Due Date: Friday, May 20, 2011 midnight
Submission Procedure:
Create a Zip file called "528-hw3-lastname-firstname" containing the following:(1) Document containing your answers to any questions asked in each exercise, as well as any figures, plots, or graphs supporting your answers,(2) Your Matlab program files,(3) Any other supporting material needed to understand/run your solutions in Matlab. Upload your Zip file to this dropbox.
1. Simplified Neuron Models (40 points): Answer the questions here about the
Fitzhugh-Nagumo neuron model discussed by Adrienne in class. More information
about the Fitzhugh-Nagumo model can be found here and here.
2. Nonlinear Recurrent Networks (60 points): Write Matlab code and
answer the questions in Exercise 4 from Chapter 7 in the textbook
as described in the file c7.pdf.
Create figures reproducing Figures 7.18 and 7.19 in the textbook using your code, and include additional example figures to illustrate the effects of varying the value of tauI. (The following files implement a nonlinear recurrent network in Matlab:
c7p5.m and c7p5sub.m. These files are for Exercise 5 in c7.pdf but you can modify them and use them for Exercise 4. For an analytical derivation of the stability matrix, see Mathematical Appendix Section A.3 in the text). Extra Credit Problem (Hopfield Networks, 20 points)
The Hopfield network is a
famous type of recurrent network with the property that if you start the
network in an initial state, the network always converges to a local minimum of
an “energy” function (or Lyapunov function) that can
be defined for that network. The local minima correspond to particular patterns
stored in the network by choosing appropriate synaptic weights. Thus, if the initial
input is an incomplete pattern, the network will converge to the closest stored
pattern, giving rise to the useful property of pattern completion. Read about Hopfield
networks in this Scholarpedia article by Hopfield.
In this problem, we will
consider Hopfield networks with Binary Neurons (output is 0 or 1).
We assume the network has
symmetric weights and no self-connections.
(a) The Lyapunov function of our
network is given in the Scholarpedia
article in the section “Binary neurons.” Suppose we pick a neuron at random
and use the update rule given in the article, i.e., the neuron’s output becomes
1 if its overall input is above the threshold of 0, and becomes 0 otherwise.
Show that this update rule necessarily decreases the value of the Lyapunov function (or leaves it unchanged).
(b) Suppose the update procedure is repeated, picking a
neuron at random at each time step and setting its output according to the
update rule. Show that the network will eventually converge to a stable state
(Hint: Use your result from (a) and ask yourself whether the value of the Lyapunov function can decrease forever).