Homework 8 (100 pts)

This problem set is due the last day of classes (12/2/2016), but can be turned in late with no penalty. Note that we will drop the lowest score assignment, so this is optional if you choose not to do it. 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 clustering through the process of spike sorting action potentials data. You should submit both commented code and a write-up for this homework assignment.

The goal of this assignment is to carry out continuous neural decoding, first just using Bayes rule and then (for extra credit) by using a particle filter to account for a prior model of animal movement.

  1. Review of generative models (30 pts) Write down simple pseudocode for the generation of simulated data for the following problems.

    Ex. Classification of data into classes using labeled training data where the data are conditionally Gaussian.

     1. Generate a random number, k from 1, ..., K as the class
     identity. (Use the class prior probabilities to weight the choices.)
    
     2. Sample a random Gaussian from a Gaussian distribution with mean
     mu_k and covariance Sigma_k.
    

    a. Clustering data into clusters using a mixture of Gaussians model. (Hint: This might not be different from the example!)

    b. Dimensionality reduction from -dimensional data to -dimensional data, using probabilistic PCA.

    c. A linear dynamical system with Gaussian initial state, innovations, and observation noise (i.e., the generative model for a Kalman filter).

  2. Convexity conditions for Poisson firing rate functions (30 pts) Consider a set of rat hippocampal neurons in which the number of spikes observed follows a Poisson distribution, , where is some function of the rat’s position . Given an observation of spike counts, , what is the log likelihood function of the rats position ? What are the first and second derivatives of the log likelihood with respect to ? What conditions on would imply that the log likelihood was concave (i.e., the second derivative was negative everywhere)?

  3. Single step of dynamical system estimation (40 pts) Consider a set of 10 neurons whose firing can be described as function of the form You will find the parameters for these neurons in the file hw8problem3.npz. The vectors MaxRates (in spikes per second), FieldCenters, and FieldWidths correspond to the values of , , and for the neurons, respectively. Spike counts for 250 milleseconds of neural activity is found in the vector NeuralObservation.

    a. Plot the log likelihood of the rat’s position, , over the range (ignore the term(s) in the likelihood function that is common to all positions). What is the maximum likelihood position of the rat?

    b. Now assume that you have prior information about the rat’s position characterized by a normal distribution, with mean 30 and standard deviation 5. On the same graph as for (a), plot the log likelihood of the rat’s position using just your prior distribution and using both the prior distribution and the observations of neural activity. What is the maximum a posteriori position of the rat?