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Published bySebastian Gorges Modified over 2 years ago

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Cable Theory CSCI

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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)? 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 --> new Our last model assumed that the axon is either a small area (nm 2 ) or it has a large are, but of uniform potential difference ( V). Our last model assumed that the axon is either a small area (nm 2 ) 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. 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. 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. 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 r m =Rm/2πaMembrane Resistance r m =Rm/2πaMembrane Resistance c m =Cm2πaCapacitance due to electrostatics c m =Cm2πaCapacitance due to electrostatics R l =R l / πa 2 Longitudinal Resistance R l =R l / πa 2 Longitudinal Resistance All these variables have been already calculated, so they are constants in our program. All these variables have been already calculated, so they are constants in our program.

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Getting to the equation Ohms Law: Ohms Law: V=I l R l x V=I l R l x Current across the membrane: Current across the membrane: i l =-i m x i l =-i m x Displacement current: Displacement current: I c =c m (V/t) I c =c m (V/t) i m =i r +i c i m =i r +i c

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Constants Space Constant Space Constant How far a current will spread along the inside of the axon, thereby influencing the voltage along that distance. How far a current will spread along the inside of the axon, thereby influencing the voltage along that distance. Time Constant Time Constant How fast the membrane potential Vm of the axon is changing in response to changes in the current injected into the cytosol. 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 V o is the potential that we get at the injection site. V lambda is the potential that is due to lambda (the space constant). V lambda is always 36.8% of V o. V o is the potential that we get at the injection site. V lambda is the potential that is due to lambda (the space constant). V lambda is always 36.8% of V o.

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PDE What we are going to need to work with is the PDE solver. Let's go to MATLAB help. 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!) Linear Cable Equation (Again!)

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Modeling the Action Potential

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

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