Dynamical Systems in Neuroscience:

Solutions to Exercises, Chap. 7 42716. The change of variables (6.5) converts the system into the formThe result follows from (6.7).ẋ = F (x) + linear termsẏ = µ(G(x) − F (x))/ω + linear terms17. The change of variables (6.5), v = x and u = µx + ωy, converts the system intoThe result follows from (6.7).ẋ = F (x) − x(µx + ωy) + linear termsẏ = µ[G(x) − F (x) + x(µx + ωy)]/ω18. The system undergoes Andronov-Hopf bifurcation when F v = −µG u and F u G v < −µG 2 u. Weperform all the steps from (6.4) to (6.7) disregarding linear terms (they do not influence a)and the terms of the order o(µ). Let ω = √ −µF u G v + O(µ), then u = (µG u x − ωy)/F u =−ωy/F u + O(µ), andandf(x, y) = F (x, −ωy/F u + O(µ)) = F (x, 0) − F u (x, 0)ωy/F u + O(µ)The result follows from (6.7).g(x, y) = (µ/ω)[G u F (x, 0) − F u G(x, 0)] + O(µ) .Solutions to Chapter 71. Take c < 0 so that the slow w-nullcline has a negative slope.2. The quasi-threshold contains the union of canard solutions.3. The change of variables z = e iωt u transforms the system into the formwhich can be averaged, yielding˙u = ε{−u + e −iωt I(t)} ,˙u = ε{−u + I ∗ (ω)} .Apparently, the stable equilibrium u = I ∗ (ω) corresponds to the sustained oscillation z =e iωt I ∗ (ω).4. The existence of damped oscillations with frequency ω implies that the system has a focusequilibrium with eigenvalues −ε ± iω, where ε > 0. The local dynamics near the focus can berepresented in the form (7.3). The rest of the proof is the same as the one for Ex. 3.5. Even though the slow and the fast nullclines in Fig. 5.21 intersect only in one point, theycontinue to be close and parallel to each other in the voltage range 10 mV to 30 mV. Such aproximity creates a tunneling effect (Rush and Rinzel 1996) that prolongs the time spent nearthose nullclines.6. (Shilnikov-Hopf bifurcation) The model is near a co-dimension-2 bifurcation having a homoclinicorbit to an equilibrium undergoing subcritical Andronov-Hopf bifurcation, as we illustratein Fig. 10.28. Many weird phenomena could happen near bifurcations of co-dimension 2or higher.