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Nens220, Lecture 3 Cables and Propagation
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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
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Current flow in membrane patch RC circuit
tm=Cm*Rm
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And now in a system of membrane patches
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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
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Solving Kirchov’s law in a neurite
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Final derivation of cable equation
divide by Dx and approach limit Dx -> 0 divide by gm membrane space constant, t is membrane time constant
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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
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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
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Electrotonic decay
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Electrotonic decay in a neuron
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Electrotonic decay in a neuron with alpha synapse
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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
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Reduced models of cells with complex morphologies
Rall analysis Bush and Sejnowski
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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
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Bush and Sejnowski
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Using Neuron Go to neuron.duke.edu and download a copy
Work through some of the tutorials
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Preview: dendritic spike generation
Stuart and Sakmann, 1994, Nature 367:69
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