Dynamical Systems in Neuroscience:

374 Burstingdepending on the type of the equilibrium. If the equilibrium of the slow subsystem isa stable node, then the canonical model has the form˙v = I + v 2 + u 1 − u 2 ,˙u 1 = −µ 1 u 1 ,˙u 2 = −µ 2 u 2 .(9.9)If the equilibrium of the slow subsystem is a stable focus, the canonical model has theform˙v = I + v 2 + u 1 ,˙u 1 = −µ 1 u 2 ,(9.10)˙u 2 = −µ 2 (u 2 − u 1 ) ,with µ 2 < 4µ 1 . In both cases, there is an after-spike resetting:if v = +∞, then v ← −1, and (u 1 , u 2 ) ← (u 1 , u 2 ) + (d 1 , d 2 ).Similarly to (9.7), the variable v ∈ R is the re-scaled membrane potential of the neuron.The positive feedback variable u 1 ∈ R describes activation of slow amplifying currentsor potential at a dendritic compartment, whereas the negative feedback variable u 2 ∈ Rdescribes activation of slow resonant currents. I, d 1 , d 2 , and µ 1 , µ 2 ≪ 1 are parameters.When µ 2 > 4µ 1 , the equilibrium of the slow subsystem in (9.10) is a stable node,so (9.10) can be transformed into (9.9) by a linear change of slow variables. If d 1 = 0,then u 1 → 0 and (9.9) is equivalent to (9.7).Both canonical models above exhibit “circle/circle” slow-wave bursting, as depictedin Fig. 9.35. When I > 0, the equilibrium of the slow subsystem is in the shadedarea corresponding to spiking dynamics of the fast subsystem. When the slow vector(u 1 , u 2 ) enters the shaded area, the fast subsystem fires spikes, prevents the vectorfrom converging to the equilibrium, and eventually pushes it out of the area. Whileoutside, the vector follows the curved trajectory of the linear slow subsystem and thenreenters the shaded area again. Such a slow wave oscillation corresponds to the thicklimit cycle attractor in Fig. 9.35, which looks remarkably similar to the one for theI Na,p +I K +I Na,slow +I K(M) -model in Fig. 9.33.9.3.3 subHopf/fold cycleWhen the resting state loses stability via subcritical Andronov-Hopf bifurcation, andthe spiking state disappears via fold limit cycle bifurcation, the burster is said to be ofthe “subHopf/fold cycle” type depicted in Fig. 9.36. Because there is a co-existence ofresting and spiking states, such bursting usually occurs via a hysteresis loop with onlyone slow variable.This kind of bursting was one of the three basic types identified by Rinzel (1987). Itwas called “elliptic” in earlier studies because the profile of oscillation of the membranepotential resembles ellipses, or at least half-ellipses; see Fig. 9.37. Rodent trigeminalinterneurons in Fig. 9.38, dorsal root ganglion and mesV neurons in Fig. 9.39 are all