Modeling the modulation of neuronal bursting: a singularity theory approach

Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. APPLIED DYNAMICAL SYSTEMS c 2014 Society for Industrial and Applied MathematicsVol. 13, No. 2, pp. 798829

Modeling the Modulation of Neuronal Bursting: A Singularity Theory Approach

Alessio Franci, Guillaume Drion, and Rodolphe Sepulchre

Abstract. Exploiting the specic structure of neuron conductance-based models, the paper investigates themathematical modeling of neuronal bursting modulation. The proposed approach combines singu-larity theory and geometric singular perturbations to capture the geometry of multiple time-scaleattractors in the neighborhood of high-codimension singularities. We detect a threetime-scale burst-ing attractor in the universal unfolding of the winged cusp singularity and discuss the physiologicalrelevance of the bifurcation and unfolding parameters in determining a physiological modulation ofbursting. The results suggest generality and simplicity in the organizing role of the winged cuspsingularity for the global dynamics of conductance-based models.

Key words. bursting, neuromodulation, singularity theory, modeling

AMS subject classications. 92B25, 37G10

DOI. 10.1137/13092263X

1. Introduction. Bursting is an important signaling component of neurons characterizedby a periodic alternation of bursts and quiescent periods. Bursts are transient, but high-frequency, trains of spikes, contrasting with the absence of spikes during the quiescent periods.Bursting activity has been recorded in many neurons, both in vitro and in vivo, and electro-physiological recordings show a great variety of bursting time series. All neuronal burstersshare nevertheless a sharp separation between three dierent time-scales: a fast time-scalefor the spike generation, a slow time-scale for the intraburst spike frequency, and an ultra-slow time-scale for the interburst frequency. Many neuronal models exhibit bursting in someparameter range, and many bursting models have been analyzed through bifurcation theory,but the exact mechanisms modulating neuronal bursting are still poorly understood, bothmathematically and physiologically. In particular, modeling the route to bursting, that is, thephysiologically observed modulation from a regular pacemaking activity to a bursting activity,has remained elusive to date. Also, many eorts have been devoted to classifying dierenttypes of bursters [29, 2, 11, 18]. But the mathematical mechanisms that allow the same neuronto be modulated across dierent types are rarely studied, despite their physiological role inhomeostatic cell regulation and development [25].

Received by the editors May 28, 2013; accepted for publication (in revised form) by M. Golubitsky January 25,2014; published electronically May 6, 2014. This paper presents research results of the Belgian Network DYSCO(Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiatedby the Belgian State, Science Policy Oce. The scientic responsibility rests with its authors.

http://www.siam.org/journals/siads/13-2/92263.htmlCorresponding author. Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK

([email protected]).Department of Electrical Engineering and Computer Science, University of Liege, Liege, Belgium, and Laboratory

of Neurophysiology, GIGA Neurosciences, University of Liege, Liege, Belgium ([email protected]).Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK ([email protected]).

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MODELING THE MODULATION OF NEURONAL BURSTING 799

As an attempt to advance the mathematical understanding of neuronal bursting, thepresent paper exploits the particular structure of conductance-based neuronal models to ad-dress with a local analysis tool the global structure of bursting attractors. Rooted in theseminal work of Hodgkin and Huxley [17], conductance-based models are nonlinear resistor-capacitor (RC) circuits consisting of one capacitance (modeling the cell membrane) in parallelwith possibly many voltage sources with voltage-dependent conductances (each modeling aspecic ionic current). The variables of the model are the membrane potential (V ) and thegating (activation and inactivation) variables that model the kinetics of each ion channel. Thevast diversity of ion channels involved in a particular neuron type leads to high-dimensionalmodels, but all conductance-based models share two central structural assumptions:

(i) A classication of gating variables in three well-separated time-scales (fast variablesare in the range of the membrane potential time scale 1ms; slow variables are 5 to 10times slower; and ultraslow variables are 10 to hundreds times slower), which roughlycorrespond to the three time scales of neuronal bursting.(ii) Each voltage-regulated gating variable x obeys the rst-order monotone dynamicsx(V )x = x + x(V ), which implies that, at steady state, every voltage-regulatedgating variable is an explicit monotone function of the membrane potential, that is,x = x(V ).

