Conductance-based models of neurons describe the dynamic interaction between the membrane potential and - possibly many - gating variables that control the ionic flow through the membrane. The gating of ion channels occurs on many different timescales. However, gating timescales can be grouped in three families, according to their influence on neuronal excitablity [8]:

In view of their importance for neuronal excitability, we focus only on slow gating variables to classify ion channels: when the slow channel provides a negative feedback on membrane potential variations, we term the associated channel a slow restorative ion channel. When the slow variable instead enhances a potential variation by positive feedback, the associated ion channel is termed slow regenerative (a characterization in terms of partial derivatives is postponed to the next sections). Ion channels that do not possess a slow gating variable are neither restorative nor regenerative and are called neutral. Neutral ion channels solely regulate the “quantity” of excitability without affecting its “quality”.

Table 1 shows a classification of many known ion channels according to this criterion. Not surprisingly, potassium channels are the main representatives of slow restorative ion channels. By increasing the total outward current, their activation induces a negative feedback on membrane potentials variations that is responsible for neuron repolarization. On the other hand, physiologically described calcium channels are all slow regenerative. Their activation induces an increase of the total post-spike inward current, in contrast to potassium channels. This is the source, for instance, of afterdepolarization potentials (ADP). Interestingly, sodium channels can be either restorative, regenerative, or neutral according to their fast transient, resurgent, or persistent behavior, respectively.

It is important to observe that the restorative (resp. regenerative) nature of channels is not solely linked to the outward (resp. inward) nature of the current. For instance, transient sodium channels (although responsible for the regenerative spike upstroke) are slow restorative, because their slow variable inactivates an inward current, inducing a negative-feedback on membrane potential variations. Similarly, potassium channels can be slow regenerative when their slow inactivation massively decreases the outward current, like in the case of A-type potassium channels.

Although elementary, the classification above seems novel. It is motivated by the central message of this paper, that the balance between regenerative and restorative ion channels in slow timescale determines its neuronal excitability type. In the remainder of the paper, we simply write restorative (resp. regenerative) for slow restorative (resp. slow regenerative) channels.

Planar models - that only consist of two state variables - have been instrumental to study excitability since the early days of neurodynamics [9], [10]. Empirical planar reductions of conductance based models only retain the (fast) voltage variable and one slow gating variable . Fast gating variables are set to steady-state (i.e. their fast variation is approximated as instantaneous), adaptation variables are treated as slowly varying parameters, and the sole gating variable aggregates all slow variables, expressing each of them as a (curve-fitted) static function of .

Motivated by the phase portrait of such an empirical reduction of the Hodgkin-Huxley model augmented with a calcium channel [11], our recent study [1] explores the neuronal excitability of the planar model(1a)(1b)whose phase portraits are reproduced in Fig. 1 for two distinct values of the parameter (an indirect representation of the calcium conductance in the high-dimensional model). The parameter characterizes the time-scale separation between and . The function has the standard sigmoid shape of conductance-based models and is the half-activation potential. The typical step responses of (1) are also reproduced in Fig. 1.

The spike generation mechanism in the phase portrait of Fig. 1 left is reminiscent of the of FitzHugh-Nagumo model and of the physiologically grounded planar reduction of Hodgkin-Huxley model by Rinzel [10]. It is associated to a reversible and sudden switch from resting to spiking and has been studied extensively, with finer distinctions depending on the mathematical nature of the underlying bifurcation [12]. For further reading, see [13], [14, Section 7.1.3], and references therein.

The phase portrait in Fig. 1 right is in sharp contrast in that the electrophysiological response to a current input exhibits spike latency, plateau oscillations, and after depolarization potential (ADP). This specific signature, experimentally observed in many families of neurons, is fundamentally associated to the bistability illustrated in the phase portrait: namely, the robust coexistence of two stable attractors (a hyperpolarized resting potential and a limit cycle of periodic action potentials) and a saddle-separatrix that sharply separates their basins of attraction. The time evolution shown in the top figure is a consequence of this phase portrait and cannot be observed in FitzHugh-Nagumo like phase portraits. The distinction between the two phase portraits, the associated excitability types, and their relation with Hodgkin's excitability classification [15] are further discussed in [1] and later in the paper.

