Computational Neuroscience Homework 4 Solution

Stimuli consisting of face images have been used in a study on visual perception. Experimen-tal stimuli are provided in the le hw4_data1.mat, which contains face images downsampled to a 32 32 square grid. The images are stored in a matrix f aces, with 1000 rows (number of di erent images) and 1024 columns (number of image pixels). Answer the questions below.


    a) The experimenter would like to t encoding models between the stimuli (i.e., face im-ages) and the measured neural responses. However, one rst needs to de ne explanatory variables (i.e., regressors) that capture important variations in stimulus properties during the experiments. To accomplish this goal, perform PCA on the 1000 faces images. Plot the proportion of variance explained by each individual PC, for the rst 100 PCs. Display the rst 25 PCs using the function dispImArray.m.

    b) The experimenter would like to know how many PCs are su cient to obtain a reasonable representation of the stimuli. Reconstruct each image in the matrix f aces using their PC projections (i.e., reconstructed images are a linear weighted combination of stimulus PCs). Obtain separate reconstructions based on rst 10, 25, and 50 PCs. Display the original images and the reconstructions using dispImArray.m, for the rst 36 images. Find the mean-squared error (MSE) between the original and reconstructed images, and report the mean and standard deviation of MSE across 1000 images. Interpret the results.

    c) Instead of PCA, nd explanatory variables to capture stimulus properties using inde-pendent component analysis (ICA). Use the FastICA package available on Moodle, and use PCA-based reduction to 50 dimensions during the calls (i.e., by setting lastEig). Note that unlike PCA, ICA results are not deterministic. Recall fastica.m to return 10 ICs, 25 ICs, and 50 ICs. Display the obtained ICs using dispImArray.m. Reconstruct face images based on their IC projections. Display the original and reconstructed images based on 10, 25, and 50 ICs. Report the mean and std of MSE in each case. Compare your results with part b.

    d) Finally, nd explanatory variables to capture stimulus properties using non-negative ma-trix factorization (NNMF). NNMF requires its input to be strictly positive, so add a single scalar constant to all entries of the matrix f aces to satisfy this constraint (but only add the minimum amount required). Recall nnmf.m to return 10, 25, and 50 MFs. Display the ob-tained MFs using dispImArray.m. Reconstruct face images based on their MF projections. Display the original and reconstructed images based on 10, 25, and 50 MFs. Report the mean and std of MSE in each case. Compare your results with parts b and c.
Question 2. [50 points]

Consider a population of 21 independent neurons with Gaussian-shaped tuning curves:


(x   i)
2
2

fi(x) = A  e


=(2  i )
(1)






The tuning curves have an amplitude of 1 and a standard deviation of i = 1, with centers i evenly spaced between -10 and 10 along the x-axis. Answer the questions below.

    a) Plot all tuning curves in the population on the same axis. Simulate the population re-sponse to the stimulus x = 1, and plot the population response as a function each neuron’s preferred stimulus value.

    b) Perform a simulated experiment with 200 trials. In each trial, sample a stimulus inten-sity uniformly from the interval [ 5 5], simulate the 21-long vector of population response ~r. Assume that each neuron’s response is corrupted by an additive Gaussian noise with zero mean and =20 standard deviation. Implement a winner-take-all decoder, and calculate the stimulus estimate xW T A for each trial. Plot the actual and estimated stimulus on the same graph. Compute the mean and standard deviation of error in stimulus estimation across 200 trials.

    c) For the experimental trials simulated in part b, implement a maximum-likelihood de-coder, and calculate the stimulus estimate xML for each trial. Plot the actual and estimated stimulus on the same graph. Compute the mean and standard deviation of error in stimulus estimation across 200 trials. (Hint: To nd xML, you can calculate the log-likelihood for the entire stimulus range. Note that the tuning curve gives you the expected value (i.e., mean) of the neural response, and there is additional variability due to additive noise that you need to consider.)

    d) For the experimental trials simulated in part b, implement a maximum-a-posteriori de-coder, and calculate the stimulus estimate xMAP for each trial. Assume that the prior of the stimulus value x follows a Gaussian distribution with a mean of 0 and a standard deviation of 2.5. Plot the actual and estimated stimulus on the same graph. Compute the mean and standard deviation of error in stimulus estimation across 200 trials. Interpret your results.

    e) Perform an experiment with 200 trials of stimulus intensity. In each trial, sample a stim-ulus intensity from the interval [ 5 5]. For the resulting stimulus vector (of length 200), separately simulate the population response vectors (~r) for i = 0:1, i = 0:2, i = 0:5, i = 1, i = 2, and i = 5. In each case, assume additive Gaussian noise with zero mean and 1=20 standard deviation. Calculate MLE estimates of the stimulus xML based on each population response separately. Compare the mean and standard deviation of error in stimulus estimation for various values. Is it better to have narrow or wide tuning curves?