Programming Question Group 1 (Fun with membranes): Download and run the following Matlab code for modeling a passive neuronal membrane as an RC-circuit: membrane.m. This code demonstrates how a membrane responds to a constant current input that is turned on for a fixed time interval and then turned off. Add a question about the steady state (if driving with an input current)?
Q1:
Change the values for the membrane's resistance and capacitance (R and C), and find out how this influences the response of the membrane. Does it reach a stable value more quickly or more slowly after multiplying R by 5? a) More quickly. +b) More slowly. c) Same.
Q2:
Change the values for the membrane's resistance and capacitance (R and C), and find out how this influences the response of the membrane. Does it reach a stable value more quickly or more slowly after dividing C by 10?
+a) More quickly.
b) More slowly.
c) Same.
Q3:
Change the values for the membrane's resistance and capacitance (R and C), and find out how this influences the response of the membrane. Does it reach a stable value more quickly or more slowly after multiplying R by 10 AND dividing C by 10?
a) More quickly.
b) More slowly.
+c) Same rate.
Q4:
An experimental method for calculating a membrane’s time constant tau (when R and/or C are not known) is to start at zero and record the time at which the membrane potential V reaches a value approximately equal to $$0.6321V_{peak} = 0.6321IR$$, where I is the constant injected current. Check if this method works by injecting different amounts of current I and changing the values for R and C. Once you’ve convinced yourself that the experimental $$\tau$$ appears to be identical to the theoretical $$\tau (= RC)$$ in all these cases, provide a theoretical justification for why this method works.
To do this, find the solution to the differential equation for the passive membrane:
$$\dfrac{\text{d}V}{\text{d}t} = - \dfrac{V}{RC} + \dfrac{I}{C}$$
$$V(0) = 0$$
Which of these equations is the solution to the given differential equation?
After solving the differential equation you should be able to use the fact that $$V_{peak} = IR$$ and $$\frac{e - 1}{e}$$ to complete the derivation and show that $$V(\tau) = 0.6321IR$$.
+a) $$V(t) = IR(1 - e^{-\frac{t}{\tau}})$$
b) $$V(t) = IRe^{-\frac{t}{\tau}}$$
c) $$V(t) = IR(1 - e^{\frac{t}{\tau}})$$ d) $$V(t) = IR$$
Programming Question Group 2 (Get fired up about Integrate-and-Fire neurons):
Run the following Matlab code for modeling an integrate-and-fire neuron: intfire.m.
Q1: What is the largest current that will fail to cause the neuron to spike?
You should vary the input current gradually from very low to high values to find this value (round your answer to the nearest 10 pA).
(would not put multiple choice..) +a) 250 pA. b) 400 pA. c) 251 pA d) 250 nA
I'm not sure if this should be multiple choice or not. If not, we should add this info: Your answer should have two significant figures. 250 pA
(Maybe add a question asking for an analytical derivation of this value) Q2: Why is this value the largest current that fails to cause a neuron to spike? a) This current causes $$V_{peak}$$ to exactly equal $$V_{thresh}$$. This means that the membrane potential will never exceed the threshold voltage and therefore never cause a spike. b)
Q3: What is the maximum firing rate (spike count/trial duration) of this neuron? Give your answer in Hz and round your answer to the nearest integer value. Do not include the units in your answer.
167
Q4: How is the maximum firing rate related to the absolute refractory period of the neuron? (TODO: reword this) a) The neuron can fire a spike once every refractory period duration and fires a spike instantaneously. +b) The neuron can fire a spike once every refractory period duration and takes 1 ms to fire spike. c) The neuron's firing rate is unaffected by the refractory period of the neuron. d) The neuron can fire a spike once every refractory period and it takes 2 ms to fire a spike.
Q5: (I'd definitely skip this, as I'd prefer to keep the terminology of "resonant" to mean capable of subthreshold oscillations..)
What is the "resonant frequency" (if one exists) of the neuron. We define the resonant frequency as the frequency that causes the neuron to "track" the input by firing exactly one spike for each peak in the input.
+a) Approximately 12 Hz. b) Approximately 10 Hz. c) Approximately 15 Hz. d) Approximately 11 Hz. e) Approximately 13 Hz. f) There is no resonant frequency.
