other channels) during action potentials Each spike  bigger  Slow, no voltage dependence! After-hyperpolarization current

31. Ca 2+ -dependent K conductances (2): I AHP Ca 2+ enters (through other channels) during action potentials Each spike  bigger , bigger m Slow, no voltage dependence! After-hyperpolarization current

32. Ca 2+ -dependent K conductances (2): I AHP Ca 2+ enters (through other channels) during action potentials Each spike  bigger , bigger m  slows down spiking Slow, no voltage dependence! After-hyperpolarization current

33. Ca 2+ -dependent K conductances (2): I AHP Ca 2+ enters (through other channels) during action potentials Each spike  bigger , bigger m  slows down spiking Slow, no voltage dependence! After-hyperpolarization current

34. Ca 2+ currents (1): low-threshold I T

35. (ohmic approximation here, but see later)

36. Ca 2+ currents (1): low-threshold I T (ohmic approximation here, but see later)

37. Ca 2+ currents (1): low-threshold I T (ohmic approximation here, but see later) Closed at rest because h nearly 0 (channel is “inactivated”) unlike HH Na channel, which is closed because m nearly 0 (channel is “not activated”)

38. Ca 2+ currents (1): low-threshold I T (ohmic approximation here, but see later) Closed at rest because h nearly 0 (channel is “inactivated”) unlike HH Na channel, which is closed because m nearly 0 (channel is “not activated”) Consequences: (1) “Post-inhibitory rebound”; fires “Ca spike” on release from hyperpolarization

39. Ca 2+ currents (1): low-threshold I T (ohmic approximation here, but see later) Closed at rest because h nearly 0 (channel is “inactivated”) unlike HH Na channel, which is closed because m nearly 0 (channel is “not activated”) Consequences: (1) “Post-inhibitory rebound”; fires “Ca spike” on release from hyperpolarization (2) Ca spikes can lead to Na spikes

40. Ca 2+ currents (2): high-threshold I L in ohmic approximation

41. Ca 2+ currents (2): high-threshold I L Persistent: in ohmic approximation

42. Ca 2+ currents (2): high-threshold I L Persistent: in ohmic approximation Lets in some Ca 2+ with each action potential

43. Ca 2+ currents (2): high-threshold I L Persistent: in ohmic approximation Lets in some Ca 2+ with each action potential This activates Ca-dependent K current

44. Ca 2+ currents (2): high-threshold I L Persistent: in ohmic approximation Lets in some Ca 2+ with each action potential This activates Ca-dependent K current Ca 2+ dynamics:

45. Non-ohmic Ca currents Current through membrane:

46. Non-ohmic Ca currents Current through membrane: Diffusive part:  = ion density

47. Non-ohmic Ca currents Current through membrane: Diffusive part: diffusion constant  = ion density

48. Non-ohmic Ca currents Current through membrane: Diffusive part: diffusion constant Drift in field:  = ion density v = velocity

49. Non-ohmic Ca currents Current through membrane: Diffusive part: diffusion constant Drift in field:  = ion density v = velocity  = mobility, F = force

50. Non-ohmic Ca currents Current through membrane: Diffusive part: diffusion constant Drift in field:  = ion density v = velocity  = mobility, F = force z = valence, e = proton charge, V = electrostatic potential

51. Non-ohmic Ca currents Current through membrane: Diffusive part: diffusion constant Drift in field:  = ion density v = velocity  = mobility, F = force z = valence, e = proton charge, V = electrostatic potential Total current:

52. Non-ohmic Ca currents Current through membrane: Diffusive part: diffusion constant Drift in field:  = ion density v = velocity  = mobility, F = force z = valence, e = proton charge, V = electrostatic potential Total current: Nernst-Planck equation

53. Can also be written

54. Nernst-Planck equation Can also be written using Einstein relation

55. Nernst-Planck equation Can also be written using Einstein relation or

56. Nernst-Planck equation Can also be written using Einstein relation or where

57. Nernst-Planck equation Can also be written using Einstein relation or where is the electrochemical potential

58. Steady state: J = const Nernst-Planck equation:

59. Steady state: J = const Nernst-Planck equation: Use integrating factor

60. Steady state: J = const Nernst-Planck equation: Use integrating factor 

61. Steady state: J = const Nernst-Planck equation: Use integrating factor  Integrate from x 0 to x 1 :

62. Steady state: J = const Nernst-Planck equation: Use integrating factor  Integrate from x 0 to x 1 :

63. Goldman-Hodgkin-Katz equation: assume constant field in membrane V = membrane potential, d = membrane thickness

64. Goldman-Hodgkin-Katz equation: assume constant field in membrane V = membrane potential, d = membrane thickness can integrate denominator x 1 = 0, x 2 = d

65. Goldman-Hodgkin-Katz equation: assume constant field in membrane V = membrane potential, d = membrane thickness can integrate denominator x 1 = 0, x 2 = d

66. Goldman-Hodgkin-Katz equation: assume constant field in membrane V = membrane potential, d = membrane thickness can integrate denominator x 1 = 0, x 2 = d Result:

67. Goldman-Hodgkin-Katz equation: assume constant field in membrane V = membrane potential, d = membrane thickness can integrate denominator x 1 = 0, x 2 = d Result: vanishes at reversal potential, by definition

68. Ohmic limit Using i.e.,

69. Ohmic limit Using i.e., 

70. Ohmic limit Using i.e.,  Now expand in V-V r :

71. Ohmic limit Using i.e.,  Now expand in V-V r :