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We use the qualitative insight of a planar neuronal phase portrait to detect an excitability switch in arbitrary conductance-based models from a simple mathematical condition. The condition expresses a balance between ion channels that provide a negative feedback at resting potential restorative channels and those that provide a positive feedback at resting potential regenerative channels. Geometrically, the condition imposes a transcritical bifurcation that rules the switch of excitability through the variation of a single physiological parameter. Our analysis of six different published conductance based models always finds the transcritical bifurcation and the associated switch in excitability, which suggests that the mathematical predictions have a physiological relevance and that a same regulatory mechanism is potentially involved in the excitability and signaling of many neurons. The condition expresses a balance between ion channels that provide a negative feedback at resting potential restorative channels and those that provide a positive feedback at resting potential regenerative channels. Geometrically, the condition imposes a transcritical bifurcation that rules the switch of excitability through the variation of a single physiological parameter. Our analysis of six different published conductance based models always finds the transcritical bifurcation and the associated switch in excitability, which suggests that the mathematical predictions have a physiological relevance and that a same regulatory mechanism is potentially involved in the excitability and signaling of many neurons. The scientific responsibility rests with its authors. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. These authors contributed equally to this work Detailed computational conductance-based models have long demonstrated their ability to faithfully reproduce the variety of electrophysiological signatures that can be recorded from a single neuron in varying physiological or pharmacological conditions. But the predictive value of a computational model is limited unless its analysis sheds light on the core mechanisms at play behind a computer simulation. Because conductance-based models are nonlinear dynamical models, their analysis often requires a drastic reduction of dimension. The reduced model is amenable to the geometric methods of dynamical systems theory, but the mathematical insight is often gained at the expense of physiological interpretability; hence the need for methodological tools that can relate mathematical predictions of low-dimensional models to physiological predictions in detailed conductance based models. More precisely, a transcritical bifurcation governed by a single parameter was shown to organize a switch from restorative excitability, extensively studied in most models inspired from the Hodgkin-Huxley model, to regenerative excitability whose distinct electrophysiological signature include spike latency, plateau oscillations, and afterdepolarizeation potentials. The main contribution of the present paper is to show that this transcritical bifurcation, and the associated excitability switch, exist in a number of high-dimensional conductance-based models and that the resulting mathematical predictions have physiological relevance. Although purely mathematical in nature, the detection of the transcritical bifurcation relies on an ansatz that leads to a simple physiological interpretation: the switch of excitability is determined by a balance between restorative those providing a negative feedback and regenerative those providing a positive feedback ion channels at the resting potential. Because this simple balance equation can take many different physiological forms, it is potentially shared by very different neurons. We use the balance equation to provide an algorithm to trace the transcritical bifurcation in arbitrary conductance-based models. In each case, the algorithm identifies a transcritical bifurcation that occurs close to the nominal model parameters and its predictions are consistent with experimental observations. As a generalization of this low-dimensional case, we mathematically construct the same bifurcation in generic conductance based models and derive the balance condition determining the regenerative or restorative nature of the model. This construction and its electrophysiological predictions are firstly Understanding the changing electrophysiological signatures of neurons in different physiological and pharmacological conditions is a central focus of experimental electrophysiology because a key component of cell signaling in the nervous system. Computational modeling may assist experimentalists in this quest by identifying core mechanisms and suggesting pharmacological targets from a mathematical analysis of the model. But a successful interplay between experiments and mathematical predictions requires new analysis tools adapted to the complexity of high-dimensional computational models nowadays available. We use bifurcation theory to propose a mathematical condition that can detect an important switch of neuronal excitability in arbitrary conductancebased neuronal models and we illustrate its physiological relevance in six published state-of-the art