Dynamical Systems in Neuroscience:

Bursting 373fast variable, v10864200sinspikingunstableatollstablerestingphase of slow oscillation,Figure 9.34: “Circle/circle” bursting inthe Ermentrout-Kopell canonical model(9.8) with r(ψ) = sin ψ and ω = 0.1.The fast variable fires spikes whilesin ψ > 0 and quiescent while sin ψ < 0.Shaded atoll is surrounded by theequilibria curves ± √ | sin ψ|. The fastsubsystem undergoes saddle-node on invariantcircle bifurcation when sin ψ = 0.Using the averaging technique described above in Sect. 9.2.3, we can reduce the fourdimensionalI Na,p + I K + I Na,slow + I K(M) -model to a simpler, two-dimensional slowI Na,slow +I K(M) -subsystem of the form (9.4). Bursting of the full model corresponds toa limit cycle attractor of the averaged slow subsystem depicted as a bold curve onthe (m slow , n slow ) plane in Fig. 9.33c. Superimposed is the projection of the burstingsolution of the full system (thin wobbly curve). In Fig. 9.33d we project a fourdimensionalbursting trajectory onto the three-dimensional subspace (V, m slow , n slow ).The I Na,p +I K +I Na,slow +I K(M) -model in Fig. 9.33 enjoys a remarkable property: Itgenerates slow-wave bursts even though its slow I Na,slow + I K(M) -subsystem consistsof two uncoupled equations, and hence cannot oscillate by itself! Another exampleof this phenomenon is presented in Ex. 12. Thus, the slow wave that drives the fastI Na,p +I K -subsystem through the two circle bifurcations is not autonomous: it needsa feedback from V . In particular, the oscillation would disappear in a voltage-clampexperiment, i.e., when the membrane potential is fixed.Now consider a “circle/circle” burster with a slow subsystem performing smallamplitudeoscillations so that the fast subsystem is always near the saddle-node oninvariant circle bifurcation. If the slow subsystem has an autonomous limit cycle attractorthat exists without feedback from V , then such a burster can be reduced to theErmentrout-Kopell (1986) canonical model˙v = v 2 + r(ψ) , if v = +∞, then v = −1, (9.8)˙ψ = ω ,which was originally written in the ϑ-form; see Ex. 13. Here, ψ is the phase of autonomousoscillation of the slow subsystem, ω ≈ 0 is the frequency of slow oscillation,and r(ψ) is a periodic function that changes sign and slowly drives the fast quadraticintegrate-and-fire neuron (9.8) back and forth through the bifurcation, as we illustratein Fig. 9.34.Alternatively, suppose that the slow subsystem cannot have sustained oscillationswithout the fast subsystem, i.e., the slow subsystem has a stable equilibrium if v isfixed. In Ex. 17 we prove that there is a piece-wise continuous change of variables thattransforms any such “circle/circle” burster into one of the two canonical models below,