Dynamical Systems in Neuroscience:

378 Bursting(a)membrane potential, V (mV)(b)0-20-40-60fold cyclesubHopfdelayed transition-800 50 100 150 200 250 300time (ms)membrane potential, V (mV)(c)0-20-40-60-80(d)limit cycle (max)delayedtransitionsubHopflimit cycle (min)fold cycle0 0.05 0.1 0.15slow K + activation gate, n slowslow K + activation gate, nslow0.20.150.10.05fold limit cyclebifurcationsubcriticalAndronov-Hopfbifurcation00 50 100 150 200 250 300time (ms)membrane potential, V (mV)0-20-40-60-800slow K + activationgate, n slow0.10.2 00.5fast K + activationgate, n1Figure 9.41: “SubHopf/fold cycle” bursting in the I Na,p +I K +I K(M) -model. Parametersof the fast I Na,p +I K -subsystem are the same as in Fig. 6.16 with I = 55. Slow K +M-current has V 1/2 = −20 mV, k = 5 mV, τ(V ) = 60 ms and g K(M) = 1.5.the fast subsystem is unstable. If the subsystem is near such an equilibrium, it slowlydiverges from the equilibrium and jumps to the large-amplitude limit cycle attractorcorresponding to spiking behavior, as one can see in Fig. 9.41a. Each spike activatesslow K + M-current, see Fig. 9.41b, and results in build-up of a net outward current thatmakes the fast subsystem less and less excitable. Geometrically, the large-amplitudelimit cycle attractor is approached by a smaller amplitude unstable limit cycle, theycoalesce, and annihilate each other via fold limit cycle bifurcation at n slow ≈ 0.14, seeFig. 9.41c. The trajectory jumps to the stable equilibrium corresponding to the restingstate. At this moment, the slow K + current starts to deactivate and the net outwardcurrent decreases. Since the activation gate n slow moves in the opposite direction, thefold limit cycle bifurcation gives birth to large-amplitude stable and unstable limitcycles, but the trajectory remains on the steady-state branch. The unstable limitcycle slowly shrinks and makes the resting equilibrium lose stability via subcriticalAndronov-Hopf bifurcation. Once the resting state becomes unstable, the trajectorydiverges from it and jumps back to the large-amplitude limit cycle, thereby closing thehysteresis loop.