Nens220, Lecture 4 Cables and Propagation

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Nens220, Lecture 4 Cables and Propagation

• John Huguenard

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Rate constants for gate n

• Derived from onset or offset of gK upon DV

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Model of gK
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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
tmCm/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 Kirchovs law in a neurite
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Final derivation of cable equation
divide by Dx and approach limit Dx -gt 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
  (pr2, cm2)
• Resistance per length (ri,pi) resistivity /
  cross sectional area Ri/pr2 (W/cm)
• For 1 mm neurite (axon) 100 W-cm/(p.00005 cm2)
  13GW/cm 1.3 GW/mm 1.3MW/mm
• For 5 mm neurite (dendrite) 100
  W-cm/(p.00025cm2) 500 MW/cm 50 MW/mm
  50kW/mm
• rm membrane resistance (units Wcm, divide by
  length to obtain total resistance)
• Rm2pr. Probably more intuitive to consider
  reciprocal resistance, or conductance
• In a neurite total conductance is Gm2prl, i.e.
  proportional to membrane area (circumference
  length)
• Normalized conductance per unit length (gm)
  Gm2pr (S/cm)
• For 1 mm neurite (axon) 0.05 mS/cm2(2p.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(2p.00025cm) 80 nS/cm 8pS/mm
• (rm 13 Mohm-cm)

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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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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 36769