Table 1.
Classification of ion channels according to their gating kinetics.
Figure 1.
The bottom figures illustrate the typical phase portrait of restorative (left) and regenerative (right) excitability.
The dark (resp. light) blue circle denotes a stable restorative (resp. regenerative) steady state . The thin full (resp. dashed) curve is the voltage (resp. slow variable) nullcline. The saddle point in the right phase portrait is represented by a cross and its separatrix as the green oriented curve. The stable limit cycle surrounding the unstable fixed point (represented as a circle) is represented by the blue oriented curve. The thick curve in the left phase portrait represents the typical trajectory associated to the generation of an action potential. The top figures illustrate the typical accompanying electrophysiological responses to step variations of current.
Figure 2.
Block diagram illustration of restorative and regenerative excitability in planar models.
Figure 3.
A continuous deformation from the restorative phase portrait of Fig. 1 left to the regenerative phase portrait of Fig. 1 right involving a transcritical bifurcation [17, Section 3.2] determined by the algebraic conditions (2) and (3).
The dark blue circle represents a restorative stable steady-state, the light blue circle a regenerative stable steady-state, and the half-filled circle represents the transcritical bifurcation which separates the restorative and regenerative regimes.
Figure 4.
Excitability types in model (1).
SN denotes the saddle-node bifurcation, TC the transcritical bifurcation. The black square denotes the pitchfork bifurcation organizing center. Varying and
the model switches between excitability types.
Figure 5.
Block diagram illustration of restorative and regenerative excitability in conductance based models.
Figure 6.
Variations of the potassium reversal potential induce excitability switches in the Hodgkin-Huxley model.
A. Bifurcation diagram of the HH model with as the bifurcation parameter. TC denotes a transcritical bifurcation, SN a saddle-node bifurcation, HB a Hopf bifurcation. Branches of stable fixed points are represented as solid curves, whereas branches of saddle points and unstable points as dashed curves. B. Electrophysiological responses of the model for three different values of
, corresponding to three different excitability types (restorative, mixed, and regenerative, from left to right). C. Bifurcation diagrams with the applied current as the bifurcation parameter for the same three values of
as in B. Black (resp. blue) full curves represent branches of stable steady-states (resp. limit cycles), black dashed curves branches of saddle and unstable steady-states. Branches of unstable limit cycle are drawn as dashed blue curves. HB denotes a Hopf bifurcation, SN a saddle-node bifurcation, and SH a saddle-homoclinic bifurcation. D. Phase portraits of reduced HH model proposed by Rinzel in [10] for the same three values of
as in B,C. Blue full curves denote the
-nullclines and black full curves the
-nullclines, where
denotes the slow variable of the reduced model. Filled circles denote stable steady-states, crosses saddle points, and circles unstable steady-states.
Table 2.
Algorithm for the detection of a transcritical bifurcation in generic conductance-based models via modulation of a regenerative ionic current and computation of the excitability switch bifurcation diagram.
Figure 7.
Modifications in the balance between restorative and regenerative channels induce excitability switches in conductance-based models.
The figure sketches excitability switches of the Hodgkin-Huxley (HH) model [2], Aplysia's R15 neuron (R15) model [6], a dopaminergic (DA) neuron model [3], thalamic reticular (RT) and relay (RE) neuron models [4], [5], and a cerebral granule cell (GC) model [7] on the excitability parameter map computed for the two-dimensional model of [1]. All these conductance-based models can switch between restorative and regenerative excitability through the physiologically relevant regulation of specific ion channels.
Figure 8.
Variations of L-type calcium channel density induce excitability switches in a model of DA neurons
[3]. A. Bifurcation diagram of the model with as the bifurcation parameter. TC denotes a transcritical bifurcation, SN a saddle-node bifurcation. Branches of stable fixed points are represented as solid curves, branches of saddle points and unstable points as dashed curves. B. Electrophysiological responses of the model to step inputs of excitatory/inhibitory current (the intracellular calcium concentration is fixed at
, which is within the physiological range). For
lower (resp. higher) than the critical value
, the model exhibits typical electrophysiological signature of restorative (resp. regenerative) excitability. The low
configuration corresponds to
, whereas the high
configuration corresponds to
.
Figure 9.
The same mathematical bifurcation in different conductance-base models causes the same switch in electrophysiological signatures.
The figure shows the electrophysiological responses of various conductance-based models to step inputs of excitatory/inhibitory current when the bifurcation parameter is lower (left) or higher (right) than the critical value
. This bifurcation parameter can be either the density or the inactivation variable of a regenerative channel. Other adaptation variables are set to constant values chosen in physiological ranges (see text). For
lower (resp. higher) than the critical value
, all models exhibit electrophysiological signatures of restorative (resp. regenerative) excitability. Numerical values of the parameter
in the different plots are as follows. Thalamic relay cell: left
, right
. Thalamic reticular cell: left
, right
. Aplysia R15 neuron: left
, right
. GC neuron: left
, right
.
Figure 10.
Joint variations of the inactivation gates of -type calcium channels and resurgent sodium channels induce excitability switches in cerebellar granule cells.
A. Two parameter bifurcation diagram of the mode with and
as bifurcation parameters. TC denotes a branch of transcritical bifurcations detected following the algorithm in Table 2. B. Electrophysiological responses of the model to step inputs of excitatory/inhibitory current: left
, right
.
Figure 11.
Three bistable phase portraits of model (1) and cartoon of the associated hysteretic bifurcation diagrams.
In the phase portraits, a solid curve denotes the -nullcline, whereas a dashed curve denotes the
-nullcline. Stable fixed points are depicted as filled circles, whereas unstable as circles and saddle points as cross. Stable limit cycles are drawn as solid oriented blue curves, whereas unstable as red dashed curves. The stable manifolds of saddle points are depicted as green oriented curves. In bifurcation diagrams, a solid curve denotes branches of stable fixed points, whereas a dashed curve denotes branches of unstable or saddle points. Branches of stable limit cycles are depicted as blue curves, whereas branches of unstable limit cycles as red dashed curves. sub.HB denotes a subcritical Hopf bifurcation, SNLC a saddle-node limit cycles bifurcation, SN a saddle-node bifurcation, and SH a saddle-homoclinic bifurcation. A–B. Restorative bistability. A. Subcritical Hopf bifurcation. Hysteresis vanishes exponentially fast as timescale separation increases. B. Restorative saddle-homoclinic bifurcation. Not physiological because it violates the time scale separation between
and
. C. Regenerative bistability ruled by a regenerative saddle-homoclinic bifurcation. Hysteresis is barely affected by time-scale separation.
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.