EEE Computational Neuroscience Homework 2 Solution

Question 1. [50 points]

The responses of a cat LGN cell to two-dimensional visual images are contained in the le c2p3.mat, data are described in Kara et al., Neuron 30:803-817 (2000). In the le, counts is a vector containing the number of spikes in each 15.6 ms bin, and stim contains the 32767, 16x16 images that were presented at the corresponding times. Speci cally, stim(x,y,t) is the stimulus presented at the coordinate (x,y) at time step t. Answer the questions below. (Note that stim is provided in integer format.)


    a) Calculate the STA images for each of the 10 time steps before each spike and show them all. For display, use imagesc with a grayscale colormap and identical display windowing for all STA images. Based on the STA derived lter, describe what type of spatio-temporal stimulus this LGN cell is selective for.

    b) Describe the changes in STA images across time. Sum the STA images over one of the spatial dimensions. You should obtain a matrix of 16 pixels by 10 time steps as a result of this process. Show this matrix (using imagesc). Based on the computed matrix, describe the temporal selectivity of the LGN cell. Is the matrix space-time separable?

    c) Project the stimulus onto the STA image at a single time step prior to the spike. Obtain the projection for each time sample by computing the Frobenius inner product between the stimulus image and the STA image. Create a histogram from all stimulus projections, and another histogram from stimulus projections at time bins where a non-zero spike count was observed. Use identical binning for the two histograms, and normalize each histogram to a maximum of 1. Compare the histograms with a bar plot. Comment on whether STA signi cantly discriminates spike-eliciting stimuli.
Question 2. [50 points] Answer the questions below. Include plots whenever applicable.


    a) Construct an on-center di erence-of-gaussians (DOG) center-surround receptive  eld
centered at 0:
1

2
2
2
1

2
+y
2
2

D(x; y) =

e
(x +y

)=2  c


e
(x


)=2  s
(1)

2  c2





2  s2






Sample this receptive eld as a 21x21 matrix, with a central Gaussian width of c = 2 pixels and a surround Gaussian width of s = 4 pixels. Display the generated receptive eld.

    b) Neurons in lateral geniculate nuclei (LGN) have DOG receptive elds. Suppose that there is a separate LGN neuron with a receptive eld centered on each pixel in the image. Compute the responses of each neuron to the image given in hw2_image.bmp. Place the neural responses topographically according to the centers of their receptive elds, and display the neural activity as an image (using imagesc). (Note: Be careful not to introduce artifacts at the image boundary.)

    c) Build an edge detector by thresholding the neural activity image (i.e., setting all values above a certain threshold to 1 and the remainder to 0.) Tune the parameters of the DOG receptive elds and the threshold to optimize the edge detector’s performance.

    d) Construct a Gabor receptive  eld on the same 21x21 pixel grid:
D(~x) = exp
~k( )  ~x

=2 l2
~k?( )  ~x

=2 w2

cos  2
?
+  !
(2)



2


2

~













k
( )
~x














~



~








~














Here, k( ) is a unit vector with the orientation , k?( ) is a unit vector orthogonal to k( ), and , l, w, and are parameters the comprise the Gabor lter. Start with assumption that = =2, l = w = 3 pixels, = 6 pixels, and = 0. Display the generated receptive eld.

    e) Simple cells in V1 have Gabor receptive elds. Suppose that there is a separate V1 neuron with a receptive eld centered on each pixel in the image. Compute the responses of each neuron to the image given in hw2_image.bmp. Place the neural responses topographically according to the centers of their receptive elds, and display the neural activity as an image (using imagesc). What is the function of this Gabor lter?

    f) Construct 4 Gabors with = 0; =6; =3; =2. Compute combined neural responses to the image hw2_image.bmp, by summing the outputs of the individual receptive elds (for di erent ). Does the edge detection performance look better in this case? What can you do with these 4 Gabors to further improve the performance?