A simple mathematical distinction between the two phase portraits shown in Fig. 1 is drawn from the Jacobian linearization of the model at the stable resting point :The product of the partial derivatives is negative on the left phase portrait (), capturing the restorative nature of the gating variable, whereas it is positive on the right phase portrait (), capturing the regenerative nature of the gating variable. This difference is schematized in the block diagrams of Fig. 2. Accordingly, excitability in planar models is called restorative (resp. regenerative) when the gating variable provides negative (resp. positive) feedback close to the resting point:

We start by grouping gating variables of a given conductance-based model according to their time scales. The family collects fast gating variables. The gating variable denotes an arbitrary member of this family. Similarly, the family collects slow gating variables , whereas collects adaptation variables . For a given ion channel type , the standard notation (resp. ) is adopted for the activation (resp. inactivation) gating variable of the associated ionic current . With these notations, a general neuron conductance-based model reads(5a)(5b)(5c)(5d)where the sum in (5a) is over all ion channels in the model, and (5b),(5c),(5d) hold for all the associated fast, slow, and adaptation variables, respectively. The activation (resp. inactivation) functions are strictly monotone increasing (resp. decreasing) sigmoids. In the forthcoming analysis, all adaptation variables are treated as constant parameters, that is, their slow evolution is neglected.

We will detect a switch from restorative to regenerative excitability by mimicking the two-dimensional algorithm of the previous section. We first impose the bifurcation condition , where denotes the Jacobian of the subsystem (5a),(5b),(5c). The algebraic condition writes(6)where the sums are over all fast and slow variables, respectively. The particular form of equation (6) is a direct consequence of the specific structure of conductance-based models, that is, parallel interconnection of two-dimensional feedback loops involving the voltage dynamics (6a) and one of the gating variable dynamics (6b),(6c).

As for the planar model (1), we track the switch between restorative and regenerative excitability by imposing the high-dimensional equivalent of the balance condition (2). We therefore look for solutions of (6) satisfying the ansatz(7)

The two conditions (6) and (7) now imply(8)where denotes the Jacobian of the fast subsystem (5a),(5b). We show in Supplementary Material S1 that the corresponding bifurcation is necessarily transcritical [17, Section 3.2].

The singularity condition (8) is the high-dimensional counterpart of the -nullcline self-intersection in the planar model. It reflects the geometric nature of the transcritical bifurcation, that is, a robust geometrical object that exists independently of the timescale separation and persists in the singular limit of an infinite timescale separation, regardless of the system dimension. Our ansatz makes the proposed analysis robust against the model time constants. The time constants are only used to classify the gating variables in the three physiological groups.

We split in two subfamilies: , which contains regenerative slow gating variables , and , which contains restorative slow gating variables . The balance condition (7) is then rewritten as(9)to express a balance between restorative and regenerative ion channels. It is the high-dimensional counterpart of (2) and it provides a rigorous high-dimensional generalization of restorative and regenerative excitability:

The Hodgkin-Huxley (HH) model [2] provides a non-physiological, but historical and experimentally verified tutorial for tracking a switch of excitability in conductance based models. The model reads(10a)(10b)(10c)(10d)where is the fast sodium channel activation while the sodium channel inactivation and the potassium channel activation are the slow gating variables. We set all time constants to one, because this simplification has no effects on the algebraic conditions (7) and (8). The Jacobian of (10) reads(11)The upper-left block is the Jacobian of the fast subsystem. Imposing the singularity condition (8) yields(12)while the balance equation (9) reads(13)Note that (12) and (13) imply the bifurcation condition in (11).

At first sight, the balance condition (13) cannot be satisfied because both sodium and potassium channels are restorative channels according to their corresponding kinetics in the model, and in agreement with our proposed classification. This is consistent with the fact that the excitability of the HH model is always restorative in physiological conditions.

However, it was long recognized [18] that potassium channels can generate an inward current at steady-state if the extracellular concentration is sufficiently large. Indeed, any change in extracellular potassium concentration induces a change in the potassium reversal potential, as expressed by the Nernst equation. This suggests to use the potassium reversal potential as a bifurcation parameter in HH model in order to satisfy the balance equation(14)where potassium now acts as a regenerative gating variable provided that . Physiologically, condition (14) imposes that the potassium Nernst potential is large enough for the regenerativity of the potassium activation to balance the restorative effects of the sodium current inactivation.

