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Final

Physical Sciences 3 Lecture 8: Lecture 4aExam


Department
Physical Sciences
Course Code
Physical Sciences 3
Professor
Roxanne Guenette
Study Guide
Final

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Physical Sciences 3: Lecture 4a
February 21, 2019
1
RC Circuits and the Cable Model
A circuit with a resistor and capacitor in series is an RC circuit. The current through the
capacitor is the time derivative of its charge:
i=dq dt
.
If the capacitor is charged, you can discharge it through the resistor:
Q(t)=Q0et RC
The time constant RC is the time required for the charge to drop to 37% of its initial
value. If the capacitor is uncharged, you can charge it with a battery:
Q(t)=Qmax 1et RC
( )
We can model the membrane as a network of resistors and capacitors. This network acts
like a bunch of RC circuits in which one circuit charges the next, and so on down the
membrane. As a result, an electrical signal can propagate passively down the membrane.
To describe signal propagation, we need to consider the membrane potential as a function
of position. This leads to the following simplified expression for the propagation of a
potential spike:
V
t
=ad
ρε
2
V
x
2
This equation looks just like the diffusion equation you studied in PS2. In this
expression,
ρ
/a represents the resistance of the cytosol to the flow of current in the
direction of the signal, while
ε
/d represents the capacitance across the membrane.
Learning objectives: After this lecture, you will be able to…
1. Describe how we can model the cell membrane as an electrical circuit containing
resistors, capacitors, and batteries. Explain the relationship between the biological
structures (membranes, ion channels, and ion pumps) and the analogous circuit elements.
2. Discuss the qualitative behavior of a simple circuit model of a cell membrane.
3. Calculate the voltage, current, and charge as a function of time for a circuit containing a
resistor, a capacitor, and (optionally) a battery. This is an RC circuit.
4. Describe the qualitative behavior of an RC circuit when it is being charged, or
discharged.
5. Construct a model of the cell membrane as a passive carrier of charge, including the fact
that the potential can vary along the length of the membrane (e.g. along the axon).
6. Derive the equation for the passive propagation of a potential along an axon, first
assuming no “leakage” of ions across the membrane, and then including “leakage.”
7. Describe quantitatively how the physical and electrical properties of the cell membrane
affect the speed of propagation of an electrical signal, and use this model to explain the
effect of myelination on the speed of nerve signals.

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