Dynamical Systems in Neuroscience:

Bursting 3970.5x 1Figure 9.58: Hopf/Hopf burstingtwithout co-existence of attractors;see Ex. 6 (modified from Hoppensteadtand Izhikevich 1997).4. (Rebound bursting) Explain the mechanism of rebound bursting in the twodimensionalFitzHugh-Nagumo oscillator (4.11,4.12) shown in Fig. 9.57.5. Can “circle/*” and “fold/*” bursters have fast subthreshold oscillations of membranepotential? Explain.6. (Hopf/Hopf bursting) The systemẋ = (y + i)x − x|x| 2 , x = x 1 + ix 2 ∈ C ,has a unique attractor for any value of the parameter y ∈ R. Ifẏ = µ(2aS( y a − a) − |x|) , µ = 0.05 , a = √ µ/20 , S(u) =11 + e −uthen the “2+1” system above can burst, as we show in Fig. 9.58. Explore thesystem numerically and explain the origin of bursting.7. (Hopf/Hopf canonical model) Consider the “2+1” fast-slow burster (9.1) andsuppose that x 0 is the supercritical Andronov-Hopf bifurcation point of the fastsubsystem when u = u 0 . Also suppose that u 0 is a stable equilibrium of theslow subsystem when x = x 0 is fixed. Show that there is a continuous change ofvariables that transforms (9.1) into the canonical modelz ′ = (u + iω)z − z|z| 2 ,u ′ = µ(±1 ± u − a|z| 2 ) ,where z ∈ C is the new fast variable, u ∈ R is a slow variable, and ω, a and µ areparameters.8. (Bursting in the I Na,t +I Na,slow -model) Take advantage of the phenomenon ofinhibition-induced spiking described in Sect. 7.2.8 to show that a slow persistentinward current, say I Na,slow , can make a spiking model burst.9. Modify the example above to obtain repetitive bursting in a model consisting offast I Na,t current, leak current, and a slow passive dendritic compartment.