Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 507d=-2d=+2s (-1+di)1100-1-1s (-1+di)-1 0 1-1 0 1Figure 10.45: Isochrons of the Andronov-Hopf oscillator; see Ex. 2.cos sin cos sinAz(t)Figure 10.46: Pulsed stimulation of theAndronov-Hopf oscillator in Fig. 10.3;see Ex. 4.3. [MATLAB] To determine isochrons of an oscillator ẋ = F (x), one can start withmany initial points near the limit cycle and integrate the system backwards, i.e.ẋ = −F (x). The images of the points at any time t lie on the same isochron.Write a MATLAB program that implements this algorithm.4. Prove that the phase response curve of the Andronov-Hopf oscillator in Fig. 10.3is{ −ψ when 0 ≤ ϑ ≤ π,PRC (ϑ, A) =(10.29)+ψ when π ≤ ϑ ≤ 2π,whereψ = arcos1 + A cos ϑ√1 + 2A cos ϑ + A2and A is the magnitude of the horizontal displacement of z(t); see Fig. 10.46.5. [MATLAB] Write a program that stimulates an oscillator at different phases anddetermines its phase response curve (PRC).6. Show that Z(ϑ) = grad ϑ, so that Winfree’s phase model (10.6) is equivalent toKuramoto’s phase model (10.8).7. Show that Z(ϑ) = Q(ϑ), so that Winfree’s phase model (10.6) is equivalent toMalkin’s phase model (10.9).