LS50B Problem Set #1
Due Friday, February 5, 2016 at 5 PM
Policies
Please turn in your problem set by the deadline to either TF. We will accept problem sets by email
([email protected] and [email protected]), in person before or after class, or dropped off
at one of our desks. (Jim’s desk is in NW 402, and Parris’ desk is in NW 457). If you’re turning in a physical
copy it must be legible and stapled into a single packet with each problem clearly labeled. Code can only
be in MATLAB and must be thoroughly commented for credit.
The problem set session this semester will be from 7:30PM-9:30PM in NW B104, a slight change from
last semester. Jim will be there to answer any questions you might have. If you run into issues at other
times, shoot us an email or stop by our office hours.
Good luck, and have fun!
1
Dissecting the Action Potential
For each of the questions below, re-draw how each perturbation qualitatively impacts both the shape of the action potential and the maximum rate of firing of the neuron. In addition, write an explanation (in complete sentences) justifying
your drawings for each item. You can reference the labels for
individual components of the action potential depicted in the
figure on the right in your explanations. Use the sheet appended to this problem set to re-draw each of the action potentials. In each template graph we have included the original trace in light-gray to serve as positional reference for your
re-drawings. Assume all perturbations are present starting at
t0 .
c
+60
+40
+20
0
mV
–20
b
–40
–60
a
–80
d
–100
–120
0
time
As an example, we have re-drawn the trace for the first scenario, but we still require your explanation of
why the trace changes as depicted.
1. A reduction in the current (i.e. the magnitude of the stimulus) coming into the focal neuron.
2. A decreased efficiency of Na+/K+ pumps.
3. An increased permeability of passive Na+ channels in the membrane.
4. An increased ratio of voltage-gated K+ channels to voltage-gated Na+ channels in the membrane.
5. A decreased ratio of voltage-gated K+ channels to voltage-gated Na+ channels in the membrane.
6. Decreased [K+] inside the cell.
1
2
Simple Models of Neurons
1. Let’s create a leaky integrator model based on the following equation:
C
V0 − V (t)
dV (t)
=
+ I(t)
dt
R
where C is the capacitance of the cell membrane, V (t) is the voltage, V0 is the resting potential, R is
the resistance of the membrane, and I(t) is the current.
(a) Using the provided MATLAB function “leaky integrator template.m” as a template, implement
a leaky integrator model by filling in the indicated line. You can use Professor de Bivort’s code
from lecture to help you, though you should note that the equation we’re using here is slightly
different. Include your code when you turn in your problem set, or just write what you filled in
for dV = . . .
(b) Inject a step function like the one shown in class into your model. The input current should be
5000 time steps long: time steps 1000 through 3000 should have a current of 5 · 10−7 A, and the
other time steps should have no current. Make plots of (1) the input current and (2) the voltage
trace that the model neuron produces in response to this input.
2. (a) Modify the leaky integrator model from part 1 to produce a leaky integrate-and-fire model of a
neuron. To do this, you’ll need to add a few lines of code. These lines should set a threshold
voltage at which the model fires an “action potential.” When the voltage hits this threshold, the
model should reset the voltage back down to an “overshoot value,” which is below the resting
potential. The firing threshold should be set at 0 mV and the overshoot voltage should be −85
mV. (The resting potential should still be −70 mV.) Include a copy of your code when you turn
in your problem set.
(b) As in part 1b, inject a step function into this new model. Make plots of both (1) the input current
and (2) the voltage trace produced by the leaky integrate-and-fire model neuron. How does the
voltage trace of the leaky integrate-and-fire neuron differ from the trace produced by the leaky
integrator?
(c) How do the spikes you see in the leaky integrate-and-fire model differ from the trace you would
expect from a real action potential? Sketch or plot the spike that the “leaky integrate and fire”
model produces, and sketch a real action potential. Your sketches should include a y-axis with
labels for 40 mV, 0 mV, -70 mV, and -85 mV. Point out the features the model’s spike is missing,
either with a few sentences or by labeling your sketch.
(d) A good metric for the overall activity of an integrate-and-fire model is the number of spikes it
produces per unit time. Play around with your model and see how each of the parameters affect
this firing rate. Then, for each of the following parameters (i-iii), briefly answer: (1) Qualitatively,
what effect does changing this parameter’s value have on the model’s behavior? (2) What is at
least one biological or physical property of a real neuron that you would expect to affect the value
of that parameter?
i. Resistance
ii. Capacitance
iii. Resting potential
3
Transportation Networks
Below are sub-graphs taken from the metropolitan transportation system of three major cities. Each graph
has 8 vertices and 10 edges, but the architecture of the graphs are slightly different. As you look at these
networks, think about what properties you might want a transportation network to have.
