Dynamical Systems in Neuroscience:

514 Solutions to Exercises, Chap. 10(in the limit h → 0)(Z1 (ϑ(x))h 1=, . . . , Z )m(ϑ(x))h mh 1h m= (Z 1 (ϑ(x)), . . . , Z m (ϑ(x))) = Z(ϑ(x)) .7. (Brown et al. 2004, Appendix A) Let x be a point on the limit cycle and z be an arbitrarynearby point. Let x(t) and z(t) be the trajectories starting from the two points, and y(t) =z(t) − x(t) be the difference. All equations below are valid up to O(y 2 ). The phase shift∆ϑ = ϑ(z(t)) − ϑ(x(t)) = grad ϑ(x(t)) · y(t) does not depend on time. Differentiating withrespect to time and taking grad ϑ(x(t)) = Z(ϑ(t)) (see previous exercise), results in0 = (d/dt) (Z(ϑ(t)) · y(t)) = Z ′ (ϑ(t)) · y(t) + Z(ϑ(t)) · Df(x(t))y(t)= Z ′ (ϑ(t)) · y(t) + Df(x(t)) ⊤ Z(ϑ(t)) · y(t)()= Z ′ (ϑ(t)) + Df(x(t)) ⊤ Z(ϑ(t)) · y(t) .Since y is arbitrary, Z satisfies Z ′ (ϑ) + Df(x(ϑ)) ⊤ Z(ϑ) = 0, i.e., the adjoint equation (10.10).The normalization follows from (10.7).8. The solution to ˙v = b − v with v(0) = 0 is v(t) = b(1 − e −t ) with the period T = ln(b/(b − 1))determined from the threshold crossing v(T ) = 1. From v = b(1−e −ϑ ) we find ϑ = ln(b/(b−v)),hencePRC (ϑ) = ϑ new − ϑ = min {ln(b/(b exp(−ϑ) − A), T } − ϑ .9. The system ˙v = 1 + v 2 with v(0) = −∞ has the solution (check by differentiating) v(t) =tan(t − π/2) with the period T = π. Since t = π/2 + atan v, we findPTC (ϑ) = π/2 + atan [A + tan(ϑ − π/2)]andPRC (ϑ) = PTC (ϑ) − ϑ = atan [A + tan(ϑ − π/2)] − (ϑ − π/2) .10. The system ˙v = b + v 2 with b > 0 and the initial condition v(0) = v reset has the solution (checkby differentiating)v(t) = √ b tan( √ b(t + t 0 ))wheret 0 =1 √batan v reset√b.Equivalently,t = 1 √batan v √b− t 0 .From the condition v = 1 (peak of the spike), we findT = √ 1 atan √ 1 − t 0 = 1 (√ atan √ 1 − atan v )reset√ ,b b b b bHencePRC (ϑ) = ϑ new − ϑ = min { 1 √batan [ A √b+ tan( √ b(ϑ + t 0 ))] − t 0 , T } − ϑ .11. Let ϑ denotes the phase of oscillator 1. Let χ n denote the phase of oscillator 2 just beforeoscillator 1 fires a spike, i.e., when ϑ = 0. This spike resets χ n to PTC 2 (χ n ). Oscillator 2 firesa spike when ϑ = T 2 −PTC 2 (χ n ), and it resets ϑ to PTC 1 (T 2 −PTC 2 (χ n )). Finally, oscillator1 fires its spike when oscillator 2 has the phase χ n+1 = T 1 −PTC 1 (T 2 −PTC 2 (χ n )).