Add files via upload

Ch7 Exercises.py

@@ -16,31 +16,180 @@
 #----------------------------Exercises------------------------------------
 def Ex1():
     '''
+    a) Consider network activities v(theta) that are steady-state solutions of
+    equation 7.36, satisfying
+    
+    v(theta) = [ h(theta) + pi/2[integral]-pi/2 ( (dtheta'/pi) * (-gamma0 +
+    gamma1*(cos(2(theta - theta')))*v(theta'))) ]
+    
+    in response to input h(theta) = Ac(1 - q + q cos(2theta)) as in equation
+    7.37. Assuming that v(theta) is symmetric about theta = 0, show that
+    v(theta) takes either the form
+    
+    v(theta) = alpha [cos(2theta) - cos(2thetaC)]+
+    
+    or the form
+    
+    v(theta) = alpha cos(2theta) + vtheta
+    
+    In the case of equation 2,which applieswhen thetaC < pi/2 and forwhich
+    thetaC defines the width of the orientation tuning curve, by calculating
+    the integral
+    
+    pi/2[integral]-pi/2 (dtheta'/pi) (-gamma0 + gamma1*(cos(2(theta - 
+    theta')))*v(theta'))
+    
+    show that alpha and thetaC must satisfy the consistency conditions
+    
+    alpha = (A*c*epsilon) / (1 - lambda1 *(thetaC - sin(4thetaC))/pi
+    
+    cos(2thetaC) = (lambda0/pi) ( sin(2thetaC)) - 2thetaC(cos(2thetaC)) - ( 
+    (1-epsilon) / epsilon) (1 - (lambda1/pi)(thetaC - ( (sin(4thetaC)) / 4))
+    
+    
+    b) In the case of equation 3, calculate alpha and v0.
+    
+    c) For values lambda0 = 7.3, lambda1 = 11, c = 1, and A = 40 Hz, use the
+    MATLAB® function fzero to find the value of thetaC that satisfies the
+    consistency condition in equation 4 as a function of q for 0 < q =< 1.
+    For q = 0.1 and c = 0.1, 0.2, 0.4, and 0.8, solve for alpha, and thereby
+    reproduce figure 7.10B. Repeat the plots for lambda1 = 0. At what value
+    of q does thetaC fall below pi/2. This corresponds to a model in which
+    feedforward orientation tuning is sharpened only by inhibition, and
+    the model lacks contrast invariant tuning.
+    
+    d) Numerically integrate equation 7.36 for the sets of parameters in 
+    (c) to confirm your results. Use 100 neurons with preferred values
+    evenly spaced between -pi/2 and pi/2.
+    
+    e) Plot thetaC - sin(4thetaC)/4 for 0 =< thetaC =< pi/2. What is its maximum
+    value? As q -> 0 (so that (1 - q)/q -> infintity), calculate (from
+    equation 4) a condition on lambda1 that ensures there will always be a
+    solution with thetaC < pi/2. This defines a marginal phase in which the
+    recurrent connections create a tuned output even from untuned input, and
+    it constitutes what is called a continuous attractor.
     '''
 
 def Ex2():
     '''
+    A Hopfield associativememory network has activities for individual
+    units, va for a = 1, 2, . . . ,N (or collectively v), that take values of 
+    either +1 or -1, and are updated at every discrete time step of the network
+    dynamics by the rule
+    
+    va(t + 1) = sgn( SIGMA(Maa' * va' (t)))
+    
+    where
+    
+    sgn(z) =  (+1 if z >= 0
+               -1 if z < 0 
+              
+    Here M is a matrix constructed from P “memory” vectors vm (m =
+    1, 2, . . .P), also composed of elements that are either +1 or -1, through
+    the sum of outer products
+    
+    Maa' = (1 - Delta,a,a') SIGMA(VamVam)
+    
+    Note that the diagonal elements ofMare set to zero by this equation.
+    Consider a 100-element network (N = 100). Construct P memory
+    states by randomly assigning +1 and -1 values with equal probabilities
+    to the N elements of each vm. Using these memory vectors, set
+    the matrix of synaptic weights according to equation 6. Then, study
+    the behavior of the network by iterating equation 5. Tomeasure how
+    close the state of the network at time t, v(t), is to a particularmemory
+    state, define the overlap function q(t) = v(t) * vm/N. This is equal to 1
+    if v(t) = vm, is near zero if v(t) is unrelated to vm, and is equal to -1 if
+    v(t) = -vm. Set the initial state v(0) so that it has a positive overlap,
+    q(0),with memory state v1. Plot q(t) as the network evolves fromthis
+    state according to equation 5. Final values of q(t) near one indicate
+    successful recovery of the memory. Do the same starting from v(0)
+    close to the inverse of the memory state -v1. What accounts for this
+    behavior? Determine the range of q(0) values (about v1) that assures
+    successfulmemory recovery for different values of P. Startwith P = 1
+    and increase it untilmemory recovery fails even for q(0) = 1. Atwhat
+    P value does this occur?
     '''
 
