Dynamical Systems in Neuroscience:

Bursting 347possibly resulting in a new burst.• Voltage-gated activation of an outward current, e.g., slow activation of persistentK + current, such as M-current. Repetitive spiking slowly activates the outwardcurrent, which eventually terminates the spiking activity. While at rest, theoutward current slowly deactivates (turns off) and unmasks inward currents thatcan depolarize the membrane potential, possibly initiating another burst.• Ca 2+ -gated inactivation of an inward current, e.g., slow inactivation of highthresholdCa 2+ -currents I Ca(L) or I Ca(N) . Calcium entry during repetitive spikingleads to its intracellular accumulation and slow inactivation of Ca 2+ -channelsthat provide an inward current needed for repetitive spiking. As a result, theneuron cannot sustain spiking activity and becomes quiescent. During this period,intracellular Ca 2+ ions are removed, Ca 2+ channels are de-inactivated, andthe neuron is primed to start a new burst.• Ca 2+ -gated activation of an outward current, e.g., slow activation of Ca 2+ -dependentK + -current I AHP . Calcium entry and buildup during repetitive spikingslowly activates the outward current and makes the neuron less and less excitable.When the spiking stops, intracellular Ca 2+ ions are removed, Ca 2+ -gatedoutward current deactivates (turns off), the neuron is no longer hyperpolarizedand ready to fire a new burst of spikes.In addition, the slow process may include Na + -, K + -, or Cl − -gated currents, suchas the “slack and slick” family of Na + -gated K + currents, or slow change of ionicconcentrations in the vicinity of the cell membrane (so called Hodgkin-Frankenhaeuserlayer), which leads to slow change of the Nernst potential for ionic species. We do notelaborate these cases in the book.Notice that in some cases, the slow process modulates fast currents responsible forspiking, while in other cases it produces an independent slow current that impedesspiking. In any case, the slow process is directly responsible for the termination ofcontinuous spiking, and indirectly for its initiation and maintenance.The four mechanisms in Fig. 9.6 and their combinations are ubiquitous in neurons,as we summarize in Fig. 9.7. However, there could be other, less obvious burstingmechanisms. In Ex. 8–10 we provide examples of bursters having slowly activatingpersistent inward current, such as I Na,p . These surprising examples show that buildupof the inward current (or any other amplifying gate) can also be responsible for thetermination of the active phase and for the repolarization of the membrane potential.To understand these mechanisms, one needs to study the geometry of bursting.9.1.3 Minimal modelsLet us follow the ideas presented in Sect. 5.1 and determine minimal models for bursting.That is, we are interested in classification of all fast-slow electrophysiologicalmodels that can exhibit sustained bursting activity, as in Fig. 9.4b, at least for some