Dynamical Systems in Neuroscience:

426 Solutions to Exercises, Chap. 611. The projection onto the v 1 -axis is described by the equationẋ = λ 1 x , x(0) = a .The trajectory leaves the square when x(t) = ae λ1t = 1; that is, whent = − 1 ln a = − 1 ln τ(I − I b ) .λ 1 λ 112. Equation (6.13) has two bifurcation parameters, b and v, and the saddle-node homoclinicbifurcation occurs when b = b sn and v = V sn . The saddle-node bifurcation curve is the straightline b = b sn (any v). This bifurcation is on an invariant circle when v < V sn and off otherwise.When b > b sn , there are no equilibria and the normal form exhibits periodic spiking. Whenb < b sn , the normal form has two equilibria,node{ }} {V sn − √ c|b − b sn |/aandsaddle{ }} {V sn + √ c|b − b sn |/a .The saddle homoclinic orbit bifurcation occurs when the voltage is reset to the saddle, i.e.,whenv = V sn + √ c|b − b sn |/a .13. The Jacobian matrix at an equilibrium is( 2v −1L =ab −a).Saddle-node condition det L = −2va + ab = 0, results in v = b/2. Since v is an equilibrium, itsatisfies v 2 − bv + I = 0, hence b 2 = 4I. Andronov-Hopf condition tr L = 2v − a = 0 results inv = a/2, hence a 2 /4 − ab/2 + I = 0. The bifurcation occurs when det L > 0, resulting in a < b.Combining the saddle-node and Andronov-Hopf conditions results in the Bogdanov-Takensconditions.14. Change of variables (6.5), v = x and u = √ µy, transforms the relaxation oscillator into theformẋ = f(x) − √ (µyfẏ = √ with the Jacobian L =′ (b) − √ )µ√µ(x − b)µ 0at the equilibrium v = x = b, u = √ µy = f(b). The Andronov-Hopf bifurcation occurs whentr L = f ′ (b) = 0 and det L = µ > 0. Using (6.7), we find that it is supercritical when f ′′′ (b) < 0and subcritical when f ′′′ (b) > 0.15. The Jacobian matrix at the equilibrium, which satisfies F (v) − bv = 0, is( )F′−1L =.µb −µThe Andronov-Hopf bifurcation occurs when tr L = F ′ − µ = 0 (hence F ′ = µ) and det L =ω 2 = µb − µ 2 > 0 (hence b > µ). The change of variables (6.5), v = x and u = µx + ωy,transforms the system into the formẋ = −ωy + f(x)ẏ = ωx + g(x)where f(x) = F (x) − µx and g(x) = µ[bx − F (x)]/ω. The result follows from (6.7).