Cable Theory

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• Cable Theory
• CSCI 2323-1

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Last time

• What did we do last time? Does anyone remember
  why our model last time did not work (other than
  getting infinity due to Rene's inabililty to
  check his units)?

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From old --gt new

• Our last model assumed that the axon is either a
  small area (nm2) or it has a large are, but of
  uniform potential difference (?V).
• This is not how axons work. The action potential
  propagates down the length of the axon via
  saltatory conduction.

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Linear Cable Equation
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The neuron as a circuit
Nelson, Philip. (2004). Biological Physics. New
York Freeman Company.
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What is cable theory?

• Mathematical Model used to calculate the flow of
  electric current along passive neuronal fibers.
• Regards axons as cables with capacitance and
  resistance. Whats different is that now the
  individual segments of membrane can be viewed as
  parallel circuits, not the flow of ions.

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Useful variables

• rmRm/2pa Membrane Resistance
• cmCm2pa Capacitance due to electrostatics
• RlRl/ pa2 Longitudinal Resistance
• All these variables have been already calculated,
  so they are constants in our program.

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Getting to the equation

• Ohms Law
• ?VIlRl?x
• Current across the membrane
• ?il-im?x
• Displacement current
• Iccm(?V/?t)?
• imiric

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Constants

• Space Constant
• How far a current will spread along the inside of
  the axon, thereby influencing the voltage along
  that distance.
• Time Constant
• How fast the membrane potential Vm of the axon is
  changing in response to changes in the current
  injected into the cytosol.

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Potential

• Vo is the potential that we get at the injection
  site. Vlambda is the potential that is due to
  lambda (the space constant). Vlambda is always
  36.8 of Vo.

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PDE

• What we are going to need to work with is the PDE
  solver. Let's go to MATLAB help.

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Linear Cable Equation (Again!)?
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Modeling the Action Potential
http//openwetware.org/wiki/BIO254AP
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Nonlinear Cable Equation

• The linear cable equation fails to relate how the
  action potential gains access to the free energy
  generated by sodium pumps.
• A nonlinear version exists to solve this problem,
  but we won't get into it because it frightens
  Rene.