The two conditions (12) and (14) can numerically be solved to determine the critical values and . The value of the applied current at the transcritical bifurcation is then determined from (10a), which givesThe numerical bifurcation diagram in Fig. 6A confirms the transcritical bifurcation at . That bifurcation diagram is drawn by varying together with applied current , following the affine reparametrization described in Supplementary Material S1. More precisely,Mathematically, this reparametrization imposes one of the defining conditions of the transcritical bifurcation. Physiologically, its effect is to keep the net current constant at steady-state (i.e. ): as is varied, the observed switch in the excitability type does not rely on changes in the net current across the membrane, but solely on changes in its dynamical properties.

The bifurcation diagram in Fig. 6A provides informations on the model excitability also far from the transcritical values. For highly hyperpolarized , the model is purely restorative and exhibits the typical excitable behavior of the original Hodgkin-Huxley model [1, Figure 8]. As is increased, a stable regenerative steady-state is born in a saddle-node bifurcation. At this transition, the system switches to a mixed excitability type. Short current pulses let the system switch between the depolarized restorative stable steady state and the hyperpolarized regenerative stable steady state (Fig. 6B,middle). The associated bifurcation diagram and phase portrait are reproduced in Fig. 6C,D,middle. Finally, a further increase of lets the restorative steady state exchange its stability with a (regenerative) saddle at the transcritical bifurcation (and, soon after, lose stability in a Hopf bifurcation) and the system switches to regenerative excitability. The regenerative steady state coexists in this case with the spiking limit cycle attractor. Current pulses switch the system asymptotic convergence between the two attractors (Fig. 6B,right). The associated bifurcation diagram and phase portrait are reproduced in Fig. 6C,D,right.

The same qualitative excitability switch was described by Rinzel in [10], who linked the appearance of a bistable behavior to the inward nature of potassium current at steady-state for sufficiently depolarized . In vitro recordings of the squid giant axon with isotonic extracellular concentration show the same transition [18].

The mathematical analysis of the previous section follows an algorithm that allows to detect a transcritical bifurcation in generic conductance based models of arbitrary dimension and to track associated excitability switches. The steps of the algorithm are summarized in Table 2. For simplicity and conciseness, we restrict our attention to the modulation of only one regenerative ionic current. However, a similar algorithm can be written for an arbitrary modulation of ionic currents (by variation of maximal conductance(s), adaptation variable(s), or reverse potential(s)) that brings the model to the balance expressed in (9).

For the sake of illustration, we apply this algorithm to a number of published conductance-based models and show that all these models can switch between restorative and regenerative excitability through a transcritical bifurcation, as sketched in Fig. 7. Figure 7 indicates two qualitatively distinct paths from restorative to regenerative excitability: one path traversing the mixed excitability region just described with Hodgkin-Huxley model (Fig. 6A) and one path switching directly from restorative to regenerative excitability through the TC bifurcation that will be illustrated on the dopaminergic neuron model.

As illustrated in Figure 4, the significance of the transcritical bifurcation is that it delineates in the parameter space the boundary of a specific type of excitability and that this boundary is determined by a simple physiological balance (Eq. 9) between restorative and regenerative channels.

Specific to regenerative excitability is the bistable phase portrait of Fig. 1, right. For the six analyzed conductance-based models, our bifurcation analysis of the full model confirms the existence of a bistable range beyond the transcritical bifurcation, where a regenerative resting state and a spiking limit cycle coexist. In each case, the bistability range is obtained for the nominal time scales of the published model and is robust to a variation of time scales. In each case, the bistabilty range is also neuromodulated, that is, determined by conductance parameters that are known to vary in slower time scales and/or across neurons of a same type.

It is important to distinguish this robust and physiologically regulated bistability from other types of bistability that can be encountered in conductance-based models. Figure 11 qualitatively illustrates three typical bistable phase portraits associated to the planar model (1) that exhibit the coexistence of a stable resting state and of a spiking limit cycle. The first two are associated to restorative excitability and are extensively studied in the literature. See, e.g. [13], [14], and references therein. Only the third one is associated to regenerative excitability.

