1. Lecture 6: Dendrites and Axons Cable equation Morphoelectronic transform Multi-compartment models Action potential propagation Refs: Dayan & Abbott, Ch 6, Gerstner & Kistler, sects 2.5-6; C Koch, Biophysics of Computation, Chs 2,6

2. Longitudinal resistance and resistivity

3. Longitudinal resistance

4. Longitudinal resistance and resistivity Longitudinal resistance Longitudinal resistivity r L ~ 1-3 k  mm 2

5. Longitudinal resistance and resistivity Longitudinal resistance Longitudinal resistivity r L ~ 1-3 k  mm 2

6. Cable equation

7. current balance:

8. Cable equation current balance: on rhs:

9. Cable equation current balance: on rhs:  Cable equation:

10. Linear cable theory Ohmic current:

11. Linear cable theory Ohmic current: Measure V relative to rest:

12. Linear cable theory Ohmic current: Measure V relative to rest: Cable equation becomes

13. Linear cable theory Ohmic current: Measure V relative to rest: Cable equation becomes Now define electrotonic length and membrane time constant:

14. Linear cable theory Ohmic current: Measure V relative to rest: Cable equation becomes Now define electrotonic length and membrane time constant: 

15. Linear cable theory Ohmic current: Measure V relative to rest: Cable equation becomes Now define electrotonic length and membrane time constant:  Note: cable segment of length has longitudinal resistance = transverse resistance:

16. Linear cable theory Ohmic current: Measure V relative to rest: Cable equation becomes Now define electrotonic length and membrane time constant:  Note: cable segment of length has longitudinal resistance = transverse resistance:

17. dimensionless units:

18. Removes,  m from equation.

19. dimensionless units: Removes,  m from equation. Now remove the hats:

20. dimensionless units: Removes,  m from equation. Now remove the hats: ( t really means t/  m, x really means x/ )

21. Stationary solutions No time dependence:

22. Stationary solutions No time dependence: Static cable equation:

23. Stationary solutions No time dependence: Static cable equation: General solution where i e = 0 :

24. Stationary solutions No time dependence: Static cable equation: General solution where i e = 0 :

25. Stationary solutions No time dependence: Static cable equation: General solution where i e = 0 : Point injection:

26. Stationary solutions No time dependence: Static cable equation: General solution where i e = 0 : Point injection: Solution:

27. Stationary solutions No time dependence: Static cable equation: General solution where i e = 0 : Point injection: Solution:

28. Stationary solutions No time dependence: Static cable equation: General solution where i e = 0 : Point injection: Solution: Solution for general i e :

29. Boundary conditions at junctions

30. V continuous

31. Boundary conditions at junctions V continuous Sum of inward currentsmust be zero at junction

32. Boundary conditions at junctions V continuous Sum of inward currentsmust be zero at junction closed end:

33. Boundary conditions at junctions V continuous Sum of inward currentsmust be zero at junction closed end: open end: V = 0

34. Green’s function Response to delta-function current source (in space and time)

35. Green’s function Response to delta-function current source (in space and time)

36. Green’s function Response to delta-function current source (in space and time) Spatial Fourier transform:

37. Green’s function Response to delta-function current source (in space and time) Spatial Fourier transform:

38. Green’s function Response to delta-function current source (in space and time) Spatial Fourier transform Easy to solve:

39. Green’s function Response to delta-function current source (in space and time) Spatial Fourier transform Easy to solve: Invert the Fourier transform:

40. Green’s function Response to delta-function current source (in space and time) Spatial Fourier transform Easy to solve: Invert the Fourier transform:

41. Green’s function Response to delta-function current source (in space and time) Spatial Fourier transform Easy to solve: Invert the Fourier transform: Solution for general i e (x,t) :

42. Pulse injection at x=0,t=0 :

44. u vs t at various x : x vs t max :

