Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 50915. Use the approaches of Winfree, Kuramoto, and Malkin to transform the quadraticintegrate-and-fire neuron ˙v = 1 + v 2 + εp(t) in Ex. 9 to its phase modelwith T = π.˙ϑ = 1 + ε (sin 2 ϑ) p(t) ,16. Use the approaches of Winfree, Kuramoto, and Malkin to transform the Andronov-Hopf oscillator ż = (1 + i)z − z|z| 2 + εp(t) with real p(t) to its phase modelwith T = 2π.˙ϑ = 1 + ε (− sin ϑ)p(t) ,17. (PRC for Andronov-Hopf) Consider a weakly perturbed system near supercriticalAndronov-Hopf bifurcation (see Sect. 6.1.3)ż = (b + i)z + (−1 + di)z|z| 2 + ɛp(t) , z ∈ C .with b > 0. Let ε = b √ b/ɛ be small. Prove that the corresponding phase modelis˙θ = 1 + d + ε Im {(1 + di)p(t)e −iθ } .When the forcing p(t) is one-dimensional, i.e., p(t) = cq(t) with c ∈ C and scalarfunction q(t), the phase model has sinusoidal form˙θ = 1 + d + εs sin(θ − ψ)q(t) ,with the strength s and the phase shift ψ depending only on d and c.18. (Delayed coupling) Show that weakly coupled oscillatorsẋ i = f(x i ) + εn∑g ij (x i (t), x j (t − d ij ))with explicit axonal conduction delays d ij ≥ 0 have the phase modelj=1ϕ ′ i = ω i +n∑H ij (ϕ j − d ij − ϕ i )j≠iwhere ′ = d/dτ, τ = εt is the slow time, and H(χ) is defined by (10.16). Thus,explicit delays result in explicit phase shifts.19. Determine the existence and stability of synchronized states in the system˙ϕ 1 = ω 1 + c 1 sin(ϕ 2 − ϕ 1 )˙ϕ 2 = ω 2 + c 2 sin(ϕ 1 − ϕ 2 )as a function of the parameters ω = ω 2 − ω 1 and c = c 2 − c 1 .