A Balance Equation Determines a Switch in Neuronal Excitability
Figure 12
The various bifurcations associated to different types of neuronal excitability.
SNIC: saddle-node on invariant circle; BT: Bogdanov-Takens; AH: Andronov-Hopf; SN: saddle-node; TC: transcritical; SH: saddle-homoclinic. See also [1] for more detailed definitions and properties of excitability Types I-V and associated transition bifurcations in a planar neuron model. Class I excitability occurs in the neighborhood of a SNIC bifurcation [12] and is purely restorative. Class II excitability can be either restorative in which case the stable equilibrium looses stability in a subcritical Hopf bifurcation (Type II in [12]) or regenerative in which case a stable equilibrium coexists with a stable limit cycle over a robust bistable range organized by a (singularly perturbed) saddle homoclinic bifurcation (Type IV in [1]). In a small parameter range, class II excitability can also exhibit a mixed type (Type Vb in [1]), where a regenerative "down" stable equilibrium coexists with a "up" restorative stable equilibrium or limit cycle. Stability of those attractors is lost either in saddle-node or Hopf bifurcations. Class III excitability can be either restorative (a monostable equilibrium) or exhibits a mixed type (Type Va in [1]), where a regenerative down stable equilibrium coexists with a restorative up stable equilibrium. Both attractors loose stability in a saddle-node bifurcation. The transition to regenerative excitability is always through a transcritical bifurcation.