Our analysis of neuronal bursting rests on these two structural assumptions. Assumption(i) suggests a threetime-scale singularly perturbed bursting model, whose singular limit pro-vides the skeleton of the bursting attractor. Assumption (ii) implies that the equilibria ofarbitrary conductance-based models are determined by Kirchos law (currents sum to zeroin the circuit), which provides a single algebraic equation in the sole scalar variable V . Thisremarkable feature calls for singularity theory [12] to understand the equilibrium structure ofthe model.

The results of jointly exploiting time-scale separation and singularity theory for neuronalbursting modeling provide the following specic contributions.

The universal unfolding of the winged-cusp singularity is shown to organize a threetime-scale burster. The three-level hierarchy of singularity theory dictates the hierarchy oftime-scales: the state variable of the bifurcation problem is the fast variable, the bifurca-tion parameter is the slow variable, and unfolding parameter(s) are the ultraslow variable(s).Because the geometric construction is grounded in the algebraic and time-scale structure ofconductance-based models, the proposed model can be related to detailed conductance-basedmodels through mathematical reduction. We provide general conditions for this mathemati-cal model to be a normal form reduction of an arbitrary conductance-based model. Both thebifurcation parameter and the unfolding parameters have a clear physiological interpretation.

The bifurcation parameter is directly linked to the balance between restorative and regen-erative slow ion channels, the importance of which was recently studied by the authors in [10].The modulation of the bifurcation parameter in the proposed threetime-scale model providesa geometrically and physiologically meaningful transition from slow tonic spiking to bursting.This route to bursting is known to play a signicant role in central nervous system activity[30, 34, 3]. Its mathematical modeling appears to be novel.

The three unfolding parameters modulate in an even slower scale the fast-slow phaseportrait of the threetime-scale burster. The ane parameter plays the classical role of anD

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800 A. FRANCI, G. DRION, AND R. SEPULCHRE

adaptation current that hysterically modulates the slow-fast phase portrait across a parameterrange where a stable resting state and a stable spiking limit cycle coexist, thereby creatingthe bursting attractor. The two remaining unfolding parameters can modulate the burstingattractor across a continuum of bursting types. As a result, transitions between dierentbursting waveforms, observed, for instance, in developing neurons [25], are geometrically cap-tured as paths in the unfolding space of the winged cusp. The physiological interpretation ofthis modulation is a straightforward consequence of the clear physiological interpretation ofeach unfolding parameter.

The existence of threetime-scale bursters in the abstract unfolding of a winged cusp ispresented in section 2. Section 3 focuses on a minimal reduced model of neuronal burstingand uses the insight of singularity theory to describe a physiological route to bursting inthis model. Section 4 shows how to trace the same geometry in arbitrary conductance-basedmodels. Section 5 discusses in a less technical way the relevance of the winged-cusp singularityfor the modeling of bursting modulation. The technical details of mathematical proofs arepresented in the appendices.

2. Universal unfolding and multipletime-scale attractors.

2.1. A primer on singularity theory. We introduce here some notation and terminologythat will be used extensively in the paper. The interested reader is referred to the main resultsof Chapters IIV in [12] for a comprehensive exposition of the singularity theory used in thispaper.

Singularity theory studies scalar bifurcation problems of the form

(1) g(x, ) = 0, x, R,where g is a smooth function. The variable x denotes the state, and is the bifurcationparameter. The set of pairs (x, ) satisfying (1) is called the bifurcation diagram. Singularpoints satisfy g(x, ) = gx(x

, ) = 0. Indeed, if gx(x, ) = 0, then the implicit function

theorem applies, and the bifurcation diagram is necessarily regular at (x, ).Except for the fold x2 = 0, bifurcations are not generic; that is, they do not persist

under small perturbations. Singularity theory is a robust bifurcation theory: it aims at classi-fying all possible persistent bifurcation diagrams that can be obtained by small perturbationsof a given singularity.

A universal unfolding of g(x, ) is a parametrized family of functions G(x, ;), where lies in the unfolding parameter space Rk, such that the following hold:

(1) G(x, ; 0) = g(x, ).(2) Given any p(x) and a small > 0, one can nd an near the origin such thatthe two bifurcation problems G(x, ;) = 0 and g(x, ) + p(x) = 0 are qualitativelyequivalent.(3) k is the minimum number of unfolding parameters needed to reproduce all per-turbed bifurcation diagrams of g(x, ). k is called the codimension of g(x, ).

