Dynamical Systems in Neuroscience:

506 Synchronization (see www.izhikevich.com)Katriel (2005) obtained many interesting results on in-phase synchronization of identicalphase oscillators.Interactions between resonant oscillators were considered by Ermentrout (1981),Hoppensteadt and Izhikevich (1997) and Izhikevich (1999) in the context of quasiperiodic(multi-frequency) oscillations. Baesens et al. (1991) undertook heroic taskof studying resonances and toroidal chaos in a system of three coupled phase oscillators.Mean-field approaches to the Kuramoto model are reviewed by Strogatz (2000)and Acebron et al. (2005). Daido (1996) extended the theory to the general couplingfunction H(χ). van Hemmen and Wreszinski (1993) were the first to find the Lyapunovfunction for the Kuramoto model, which was generalized (independently) byHoppensteadt and Izhikevich (1997) to arbitrary coupling function H(χ).Ermentrout (1985), Aronson et al. (1990) and Hoppensteadt and Izhikevich (1996,1997) studied weakly coupled Andronov-Hopf oscillators, and discussed the phenomenaof self-ignition (coupling-induced oscillations) and oscillator death (coupling-inducedcessation of oscillation). Collins and Stewart (1993, 1994) and Golubitsky and Stewart(2002) applied group theory to study synchronization of coupled oscillators in networkswith symmetries.In this chapter we consider either strong pulsed coupling or weak continuous coupling.These limitations are severe, but they allow us to derive model-independentresults. Studying synchronization in networks of strongly coupled neurons is an activearea of research, however, most of such studies fall into two categories: (1) simulationsand (2) integrate-and-fire networks. In both cases, the results are model-depended. Ifthe reader wants to pursue this line of research, he or she will definitely need to readthe following papers: Mirollo and Strogatz (1990), van Vreeswijk et al. (1994), Chowand Kopell (2000), Rubin and Terman (2000, 2002), Bressloff and Coombes (2000),van Vreeswijk (2000), van Vreeswijk and Hansel (2001), Pfeuty et al. (2003), Hanseland Mato (2003).Exercises1. Find the isochrons of the Andronov-Hopf oscillatorż = (1 + i)z − z|z| 2 , z ∈ C ,in Fig. 10.3.2. Prove that the isochrons of the Andronov-Hopf oscillator in the formż = (1 + i)z + (−1 + di)z|z| 2 , z ∈ C ,are the curvesz(s) = s (−1+di) e χi , s > 0 ,where χ is the phase of the isochron; see Fig. 10.45.