a) Interpret the terms in this model in words. b) Sketch phase portaits for the following values of I and b: I = 1, b = 2; I = 1, b = 4; I = 1, b = 8. Discuss what you observe. What type of bifurcation do you see? c) Show that there is always a symmetric fixed point at x* = x*1 = x*2 for any value of I, b. Do this by using the intermediate value theorem applied to a certain function (compare values x = 0 and x > max(1, I/b). Argue that this symmetric fixed point is unique by observing that the function from earlier is strictly increasing.
Normally when you look at something, your left and right eyes take in very similar images, and your brain blends them together, However, if totally different images are shown to each eye, the brain does something wild: it perceives one image for a brief time, then switches again and again. To model this process, we posit that there are two populations of neurons with average firing rates x1 and x2. These two populations battle for dominance, resulting in a model for neuronal competition:
x'1 = -x1 + F(I - bx2)
x'2 = -x2 + F(I - bx1)
Here F(x) = 1/(1 + e-x) is a "gain function", I > 0 is the strength of the input stimulus, and b > 0 is the strength of the antagonism between neuron groups.
a) Interpret the terms in this model in words.
b) Sketch phase portaits for the following values of I and b: I = 1, b = 2; I = 1, b = 4; I = 1, b = 8. Discuss what you observe. What type of bifurcation do you see?
c) Show that there is always a symmetric fixed point at x* = x*1 = x*2 for any value of I, b. Do this by using the intermediate value theorem applied to a certain function (compare values x = 0 and x > max(1, I/b). Argue that this symmetric fixed point is unique by observing that the function from earlier is strictly increasing.
d) Show that the Jacobian at this symmetric fixed point has delta = 1 - b2(x* - (x*)2)2 and Tau = -2. Argue that for large enough b the stability of the fixed points ches from a stable node to a saddle.
e) Does this model exhibit the binocular rivalry switching behavior observed in experiments?
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