The three bistable phase portraits share the common feature of “hard excitation”: as the amplitude of a step input depolarizing current is increased, the response of the neuron abruptly switches from no oscillation to high frequency spiking. Following the historical classification of Hodgkin [15], the three situations correspond to Class II neurons, as opposed to Class I neurons for which the spiking frequency gradually increases with the depolarizing current amplitude.

Hard excitation can be a manifestation either of a switch-like monostable bifurcation diagram or of a hysteretic bistable bifurcation diagram. By definition, the three bistable phase portraits in Figure 11 give rise to hysteretic bifurcation diagrams. But for the two bistable phase portraits associated to restorative excitability, the hysteresis is highly dependent on the time scale separation, i.e., the ratio between the fast and slow time constants. In the case of the first phase portrait (subcritical Hopf bifurcation), asymptotic analysis shows that the hysteresis vanishes as . In the case of second phase portrait (saddle-homoclinic bifurcation), the situation is even worse because for small the system necessarily undergoes a monostable saddle-node on invariant circle bifurcation. In fact, the second phase portrait is not physiological for neuron conductance based models. For instance, in the hypothetical conductance-based model considered in [14, Fig. 6.44], the time constant of the potassium activation must be set below to create a saddle-homoclinic bifurcation, which is roughly 40 times smaller than its physiological value and even smaller than the fast time constant. A geometric proof of the generality of this fact is provided in [1]. The conclusion is that hysteresis associated to restorative excitability is at best very small (if any) in physiologically plausible conductance based models, which makes their electrophysiological signatures similar to those associated to a switch-like monostable bifurcation diagram.

In sharp contrast, the hysteresis associated to regenerative excitability is barely affected by the time-scale separation. Instead it is regulated by conductance parameters whose modulation is physiological (for instance, a regenerative ion channel density). The extended hysteresis is what determines the specific electrophysiological signature of regenerative excitability: a pronounced spike latency, a possible plateau oscillation, and an after depolarization potential. As a consequence, those features cannot be robustly reproduced in physiologically plausible conductance based models of restorative excitability. Because those features are important markers of modern electrophysiology [22], [23], the distinction between restorative and regenerative excitability seems physiologically relevant, beyond the possible shared feature of hard excitation.

In conclusion, the bistability associated to regenerative excitability is specific in that it produces a robust electrophysiological signature in physiologically plausible parameter ranges and consistent with many experimental observations. It is in that sense that the balance equation delineates a switch of excitability of physiological relevance.

Early in the history of neurodynamics [15], Hodgkin proposed a classification of excitability in three different classes:

Class I: The spiking frequency vs. input current amplitude (f/I) curve is continuous, i.e., the spiking frequency continuously increases from zero to high-frequency firing as the input current amplitude rises. Class I excitability is also referred to as “soft” excitation.

Class II: The f/I is discontinuous, i.e., the spiking frequency abruptly switches from zero to high-frequency firing as the amplitude of the applied current is raised above a certain threshold. Class II excitability is also referred to as “hard” excitation.

Class III: The spiking frequency is zero for all amplitudes of the applied current. Transient action potentials can be generated in response to high-frequency stimuli.

Because regenerative excitability exhibits hard excitation, it is a physiologically distinct subtype of Class II excitability.

Bifurcation theory helps relating this physiological classification to mathematical signatures of the associated neuron models. Distinct bifurcations delineate the different excitability classes as well as the different excitability mechanisms within a given class. They are summarized in Fig. 12.

Ermentrout [12] showed that Class I excitability arises from a saddle-node on invariant circle bifurcation, whereas Class II excitability arises from a Hopf bifurcation. Both bifurcations correspond to examples of restorative excitability in the terminology of the present paper and the transition between Class I and II is governed by a Bogdanov-Takens bifurcation.

Our recent paper [1] further expands this classification to account for regenerative excitability. Regenerative excitability (called Type IV in [1]) arises from a (singularly perturbed) saddle-homoclinic bifurcation and the transition from restorative to regenerative excitability always involves a transcritical bifurcation.