Download and run the following Matlab code for modeling a passive neuronal membrane as an RC-circuit: membrane.m. This code demonstrates how a membrane responds to a constant current input that is turned on for a fixed time interval and then turned off. Add a question about the steady state (if driving with an input current)?
Q1:
Change the values for the membrane's resistance and capacitance (R and C), and find out how this influences the response of the membrane. Does it reach a stable value more quickly or more slowly after multiplying R by 5?
a) More quickly.
+b) More slowly.
c) Same.
Q2:
Change the values for the membrane's resistance and capacitance (R and C), and find out how this influences the response of the membrane. Does it reach a stable value more quickly or more slowly after dividing C by 10?
+a) More quickly.
b) More slowly.
c) Same.
Q3:
Change the values for the membrane's resistance and capacitance (R and C), and find out how this influences the response of the membrane. Does it reach a stable value more quickly or more slowly after multiplying R by 10 AND dividing C by 10?
a) More quickly.
b) More slowly.
+c) Same rate.
Q4:
An experimental method for calculating a membrane’s time constant tau (when R and/or C are not known) is to start at zero and record the time at which the membrane potential V reaches a value approximately equal to $$0.6321V_{peak} = 0.6321IR$$, where I is the constant injected current. Check if this method works by injecting different amounts of current I and changing the values for R and C. Once you’ve convinced yourself that the experimental $$\tau$$ appears to be identical to the theoretical $$\tau (= RC)$$ in all these cases, provide a theoretical justification for why this method works.
To do this, find the solution to the differential equation for the passive membrane:
$$\dfrac{\text{d}V}{\text{d}t} = - \dfrac{V}{RC} + \dfrac{I}{C}$$
$$V(0) = 0$$
Which of these equations is the solution to the given differential equation?
After solving the differential equation you should be able to use the fact that $$V_{peak} = IR$$ and $$\frac{e - 1}{e}$$ to complete the derivation and show that $$V(\tau) = 0.6321IR$$.
+a) $$V(t) = IR(1 - e^{-\frac{t}{\tau}})$$
b) $$V(t) = IRe^{-\frac{t}{\tau}}$$
c) $$V(t) = IR(1 - e^{\frac{t}{\tau}})$$
d) $$V(t) = IR$$
Programming Question Group 2 (Get fired up about Integrate-and-Fire neurons):
Run the following Matlab code for modeling an integrate-and-fire neuron: intfire.m.
Q1:
What is the largest current that will fail to cause the neuron to spike?
You should vary the input current gradually from very low to high values to find this value (round your answer to the nearest 10 pA).
(would not put multiple choice..)
+a) 250 pA.
b) 400 pA.
c) 251 pA
d) 250 nA
I'm not sure if this should be multiple choice or not. If not, we should add this info:
Your answer should have two significant figures.
250 pA
(Maybe add a question asking for an analytical derivation of this value)
Q2:
Why is this value the largest current that fails to cause a neuron to spike?
a) This current causes $$V_{peak}$$ to exactly equal $$V_{thresh}$$. This means that the membrane potential will never exceed the threshold voltage and therefore never cause a spike.
b)
Q3:
What is the maximum firing rate (spike count/trial duration) of this neuron? Give your answer in Hz and round your answer to the nearest integer value. Do not include the units in your answer.
167
Q4:
How is the maximum firing rate related to the absolute refractory period of the neuron? (TODO: reword this)
a) The neuron can fire a spike once every refractory period duration and fires a spike instantaneously.
+b) The neuron can fire a spike once every refractory period duration and takes 1 ms to fire spike.
c) The neuron's firing rate is unaffected by the refractory period of the neuron.
d) The neuron can fire a spike once every refractory period and it takes 2 ms to fire a spike.
Q5: (I'd definitely skip this, as I'd prefer to keep the terminology of "resonant" to mean capable of subthreshold oscillations..)
What is the "resonant frequency" (if one exists) of the neuron. We define the resonant frequency as the frequency that causes the neuron to "track" the input by firing exactly one spike for each peak in the input.
+a) Approximately 12 Hz.
b) Approximately 10 Hz.
c) Approximately 15 Hz.
d) Approximately 11 Hz.
e) Approximately 13 Hz.
f) There is no resonant frequency.