Dynamical Systems in Neuroscience:

498 Synchronization (see www.izhikevich.com)displaces the trajectory to the exterior of the limit cycle, as in Fig. 10.39b, pulse A,then the trajectory becomes closer to the saddle equilibrium when it reenters a smallneighborhood of the saddle, thereby leading to a significant phase delay. Displacementsto the interior of the cycle, as in Fig. 10.39b, pulse B, push away from the saddleand lead to phase advances. The direction and the magnitude of displacements isdetermined by the derivative of the slow variable n ′ along the limit cycle.The region of disagreement between theoretical and numerical PRCs becomes infinitesimalrelative to T → ∞ near the bifurcation. Theoretical PRC can be used tostudy anti-phase and out-of-phase synchronization of pulse-coupled oscillators, but notin-phase synchronization, because the region of breakdown is the only important regionin this case. Finally notice that as T → ∞, the spiking limit cycle fails to be exponentiallystable, and the theory of weakly coupled oscillators is no longer applicable toit.Though the PRC in Fig. 10.39 is quite different from the one corresponding to SNICoscillators in Fig. 10.36, there is an interesting similarity between these two cases: Bothcan be reduced to quadratic integrate-and-fire neurons, both have cotangent-shapedperiodic spiking solutions and sine-squared-shape PRCs, except they are “regular” inthe SNIC case and hyperbolic in the homoclinic case; see also Ex. 26.10.4.4 Relaxation oscillators and FTMConsider two relaxation oscillators having weak fast → fast connectionsµẋ i = f(x i , y i ) + εp i (x i , x k ) ,ẏ i = g(x i , y i ) ,(10.26)where i = 1, 2 and k = 2, 1. This system can be converted to a phase model inthe relaxation limit ε ≪ µ → 0 (Izhikevich 2000). The connection functions H i (χ)have a positive discontinuity at χ = 0, which occurs because the x-coordinate of therelaxation limit cycle is discontinuous at the jump points. Hence, the phase differencefunction G(χ) = H 2 (−χ) − H 1 (χ) has a negative discontinuity at χ = 0 depicted inFig. 10.31. This reflects the profound difference between behaviors of weakly coupledoscillators of relaxation and non-relaxation type, discussed in Sect. 10.3.1: The inphasesynchronized solution, χ = 0, in the relaxation limit µ → 0 is stable, persistentin the presence of frequency mismatch ω, and it has a rapid rate of convergence.The reduction to a phase model breaks down when ε ≫ µ → 0, that is, when theconnections are relatively strong. One can still analyze such oscillators in the specialcase considered below.Fast threshold modulationConsider (10.26) and suppose that p 1 = p 2 = p is a piece-wise constant function: p = 1when the pre-synaptic oscillator, x k , is on the right branch of the cubic x-nullclinecorresponding to an active state, and p = 0 otherwise; see Fig. 10.40a. Somers and