models of different neurons. An algorithm for generic conductance-based models is subsequently derived and different models analysed. Slow restorative and slow regenerative ion channels Conductance-based models of neurons describe the dynamic interaction between the membrane potential V and - possibly many - gating variables that control the ionic flow through the membrane. The gating of ion channels occurs on many different timescales. They generate the rapid regenerative upstroke of an action potential. Prominent representatives of this family are the activation gating variables of fast voltage-gated sodium channels NaV1:1 to NaV1:9. Slow gating variables: These variables have a time constant 5 to 10 times larger than fast gating variables. They influence the spike initiation, downstroke, and the afterspike period. They are key players of neuronal excitability. Prominent representatives of this family are the activation gating variables of delayed rectifier potassium channels KV1:1 to KV1:3, KV1:5 to KV1:8, KV2:1, KV2:2, KV3:1, KV3:2, KV7:1 to KV7:5, KV10:1 and the activation gating variables of all calcium channels CaV1:x, CaV2:x, CaV3:x. Ultra-Slow adaptation variables: These variables gate too slowly to be strongly activated by single action potentials. They modulate neuronal excitability only over periods of many action potentials. Prominent representative of this family are the inactivation gating variables of transient calcium channels CaV2:x, CaV3:x. Ultra-slow variables might also include non gating variables. For instance, the intracellular calcium concentration Ca2z in, which modulates the conductance of calcium-regulated channels. 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. Activation and inactivation variables are distributed in three groups: fast, slow, and ultra-slow adaptation. Slow variables are defined as restorative resp. An ion channel that posses a slow restorative variable is called slow restorative channel, and similarly for slow regenerative channels. Channels that do not posses a slow variable are called neutral channels. This classification might change for a given channel for some channel subtypes. FA: fast activation, FI: fast inactivation, SA: slow activation, SI: slow inactivation, USA: utraslow activation, USI: ultraslow inactivation. 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. Empirical planar reductions of conductance based models only retain the fast voltage variable V and one slow gating variable n. Fast gating variables are set to steadystate i. V {V0zn0{n whose phase portraits are reproduced in Fig. The parameter ew0 characterizes the time-scale separation between V and n. The typical step responses of 1 are also reproduced in Fig. The spike generation mechanism in the phase portrait of Fig. The phase portrait in Fig. 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. A simple mathematical distinction between the two phase portraits shown in Fig. V {V0 is negative on the left phase portrait nw0 , LV capturing the restorative nature of the gating variable, whereas it is positive on the right phase portrait nv0 , capturing the regenerative nature of the gating variable. This difference is schematized in the block diagrams of Fig. Accordingly, excitability in planar models is called restorative resp. The model is said to be restorative at steady state if: Planar regenerative excitability. This analysis results in five different types of excitability obtained by varying the two parameters V0, n0 around the singular phase portrait of Fig. The parameter n0 acts in particular as the sole regulator of the balance between regenerative and restorative excitability by shifting the n-nullcline up and down: a positive n0 corresponds to a phase portrait as in Fig. Figure 4 delineates the two types of excitability in a two parameter chart. It contains the two types of excitability discussed above. The transition from restorative to regenerative excitability is always through a transcritical bifurcation. In addition, some paths traverse a small mixed region where a down regenerative steady state and an up restorative steady state coexist. Our main contribution in the present paper is to show that the diagram in Fig. Restorative and regenerative excitability in conductance based models We start by grouping gating variables of a given conductancebased model according to their time scales. The family GF ~fmNa,f , mNa,p, mK,A,. Similarly, the family GS~fhNa,f , mK,DR, mCa,L,. For a given ion channel type i, the standard notation mi resp. With these notations, a general neuron conductance-based model reads Figure 2. Block diagram illustration of restorative and regenerative excitability in planar models. 