Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 505Figure 10.44: Nancy Kopell in her Boston University office in 1990 (photograph providedby Dr. Kopell).As a result, we avoided the annoying terms 2π/T and 2π/Ω in the formulae. The onlydrawback is that some of the results may have an “unfamiliar look”, such as sin 2 ϑ withϑ ∈ [0, π] for the PRC of Class 1 neurons, as opposed to 1 − cos ϑ with ϑ ∈ [0, 2π] usedby many authors before.Hansel, Mato, and Meunier (1995) were the first to notice that the shape of PRCdetermines the synchronization properties of synaptically coupled oscillators. Ermentrout(1996) related this result to the classification of oscillators and proved that PRCsof all Class 1 oscillators have the form of 1−cos ϑ, though the proof can be found in hisearlier papers with Kopell (Ermentrout and Kopell 1986a,b). Reyes and Fetz (1993)measured PRC of a cat neocortical neuron and largely confirmed the theoretical predictions.The experimental method in Sect. 10.2.4 is related to that of Rosenblum andPikovsky (2001). It needs to be developed further, e.g., by incorporating the measurementuncertainty (error bars). In fact, most experimentally obtained PRCs, includingthe one in Fig. 10.24, are so noisy that nothing useful could be derived from them.This issue is the subject of active research.Our treatment of the FTM theory follows closely that of Somers and Kopell (1993,1995). Anti-phase synchronization of relaxation oscillators is analyzed using phasemodels by Izhikevich (2000) and FTM theory by Kopell and Somers (1995). Ermentrout(1994) and Izhikevich (1998) considered weakly coupled oscillators with axonalconduction delays and showed that delays result in mere phase shifts (see Ex. 18).Frankel and Kiemel (1993) observed that slow coupling can be reduced to weak coupling.Izhikevich and Hoppensteadt (2003) used Malkin’s theorem to extend the resultsto slowly coupled networks, and to derive useful formulae for the coupling functions andcoefficients. Ermentrout (2003) showed that the result could be generalized to synapseshaving fast-rising and slow-decaying conductances. Goel and Ermentrout (2002) and