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Lecture 6: Dendrites and Axons

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. Longitudinal resistance and resistivity.

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Lecture 6: Dendrites and Axons

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  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 and resistivity Longitudinal resistance

  4. Longitudinal resistance and resistivity Longitudinal resistance Longitudinal resistivity rL ~ 1-3 kW mm2

  5. Longitudinal resistance and resistivity Longitudinal resistance Longitudinal resistivity rL ~ 1-3 kW mm2

  6. Cable equation

  7. Cable equation 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 l 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 l has longitudinal resistance = transverse resistance:

  17. dimensionless units:

  18. dimensionless units: Removes l, tm from equation.

  19. dimensionless units: Removes l, tm from equation. Now remove the hats:

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

  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 ie = 0:

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

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

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

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

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

  29. Boundary conditions at junctions

  30. Boundary conditions at junctions V continuous

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

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

  33. Boundary conditions at junctions V continuous Sum of inward currents must 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 ie(x,t) :

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

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

  44. Pulse injection at x=0,t=0: u vs t at various x: x vs tmax:

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

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

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

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

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

  50. Pulse injection at x=0,t=0: u vs t at various x: x vs tmax: At what t does u peak?   Restoring l, tm: 

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