Dynamical Systems in Neuroscience:

Bursting 389spike synchronizationburst synchronizationFigure 9.51: Various regimes of synchronization of bursters.neurons but not for others. In contrast, other topological types of bursters have widelyvarying instantaneous interspike frequencies, as in Fig. 9.49a, that scan or sweep abroad frequency range going all the way to zero.When the bifurcation from resting to spiking state is of the saddle-node on invariantcircle type, i.e., the system is Class 1 excitable, the frequency of emerging spiking isfirst small, and then increases. Therefore, all “circle/*” bursters generate chirps withinstantaneous interspike frequencies increasing from zero to a relatively large value,at least at the beginning of the burst. Similarly, when the bifurcation of the spikingstate is of the saddle-node on invariant circle or saddle homoclinic orbit type, thefrequency of spiking at the end of the burst decreases to zero, so all “*/circle” and“*/homoclinic” bursters also generate chirps, as in Fig. 9.49a. In summary, all shadedbursters in Fig. 9.50 have sweeping interspike frequencies, so that one part of the burstis resonant for one neuron and another part of the same burst is resonant for anotherneuron.9.4.5 SynchronizationConsider two coupled bursting neurons of the fast-slow type. Since each burster hastwo times scales, one for rhythmic spiking and one for repetitive bursting, there aretwo synchronization regimes:• Spike synchronization, as in Fig. 9.51, left.• Burst synchronization, as in Fig. 9.51, right.One of them does not imply the other. Of course, there is an additional regime whenspikes and bursts are synchronized. We will study synchronization phenomena in detailin Chap. 10; here we just mention how they depend on the topological type of bursting.Let us consider spike synchronization first. Since we are interested in the fast timescale, we neglect the slow variable dynamics for a while and treat two bursters ascoupled oscillators. A necessary condition for synchronization of two weakly coupledoscillators is that they have nearly equal frequencies. How near is “near” dependson the strength of the coupling. Thus, spike synchronization depends crucially on