Math 5447, Take Home Midterm, Fall 2022
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Math 5447, Take Home Midterm, Fall 2022
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1. Two linear neurons. Consider a network of two neurons, with firing rates given by x and y, respectively. If the neurons have linear dynamics, we can write the system of equations describing their activity as
dt τ北 dy 1 dt τy I北 is the input to neuron x and Iy is the input to neuron y . The coupling strength from y to x is a which could be either positive (excitatory) or nega- tive (inhibitory). Similarly, the coupling strength from x to y is b. The time constants τ北 and τy are positive numbers. |
a |
I x
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(a) General case
i. (15 points) Assuming all parameters are fixed, how many steady states does the system have? Solve for the steady state firing rates xss and yss as a function of the parameters. Which parameters determine the steady state location?
ii. (5 points) Write down the matrix that determines the stability of the steady state.
iii. (5 points) Write down expressions for the trace and determinant of that matrix.
iv. (5 points) What are conditions for the equilibrium to be asymptotically stable?
v. (10 points) Given the above assumptions on the parameter values, simplify the condition for asymptotic stability to a condition just involving a small number of parameters. Which parameters determine the stability?
(b) Mutual excitation. Let the neurons excite each other so that a > 0 and b > 0.
i. (25 points) Let I北 = 0.4, Iy = 0.1, a = 0.5, b = 1.8, τ北 = 15 ms, and τy = 25 ms. If x(0) = 0.4 and y(0) = 0.4, calculate the firing rates x(t) and y(t) for t > 0. What are the eigenvalues of the system? Classify the type of steady state.
ii. (25 points) If we increase the strength of the excitation onto neuron 1 to a = 0.6, what is the steady state? Does the steady state make sense physically? Even with τ北 and τ北 as unknown positive parameters, classify the steady state. Starting at any physically possible initial conditions, what happens to x(t) and y(t) as t increases?
(c) Mutual inhibition. (20 points) Let the neurons inhibit each other so that a < 0 and b < 0, and set the time constants to be the same τ北 = τy = τ > 0. Determine how the type of steady state depends on the values of a and b, i.e., give conditions on a and b for different types of steady states.
(d) Feedback inhibition. (20 points) Let the neuron x be excitatory and neuron y be inhibitory so that a < 0 and b > 0, and set the time constants to be the same τ北 = τy = τ > 0. No matter what the I北 and Iy are, classify the type of steady state.
2. (10 points) Solve 
= −2w − 3 subject to the condition that w(2) = −4.
3. (25 points) Solve
u\ (t) = 2 − u + 4v
v\ (t) = 3 − 2v + u
subject to the condition that u(0) = 5 and v(0) = −3.
Plot the solution (u vs. t and v vs. t) for 0 ≤ t ≤ 5
4. Consider the differential equation
dx 2
dt
dy
(a) (10 points) Determine the nullclines and plot them on the phase plane.
(b) (20 points) Determine the equilibria and classify them.
(c) (5 points) The nullclines should divide the phase plane into five regions. In each region, calculate the sign of d(d)t(北) and 
and sketch a consistent direction vector in the region.
(d) (5 points) The equilibria should divide each nullcline into three pieces. For each piece, calcu- late d(d)t(北) or 
(whichever isn’t determined by the nullcline definition) and sketch a direction vector on the nullcline piece that is consistent with your findings.
(e) (10 points) Starting with initial condition x(0) = 0.6, y(0) = 0.8, sketch a plausible solution (x(t),y(t)) on the phase plane. Also sketch this solution x(t) and y(t) versus time.
5. Two nonlinear neurons with feedback inhibition. Let x and y be the firing rate of two neurons that are coupled together in a network, as in problem number 1. This time the neurons have nonlinear dynamics, and we can write the system of equations describing their activity as
dx 1
dt τ北
dy 1
dt τy
where S(x) is a sigmoidal function with S\ (x) ≥ 0, lim北→ −∞ S(x) = 0, lim北→∞ S(x) = M . If it helps, you may think of S as a Naka-Rushton function, but your answer shouldn’t rely on that. Your answers should be in terms of the function S .
Let the neuron x be excitatory and neuron y be inhibitory so that the coupling parameters satisfy a < 0 and b > 0, and set the time constants to be the same τ北 = τy = τ > 0. No matter what the input rates I北 and Iy are, our goal is to classify the types of steady states.
(a) (25 points) Show that there is only one steady state. Hint, what does the graph of the nullcline x = S(ay + I北 ) look like? What does the graph of the nullcline y = S(bx + Iy) look like? How many times can these shapes intersect?
(b) (30 points) Show that the trace of the Jacobian matrix A of the system is negative and its determinant is positive, independent of the point where you evaluate the Jacobian matrix.
(c) (30 points) Given what you know about the trace and determinant of the Jacobian matrix, what is the sign of the real part of its eigenvalues? Is the steady state stable or unstable? Can you go further and classify the equilibrium as a saddle, node, or spiral?
Total points: 300
Show your work.
Hint: don’t lose needless points by forgetting to answer all what is asked!
2022-10-22