Motivated by a geometric analysis of a qualitative phase portrait, we have proposed an algorithm that easily detects a transcritical bifurcation in arbitrary conductance based models. Owing to the special structure of such models, the algorithm leads to solving an algebraic equation of remarkable simplicity and physiological relevance: a balance between slow restorative and slow regenerative ion channels. The condition is also robust because the balance is independent of the detailed kinetics, even though it critically relies on a classification of variables in three well separated time-scales, in good agreement with what is known on ion channels kinetics [8].

The detection of the transcritical bifurcation relies on the sole existence of a physiological balance between restorative and regenerative ion channels. Given that all neuronal models possess restorative sodium and potassium channels, this implies that a transcritical bifurcation exists in every conductance-based model that possesses at least one regenerative ion channel. Moreover, the channel balance, and therefore the TC bifurcation, are readily detectable in a model of arbitrary dimension (both in the state and parameters): the balance (9) simply defines a hypersurface in the parameter space that can algebraically be tracked under arbitrary parameter variations. An illustration was provided on the GC model above.

In spite of its ubiquity and of its physiological significance, we are not aware of an earlier reference to a transcritical bifurcation in conductance based models. A reason for this omission might be accidental: there are no regenerative channels in the seminal model of Hodgkin and Huxley (unless one modifies the potassium resting potential ) and this model has been the inspiration of most mathematical analyses of conductance-based models.

For the same reason, it seems physiologically relevant to distinguish between restorative and regenerative excitability beyond Hodgkin's classification of Class I (“soft”) and Class II (“hard”) excitability. Regenerative (and restorative) excitability faithfully capture the presence (or the absence) of specific electrophysiological signatures of modern electrophysiology such as spike latency, afterdepolarization potentials, or robust coexistence of resting state and repetitive spikes.

Although purely mathematical in nature, the transcritical bifurcation has a remarkable predictive value in several published conductance based models. In each of the six analysed models, the proposed algorithm identifies a physiological parameter that acts as a tuner of neuronal excitability in a physiologically plausible range and in full agreement with existing experimental data. At the same time, the distinct nature of the regulating parameter, which can be either the maximal conductance or the inactivation gating variable of a regenerative ion channel depending on the neuron model, is associated to distinctly different regulation mechanisms.

The classification of gating variables in three distinct time scales is an essential modeling step both for the proposed algorithm and for the reduction of full conductance-based models to low-dimensional models that can be used in population studies [24]. Despite the inherent robustness of time-scale separation analysis, this classification is a limitation of the proposed approach if the model contains ion channels with poorly known kinetics. When all slow ion channels are properly identified, they can be aggregated in a single slow variable to lead to a second order model of the type (1), where the single parameter captures the restorative or regenerative nature of the aggregated slow variable. Further reduction to a one-dimensional hybrid model with reset is possible thanks to the time-scale separation between the voltage and the slow variable . This reduction is illustrated in [11] on the thalamic TC neuron and the reduced model remarkably retains the switch of excitability of its high-dimensional counterpart. In contrast, a reduced model will lose the switch of excitability of the full conductance-based model when a regenerative ion channel is treated as a fast gating variable. It is for instance common in model reduction to set the activation of a calcium channel to steady state. This amounts to treat the calcium activation as a fast variable, which makes the channel either “slow restorative” or “neutral” in the terminology of this paper. If the calcium channel is the only source of regenerative excitability, then the reduced model will not retain features of regenerative excitability.

In each of the analysed conductance-based models, the balance equation responsible for the switch of excitability is satisfied for a set of parameters that is close to the published parameter values. This observation supports the hypothesis that neuronal excitability is tightly regulated by molecular mechanisms and that the influence of the channel balance condition on neuronal excitability might play a role in neuronal signaling.

Numerical temporal traces of the different neurons (Figs. 1, 6, 8, 9) were reproduced by implementing in MATLAB (available at http://www.mathworks.com) the original models as described in the associated papers. The phase portraits in Figures 1 and 3 were hand-drawn using the Open Source vector graphics editor Inkscape (http://inkscape.org). The phase portraits in Figure 6 were numerically drawn with MATLAB by implementing the planar model in [10] and subsequently modified with Inkscape. The bifurcation diagrams in Figures 6, 7, and 8 were drawn by implementing the algorithm of Table 2 in MATLAB. No figure or part of figure was reproduced from other published works.