45. Pulse injection at x=0,t=0 : u vs t at various x : x vs t max : At what t does u peak?

46. Pulse injection at x=0,t=0 : u vs t at various x : x vs t max : At what t does u peak?

47. Pulse injection at x=0,t=0 : u vs t at various x : x vs t max : At what t does u peak? 

48. Pulse injection at x=0,t=0 : u vs t at various x : x vs t max : At what t does u peak? 

49. Pulse injection at x=0,t=0 : u vs t at various x : x vs t max : At what t does u peak?  

50. Pulse injection at x=0,t=0 : u vs t at various x : x vs t max : At what t does u peak?   Restoring ,  m :

51. Compare with no-leak case:

52. Just diffusion, no decay

53. Compare with no-leak case: Just diffusion, no decay Now when does it peak for a given x ?

54. Compare with no-leak case: Just diffusion, no decay Now when does it peak for a given x ?

55. Compare with no-leak case: Just diffusion, no decay Now when does it peak for a given x ? 

56. Compare with no-leak case: Just diffusion, no decay Now when does it peak for a given x ? 

57. Compare with no-leak case: Just diffusion, no decay Now when does it peak for a given x ?  Restoring ,  m :

58. Finite cable Method of images:

59. Finite cable Method of images:

60. Finite cable Method of images: General solution:

61. Finite cable Method of images: General solution:

62. Morphoelectronic transform

63. Frequency-dependent morphoelectronic transforms

64. Multi-compartment models

65. Discrete cable equations

66. Resistance between compartments:

67. Discrete cable equations Resistance between compartments: Current between compartments:

68. Discrete cable equations Resistance between compartments: Current between compartments: Current per unit area:

69. Discrete cable equations Resistance between compartments: Current between compartments: Current per unit area: 

70. Action potential propagation

72. a “reaction-diffusion equation”

73. Action potential propagation a “reaction-diffusion equation” Moving solutions: look for solution of the form

74. Action potential propagation a “reaction-diffusion equation” Moving solutions: look for solution of the form  Ordinary DE

75. Action potential propagation a “reaction-diffusion equation” Moving solutions: look for solution of the form  Ordinary DE

76. Action potential propagation a “reaction-diffusion equation” Moving solutions: look for solution of the form  Ordinary DE HH solved iteratively for s (big success of their model)

77. Propagation speed a/s 2 must be independent of a

78. Propagation speed a/s 2 must be independent of a

79. Propagation speed a/s 2 must be independent of a This is probably why the squid axon is so thick.

80. Multi-compartment model

81. Multicompartment calculation

82. Myelinated axons Nodes of Ranvier: active Na channels

83. Myelinated axons Nodes of Ranvier: active Na channels

84. Myelinated axons Treat as multilayer capacitor each layer of thickness  a : Nodes of Ranvier: active Na channels

85. Myelinated axons Treat as multilayer capacitor each layer of thickness  a : Nodes of Ranvier: active Na channels

86. Myelinated axons Treat as multilayer capacitor each layer of thickness  a : Integrate up from a1 to a2 (inverse capacitances add) Nodes of Ranvier: active Na channels

87. Myelinated axons Treat as multilayer capacitor each layer of thickness  a : Integrate up from a1 to a2 (inverse capacitances add) Nodes of Ranvier: active Na channels

88. Myelinated axons Treat as multilayer capacitor each layer of thickness  a : Integrate up from a1 to a2 (inverse capacitances add) Negligible leakage between nodes: cable equation becomes Nodes of Ranvier: active Na channels

89. Myelinated axons Treat as multilayer capacitor each layer of thickness  a : Integrate up from a1 to a2 (inverse capacitances add) Negligible leakage between nodes: cable equation becomes Diffusion constant: Nodes of Ranvier: active Na channels

90. How much myelinization? optimal a 1 /a 2 Find the value of y = a 1 /a 2 that maximizes D

91. How much myelinization? optimal a 1 /a 2 Find the value of y = a 1 /a 2 that maximizes D

92. How much myelinization? optimal a 1 /a 2 Find the value of y = a 1 /a 2 that maximizes D 

93. How much myelinization? optimal a 1 /a 2 Find the value of y = a 1 /a 2 that maximizes D  Agrees with experiment

94. Speed of propagation Diffusion equation with diffusion constant

95. Speed of propagation Diffusion equation with diffusion constant 

96. Speed of propagation Diffusion equation with diffusion constant   Speed of propagation proportional to a 2

97. Speed of propagation Diffusion equation with diffusion constant   Speed of propagation proportional to a 2 (cf a 1/2 for unmyelinated axon)