Description
1. (15 pts) True / False. Determine if the following statements are true or false. If a statement
is false, please correct the statement to receive full credit. Note that we use “spike” and
“action potential” interchangeably. Each statement is worth 1 point.
(a) When an action potential is fired, there is first a larger current attributable to K+
channels opening before Na+ channels.
(b) During an action potential, Na+ currents and K+ currents both serve to depolarize the
cell.
(c) The patch-clamp allows experimenters to measure the current flowing through a single
ion channel.
(d) It is possible to record action potentials with electroencephalograms (EEG).
(e) Imagine a neuron perfectly modeled by a Poisson process with a homogeneous firing rate
of 1 spike per second. Consider two scenarios. In scenario (a), the last spike occurred
100 ms ago. In scenario (b), the last spike occurred 1.2 s ago. It is more likely that in
the next 100 ms, a spike will fire in scenario (b) than it will in scenario (a).
(f) If the Fano factor of a neuron is greater than 1, then its firing rate mean is greater than
its firing rate variance.
(g) A Poisson process will always have a Fano factor of 1.
(h) An exponential interspike interval distribution models the refractory period well.
(i) Chronic, multi-site electrode arrays allow the measurement of action potentials from
several neurons at millisecond resolution.
(j) There is stimulation electrode technology that can remain functional and effective in
humans for years.
(k) Using single-electrode technology, it is possible to record spikes from different neurons
at the same time.
(l) To accurately and reliably detect action potentials from a single neuron using a static
threshold, a high-pass filter, which removes DC and low-frequency components of raw
electrode voltage waveforms, is required.
(m) Convolving spike trains with a Gaussian kernel to approximate a spike rate, r(t), is a
type of high-pass filtering.
(n) Tuning curves describe neural activity in the visual and motor systems well.
(o) During the relative refractory period, it is impossible for a spike to be generated.
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2. (30 points) Tuning curves. One way to model the firing rate of motor cortical neurons is with
tuning curves. The tuning curve models the average firing rate, f(θ), for a neuron when a
reach is made in the direction θ. The cosine tuning model asserts that:
f(θ) = c0 + c1 cos(θ − θ0)
In this question, we will learn how to derive values for the parameters c0, c1, and θ0 given
that we know the average firing rates for reaches in certain directions.
(a) (1 point) Show that θ0 is the preferred direction of the neuron. The preferred direction
is the direction for which the neuron fires most.
(b) (3 points) Your colleague writes a script to find the values of c0, c1, and θ0 given some
neural data. He runs it and finds that c0 = −11, c1 = 8, and θ0 = 125◦
. Do you tell
him “Great job! This is a completely reasonable model!” or do you tell him “You’ve
made a mistake.” Why?
(c) (3 points) You decide to take a stab at writing a script that will find the parameters
of the tuning model. However, you’re going to go about it a different way than your
colleagues. You’re first going to simplify the term cos(θ − θ0). Show that:
cos(θ − θ0) = cos(θ) cos(θ0) + sin(θ) sin(θ0).
(d) (3 points) To simplify our parameter estimation, we will re-write f(θ) as
f(θ) = k0 + k1 sin(θ) + k2 cos(θ).
Find k0, k1, and k2 in terms of c0, c1, and θ0.
(e) (5 points) We define yθ to be the measured average firing rates for a reach in direction
θ. For simplicity in this question, assume that we’ve only measured the firing rate when
the monkey reaches to three unique directions: y0 for the reach at an angle of 0◦
(i.e.,
to the right), y120 for the reach to 120◦
(up and to the left) and y240 for the reach to
240◦
(down and to the left). Find k0, k1, and k2 in terms of y0, y120 and y240.
(f) (5 points) Plot the tuning curve, f(θ), when y0 = 25, y120 = 70, and y240 = 10. Also
include y0, y120, and y240 on the plot. Finally, provide the values of c0, c1, and θ0.
(g) (10 points) Now consider that we sampled the workspace much more effectively. We
now have the following data:
y0 = 25, y60 = 40, y120 = 70, y180 = 30, y240 = 10 and y300 = 15.
Report the values of c0, c1, and θ0 that minimize the mean-square error between the
tuning curve and the observed data.
3. (5 points) Refractory periods
(a) (2 points) In class, we introduced the exponential distribution to model inter-spike
intervals (ISI). Does the exponential distribution incorporate the concept of refractory
periods? Please explain.
(b) (3 points) If a model neuron spikes at 50 spikes per second according to a homogeneous
Poisson process, what percentage of spikes would violate a 1 ms refractory period?
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4. (34 points) A neuron spikes according to a homogeneous Poisson process with rate λ.
(a) (2 points) What is the mean ISI of this neuron?
(b) (4 points) What is the probability that a given ISI is greater than the mean ISI?
(c) (7 points) What is the expected ISI given that it is larger than the mean ISI?
(d) (7 points) What is the expected ISI given that it is smaller than the mean ISI?
(e) (7 points) What is the expected number of spikes that will be fired before one sees an
ISI greater than the mean ISI?
(f) (7 points) What is the expected waiting time until (and including) an ISI greater than
the mean ISI?
5. (16 points) You insert a pair of electrodes into the brain. Unbeknownst to you, electrode 1
sits next to a neuron with mean ISI of 20 ms, and electrode 2 sits next to a different neuron
with mean ISI of 30 ms. Each neuron spikes independently according to a homogeneous
Poisson process. A neuron is “detected” when it fires its first spike.
(a) (4 points) What is the probability that no neurons are detected (on either electrode)
during the first 60 ms?
(b) (4 points) Given that no neurons are detected in the first s seconds, what is the probability that no neurons are detected in the first s + t seconds?
(c) (8 points) A single spike comes in, and thus a neuron is detected. What is the probability
that a neuron is detected on electrode 1 before electrode 2? (Hint: your answer should
not be a function of time.)
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