Dynamical Systems in Neuroscience:

510 Synchronization (see www.izhikevich.com)20. Consider the Kuramoto modelϕ i = ω +n∑c ij sin(ϕ j + ψ ij − ϕ i ) ,j=1where c ij and ψ ij are parameters. What can you say about its synchronizationproperties?21. Derive the self-consistency equation (10.22) for the Kuramoto model (10.20).22. Consider the phase deviation modelϕ ′ 1 = ω + c 1 H(ϕ 2 − ϕ 1 )ϕ ′ 2 = ω + c 2 H(ϕ 1 − ϕ 2 )with an even function H(−χ) = H(χ). Prove that the in-phase synchronizedstate, ϕ 1 = ϕ 2 , if it exists, cannot be exponentially stable. What can you sayabout the anti-phase state ϕ 1 = ϕ 2 + T/2?23. Prove that the in-phase synchronized state in a network of three or more pulsecoupledquadratic integrate-and-fire neurons is unstable.24. Prove (10.25).25. (Brown et al. 2004) Show that PRC for an oscillator near saddle homoclinic orbitbifurcation scales as PRC (ϑ) ∼ e λ(T −ϑ) , where λ is the positive eigenvalue of thesaddle and T is the period of oscillation.26. Consider the quadratic integrate-and-fire neuron ẋ = ±1 + x 2 with the resetting“ if x = +∞, then x ← x reset ”. Prove thatregime SNIC homoclinicmodel x ′ = +1 + x 2 x ′ = −1 + x 2 , (x reset > 1)x reset x x -1 1 x resetperiod T π/2 − atan x reset acoth x resetsolution x(t) − cot(t − T ) − coth(t − T )PRC Q(ϑ) sin 2 (ϑ − T ) sinh 2 (ϑ − T )15x reset =+1.1x reset =-1.100T00T