Cable TheoryCable TheoryCSCI 2323-1CSCI 2323-1

Last timeLast time

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

From old --> newFrom old --> new

Our last model assumed that the axon Our last model assumed that the axon is either a small area (nmis either a small area (nm22) or it has a ) or it has a large are, but of uniform potential large are, but of uniform potential difference (difference (∆∆V).V).This is not how axons work. The This is not how axons work. The action potential propagates down the action potential propagates down the length of the axon via saltatory length of the axon via saltatory conduction.conduction.

Linear Cable EquationLinear Cable Equation

The neuron as a circuitThe neuron as a circuit

Nelson, Philip. (2004). Biological Physics. New York: Freeman & Company.

What is cable theory?What is cable theory?

Mathematical Model used to Mathematical Model used to calculate the flow of electric current calculate the flow of electric current along passive neuronal fibers.along passive neuronal fibers.

Regards axons as cables with Regards axons as cables with capacitance and resistance. What’s capacitance and resistance. What’s different is that now the individual different is that now the individual segments of membrane can be segments of membrane can be viewed as parallel circuits, not the viewed as parallel circuits, not the flow of ions.flow of ions.

Useful variablesUseful variables

rrmm=Rm/2πa=Rm/2πa Membrane ResistanceMembrane Resistance

ccmm=Cm2πa=Cm2πa Capacitance due to Capacitance due to electrostaticselectrostatics

RRll=R=Rll/ πa/ πa22 Longitudinal Longitudinal ResistanceResistance

All these variables have been All these variables have been already calculated, so they are already calculated, so they are constants in our program.constants in our program.

Getting to the equationGetting to the equation

Ohm’s Law: Ohm’s Law: ∆∆V=IV=IllRRll∆x∆x

Current across the membrane:Current across the membrane: ∆∆iill=-i=-imm∆x∆x

Displacement current:Displacement current: IIcc=c=cmm(∂V/∂t)(∂V/∂t)

iimm=i=irr+i+icc

ConstantsConstants

Space ConstantSpace Constant How far a current will spread along the inside How far a current will spread along the inside

of the axon, thereby influencing the voltage of the axon, thereby influencing the voltage along that distance.along that distance.

Time ConstantTime Constant How fast the membrane potential Vm of the How fast the membrane potential Vm of the

axon is changing in response to changes in the axon is changing in response to changes in the current injected into the cytosol.current injected into the cytosol.

PotentialPotential

VVoo is the potential that we get at the is the potential that we get at the injection site. Vinjection site. Vlambdalambda is the potential is the potential that is due to lambda (the space that is due to lambda (the space constant). Vconstant). Vlambdalambda is always 36.8% of is always 36.8% of VVoo..

PDEPDE

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

Linear Cable Equation Linear Cable Equation (Again!)(Again!)

Modeling the Action Modeling the Action PotentialPotential

http://openwetware.org/wiki/BIO254:AP

Nonlinear Cable Nonlinear Cable EquationEquation

The linear cable equation fails to The linear cable equation fails to relate how the action potential gains relate how the action potential gains access to the free energy generated access to the free energy generated by sodium pumps.by sodium pumps.

A nonlinear version exists to solve A nonlinear version exists to solve this problem, but we won't get into it this problem, but we won't get into it because it frightens Rene.because it frightens Rene.