18-698 / 42-632
Neural Signal Processing
Spring 2025
Problem Set 1
This problem set is due on Thursday, February 13, 11:59pm.
Please show your work by writing down each step to get from the problem to your answer. If you have questions, please post them on the Piazza Q&A webpage, rather than emailing the course staff. This will allow other students with the same question to see the response and any ensuing discussion.
Please submit your work as a single PDF file on Gradescope, which is linked from Canvas. When preparing your solutions, please complete each problem on a separate page. Grade-scope will ask you select the pages that contain the solution to each problem.
Submissions can be written in LaTeX or they can be handwritten and photocopied using a scanner or smartphone camera. Handwritten work should be clearly labeled and legible.
1. A neuron spikes according to a homogeneous Poisson process at 10 spikes per sec-ond. The recording equipment is broken and drops each spike independently with 50% probability.
(a) (5 points) What is the expected time until the second spike is recorded?
(b) (10 points) What is the probability that the second recorded spike occurs after one second?
2. You insert an electrode into the brain. Unbeknownst to you, the electrode sits next to two neurons. Each neuron spikes independently according to a homogeneous Poisson process with rate λ1 and λ2, respectively.
(a) (5 points) What is the probability that no spikes are observed during the first t seconds?
(b) (5 points) Given that no spikes are observed in the first s seconds, what is the probability that no spikes are observed in the first s + t seconds?
(c) (10 points) What is the probability that the first spike observed came from neuron 1?
(d) (10 points) What is the probability that 2 of the first 3 spikes came from neuron 1?
(e) (5 points) At some point during your experiment, the recording equipment breaks down and begins dropping spikes randomly with probability p. What is the probability that the first spike observed (after breakdown) came from neuron 1?
3. Neurons 1 and 2 each spike according to a homogeneous Poisson process with rate λ1 and λ2, respectively.
(a) (20 points) What is the expected time of the first spike of neuron 2, given that neuron 1 spikes before neuron 2?
(b) (15 points) What is the probability that neuron 1 spikes m times before neuron 2 spikes n times? (Hint: your final answer can have a summation in it.)
4. (15 points) A neuron spikes according to a homogeneous Poisson process with rate λ. The duration of the nth ISI is tn (n = 1, ..., N). Estimate the neuron’s firing rate by finding the λ that maximizes (Hint: Work with log probabilities. Your final answer can have a summation in it.)