Let there be two identical, symmetrically coupled oscillators, denote the phase as [tex]\phi_1, \phi_2[/tex]. Let us denote the phase response curve (PRC) as [tex]Z(\phi)[/tex]. The PRC represents the change in phase for infinitesimal perturbation at phase [tex]\phi[/tex].

If the coupling is pulse coupled with some sort of transfer function [tex]\p(\phi, \Delta\phi)[/tex], the effective coupling function (or effective PRC in [3]), [tex]\Gamma(\Delta\phi) = \frac{1}{2\pi} \int_0^{2\pi} Z(\phi) p(\phi, \Delta\phi) d\phi[/tex].

The phase dynamics can be described by,

[tex]\frac{d\phi_1}{dt} = \omega + \Gamma(\Delta\phi)[/tex],

[tex]\frac{d\phi_2}{dt} = \omega + \Gamma(-\Delta\phi)[/tex].

Now, in case of phase synchrony, [tex]\frac{d\phi_1}{dt} = \frac{d\phi_2}{dt}[/tex].

Therefore,

[tex]\Gamma_{-}(\Delta\phi) = \Gamma(\Delta\phi)-\Gamma(-\Delta\phi)[/tex].

is the key function to consider. If this function is 0 at 0, it means there exist a stable synchronizing solution. And 0 at other phase differences implies phase synchrony.

Note that this function has the same sign and solution as the odd portion of [tex]\Gamma(\Delta\phi)[/tex].

As [tex]\Delta\phi \rightarrow 0[/tex], the alternative expression,

[tex](\Gamma(\Delta\phi)-\Gamma(0))-(\Gamma(-\Delta\phi)-\Gamma(0))[/tex],

says the second derivative of [tex]\Gamma(\Delta\phi)[/tex] has the same sign and solution.

[1] Dong-Uk Hwang, Sang-Gui Lee, Seung K Han, Hyungtae Kook, "Phase-model analysis of coupled neuronal oscillators with multiple connections", Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), Vol. 74, No. 3. (2006)

[doi][2] Hyungtae Kook , Sang-Gui Lee , Dong-Uk Hwang and Seung Kee Han
Synchronization of a Neuronal Oscillator Network with multiple connections of time delays Journal of the Korean Physical Society, 2007 50:341-345.

[3] Hideyuki Cateau, Katsunori Kitano, Tomoki Fukai, "Interplay between a phase response curve and spike-timing-dependent plasticity leading to wireless clustering",
Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), Vol. 77, No. 5. (2008)

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