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Ch2 Exercises.py

@@ -124,46 +124,176 @@ def Ex5():
 
 def Ex6():
     '''
+    Performthe integrals in equations 2.31 and 2.32 for the case sigmax =
+    sigmay = sigma to obtain the results
+    
+    (A/2) * e**(- (sigma**2 (k**2 + k**2)) / 2) * (cos(phi - psi) * e**(sigma**2 * k * K * cos(sigma)) + cos(phi - psi) * e**(-simga**2 * k * K * cos(sigma)))
+    
+    and 
+    
+    Lt(t) = ( (alpha**6 * |w| * root(w**2 + 4alpha**2) / (w**2 + alpha**2)**4) ) * cos(w*t - delta)
+    
+    with
+    
+    delta = 8 * arctan (w / alpha) + arctan (2alpha/w) - PI
+    
+    From these results, verify the selectivity curves in figures 2.15 and
+    2.16. In addition, plot delta as a function of w.
     '''
 
 def Ex7():
     '''
+    Numerically compute the spatial part of the linear response of a
+    simple cellwith a separable space-time receptive field to a sinusoidal
+    grating, as given by equation 2.31. Use a stimulus oriented with
+    Phi = 0. For the spatial receptive field kernel, use equation 2.27 with
+    sigma*x = sigma*y = 1degrees, phi = 0, and 1/k = 0.5degrees. Plot Ls as a
+    function of K taking Phi = 0 and A = 50. This determines the 
+    frequency selectivity of the cell. What is its preferred spatial
+    frequency? Plot Ls as a function of Phi taking 1/K = 0.5degrees and A = 50. This determines the spatial phase selectivity of the cell. What is its 
+    preferred spatial phase?
     '''
 
 def Ex8():
     '''
-    '''
+    Consider a complex cell with the spatial part of its response given by
+    L2,1 + L2,2, where L1 and L2 are linear responses determined by equation
+    2.31 with kernels given by equation 2.27 with sigmax = sigmay = 
+    1degrees, and 1/k = 0.5degrees; and with phi = 0 for L1 and phi =
+    -PI/2 for L2. Use a stimulus oriented with Phi = 0. Compute and plot
+    L2,1 + L2,2 as a function of K taking Phi = 0 and A = 5. This
+    determines the spatial frequency selectivity of the cell. Compute and
+    plot L2,1 + L2,2 as a function of Phi taking 1/K = 0.5degrees and A =
+    5. This determines the spatial phase selectivity of the cell. Does the
+    spatial phase selectivity match what you expect for a complex cell?
+    '''    
+
 
 def Ex9():
     '''
+    Consider the linear temporal response for a simple or complex cell
+    given by equation 2.32with a temporal kernel given by equation 2.29
+    with 1/alpha = 15 ms. Compute and plot Lt(t) for w = 6PI/s. This
+    determines the temporal response of the simple cell. Do not plot the
+    negative part of Lt(t) because the cell cannot fire at a negative rate.
+    Compute and plot L2t(t) for w = 6PI/s. This determines the temporal
+    response of a complex cell. What are the differences between the
+    temporal responses of the simple and complex cells?
     '''
 
 def Ex10():
     '''
+    Compute the response of amodel simple cellwith a separable spacetime
+    receptive field to a moving grating
+
+    s(x, y, t) = cos(Kx - w*t) .
+    
+    For Ds, use equation 2.27with sigmax = sigmay = 1degrees, Phi = 0, and 1/k = 0.5degrees. For
+    Dt, use equation 2.29 with 1/alpha = 15 ms. Compute the linear estimate
+    of the response given by equation 2.24 and assume that the actual
+    response is proportional to a rectified version of this linear response
+    estimate. Plot the response as a function of time for 1/K = 1/k = 0.5degrees
+    and w = 8PI/s. Plot the response amplitude as a function of w for
+    1/K = 1/k = 0.5degrees and as a function of K for w = 8PI/s.
     '''
 
