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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 I L gm (V VL )x dV I c cm x dt Im Ic I L gm normalized leak conductance per unit length of neurite V ( x x) V ( x) I ( x) ri x cm normalized membrane capacitance per unit length of neurite I ( x x) ri normalized internal resistance per unit length of neurite V ( x) V ( x x) ri x I C I L I ( x) I ( x x) 0 Solving Kirchov’s law in a neurite I L gm (V VL )x dV I c cm x dt ^ gL Im Ic I L cm dV V ( x x) V ( x) V ( x) V ( x x) x g m (V VL )x 0 dt ri x ri x V ( x x) V ( x) I ( x) ri x I ( x x) V ( x) V ( x x) ri x I C I L I ( x) I ( x x) 0 Final derivation of cable equation ^ gL cm dV V ( x x) V ( x) V ( x) V ( x x) x g m (V VL )x 0 dt ri x ri x divide by x and approach limit x -> 0 2 dV V ( x) 2 t (V VL ) 0 2 dt x dV 1 2V ( x) cm g m (V VL ) 0 2 dt ri x divide by gm dV 1 2V ( x) t (V VL ) 0 dt ri g m x 2 1 2 ri g m 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, Wcm) * 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 Mohmcm) – 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 2Vm Vm Vm 0 2 X T where X x/ , and T t /t m rm ri rRm 2 Ri 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 V 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