Dynamical Systems in Neuroscience:

496 Synchronization (see www.izhikevich.com)nI Na +I K -modelV(t)Vquadratic integrate-and-fire modelx(t)=-coth 2 (t-T)resetspikex reset-1 0 +1xresetresetresetFigure 10.38: Top: Periodic spiking of the I Na +I K -neuron near saddle-node homoclinicorbit bifurcation (parameters as in Fig. 4.1a with τ(V ) = 0.167 and I = 4.49). Bottom:Spiking in the corresponding quadratic integrate-and-fire model.and it can be computed explicitly for some simple p(t).The in-phase synchronized solution, χ = 0, is stable whenH ′ (0) = 1 π∫ π0sin 2 t p ′ (t) dt > 0 .Since the function sin 2 t depicted in Fig. 10.37 is small at the ends of the interval andlarge in the middle, the integral is dominated by the sign of p ′ in the middle. Fast risingand slowly decaying excitatory (p > 0) synaptic transmission has p ′ < 0 in the middle,as in the figure, so the integral is negative and the in-phase solution is unstable. Incontrast, fast rising slowly decaying inhibitory (p < 0) synaptic transmission has p ′ > 0in the middle, so the integral is positive and the in-phase solution is stable. Anotherway to see this is to integrate the equation by parts, reduce it to − ∫ p(t) sin 2t dt,and notice that p(t) is concentrated in the first (left) half of the period where sin 2tis positive. Hence, positive (excitatory) p results in H ′ (0) < 0, and vice versa. Bothapproaches confirm the theoretical results independently obtained by van Vreeswijk etal. (1994) and Hansel et al. (1995) that inhibition, not excitation synchronizes Class1 (SNIC) oscillators. The relationship is inverted for the anti-phase solution χ = π/2(prove it), and no relationships is known for other types of oscillators.10.4.3 Homoclinic oscillatorsBesides the SNIC bifurcation considered above, low frequency oscillations may also beindicative of the proximity of the system to a saddle homoclinic orbit bifurcation, as in