Nens220, Lecture 3 Cables and Propagation

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

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

And now in a system of membrane patches

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

Solving Kirchov’s law in a neurite

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

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

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

Electrotonic decay

Electrotonic decay in a neuron

Electrotonic decay in a neuron with alpha synapse

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

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

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

Bush and Sejnowski

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

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