Homework 2

Homework 2 (140 pts)

This problem set is due Tuesday (9/21/2021) at 11:59 pm. 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.

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?
Bayesian Analysis of the Poisson Process (30 pts)

(Murphy, 2012, problems 3.6 and 3.7)

a. The Poisson pmf is defined as \(\text{Poi}(x \mid \lambda) = e^{-\lambda} \frac{\lambda^x}{x!}\), for \(x \in \lbrace 0, 1, 2, \ldots\rbrace\) where \(\lambda > 0\) is the rate parameter. Derive the Maximum Likelihood Estimate of the rate parameter when given a series of observations \(D = \lbrace x_1, x_2, \ldots, x_N\rbrace\). (Hint: solve for the maximum of the log of the likelihood, \(p(D \mid \lambda)\).)

b. (Bayesian parameter estimation and a “conjugate prior”) Derive the posterior \(p(\lambda \mid D)\) assuming a prior distribution \(p(\lambda) = \text{Ga}(\lambda \mid \alpha, \beta) \propto \lambda^{\alpha-1} e^{-\lambda \beta}\). Hint: the posterior is also a Gamma distribution!! This is the definition of a “conjugate prior” - the posterior has the same form (but with different parameters) as the prior.

c. What does the posterior mean tend to as \(\alpha \to 0\) and \(\beta \to 0\). Hint: Recall that the mean of a \(\text{Ga}(a,b)\) distribution (defined as an integral) is \(a/b\). (So if you’re doing a crazy sum, you’re doing it wrong.)

Point Process Model Validation (40 pts)

We can describe the number of spikes that a neuron produces in a one second window using a wide variety of random process models. Let’s refer to the parameters of a given model as \(\theta\), and a set of neural counts as \(\mathbf{n}\). We can evaluate the quality of the model by asking for which model/parameters the likelihood of the data, \(P(\theta \mid \mathbf{n})\), is maximized. In this problem you will try to determine the right model for the spikes from a synthetic neuron.

a. Model Comparison (10 pts) Look at the data in the numpy data file hw2problem2A.npy. This file contains two variables, SpikeTimes and SpikeCounts. SpikeTimes is a list of spike times recorded in one experimental trial (each trial is 1 s for simplicity) from a neuron. For each trial, for convenience, the number of spikes is given in the corresponding element of SpikeCount. You can load the data into python using numpy.load():

import numpy as np
[SpikeTimes, SpikeCount] = np.load('hw4problem2A.npy')

Note that you can find these data in Matlab format in hw2problem2A.mat. Consider two possible models for the number of spikes per trial:

Model	Parameters
Gaussian	Mean = 10.2, Variance = 9.5
Poisson	Rate = 9.8

Using the data provided evaluate which set of parameters best describes the data. (Hint: You should calculate the likelihood of each model. In other words, calculate the probability of, e.g., data from 1000 trials of the simulated data given one model or the other?)

b. (10 pts) Now split the data into training and validation sets, use the training sets to train a Gaussian model and a Poisson model, and use the validation set to assess which model fits the data best. (Hint: You should use the maximum likelihood estimate for the parameters of the models, which will be the sample mean and variance of the training data for the Gaussian and the sample mean for the Poisson.)

c. Point Processes (10 pts) Consider the actual spike times from (a) (given in the SpikeTimes list). If you were to model this neuron with a Poisson process, would you use a constant or a time-varying rate over the 1 second window? (Justify your answer with one or more plots.)

d. Fano Factor / ISI Distribution (10 pts) Consider the spike times returned by a different neuron whos data are given in hw2problem2D.npy (the same data are given in Matlab format in hw2problem2D.mat). Again, you can load the data into python using numpy.load():

import numpy as np
[SpikeTimes, SpikeCount] = np.load('hw4problem2D.npy')

If you were to model this neuron with a Poisson process, would you use a constant or a time- varying rate? Do the variance and mean of the count distribution have the relationship you would expect for a Poisson process? What physiological process might you observe in the spike times that would explain this? (Hint: You’ll want to actually look at the spike times. Generating a peri-stimulus time histogram (PSTH) showing the average number of spikes in small bins of time will help explain your observations.)

Describing Real Data (40 pts)

In this problem you are going to characterize the neural activity recorded on a multielectrode array in visual cortex, found in hw2problem3.npy (or hw2problem3.mat). Spike times (in microseconds) for 10 neurons are given in spiketimes, a 10 element numpy array, where each element is a numpy vector of spiketimes. The time-varying stimulus is described in the stimulus numpy array, where the first column is timestamps (which follow regular 5 ms steps) and the second column is the direction of motion (in degrees) of a moving bar. Stimulus directions are randomized, each direction is maintained for 4 s, and directions are repeated 8 times. You can load the data into python using numpy.load():
```
import numpy as np
[stimulus, spiketimes] = np.load('hw2problem3.npy')
```
a. (10 pts) Neural activity in visual cortex was recorded while the cat was viewing drifting bars (like in the Hubel and Weisel video) that cycled through different angles. The visual stimulus was constant for 4s, and then changed (the angle of the stimulus in 5 ms bins is given in the stimulus vector). Plot the “tuning curve” for the activity from the 10th neuron, show how the mean number of spikes varies as a function of the direction of the moving bar. (The x-axis should be stimulus direction and the y-axis should be “average spikes per second”.) Does this neuron have one “preferred direction”? Why might the directional tuning be shaped the way it is? (Hint: Is there some sort of symmetry for opposite directions?)

b. (10 pts) Calculate the tuning curves for all the neurons. Plot a histogram of the “preferred direction” (i.e., direction of stimulus which produces maximum average spiking) of the neurons. Plot a histogram of the “maximum firing rate” (i.e., the average firing rate for the preferred direction).

c. (10 pts) Select the neuron with the largest maximum firing rate. For each stimulus direction, calculate the mean spiking rate and the variance from this mean in the number of spikes observed in one stimulus presentation. Calculate the Fano factor for each stimulus direction and compare to what is expected for a Poisson process. Repeat for the neuron with the largest average (i.e., across all directions) firing rate.

d. (10 pts) For the neurons in part (c), create ISI distributions for spikes during stimuli in the preferred direction. Compare this to the ISI distribution expected under a Poisson approximation.

e. (Extra Credit) For the neurons in part (c), use the time-rescaling theorem to renormalize the ISIs for all stimulus directions (i.e., assuming a piece-wise constant inhomogeneous Poisson process). Plot the resulting ISI distribution. Conduct a goodness-of-fit test on the spikes. Is the piece-wise Poisson model a good one?

Data Source Note:

This data is courtesy of Tim Blanche, UC Berkeley. He captured single and multiunit recordings in the primary visual cortex of cats during viewing of simple and complex stimuli. It has been shared for public use on the CRCNS.org website as data set “pvc3”, and we are using the data from the “drifting bar” section.