Dynamical Systems in Neuroscience:

356 Bursting0maxmembrane potential, V (mV)-30-60-65minspikingV spike (t,n slow )V rest (n slow )-70averaged function g0.0150.0100.0050-0.005-0.02spikingg(n slow )resting0 0.02 0.04 0.06 0.08 0.1slow K + gating variable, n slowFigure 9.14: Spiking solutions V (t) = V spike (t, u slow ), resting membrane potential V =V rest (n slow ), and the reduced slow subsystem ṅ slow = ḡ(n slow ) of the I Na,p +I K +I K(M) -model. The reduction is not valid in the shaded regions.an infinite spike train of the fast subsystem when u is frozen. Slices of this function areshown in Fig. 9.14, top. Let T (u) be the period of spiking oscillation. The periodicallyforced slow subsystem˙u = µg(x spike (t, u), u) (slow subsystem) (9.3)can be averaged and reduced to a simpler modelẇ = µḡ(w) (averaged slow subsystem) (9.4)by a near-identity change of variables w = u + o(µ), where o(µ) denotes small terms oforder µ or less. Hereḡ(w) = 1T (w)∫ T (w)0g(x spike (t, w), w) dtis the average of g, shown in Fig. 9.14, bottom, for the I Na,p +I K +I K(M) -model. Checkthat ḡ(w) = g(x rest (w), w) when the fast subsystem is resting. Limit cycles of the