2
A
B
C
1. For the first part of this problem, think of a network-wide property that can be calculated to reveal
something about the efficiency of traversing the graph (hint: from any given point to any other point;
for these graphs you should be able to do this by hand). You have already seen an example of such an
efficiency metric calculated at the level of a single vertex called ”closeness”:
Cx = X
1
d(y, x)
y
Now consider calculating this metric for each vertex in the graph so that the efficiency of the graph
can be summarized (e.g. by the sum or average of Cxi for all i vertexes in the graph). Describe and
justify one of these possible whole-graph metrics. Show how you compute the whole-graph metric, for
each of the graphs, and then use this quantity to rank the graphs according to efficiency. You must
show your work for how you carried out your calculations. For now, assume all edge weights = 1 (i.e.
ignore differences in line lengths) and that all of the edges are bidirectional.
2. Now assume now that the edge weights in the following figure correspond to travel time burdens associated with rush hour. Re-compute efficiency in each graph and re-rank them. Assume un-labeled
edges have weights of 1.
D
E
F
2
2
2
2
2
2
4
2
2
4
2
2
2
3. Suppose the mayor of each city has hired you to build a single additional edge within these graphs.
Where would you put the edge if you wanted to make a substantial increase in efficiency of the graph at
rush hour? Is there any rule of thumb that would easily allow you to make your choice? Indicate your
choices in the notation of graph theory (e.g. {v1,v2}) and justify them by calculating the difference in
the metric utilized above before and after your edge addition. There may be more than one equivalent
best choice for each graph.
4. Now suppose that the mayor contracts you again to assess the vulnerability of the transit system to
disruption of a station (i.e. a vertex). They want you to determine a vertex in the graph which, if
knocked-out, would substantially decrease overall network function. One good metric for this would
be the ”between-ness,” Bx , which captures the proportion of all shortest walks from pairs of all other
vertexes that include a focal vertex, x:
3
Bx =
X σw,y (x)
σw,y
w6=y6=x
where σw,y is the number of shortest paths betwen two vertexes w, y and σw,y (x) is the number of
these which contain vertex x. The vertex with the highest B will disrupt the most shortest paths in
the network if it were to become disrupted; it may also render the graph un-connected! For one of the
graphs above (choose from among the un-weighted graphs A–C), convince the mayor to target their
investment towards the particular vertex you have chosen by comparing B for a number of candidate
vertexes. There may be more than one good choice in your transit system.
5. Finally, can you guess which cities these transportation systems sub-graphs belong to? (this part is
un-graded).
4
Yeast Graph
In this problem, we’ll look at the yeast protein-protein interaction network. We’ll represent each protein
as a vertex and an interaction between two proteins as an edge. The graph we’re providing to you for this
problem was retrieved from www.thebiogrid.org. (This graph is larger than the yeast network discussed
in class because it is based on a larger set of data.)
First, download the file “yeast graph.zip” and unpack it. Inside, you’ll find three files: vertices.txt, which
contains a list of vertex names; edges.txt, which contains a list of the (undirected) edges for this graph; and
essential.txt, which lists the vertices that represent proteins encoded by essential genes.
1. How many edges are possible in a graph of this many (6033) vertices? How many edges are there in
the yeast graph? Would you describe the yeast graph as dense or sparse?
2. Is the yeast graph we gave you connected? Complete? Very briefly justify your answers.
3. What vertex is of highest degree? What is its degree? Look that protein up on www.yeastgenome.org
and briefly describe (in a sentence) what it does.
4. Plot the degree distribution for this graph. Label the axes of this graph.
5. We gave you a list of essential genes (“essential.txt”) that we obtained by downloading all genes
annotated with an inviable null phenotype in YeastMine (yeastmine.yeastgenome.org). What is
the median degree for vertices included in this list of essential genes? For vertices not on that list?
According to your results, do proteins that are encoded by essential genes tend to have more or fewer
interactions than those encoded by non-essential genes? Briefly comment on why this might be.
6. Compare the degree distribution of this graph to the degree distribution expected of a random graph
generated using (1) the Bernoulli model and (2) the Barabasi-Albert model.
7. Is the yeast graph we gave you scale-free? Justify your answer using a log-log plot (with labeled axes)
and a sentence or two. (Hint: on the log-log plot that you’ll create, the trend is clearest if you use
around 500 bins.)
4
1.
mV
2.
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3.
mV
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+60
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5.
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mV
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6.
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+20
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mV
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–20
–40
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–80
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5