1. Lecture 2 Membrane potentials Ion channels Hodgkin-Huxley model References: Dayan and Abbott, 5.1-5.6 Gerstner and Kistler, 2.1-2.2

2. Cell membranes

3. Lipid bilayer, 3-4 nm thick  capacitance c = C/A ~ 10 nF/mm 2

4. Cell membranes Lipid bilayer, 3-4 nm thick  capacitance c = C/A ~ 10 nF/mm 2 Ion channels  conductance

5. Cell membranes Lipid bilayer, 3-4 nm thick  capacitance c = C/A ~ 10 nF/mm 2 Ion channels  conductance Typical A =.01 -.1 mm 2  C ~.1 – 1 nF

6. Cell membranes Lipid bilayer, 3-4 nm thick  capacitance c = C/A ~ 10 nF/mm 2 Ion channels  conductance Typical A =.01 -.1 mm 2  C ~.1 – 1 nF Q=CV, Q= 10 9 ions  |V| ~ 65 mV

7. Membrane potential Fixed potential  concentration gradient

8. Membrane potential Fixed potential  concentration gradient Concentration difference  Potential difference Concentration difference maintained by ion pumps, which are transmembrane proteins

9. Nernst potential Concentration ratio for a specific ion (inside/outside):  = 1/k B T  ( q = proton charge, z = ionic charge in units of q )

10. Nernst potential Concentration ratio for a specific ion (inside/outside):  = 1/k B T  ( q = proton charge, z = ionic charge in units of q ) No flow at this potential difference Called Nernst potential or reversal potential for that ion

11. Reversal potentials Note: V T = k B T/q = (for chemists) RT/F ~ 25 mv Some reversal potentials: K: -70 - -90 mV Na: +50 mV Cl: -60 - -65 mV Ca: 150 mV Rest potential: ~ -65 mV ~2.5 V T

12. Effective circuit model for cell membrane

13. ( C, g i, I ext all per unit area) (“point model”: ignores spatial structure)

14. Effective circuit model for cell membrane ( C, g i, I ext all per unit area) (“point model”: ignores spatial structure) g i can depend on V, Ca concentration, synaptic transmitter binding, …

15. Ohmic model One g i = g = const or

16. Ohmic model One g i = g = const or membrane time const

17. Ohmic model One g i = g = const or Start at rest: V= V 0 at t=0 membrane time const

18. Ohmic model One g i = g = const or Final state: Start at rest: V= V 0 at t=0 membrane time const

19. Ohmic model One g i = g = const or Final state: Start at rest: V= V 0 at t=0 Solution: membrane time const

20. channels are stochastic

21. Many channels: effective g = g open * (prob to be open) * N

22. Voltage-dependent channels

23. K channel Open probability: 4 independent, equivalent, conformational changes

24. K channel Open probability: 4 independent, equivalent, conformational changes Kinetic equation:

25. K channel Open probability: 4 independent, equivalent, conformational changes Kinetic equation: Rearrange:

26. K channel Open probability: 4 independent, equivalent, conformational changes Kinetic equation: Rearrange: relaxation time: asymptotic value

27. Thermal rates: u 1, u 2 : barriers

28. Thermal rates: u 1, u 2 : barriers Assume linear in V :

29. Thermal rates: u 1, u 2 : barriers Assume linear in V : 

30. Thermal rates: u 1, u 2 : barriers Assume linear in V :  Simple model: a n =b n, c 1 =c 2

31. Thermal rates: u 1, u 2 : barriers Assume linear in V :  Simple model: a n =b n, c 1 =c 2 Similarly,

32. Hodgkin-Huxley K channel

33. (solid: exponential model for both  and  Dashed: HH fit)

34. Transient conductance: HH Na channel 4 independent conformational changes, 3 alike, 1 different (see picture)

35. Transient conductance: HH Na channel 4 independent conformational changes, 3 alike, 1 different (see picture) HH fits:

36. Transient conductance: HH Na channel 4 independent conformational changes, 3 alike, 1 different (see picture) HH fits:

37. Transient conductance: HH Na channel 4 independent conformational changes, 3 alike, 1 different (see picture) HH fits: m is fast (~.5 ms) h,n are slow (~5 ms)

38. Hodgkin-Huxley model

39. Parameters: g L = 0.003 mS/mm 2 g K = 0.36 mS/mm 2 g Na = 1.2 ms/mm 2 V L = -54.387 mV V K = -77 V Na = 50 mV

40. Spike generation Current flows in, raises V  m increases (h slower to react)  g Na increases  more Na current flows in  …  V rises rapidly toward V Na Then h starts to decrease  g Na shrinks  V falls, aided by n opening for K current Overshoot, recovery

41. Spike generation Current flows in, raises V  m increases (h slower to react)  g Na increases  more Na current flows in  …  V rises rapidly toward V Na Then h starts to decrease  g Na shrinks  V falls, aided by n opening for K current Overshoot, recovery Threshold effect

42. Spike generation (2)

43. Regular firing, rate vs I ext

44. Step increase in current

45. Noisy input current, refractoriness