Homework 3 (60 pts)

This problem set is due Tuesday (10/4/2016) at 5pm. Please turn in your work by uploading to Canvas. f you have questions, please post them on the course forum, rather than emailing the course staff. This will allow other students with the same question to see the response and any ensuing discussion. The goal of this problem set is to review the basics of point processes in general and the Poisson process in particular.

1. Properties of the Poisson Random Processes (30 pts)

   It is not uncommon for the action potential waveforms from multiple neurons to be recorded on a single electrode. Imagine that exactly two neurons are contributing spikes to the signal you are recording. Each neuron is spiking independently according to a Poisson process with rate $\lambda_1$ and $\lambda_2$ respectively.

   a. The raw spike train recorded on the electrode is a Poisson process. Show that it has an exponential ISI distribution. (Hint: What is Pr(combined signal > t), which is 1 – CDF, in terms of the same quantity for each of the neurons?)

   b. What is the rate of this Poisson process?

   c. What is the probability that each neuron will spike once within the same 1 ms interval?

2. Describing Simulated Data (30 pts).

   In this problem, we will investigate how spikes generated by the Connor-Stevens neuron from the previous homework compare to a Poisson process. Use the Connor-Stevens neuron with smoothed–Gaussian random input as in HW2 (but with new parameters). Generate 100 trials per parameter combination rather than 25. For the mean input, use {0.1 μA, 0.2 μA}; for the standard deviation (after smoothing!), use {0.2 μA , 0.5 μA). Under what input conditions (mean and variance) are the spikes that are produced “most” like a Poisson process? (Show spike count and ISI distributions, comparing data to expected distributions.) Under what input conditions are the spikes “least” like a Poisson process?