Dynamical Systems in Neuroscience:
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Dynamical Systems in Neuroscience:

Dynamical Systems in Neuroscience: Dynamical Systems in Neuroscience:

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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

Bursting 37510stable node4fast variable, vslow variables86420-24resetslow variable, u220restingu 2 =I+u 1equilibirum ofslow subsystemspiking2 u 20 u 1-20 2 40 50 100 150 200 250timeslow variable, u 1108stable focus4restingspikingfast variable, v6420resetslow variable, u220slow variablesu22u 10-20 20 40 60 80time-2u 1 =-I-4 -2 0 2 4slow variable, u 1Figure 9.35: “Circle/circle” bursting in the canonical models (9.9) (top, parameters:I = 1, µ 1 = 0.1, µ 2 = 0.02, d 1 = 1, d 2 = 0.5) and (9.10) (bottom, parameters:I = 1, µ 1 = 0.2, µ 2 = 0.1, d 1 = d 2 = 0.5).“subHopf/fold cycle” bursters, yet the bursting profiles do not look like ellipses. Manymodels of “subHopf/fold cycle” bursters do not generate elliptic profiles either, hencereferring to this type of bursting by its shape is misleading and should be avoided.It is quite easy to transform the I Na,p +I K -model into a “subHopf/fold cycle” burster.First, we chose the parameters of the model as in Fig. 6.16 so that the phase portraitdepicted in Fig. 9.40 is the same as in Fig. 9.36, bottom. The co-existence of the stableequilibrium, an unstable limit cycle and a stable limit cycle is essential for producing thehysteresis loop oscillation. Then, we add a slow K + M-current that activates while thefast subsystem fires spikes and deactivates while it is resting. Such a resonant currentprovides a negative feedback to the fast subsystem, and the full I Na,p +I K +I K(M) -modelexhibits “subHopf/fold cycle” bursting, shown in Fig. 9.41.As in the previous examples, the burster in this figure is conditional: It needs aninjection of a dc-current I, so that the equilibrium corresponding to the resting state of

Burst<strong>in</strong>g 37510stable node4fast variable, vslow variables86420-24resetslow variable, u220rest<strong>in</strong>gu 2 =I+u 1equilibirum ofslow subsystemspik<strong>in</strong>g2 u 20 u 1-20 2 40 50 100 150 200 250timeslow variable, u 1108stable focus4rest<strong>in</strong>gspik<strong>in</strong>gfast variable, v6420resetslow variable, u220slow variablesu22u 10-20 20 40 60 80time-2u 1 =-I-4 -2 0 2 4slow variable, u 1Figure 9.35: “Circle/circle” burst<strong>in</strong>g <strong>in</strong> the canonical models (9.9) (top, parameters:I = 1, µ 1 = 0.1, µ 2 = 0.02, d 1 = 1, d 2 = 0.5) and (9.10) (bottom, parameters:I = 1, µ 1 = 0.2, µ 2 = 0.1, d 1 = d 2 = 0.5).“subHopf/fold cycle” bursters, yet the burst<strong>in</strong>g profiles do not look like ellipses. Manymodels of “subHopf/fold cycle” bursters do not generate elliptic profiles either, hencereferr<strong>in</strong>g to this type of burst<strong>in</strong>g by its shape is mislead<strong>in</strong>g and should be avoided.It is quite easy to transform the I Na,p +I K -model <strong>in</strong>to a “subHopf/fold cycle” burster.First, we chose the parameters of the model as <strong>in</strong> Fig. 6.16 so that the phase portraitdepicted <strong>in</strong> Fig. 9.40 is the same as <strong>in</strong> Fig. 9.36, bottom. The co-existence of the stableequilibrium, an unstable limit cycle and a stable limit cycle is essential for produc<strong>in</strong>g thehysteresis loop oscillation. Then, we add a slow K + M-current that activates while thefast subsystem fires spikes and deactivates while it is rest<strong>in</strong>g. Such a resonant currentprovides a negative feedback to the fast subsystem, and the full I Na,p +I K +I K(M) -modelexhibits “subHopf/fold cycle” burst<strong>in</strong>g, shown <strong>in</strong> Fig. 9.41.As <strong>in</strong> the previous examples, the burster <strong>in</strong> this figure is conditional: It needs an<strong>in</strong>jection of a dc-current I, so that the equilibrium correspond<strong>in</strong>g to the rest<strong>in</strong>g state of

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