Unfolding parameters are not bifurcation parameters. Instead, they change the qualitativebifurcation diagram of the perturbed bifurcation problem G(x, ;) = 0. That is why is a distinguished parameter in the theory. Historically, this parameter was associated to aslow time, whose evolution lets the dynamics visit the bifurcation diagram in a quasisteadyD

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state manner. It will play the same role in the present paper, where we consider only twosingularities and their universal unfolding.

The rst singularity we consider is the codimension 1 hysteresis

(2) gshy(x, ) = x3 ,

whose universal unfolding is shown to be [12, Chapter IV]

(3) Gshy(x, ; ) = x3 + x.

The second singularity we consider is the codimension 3 winged cusp

(4) gswcusp(x, ) = x3 2,

whose universal unfolding is shown to be [12, section III.8 and Chapter IV]

(5) Gswcusp(x, ; , , ) = x3 2 + x x .

The universal unfolding of codimension 1 bifurcations contains some codimension 1bifurcation. For instance, the universal unfolding of the winged cusp possesses hysteresisbifurcations on the unfolding parameter hypersurface dened by 2 + = 0, 0. Eventhough such bifurcation diagrams are not persistent, they dene transition varieties thatseparate equivalence classes of persistent bifurcation diagrams, hence providing a completeclassication of persistent bifurcation diagrams.

An unperturbed bifurcation problem assumes the suggestive role of organizing center:all the perturbed bifurcation diagrams are determined and organized by the unperturbedbifurcation diagram, which constitutes the most singular situation. Via the inspection of localalgebraic conditions at the singularity, an organizing center provides a quasi-global descriptionof all possible perturbed bifurcation diagrams.

2.2. The hysteresis singularity and spiking oscillations. The hysteresis singularity hasa universal unfolding x3 + x with persistent bifurcation diagram plotted in Figure 1Afor > 0. We use this algebraic curve to generate the phase portrait in Figure 1B of thetwotime-scale model

x = Gshy(x, + y; )(6a)

= x3 + x y,y = (x y).(6b)

Because y is a slow variable, it acts as a slowly varying modulation of the bifurcation parameterin the fast dynamics (6a). As a consequence, the global analysis of system (6) reduces to aquasi-steady state bifurcation analysis of (6a), whence the relationship between Figure 1Aand Figure 1B.

The following (well-known) theorem characterizes a global attractor of (6), that is, theexistence of Van der Pol type relaxation oscillations in the universal unfolding of the hysteresis.D

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x

y

x

A) B)

Figure 1. Relaxation oscillations in the universal unfolding of the hysteresis bifurcation. A. A persistentbifurcation diagram of the hysteresis singularity. Branches of stable (resp., unstable) xed points are depictedas full (resp., dashed) lines. B. Through a slow adaptation of the bifurcation parameter, the bifurcation diagramin A is transformed into the phase plane of a two-dimensional dynamical system, which still denes a universalunfolding of the hysteresis singularity. The thick full line is the fast subsystem nullcline. The thin full line isthe slow subsystem nullcline. The circle denotes an unstable xed point. For small > 0, the model exhibitsexponentially stable relaxation oscillations (depicted in blue).

Theorem 1 (see [13],[26],[24]). For = 0 and for all 0 < < 1, there exists > 0 suchthat, for all (0, ], the dynamical system (6) possesses an exponentially stable relaxationlimit cycle, which attracts all solutions except the equilibrium at (0, 0).

The reader familiar with neurodynamics will recognize in (6) a famous model introducedby FitzHugh [8]. It is the prototypical planar reduction of spiking oscillations. There istherefore a close relationship between the hysteresis singularity and spike generation.

It is worth emphasizing that the relationship between singularity theory (Figure 1A) andthe twotime-scale phase portrait (Figure 1B) imposes choosing the bifurcation parameter,not an unfolding parameter, as the slow variable. It should also be observed that the slowvariable is a deviation from the unfolding parameter rather than the bifurcation parameteritself. Keeping as the bifurcation parameter of the two-dimensional dynamics (6) allowsus to shape its equilibrium structure according to the universal unfolding of the organizingsingularity (in this case, the hysteresis) and will play an important role in the next section.

2.3. The winged cusp singularity and rest-spike bistability. We repeat the elementaryconstruction of section 2.2 for the codimension-three winged cusp singularity x3 2. Itdiers from the hysteresis singularity in the nonmonotonicity of g(x, ) in the bifurcation

parameter; that is, (x32) = 2 changes sign at the singularity.

