1. Nens220, Lecture 3 Cables and Propagation

2. Cable theory
Developed by Kelvin to describe properties of current flow in transatlantic telegraph cables.
The capacitance of the “membrane” leads to temporal and spatial differences in transmembrane voltage.
From Johnston & Wu, 1995

3. Current flow in membrane patch RC circuit
tm=Cm*Rm

4. And now in a system of membrane patches

5. Components of current flow in a neurite
normalized leak conductance per unit length of neurite
normalized membrane capacitance per unit length of neurite
normalized internal resistance per unit length of neurite

6. Solving Kirchov’s law in a neurite

7. Final derivation of cable equation
divide by Dx and approach limit Dx -> 0
divide by gm
membrane space constant, t is membrane time constant

8. Cable properties, unit properties
For membrane, per unit area
Ri = specific intracellular resistivity (~100 W-cm)
Rm = specific membrane resistivity (~20000 W-cm2)
Gm =specific membrane conductivity (~0.05 mS/cm2)
Cm = specific membrane capacitance (~ 1 mF/cm2)
For cylinder, per unit length:
ri = axial resistance (units = W/cm)
Intracellular resistance (W)= resistivity (Ri, W-cm) * length (l, cm)/ cross sectional area (πr2, cm2)
Resistance per length (ri) = resistivity / cross sectional area = Ri/πr2 (W/cm)
For 1 mm neurite (axon) = 100 W-cm/(π* cm2) = ~13GW/cm = 1.3 GW/mm = 1.3MW/mm
For 5 mm neurite (dendrite) = 100 W-cm/(π*.00025cm2) = ~ 500 MW/cm = 50 MW/mm = 50kW/mm
rm = membrane resistance (units: Wcm, divide by length to obtain total resistance)
Rm2πr. Probably more intuitive to consider reciprocal resistance, or conductance:
In a neurite total conductance is Gm2πrl, i.e. proportional to membrane area (circumference * length)
Normalized conductance per unit length (gm) = Gm2πr (S/cm)
For 1 mm neurite (axon) = 0.05 mS/cm2(2π*.00005cm) = ~16 nS/cm ~ 1.6pS/mm
(equivalent normalized membrane resistance, rm obtained via reciprocation is ~60 Mohm-cm)
For 5 mm neurite (dendrite) = 0.05 mS/cm2(2π*.00025cm) = ~80 nS/cm ~ 8pS/mm
(rm ~ 13 Mohm-cm)
cm = membrane capacitance (units: F/cm)
Derived as for gm, normalized capacitance per unit length = Cm2πr (F/cm)
For 1 mm neurite (axon) = 1 mF/cm2(2π*.00005cm) = ~300 pF/cm ~ 30 fF/mm
For 5 mm neurite (dendrite) = 1 mF/cm2(2π*.00025cm) = ~1.6 nF/cm ~160 fF/mm

9. Cable equation Solved for different boundary conditions
Infinite cylinder
Semi infinite cylinder (one end)
Finite cylinder
l scales with square root of radius
For 1 mm neurite (axon) =sqrt(64e6/13e9) = 0.07 cm, 700 mm
For 5 mm neurite (dendrite) =sqrt(13e6/79e9) = 0.16 cm, 1600 mm

10. Electrotonic decay

11. Electrotonic decay in a neuron

12. Electrotonic decay in a neuron with alpha synapse

13. Compartmental models
Can be developed by combining individual cylindrical components
Each will have its own source of current and EL via the parallel conductance model
Current will flow between compartments (on both ends) based on DV and Ri

14. Reduced models of cells with complex morphologies
Rall analysis
Bush and Sejnowski

15. Collapsing branch structures
From cable theory
conductance of a cable =
(p/2) (RmRi)-1/2(d)3/2
When a branch is reached the conductances of the two daughter branches should be matched to that of the parent branch for optimal signal propagation
This occurs when the sum of the two daughter g’s are equal to the parent g, which occurs when
d03/2 = d13/2 + d23/2
This turns out to be true for many neuronal structures

16. Bush and Sejnowski

17. Using Neuron Go to neuron.duke.edu and download a copy
Work through some of the tutorials

18. Preview: dendritic spike generation
Stuart and Sakmann, 1994, Nature 367:69