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 det J~0, where J denotes the Jacobian of the subsystem 5a , 5b , 5c. The algebraic condition writes 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 highdimensional equivalent of the balance condition 2. We therefore look for solutions of 6 satisfying the ansatz 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 n0 and V0 the model switches between excitability types. The singularity condition 8 is the high-dimensional counterpart of the V -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 GS in two subfamilies: GSz, which contains regenerative slow gating variables xsz, and GS{, which contains restorative slow gating variables xs{. The balance condition 7 is then rewritten as TC TC 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: Restorative excitability. The model is said to be restorative at steady state if: Regenerative excitability. The insight provided by the planar model of the previous section predicts that the switch of excitability detected by the balance equation 9 will lead to the accompanying distinct electrophysiological signatures of Fig. Note that 12 and 13 imply the bifurcation condition det J~0 in 11. At first sight, the balance condition 13 cannot be satisfied because both sodium and potassium channels are restorative Figure 5. Block diagram illustration of restorative and regenerative excitability in conductance based models. V {n, where m is the fast sodium channel activation while the sodium channel inactivation h and the potassium channel activation n 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 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. 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 VK as a bifurcation parameter in HH model in order to satisfy the balance equation where potassium now acts as a regenerative gating variable provided that VK wVSS. 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 VKc and V c. The value of the applied current at the transcritical bifurcation is then determined from 10a , which gives Iapp~gK n4? V c V c{VKc zgNam3? V c V c{VNa z c The numerical bifurcation diagram in Fig. That bifurcation diagram is drawn by varying VK together with applied current Iapp, following the affine reparametrization described in Supplementary Material S1. More precisely, Iapp VK ~Iacpp{gK n4? V c VK {VKc : 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 V c i. Iion VK zIapp VK ~0 : as VK 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. As VK 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. The associated bifurcation diagram and phase portrait are reproduced in Fig. Finally, a further increase of VK 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. The associated bifurcation diagram and phase portrait are reproduced in Fig. Tracking excitability switches in conductance-based models 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. 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. Dopaminergic DA neuron model. Classification of gating variables as fast GF , slow GS , and adaptation GA variables. The model includes fast sodium channels INa,f , delayed-rectifier potassium channels IK,DR , L-type calcium channels ICa,L , small conductance calcium-activated potassium SK channels IK,Ca , and calcium pumps ICa,pump. The intracellular calcium concentration is fixed at Ca2z in~300 nM. Balance equation and choice of the bifurcation parameter. The only source of regenerative excitability is provided by L-type calcium channels. The balance equation reads i Classification of gating variables as fast GF , slow GS , and adaptation GA variables i-a Following Tab. If Ireg has no adaptation variable, pick l~greg iii Singularity condition and fixed point equation iii-a Solve b. For numerical implementation, recall that the left hand side of 8 is proportional to iv Tracking of excitability switches Change the applied current according to the equation and compute the model bifurcation diagram with V as the variable and l as the bifurcation parameter. LV LV We use the L-type calcium channel density gCa,L as the bifurcation parameter i. The resulting bifurcation diagram is drawn in Fig. In addition to confirming the existence of a transcritical bifurcation for the computed values, it reveals the excitability switches induced by changes in L-type calcium channel density in this model: in the absence of L-type calcium channels, the model exhibits restorative excitability. As gCa,L increases, a saddle point and an unstable node emerge at a saddle-node bifurcation, which induces no excitability switch. Further increase of gCa,L causes a transcritical bifurcation, where the stable point and the saddle exchange their stability. At this point, the stable steadystate becomes regenerative, and the model switches to regenerative excitability. These excitability switches induce the predicted changes in the electrophysiological signatures, as illustrated in Fig. Whereas the DA neuron model instantaneously reacts to a step input of c depolarizing current for gCa,Lv gCa,L, it exhibits spike latency, plateau oscillations and ADP as soon as gCa,L becomes higher than c gCa,L. In addition, the model becomes strongly bistable. Thalamic relay RE neuron model. The model includes fast sodium channels INa,f , delayed-rectifier potassium channels IK,DR and T-type calcium channels ICa,T. We classify model variables as follows N GS,{~fhNa,f , mK,DRg and GS,z~fmCa,T g Balance equation and choice of the bifurcation parameter. The only source of regenerative excitability is includes fast sodium channels INa,f , delayed-rectifier potassium channels IK,DR, slow L-type calcium channels ICa,L and calcium-activated potassium channels IK,Ca. We classify model variables as follows N GS,{~fhNa,f , mK,DRg and GS,z~fmCa,Lg Balance equation and choice of the bifurcation parameter. As in the case of DA neurons, the source of