School

Harvard UniversityDepartment

Physical SciencesCourse Code

Physical Sciences 3Professor

Roxanne GuenetteStudy Guide

FinalThis

**preview**shows pages 1-3. to view the full**10 pages of the document.**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)=Q0e−t 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 1−e−t 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.

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

9

9

Activity 5 - Gauss's law with conductors

£O

inside �

surce perpendicular

q;nside E p

E p

€0

5�

=

0

�

�J. If

0

0

- � ?� /

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