Nens 220 Lecture 4 Cables and Propagation Rate

Nens 220, Lecture 4 Cables and Propagation

Rate constants for gate n • Derived from onset or offset of g. K upon DV

Model of g. K

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-cm 2) • Gm =specific membrane conductivity (~0. 05 m. S/cm 2) • – Cm = specific membrane capacitance (~ 1 m. F/cm 2) For cylinder, per unit length: – ri = axial resistance (units = W/cm) • Intracellular resistance (W)= resistivity (Ri, W-cm) * length (l, cm)/ cross sectional area (πr 2, cm 2) • Resistance per length (ri, pi) = resistivity / cross sectional area = Ri/πr 2 (W/cm) – For 1 mm neurite (axon) = 100 W-cm/(π*. 00005 cm 2) = ~13 GW/cm = 1. 3 GW/mm = 1. 3 MW/mm – For 5 mm neurite (dendrite) = 100 W-cm/(π*. 00025 cm 2) = ~ 500 MW/cm = 50 MW/mm = 50 k. W/mm – rm = membrane resistance (units: Wcm, divide by length to obtain total resistance) • Rm 2πr. Probably more intuitive to consider reciprocal resistance, or conductance: – In a neurite total conductance is G m 2πrl, i. e. proportional to membrane area (circumference * length) • Normalized conductance per unit length (gm) = Gm 2πr (S/cm) – For 1 mm neurite (axon) = 0. 05 m. S/cm 2(2π*. 00005 cm) = ~16 n. S/cm ~ 1. 6 p. S/mm » (equivalent normalized membrane resistance, rm obtained via reciprocation is ~60 Mohm-cm) – For 5 mm neurite (dendrite) = 0. 05 m. S/cm 2(2π*. 00025 cm) = ~80 n. S/cm ~ 8 p. S/mm » (rm ~ 13 Mohm-cm) – cm = membrane capacitance (units: F/cm) • Derived as for gm, normalized capacitance per unit length = Cm 2πr (F/cm) – For 1 mm neurite (axon) = 1 m. F/cm 2(2π*. 00005 cm) = ~300 p. F/cm ~ 30 f. F/mm – For 5 mm neurite (dendrite) = 1 m. F/cm 2(2π*. 00025 cm) = ~1. 6 n. F/cm ~160 f. F/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(64 e 6/13 e 9) = 0. 07 cm, 700 mm For 5 mm neurite (dendrite) =sqrt(13 e 6/79 e 9) = 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) (Rm. Ri)-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 • d 03/2 = d 13/2 + d 23/2 • 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