Nens220, Lecture 3 Cables and Propagation

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*.00005 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