 
 def Ex3():
     '''
+    Repeat exercise 2 with the matrixM replaced by
+    Maa' = (1 - deltaaa' )
+
+    Maa' = (1 - deltaaa') Sigma(Vam * Cmm' * Va'm')
+
+    where Cmm' is the m,m' element of the inverse of the matrix
+    
+    Sigma(Vam*Vam')
+    
+    Compare the performance and capacity of the associative memory
+    constructed using this matrix with that of the associative memory in
+    exercise 2.
     '''
 
 def Ex4():
     '''
+    Build and study a simple model of oscillations arising from the interaction
+    of excitatory and inhibitory populations of neurons. The
+    firing rate of the excitatory neurons is rE, and that of the inhibitory
+    neurons is rI and these are characterized by equations 7.50 and 7.51.
+    Set MEE = 1.25, MIE = 1, MII = 0, MEI = -1, gammaE = -10 Hz, gammaI = 10
+    Hz, tauE = 10 ms, and vary the value of tauI. The negative value of gammaE
+    means that this parameter serves as a source of background activity
+    (activity in the absence of excitatory input) rather than as a threshold.
+    Show what happens for tauI = 30ms and for tauI = 50 ms. Find the value
+    of tauI for which there is a transition between fixed-point and oscillatory
+    behavior, thereby verifying the results obtained analytically in
+    chapter 7 on the basis of equation 7.53.
     '''
 
 def Ex5():
-    '''
+    '''MATLAB files c7p5.m and c7p5sub.m perform a numerical integration
+    of a two-unit, nonlinear, symmetric recurrent network with a
+    threshold linear activation function F(I) = beta[I]+ and plot the results.
+    Here, the dynamics come from
+    
+    dv/dt = -v + F(M * v + h)
+    
+    with v = (v1, v2) and h1 = h2 = 1. The weight matrix in this example
+    isM = [0 -1 ; -1 0],which tends tomake v1 and v2 compete. Execute
+    c7p5.m to see the consequences of regimes of high (beta = 2) and low
+    (beta = 0.5) activation (which is equivalent to large and small recurrent
+    weights). For these two values of beta, plot the nullclines (the locations
+    in the v1-v2 phase planewhere dv1/dt = 0 and dv2/dt = 0). You should
+    find one fixed point for beta = 0.5 and three for beta = 2. Linearize the
+    network about the fixed point with v1 = v2 and derive a condition on
+    beta for this fixed point to be stable. (Based on a problem from Dawei
+    Dong.)
     '''
 
 def Ex6():
     '''
+    Plot the results of exercise 5 for the inputs h = (0.75, 1.25) and h =
+    (0.5, 1.5). By plotting nullclines for these values of h, explain the
+    resulting behavior. (Based on a problem from Dawei Dong.)
     '''
 
 def Ex7():
     '''
+    Use the expression
+    
+    fu(s - epsilon, g - mu) = A * e** ( -(s-epsilon)**2 / 2sigmas**2) N ( (g
+    - gamma) / sigmag)
+    
+    where A, epsilon, sigmas, mu, and sigmag are parameters and N is the
+    (sigmoidal) cumulative normal function
+    
+    N(z) = integral[z, -infinity] (dx (1/root(2pi)) e**(-x**2 / 2 ) = 1 -
+    (-1/2) erfc (z / root(2) )
+    
+    Plot fu(s - epsilon, g - mu) and find values of the parameters that make it
+    roughly match the gain-modulated response of figure 7.6B. Using
+    w(epsilon, mu) = exp(-(epsilon + ,mu)2/2sigma2
+    w), evaluate the integral in equation 7.15
+    in terms of a single cumulative normal function to show that the
+    resulting tuning curves are functions of s + g, and assess how the
+    tuning width depends on sigmas, sigmag and sigmaw.
     '''