Dynamical Systems in Neuroscience:

438 Solutions to Exercises, Chap. 94slow variable u220-2averagedfull-4-4 -2 0 2 4slow variable u 1Figure 10.40: Ex. 18.The interspike period, T , is defined by v(T ) = +∞, given by the formulaT (u) = √ 1 ( πu 2 + atan √ 1 ). uThe result follows from the integraland the relationships1T (u)∫ T (u)0d i δ(t − T (u)) dt = d 1T (u)f(u) = 1T (u)and atan 1 √ u= arcot √ u .Periodic solutions of the averaged system (focus case) and the full system are depicted inFig. 10.40. The deviation is due to the finite size of the parameters µ 1 and µ 2 in Fig. 9.35.19. There are only two co-dimension-1 bifurcations of an equilibrium that result in transitions toanother equilibrium: saddle-node off limit cycle and subcritical Andronov-Hopf bifurcation.Hence, there are four point-point hysteresis loops, depicted in Fig. 10.41. More details areprovided in Izhikevich (2000).20. This figures are modified from (Izhikevich 2000), where one can find two models exhibitingthis phenomenon. The key feature is that the slow subsystem is not too slow, and the rateof attraction to the upper equilibrium is relatively weak. The spikes are actually dampedoscillations that are generated by the fast subsystem while it converges to the equilibrium.Periodic bursting is generated via the point-point hysteresis loop.21. There are only two co-dimension-1 bifurcations of a small limit cycle attractor (subthresholdoscillation) on a plane that result in sharp transitions to a large-amplitude limit cycle attractor(spiking): Fold limit cycle bifurcation and saddle-homoclinic orbit bifurcation; see Fig. 10.42.These two bifurcations paired with any of the four bifurcations of the large-amplitude limitcycle attractor result in 8 planar co-dimension-1 cycle-cycle bursters; see Fig. 10.43. Moredetails are provided by Izhikevich (2000).