## Description

## 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).