Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 503in-phase anti-phase out-of-phaseBibliographical NotesFigure 10.42: Different types of synchronization.Surprisingly, this chapter turned out to be quite different from Chap. 9 (Weakly ConnectedOscillators) of the book Weakly Connected Neural Networks by Hoppensteadtand Izhikevich (1997) or from the book Synchronization: A universal concept in nonlinearsciences by Pikovsky, Rosenblum and Kurths. All three texts, though devotedto the same subject, do not repeat, but rather complement each other. The latter providesan excellent historical overview of synchronization, starting with the work of thefamous Dutch mathematician, astronomer, and physicist Christiaan Huygens (1629–1695), who was the first one to describe synchronization of a couple of pendulum clockshanging from a common support, which was incidentally anti-phase. While providingmany examples of synchronization of biological, chemical, and physical systems, thebook by Pikovsky et al. also discusses the definition of a phase and synchronization ofnon-periodic, e.g., chaotic, oscillators, a topic not covered here. A major part of SpikingNeuron Models by Gerstner and Kistler (2002) is devoted to synchronization ofspiking neurons, with the emphasis on the integrate-and-fire model and spike-responsemethod.The formalism of phase response curve (PRC) was introduced by Hastings andSweeney (1958), and it has been used extensively in the context of resetting the circadianrhythms. Forty Years of PRC — What Have We Learned? by Johnson (1999)gives an historical overview of this idea and some recent developments. John Guckenheimer(1975) used the theory of normally hyperbolic invariant manifolds to provide amathematical foundation for the existence of isochrons, and their geometrical properties.An encyclopedic exposition of isochrons and phase resettings in nature, as well asnumerous anecdotes, can be found in Arthur Winfree’s remarkable book The Geometryof Biological Time (first edition 1980, second edition 2001). In particular, Winfree describesthe work of George R. Mines (1914) who was doing phase resetting experimentsby shocking rabbits at various phases of their heartbeat. He found the phase and shockthat could stop a rabbit’s heart (black hole in Fig. 10.8), and then applied it to himself.He died.Glass and MacKey (1988) provide a detailed exposition of circle phase maps. Althoughthe structure of phase-locking regions in Fig. 10.15 was discovered by Cartwrightand Littlewood (1945), it is better known at present as Arnold tongues (Arnold 1965).