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Nens220, Lecture 4 Cables and Propagation John Huguenard Rate constants for gate n • Derived from onset or offset of gK upon DV n n / n , n (1 n ) / n Model of gK g K g k n4 n nc no n dn n (1 n) n n dt If n no when t 0 : n n (n no ) exp(t / n ) where n 1/( n n ) Wit h dn / dt 0, obtain n to give best fit toI K i.e. at steady state, obtain n fromg K /g K max n n n n T hen n n / n , and n (1 n ) / n 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 m=Cm/Rm And now in a system of membrane patches Components of current flow in a neurite I L gm (V VL )Dx dV I c cm Dx dt Im Ic I L gm normalized leak conductance per unit length of neurite V ( x Dx) V ( x) I ( x) ri Dx cm normalized membrane capacitance per unit length of neurite I ( x Dx) ri normalized internal resistance per unit length of neurite V ( x) V ( x Dx) ri Dx I C I L I ( x) I ( x Dx) 0 Solving Kirchov’s law in a neurite I L gm (V VL )Dx dV I c cm Dx dt ^ gL Im Ic I L cm dV V ( x Dx) V ( x) V ( x) V ( x Dx) Dx g m (V VL )Dx 0 dt ri Dx ri Dx V ( x Dx) V ( x) I ( x) ri Dx I ( x Dx) V ( x) V ( x Dx) ri Dx I C I L I ( x) I ( x Dx) 0 Final derivation of cable equation ^ gL cm dV V ( x Dx) V ( x) V ( x) V ( x Dx) Dx g m (V VL )Dx 0 dt ri Dx ri Dx divide by Dx and approach limit Dx -> 0 2 dV V ( x) 2 (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) (V VL ) 0 dt ri g m x 2 1 2 ri g m membrane space constant, 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,pi) = 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 / 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 DV and Ri 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