regenerative excitability is provided by L-type calcium channels, and we take their maximal conductance gCa,L as the bifurcation parameter. The associated balance equation has the same structure as for the DA neuron model considered above. Switches of electrophysiological signatures are illustrated in Fig. Cerebellar granule cell GC model. The associated balance equation reads LV LV T-type calcium channels are dynamically regulated by a slow voltage-gated inactivation hCa,T. We use this variable as the bifurcation parameter i. Thalamic reticular RT neuron model. Classification of gating variables as fast GF , slow GS , and adaptation GA variables The model includes fast sodium channels INa,f , delayed-rectifier potassium channels IK,DR and T-type calcium channels ICa,T. We classify model variables as follows N GS,{~fhNa,f , mK,DRg and GS,z~fmCa,T g Balance equation and choice of the bifurcation parameter The only source of regenerative excitability is provided T-type calcium channels. The associated balance equation has the same structure as for the thalamic relay neuron model considered above. Along the same line, we choose the T-type calcium channel inactivation hCa,T as the bifurcation parameter i. Aplysia R15 neuron model. Classification of gating variables as fast GF , slow GS , and adaptation GA variables. The model Classification of gating variables as fast GF , slow GS , and adaptation GA variables. The model includes the following ion channels: fast INa,f , persistent INa,P and resurgent sodium channels INa,R ; N-type calcium channels ICa,N ; delayed rectifier IK,DR , A-type IK,A , inward rectifier IK,IR , calcium activated IK,Ca and slow potassium channels IK,slow. The inactivation of the A-type potassium current is fixed at hK,A~0:02 and the activation of the slow potassium current is fixed at mK,slow~0:13. Balance equation and choice of the bifurcation parameter. The neuron model possesses two sources of regenerative excitability: resurgent sodium channels and Ntype calcium channels. As in the case of T-type calcium channels mentioned above, these two channels possess an inactivation gate, which is used as the bifurcation parameter. We apply our algorithm by varying the parameter of one regenerative current while fixing the other at different values. This permits to draw an approximated hypersurface in the hCa,N ,hNa,R plane at which the balance equation is satisfied and the model undergoes the transcritical bifurcation and the hCa,N ~0:3, hNa,R~0:1. The two balance equations read, respectively: Solving Steps iii and iv of the algorithm, one obtains the parameter chart in Fig. These results show that both channels can induce a dynamical switch in excitability. On the contrary, when hCa,N ~0 the inactivation of resurgent sodium channel should be reduced at least by a factor two for the model to exhibit regenerative excitability. The balance equation determines a switch from restorative to regenerative excitability 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. Specific to regenerative excitability is the bistable phase portrait of Fig. 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. 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. 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. 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. In the case of second phase portrait saddlehomoclinic bifurcation , the situation is even worse because for small ew0 the system necessarily undergoes a monostable saddlenode on invariant circle bifurcation. In fact, the second phase portrait is not physiological for neuron conductance based models. 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. 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. Class I excitability is also referred to as soft excitation. 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. 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. A simple and robust balance equation identifies a transcritical bifurcation in arbitrary conductance based models 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 ubiquity of a transcritical bifurcation in conductancebased models 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 Vk 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 Hodgkins 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. A single mathematical prediction applies to many distinct physiological observations 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. 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 n0 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 V and the slow variable n. 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. Neuronal excitability is regulated 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. Materials and Methods Numerical analysis Numerical temporal traces of the different neurons Figs. 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. Supporting Information The reviewers are gratefully acknowledged for insightful comments and suggestions about the original version of the manuscript. Franci A , Drion G , Sepulchre R 2012 An organizing center in a planar model of neuronal excitability. SIAM J Appl Dyn Syst 11 : 1698 - 1722. Hodgkin A , Huxley A 1952 A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117 : 500 - 544. 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