Question 4

Start by replacing each symbol with its value and units enclosed in parentheses. If the symbol has an exponent, the exponent should remain outside the parentheses!

For this problem, you will be predicting the membrane potential for several diameters. In this example, let's assume $d = 2\ \mu\text{m}$.

$

 \begin{align}
   V_{0} &= I \frac{1}{\pi} \sqrt{\frac{R_{i}}{g_{m} d^3}} \\
         &= (10\ \text{nA})\frac{1}{\pi}\sqrt{\frac{(36\ \Omega \text{cm})}{(36\ \text{mS}/\text{cm}^2)(2\ \mu\text{m})^3}} \\
 \end{align}

$

It is often useful at this point to factor out the unit prefixes. That is, convert centimeters to meters, nanoamperes to amperes, millisiemens to siemens, etc.

To do this, replace each SI prefix with $10^x$, where $x$ is the appropriate exponent. If you are unsure, you can refer to this Wikipedia article to look up the right exponent for each prefix.

For example, the prefix of the first unit encountered in the equation can be eliminated by replacing "$\text{nA}$" with "$10^{-9} \text{A}$":

$

 = (10 \cdot 10^{-9} \text{A})\frac{1}{\pi}\sqrt{\frac{(36\ \Omega \text{cm})}{(36\ \text{mS}/\text{cm}^2)(2\ \mu\text{m})^3}}

$

This is most difficult for the membrane conductance, $36\ \text{mS}/\text{cm}^2$. Try to get it into the form $36 \cdot (10^{x}\text{S})/(10^{y}\text{m})^2$, where $x$ and $y$ are the appropriate exponents. Notice that the power of 2 applies to both parts of the denominator.

Once you have eliminated all the prefixes so that the remaining units are amperes, ohms, meters, and siemens, you can use algebra to simplify both the numbers and the units. Remember the following:

• The reciprocal of siemens is ohms
• The product of ohms and amperes is volts
• Exponents add when multiplying powers of 10, e.g., $10^{4} \cdot 10^{-9} = 10^{-5}$
• Exponents subtract when dividing powers of 10, e.g., $10^{4} / 10^{-9} = 10^{13}$
• Exponents multiply when raising powers of 10 to other powers, e.g., $(10^{-5})^3 = 10^{-15}$
• The square root is equivalent to the power 1/2, i.e., $\sqrt{3} = 3^{1/2}$

If you are successful, you will get an answer in volts. As a final step, convert this to millivolts so that you can directly compare it to the output of the simulation.