BME 502, Spring 2001

Assignment 2

Name:______

Student ID:______


Problem 1.

Consider the following equivalent circuit:

A. In the steady-state, what value of valtage (in mV) is expected for Vtest-Vreference? Explain your reasoning.

Voltage across the capacitor must equal voltage across the resistor in steady state. So (Vreference – Vtest) =60mV

B. In the steady-state, what is the direction of current flow (if any) in the circuit? Explain your reasoning.

In steady-state, so there’s no current.


Problem 2.

You are investigating an axon, shown above, not drawn to scale. In this axon, you have inserted three intracellular electrodes, A, B and C. Electrodes A and C are inserted into cell at the same place, which is 0.1 cm apart from electrode B. You have measured the resting potential, which is –70mV. You inject a current into the cell through electrode A, then immediately you shut off your current injection. This time, you do not observe an action potential. 0.1 seconds later, you measure the voltage through electrode B, and you get –65mV.

You wait for time long enough and you measure the resting potential, which is –70mV again. You inject a current into the cell through electrode A. The current is exactly the same with the current you inject last time. You shut off your injection immediately. At the same time, you measure the voltage through electrode C. What is voltage you measure through electrode C this time? Include your reasoning and show your calculation.

Following values maybe useful.

Axon diameter / 20 μ
Rm / 10,000 Ωcm2
Ri / 100 Ωcm
Cm / 10 μF/cm2


First we calculate the τ and λ :

The second time, what you measured through electrode C is exactly the potential at point A when you inject that current through electrode A the first time.

So we can first calculate the potential at point B at the time when you inject the current according to the time-decay property of potential:

Next we calculate the potential at point A at the time when you inject the current according to the spatial-decay property of potential:

So at last we know the voltage change of point A at the moment of injection is 21.2mV, at that time the voltage is 21.2+(-70)= -48.9mV.

Problem 3:

You are studying the branching neural process above. The larger branch divides into two smaller branches. The diameters of the branches are shown. In terns of linear distance along the dendrite, the distance between B and C is the same as the distance between B and D, the distance between C and D, and the distance between A and B.

A. You inject a current into electrode B, and measure from electrodes C and D. What will you observe? How will what you measure at C compare with what you measure at D? Explain your reasoning, showing your calculations, if any.

The two daughter processes are identical; therefore, the measured voltage at each electrode will be identical.

B. You inject a current into electrode B, and measure from electrodes A and D. What will you observe? How will what you measure at A compare with what you measure at D? Explain your reasoning, showing your calculations, if any.

The two daughter processes are, together, electronically equivalent to the parent process. This means that the same amount of current will flow toward electrode A as toward the combination of C and D. Since the daughter processes are identical, 50% of the current will flow down one process, and 50% down the other, according to Kirchoff’s current law.

Each daughter process is significantly smaller in diameter than the parent process. Consequently, the length constant will be smaller, and the voltage will decay more rapidly as a function of distance.

We can conclude that the voltage at D will be less than the voltage measured at A.

C. You inject a current into electrode C, and measure from electrodes B and D. What will you observe? How will what you measure at B compare with what you measure at D? Explain your reasoning, showing your calculations, if any.

The input resistance to branch D is much higher than that of branch B. More current will flow into the larger branch.

As in part B, the length constant is smaller in branch D than in branch B. Voltage will decay faster in branch D.

The voltage measured at D will be lower than that measured at B.


D. Are these processes electronically equivalent? Explain your reasoning and show your work, including any equations used.

By definition, the two daughter branches are electronically equivalent to the larger branch if the following equation is satisfied:

Since

we may say that the branches are electronically equivalent. A few folks pointed out that

and argued that because of this small inequality, the branches are not exactly electronically equivalent. This was also an acceptable answer.