Homework 2 (120 pts)

This problem set is due Tuesday (9/20/2016) at 5pm. Please turn in your work by uploading to Canvas. For extra credit, create a Jupyter Notebook containing your answers and your python code for question 4, and push this to a github repository (the commit time will be taken as the submission time). If you have questions, please post them on the course forum, rather than emailing the course staff. This will allow other students with the same question to see the response and any ensuing discussion. The goal of this problem set is to review the fundamentals of electrophysiology as well as providing some “warmup” Python work.

  1. Membrane Potentials and the Nernst Equation (20 pts)

    As discussed in class, a neuron’s resting potential is determined by the relative intra- and extra-cellular ion concentrations and the corresponding conductivities of the respective ion channels.

    a. Use the Nernst equation to fill in the equilibrium potentials for potassium, sodium, and chloride in the table below. (3 pts)

    Ion Extracellular Concentration Intracellular Concentration Permeability Equilibrium Potential
    K+ 20 mM 400 mM 1  
    Na+ 440 mM 50 mM 0.04  
    Cl- 560 mM 52 mM 0.45  
    Mn++ 2 mM 0.01 mM 0.01  

    b. Taking into account the potassium, sodium, and chloride concentrations (ignoring manganese), calculate the resting potential of a neuron with the characteristics in the table above. (2 pts)

    c. How could you adjust the potassium level to raise the resting potential? Why does this work? (3 pts)

    d. Calculate the new extracellular potassium concentration that would be needed to bring the resting membrane potential to -65mV. To counteract the decrease in potassium concentration, the cell could insert or remove ion channels into the membrane thus changing the permeability ratio. How might the cell recover the original resting membrane potential and calculate the new permeability ratio? (6 pts)

    e. Cells have mechanisms for active transport of sodium, potassium and chloride ions. What would happen to the information in the table above (ignoring Mn++) if the chloride transporter protein was turned off and only passive forces affected chloride ions? What about the resting membrane potential? (6 pts)

  2. GHK equation with divalent ions (35 pts).

    The ion channels which are active at the cell’s resting potential (“resting channels”) are in real-life fairly selective and not particularly permeable to ions other than sodium and potassium, and in particular not to divalent ions. Let’s imagine that that is not the case, and that the resting membrane is also permeable to manganese. Further, let’s assume intra- and extracellular concentrations as in the table above.

    a. Use the Nernst equation to specify the equilibrium potential for Mn++. (2 pts)

    b. Derive the GHK equation for 3 monovalent ions and a divalent ion. (20 pts)

    c. Consider an unusual cell with the membrane permeabilities and internal and external ionic concentrations in the table. What is the resting membrane potential of this cell? (13 pts)

  3. Simulating a neuron (65 pts)

    For the following problems, you will make use of Python code that simulates a neuron. You can find the hh() function in a Jupyter notebook in the class github repository. You should use a time step, Δt = 0.01 ms (except for 3b).

    a. Consider 1 s of responses of the HH neuron. Plot the responses to a 0.2 uA/mm2 current step at time 0.2 s that lasts for 0.2 s. How many spikes are generated by this input? (4 pts)

    b. Consider the input in 3a with Δt = 0. 1 ms, Δt = 0.05 ms, and Δt = 0.001 ms. How are the results similar/different? (4 pts)

    c. Evaluate 0.2 s current injections at time 0.2 s. What is the “threshold voltage” for this neuron? (You can either look for a “knee” in the voltage curve or use test currents of increasing size to probe when an AP is triggered.) Tweak the parameters to decrease and increase the threshold voltage by ~5 mV. What did you change, and by how much? (8 pts)

    d. Using the same time/length of current injection (0.2 s at 0.2 s), consider depolarizations of 0 to 0.50 uA/mm2 (in 0.025 uA/mm2 steps). Count and plot the number of spikes generated for each level. This is an “input-output” curve. How could you change the slope of this curve? (8 pts)

    e. Now modify the HH neuron to become a CST neuron (TCN Ch. 6 and/or included paper for parameters). Plot the input-output curve of the CST neuron as in 3d. How is it different? (10 pts)

    f. Plot and compare the response of an HH neuron and a CST neuron to an injected current profile of 0 for 0.2 s, −0.050 uA/mm2 for 0.2 s, and then 0.15 uA/mm2 for 0.1 s (i.e., an input current that goes low then high). How are they similar? How are they different? (6 pts)

    g. (Random input.) Now you will consider input current profiles for the CST neuron (your choice!) that are random. The input should be 0.5 s of a smoothed Gaussian random variable, where the smoothing is done using a 100 point moving average filter. After smoothing, the input should have mean and standard deviation , where and . For each pair of parameters, construct 25 input current profiles. (For the zero variability input, it is only necessary to do one.) For each of these 3 x 5 x 25 + 5 = 380 input “experiments”, count the number of action potentials produced by the neuron. Make a plot showing the the average number of generated spikes as a function of the mean depolarization for each of the standard deviations (i.e., 4 lines)? How does the variance in the number of spikes per trial change with these different parameters?

    Implementation Hint: You will probably want to use the functions np.random.randn (for the Gaussian) and np.conv (to calculate the smoothed signal remembering that you’ll need to truncate the convolution output to be the right length). Recall that if , then and .

    (25 pts)