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AMATH 342 Problem set 3 solved

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I Filtering of inputs:

what matters in driving the membrane response? Let’s say
we know the voltage right now: at a time tnow. (For example, we could have just seen
a spike, so we know V (tnow) has just risen above threshold.)

And – we’d like to know
what current I(t), defined over a time interval [0, tnow], could have driven the neuron to
this voltage. For the current-driven (RC) circuit model studied in class,
• Choose tnow = 30ms, V (tnow) = 10mV , and R⇤C = 10ms. Find two input currents
I1(t) and I2(t), that look very di↵erent but produce the same value of Vtnow . For
both cases demonstrate your results by plotting the relevant voltage and current
traces using MATLAB (you may start with the relevant code provided / studied in
class).

• Answer this question: what is it about the explicit solution for V (t) from class
that indicates (a) that you would be able to find more than one di↵erent current
trace that leads to any V (tnow), and (b) why the two specific traces you took while
di↵erent, both led to the same V (tnow).

II Summation of simultaneous impulses:

do impluses summate linearly, sublinearly, or superlinearly?
• Consider the current-driven (RC) circuit, with R⇤C = 10ms. Consider an incoming
current impulse, with magnitude ¯I µA and width 1 ms (choose whatever values
you wish for these constants).

Starting from V (0) = 0 mV , what is the peak voltage
achieved over time in response to this impulse? If the threshold for spike generation
is 10 mV, what fraction of the way to threshold does this impulse drive the voltage
response (if your impulse take the cell over threshold, reduce ¯I and repeat)? Call this
fraction f.

Next consider the case in which N such impulses arrive simultaneously
(equivalent to taking the amplitude ¯I ! N ¯I). (NOTE: this is di↵erent from having
two pulses arrive one after another in a temporal sequence, they should arrive at the
same time.) What is the lowest value N that will drive the voltage over threshold?
How are f and N related?

Solve this question using BOTH MATLAB code, AND
the explicit solution from in integral form from class (check your work, you should
get the exact same results for both approaches)!

• Now study the same question, for a conductance-based input model. Now, the
impulses should be in g(t) conductance instead of the current I(t), but otherwise
be formed in the same way as for the previous problem. Choose E = 11 mV.

First, using MATLAB only, experiment with a number of magnitudes ¯g for the
conductance impulses and explain your findings for how pulses combine. Second,
use the form of the explicit solution from class, or other arguments from class, to
write down a two-sentence explanation of why you found what you did.

III HH model

• Use and / or modify the appropriate codes provided in the HH directory to plot
the firing rate – current tuning curve for the Hodgkin Huxley model. That is: as
a function of the constant value of applied current (i.e., IA(t) = ¯I), plot the firing
frequency in Hz.

Hint: start at ¯I = 0 µA and gradually test more negative currents.
• Now repeat, but with a sinusoidal background current of frequency ! kHz. That is,
plot firing frequency as a function of ¯I for the applied current IA(t) = ¯I+✏ sin(2⇡t!),
where you choose values for ✏ and ! (try several di↵erent values). How does your
firing rate – current tuning curve change? Can you provide a qualitative explanation?

Hint:

look for changes around the ¯I value near the threshold for repetitive firing.
• Finally, think about how to add noise to your applied current. Next, implement
this: include a noise term in your MATLAB code any way you wish (describe what
you have done in a sentence or two!), and compute the fano factor – as defined in
class.

This will require repeating your simulation with multiple realizations of the
noise current. For at least one value of ¯I, plot the fano factor vs. a measure of the
noise amplitude.

NOTE!

Those taking NBIO 301 will have the opportunity to do almost exactly
the experiment above with the snail neuron in lab. The only di↵erence is that
the sin wave will be replaced by periodic pulses of frequency ! and strength ✏ (and some
width in time).