Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 501restinga 2 'slow variable, yb 2a 1slownullcline, g=0spikingX equiv (y, e)b 1 'C 2 = |g(a 2 ')||g(b 2 )|C 1 = |g(a 1 )||g(b 1 ')|y(t)X equiv (y, 0)fast variable, xFigure 10.41: Reduction of the I Na,p +I K +I K(M) -burster to a relaxation oscillator. Theslow variable exhibits “scalloped” oscillations needed for stability of in-phase burstsynchronization. C 1 and C 2 are compression functions at the two jumps.Burst synchronizationIn Chap. 9 we presented two methods, averaging and equivalent voltage, that removefast oscillations and reduce bursters to slow relaxation oscillators. Burst synchronizationis then reduced to synchronization of such oscillators, and it can be studied usingphase reduction or fast threshold modulation (FTM) approaches.To apply FTM, we assume that the coupling in (10.27) is piece-wise constant, thatis, p(x i , x j ) = 0 when the presynaptic burster x j is resting, and p(x i , x j ) = 1 (or anyfunction of x i ) when the presynaptic burster is spiking. We also assume that the slowsubsystem (10.28) is one-dimensional so that we can use the equivalent voltage method(Sect. 9.2.4) and reduce the coupled system to0 = X equiv (y i , εp) − x i ,y ′ i = g(x i , y i ) .When the burster is of the hysteresis loop type, i.e., the resting and spiking statescoexist, the function x = X equiv (y, εp) often, but not always, has a Z-shape on theslow/fast plane, as in Fig. 9.16, so that the system corresponds to a relaxation oscillatorwith nullclines as in Fig. 10.41. Fast threshold modulation occurs via the constant εp,which shifts the fast nullcline up or down. The compression criterion for stability ofthe in-phase burst synchronization, presented in the previous section, has a simplegeometrical illustration in the figure. The slow nullcline has to be sufficiently closeto the jumping points so that y(t) slows down before each jump and produces the“scalloped” shape curve. Many hysteresis loop fast/slow bursters do generate suchshape. In particular, “fold/*” bursters exhibit robust in-phase burst synchronizationwhen they are near the bifurcation from quiescence to bursting, since the slow nullclineis so close to the left knee that the compression during the resting → spiking jump (C 1in Fig. 10.41) dominates the expansion, if any, during the spiking → resting jump.