Dynamical Systems in Neuroscience:

494 Synchronization (see www.izhikevich.com)Weak pulsesNow consider two weakly pulse-coupled oscillatorsx ′ 1 = 1 + x 2 1 + ε 1 δ(t − t 2 ) ,x ′ 2 = 1 + x 2 2 + ε 2 δ(t − t 1 ) ,where t 1 and t 2 are the moments of firing (x(t) = ∞) of the first and the secondoscillator, respectively, and ε 1 and ε 2 are the strengths of synaptic connections. Thecorresponding phase model (10.12) has the formSinceH(χ) = 1 πϑ ′ 1 = 1 + ε 1 (sin 2 ϑ 1 )δ(t − t 2 )ϑ ′ 2 = 1 + ε 2 (sin 2 ϑ 2 )δ(t − t 1 ) .∫ π0sin 2 t δ(t + χ) dt = 1 π sin2 χ ,the corresponding phase deviation model (10.15) isϕ ′ 1 = ε 1π sin2 (ϕ 2 − ϕ 1 ) ,ϕ ′ 2 = ε 2π sin2 (ϕ 1 − ϕ 2 ) .The phase difference χ = ϕ 2 − ϕ 1 satisfies the equationχ ′ = ε 2 − ε 1πsin 2 χ,which becomes χ ′ = 0 when the coupling is symmetric. In this case, the oscillatorspreserve (on average) the initial phase difference. When ε 1 ≠ ε 2 , the in-phase synchronizedstate χ = 0 is only neutrally stable. Interestingly, it becomes exponentiallyunstable in a network of three or more pulse-coupled Class 1 oscillators; see Ex. 23.Weak pulses with delaysThe synchronization properties of weakly pulse-coupled oscillators could change significantlywhen explicit axonal conduction delays are introduced. As an example, considerthe systemx ′ 1 = 1 + x 2 1 + εδ(t − t 2 − d) ,x ′ 2 = 1 + x 2 2 + εδ(t − t 1 − d) ,where d ≥ 0 is the delay. Ex. 18 shows that delays introduce simple phase shifts, sothat the phase model has the formϕ ′ 1 = ε π sin2 (ϕ 2 − ϕ 1 − d) ,ϕ ′ 2 = ε π sin2 (ϕ 1 − ϕ 2 − d) ,