Dynamical Systems in Neuroscience:

Bursting 399˙u = µ(u − u 3 − w) ,ẇ = µ(|z| 2 − 1) ,is a slow-wave burster, even though the slow subsystem cannot oscillate for anyfixed value of the fast subsystem z.13. (Ermentrout and Kopell 1986) Consider the system˙ϑ = 1 − cos ϑ + (1 + cos ϑ)r(ψ) ,˙ψ = ω ,with ϑ and ψ being phase variables on the unit circle S 1 and r(ψ) being anycontinuous function that changes sign. Show that this system exhibits burstingactivity when ω is sufficiently small but positive. What type of bursting is that?14. Prove that the canonical model for “subHopf/fold cycle” bursting (9.11) exhibitssustained bursting activity when 0 < a < 1. What happens when a approaches0 or 1?15. Show that the canonical model for “fold/homoclinic” bursting (9.7) is equivalentto a simpler model (eq.27 in Izhikevich 2000 and Chap. 8)˙v = v 2 + w ,ẇ = µ ,with after-spike (v = +∞) resetting v ← 1 and w ← w − d, when I is sufficientlylarge and µ and d are sufficiently small.16. Derive the canonical model for “fold/homoclinic” bursting (9.7) assuming thatthe fast subsystem is near a saddle-node homoclinic orbit bifurcation point atsome u = u 0 , which is an equilibrium of the slow subsystem.17. Derive the canonical models (9.9) and (9.10) for “circle/circle” bursting.18. Show that the averaged slow subsystems of the canonical models for “circle/circle”bursters (9.9) and (9.10) have the formandrespectively, where˙u 1 = −µ 1 u 1 + d 1 f(I + u 1 − u 2 ) ,˙u 2 = −µ 2 u 2 + d 2 f(I + u 1 − u 2 ) ,˙u 1 = −µ 1 u 2 + d 1 f(I + u 1 ) ,˙u 2 = −µ 2 (u 2 − u 1 ) + d 2 f(I + u 1 ) ,f(u) =√ uπ/2 + arcot √ uis the frequency of spiking of the fast subsystem (in Hz).