 def Ex11():
     '''
+    Compute the response of a model complex cell to the moving grating
+
+    s(x, y, t) = cos(Kx - w*t)
+
+    The complex cell should be modeled by squaring the unrectified
+    linear response estimate of a simple cells with a spatial receptive
+    field given by equation 2.27 with sigmax = sigmay = 1degrees, Phi = 0, and 1/k = 0.5degrees,
+    and adding this to the square of the unrectified linear response of
+    a second simple cell with identical properties except that its spatial
+    phase preference is Phi = -PI/2 instead of Phi = 0. Both linear responses
+    are computed fromequation 2.24. For both of these, use equation 2.29
+    with 1/alpha = 15 ms for the temporal receptive field. Plot the complex
+    cell response as a function of time for 1/K = 1/k = 0.5degrees and w = 8PI/s.
+    Plot the response amplitude as a function of w for 1/K = 1/k = 0.5degrees
+    and as a function of K for w = 8PI/s.
     '''
 
 def Ex12():
     '''
+    Construct a model simple cell with the nonseparable space-time receptive
+    field described in the caption of figure 2.21B. Compute its
+    response to the moving grating
+
+    s(x, y, t) = cos(Kx - w*t) .
+    
+    Plot the amplitude of the response as a function of the velocity of the
+    grating, w/K, using w = 8PI/s and varying K to obtain a range of both
+    positive and negative velocity values (use negative K values for this).
+    Show that the response is directionally selective.
     '''
 
 def Ex13():
     '''
+    Construct amodel complex cell that is disparity tuned but insensitive
+    to the absolute position of a grating. The complex cell is constructed
+    by summing the squares of the unrectified linear responses of two
+    simple cells, but disparity effects are now included. For this exercise,
+    we ignore temporal factors and only consider the spatial dependence
+    of the response. Each simple cell response is composed of two terms
+    that correspond to inputs coming from the left and right eyes. Because
+    of disparity, the spatial phases of the image of a grating in the
+    two eyes, PhiL and PhiR, may be different. We write the spatial part of
+    the linear response estimate for a grating with the preferred spatial
+    frequency (k = K) and orientation (Psi = theta = 0) as
+    
+    L1 = A/2 * (cos(PhiL) + cos(PhiR)) ,
+    
+    assuming that phi = 0 (this equation is a generalization of equation
+    2.34). Let the complex cell response be proportional to L2,1 + L2,2,
+    where L2 is similar to L1 but with the cosine functions replaced by
+    sine functions. Show that the response of this neuron is tuned to the
+    disparity, PhiL - PhiR, but is independent of the absolute spatial phase
+    of the grating, PhiL + PhiR. Plot the response tuning curve as a function
+    of disparity. (See DeAngelis, GC, Ohzawa, I, & Freeman, RD (1991)
+    Depth is encoded in the visual cortex by a specialized receptive field
+    structure. Nature 352:156–159.)
     '''
 
 def Ex14():
     '''
+    Determine the selectivity of the LGN receptive field of equation 2.45
+    to spatial frequency by computing its integrals when multiplied by
+    the stimulus
+    
+    s = cos(Kx)
+    
+    for a range of K values. Use sigmac = 0.3degrees, sigmas = 1.5degrees, B = 5, 1/alpha = 16
+    ms, and 1/Beta = 64 ms, and plot the resulting spatial frequency tuning
+    curve.
     '''
 
 def Ex15():
     '''
+    Construct the Hubel-Wiesel model of a simple-cell spatial receptive
+    field, as depicted in figure 2.27A. Use difference-of-Gaussian functions
+    (equation 2.45) to model the LGN receptive fields. Plot the
+    spatial receptive field of the simple cell constructed by summing
+    the spatial receptive fields of the LGN cells that provide its input.
+    Compare the result of summing appropriately placed LGN centersurround
+    receptive fields (figure 2.27A) with the results of an appropriately
+    adjusted Gabor filter model of the simple cell that uses the
+    spatial kernel of equation 2.27.
     '''
 
 def Ex16():
     '''
+    Construct the Hubel-Wiesel model of a complex cell, as depicted in
+    figure 2.27B. UseGabor functions (equation 2.27) to model the simple
+    cell responses, which should be rectified before being summed. Plot
+    the spatial receptive field of the complex cell constructed by summing
+    the different simple cell responses. Compare the responses of a complex
+    cell constructed by linearly summing the outputs of simple cells
+    (figure 2.27B) with different spatial phase preferences with the complex
+    cell model obtained by squaring and summing two unrectified
+    simple cell responses with spatial phases 90degrees apart as in exercise 8.
     '''