Dynamical Systems in Neuroscience:

esting390 Burstingspikesynchronizationspikede-synchronizationspikingABexcitatory couplingburster BinhibitorystimulationexcitatorystimulationABinhibitory couplingFigure 9.52: Burst synchronization and de-synchronization of two coupled“fold/homoclinic” bursters (modified from Izhikevich 2000).the instantaneous interspike frequency, which may vary substantially during a burst.Indeed, a small perturbation of the slow variable may result in large perturbations ofthe interspike frequency in any shaded burster in Fig. 9.50, hence such a burster wouldbe reluctant to exhibit spike synchronization, unless the coupling is strong.Studying burst synchronization of weakly coupled neurons involves the same mathematicalmethods as studying synchronization of strongly coupled relaxation oscillators,which we consider in detail in Chap. 10. The mechanisms of synchronization dependon whether the bursting is of the hysteresis loop type or of the slow wave type, andwhether the resting state is an integrator or a resonator.In Fig. 9.52 we illustrate the geometry of burst synchronization of two coupled“fold/homoclinic” bursters of hysteresis loop type. Burster A is slightly ahead ofburster B so that A starts the spiking phase while B is still resting. If the synapticconnections between the bursters are excitatory, firing of A causes B to jump to thespiking state prematurely, thereby shortening the time difference between the bursts.In addition, the evoked burst of B is shorter, which also speeds up the synchronizationprocess. In contrast, when the connections are inhibitory, firing of A delays thetransition of B to the spiking state, thereby increasing the time difference betweenthe bursts and desynchronizing the bursters. Thus, the “fold/homoclinic” burster behavesaccording to the principle excitation means synchronization, inhibition meansde-synchronization. Since the instantaneous interspike frequency of “fold/homoclinic”bursting decays to zero, small deviations of the slow variable result in large deviationsof the period of oscillation. Typically, the periods of fast oscillations of the two bursterscan slowly diverge from each other. As a result, spikes start synchronized and thende-synchronize during the burst, as we indicate in the figure.If the bursting neuron is a resonator, i.e., it is of the “Hopf/*” or “subHopf/*” type,then both excitation and inhibition may evoke premature spiking, as we have shown