15-883 Midterm Exam

This exam is open book, open notes. You can also use Matlab, the library, or the Web. And you can ask me if you have questions about any of the problems. But you may not discuss the exam with anyone else.

Please type your answers; handwritten answers will not be accepted. Hand in hardcopy at the start of class on October 17.

1. Let a = <2, 0, 3> and b = <-1, 0, 5> be two vectors. Answer the following questions, showing how you derived or verified each answer:
   • What is the dot product of a and b?
   • What is the angle in degrees between a and b?
   • Give a unit vector c that is orthogonal to both a and b.
   • Give a unit vector d such that the vectors {a, b, d} are linearly independent but not mutually orthogonal.

2. The O'Reilly and McClelland model and the Hasselmo and Schnell model both examined a problem of managing between two opposing phenomena.
   • For O'Reilly and McClelland, it was _______________ vs. _______________,
     while for Hasselmo and Schnell, it was _______________ vs. _______________.

   • In a few paragraphs, discuss how these two papers relate to each other, e.g., what common problems do they consider, how might their proposed mechanisms interact, etc. There is no "right" answer; what I'm looking for is a half page of intelligent discussion that demonstrates your understanding of the models and the issues they address.

3. Theta rhythm question:
   1. What is hippocampal theta rhythm?
   2. Describe the phenomenon of theta phase precession.
   3. What role does theta rhythm play in the Burgess et al. grid cell model?

4. How do attractor neural networks explain place fields? Consider issues such as the response to the removal of landmarks, the persistence of place fields in the dark, and the remapping of place fields observed when landmarks are scrambled or the rat moves to another room. One page should suffice.

5. Write Matlab code to demonstrate the Hebbian covariance learning rule. Assume two neurons, x1 and x2, that each project to a linear neuron y using weights w1 and w2, respectively. Use the following parameters:
   • x1(t) = sin(t/10)
   • x2(t) = cos³(0.13t)
   • t ranges from 1 to 350 in steps of 1
   • Moving average constant kappa = 0.05
   • Compute the exponentially weighted moving average of x1 as mu1(t+1) = (1-kappa)*mu1(t) + kappa*x1(t). Similarly for the averages of x2 and y. Initialize all moving averages to zero.
   • Initial weights w1(1) = 1, w2(1) = 1
   • Learning rate eta = 0.01

   Train the weights w1 and w2 using the Hebbian covariance learning rule. Generate four plots: x1(t) and mu1(t), x2(t) and mu2(t), y(t) and muy(t), w1(t) and w2(t). Hand in your code and the plots. Hint: you can put all four plots in one window using the subplot function. To keep a record of the values you need to plot, use code like this:

   clear myhist
   for t = 1 : 350
     ... compute some stuff, then:
     myhist(t,1) = x1(t);
     myhist(t,2) = mu1(t);
     myhist(t,3) = x2(t);
     myhist(t,4) = mu2(t);
     myhist(t,5) = ... etc.
   end
   
   plot(myhist(:,1:2))   % plot x1(t) and mu1(t)
   legend('x1','mu1')
   

   Note: if you're handing in black-and-white printouts, use a solid line for one variable and a dashed line for the other.