Figure 2A illustrates an important persistent bifurcation diagram in the unfolding ofthe winged cusp, obtained for = 0, > 0, and < 2(3 )3/2. We call it the mirroredhysteresis bifurcation diagram. The right part ( > 0) of this bifurcation diagram is essentiallythe persistent bifurcation diagram of the hysteresis singularity in Figure 1A. In that region,Gswcusp

< 0. The left part ( > 0) is the mirror of the hysteresis, and, in that region,Gswcusp

> 0. For = 0, the mirroring eect is not perfect, but the qualitative analysis doesnot change. The hysteresis and its mirror collide in a transcritical singularity for = 2(3 )3/2.Do

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This singularity belongs to the transcritical bifurcation transition variety in the winged cuspunfolding (see Appendix A). The transcritical bifurcation variety plays an important role inthe forthcoming analysis.

x

y

x

A) B)

0

NsS

Nu

eWs

Figure 2. Singularly perturbed rest-spike bistability in the universal unfolding of the winged cusp. A.Mirrored hysteresis persistent bifurcation diagram of the winged cusp for > 0, < 2(

3)3/2, and = 0. B.

A phase plane of (7).

We use the algebraic curve in Figure 2A to generate the phase portrait in Figure 2B ofthe two-dimensional model

x = Gswcusp(x, + y; , , )(7a)

= x3 + x (+ y)2 (+ y)x ,y = (x y).(7b)

Its xed point equation

(8) F (x, , , , ) := x3 + x (+ x)2 (+ x)x is easily shown to be again a universal unfolding of the winged cusp around xwcusp :=

13 ,

wcusp := 0, wcusp := 127 , wcusp := 13 , wcusp := 2. The phase portrait in Figure 2B isa prototype phase portrait of rest-spike bistability: a stable xed point coexists with a stablerelaxation limit cycle.

Similarly to the previous section, the analysis of the singularly perturbed model (7) iscompletely characterized by the bifurcation diagram of Figure 2A. This bifurcation diagramprovides a skeleton for the rest-spike bistable phase portrait in Figure 2B, as stated in thefollowing theorem. Its proof is provided in section B.1.

Theorem 2. For all > wcusp, there exist open sets of bifurcation () and unfolding (, )parameters near the pitchfork singularity at (, , ) = (PF (), PF (), PF ()), in which,for suciently small > 0, model (7) exhibits the coexistence of an exponentially stable xedpoint Ns and an exponentially stable spiking limit cycle

. Their basins of attraction areseparated by the stable manifold W s of a hyperbolic saddle S (see Figure 2B).

Figure 3 shows the transition in (7) from the hysteresis phase portrait in Figure 1B to thebistable phase portrait in Figure 2B through a transcritical bifurcation. Both phase portraitsD

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x

y

Pitchfork

Transcritical

Figure 3. An unfolding of the pitchfork bifurcation variety in (7). The phase portraits in Figures 1 and 2both belong to the unfolding of the pitchfork singularity in center. A smooth deformation of the phase portraitof Figure 1 into the phase portrait of Figure 2 involves a transcritical bifurcation, which degenerates into apitchfork for a particular value of the unfolding parameter .

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are generated by unfolding the degenerate portrait in Figure 3 (center), which belongs to thepitchfork bifurcation variety (, ) = (PF (), PF ()), > wcusp (see Appendix A). Thetranscritical bifurcation variety = TC(, ) is obtained through variations of the unfoldingparameter away from the pitchfork variety. It provides the two phase portraits in Figure3 (center top and bottom). By increasing or decreasing the bifurcation parameter anddecreasing the unfolding parameter out of the transcritical bifurcation variety, these phaseportraits perturb to the generic phase portraits in the corner, corresponding to the qualitativephase portraits in Figures 1B and 2B, respectively. The reader of [9] will recognize the sameorganizing role of the pitchfork bifurcation in a planar model of neuronal excitability.

2.4. A threetime-scale bursting attractor in the winged cusp unfolding. The coexis-tence of a stable resting state and stable spiking oscillation, or singularly perturbed rest-spikebistability, makes (7) a good candidate for the slow-fast subsystem of a threetime-scale min-imal bursting model:

x = Gswcusp(x, + y; + z, , )(9a)

= x3 + x (+ y)2 (+ y)x z,y = 1(x y),(9b)z = 2(z + ax+ by + c),(9c)

where 0 < 2 1 1 and a, b, c R. The z-dynamics model the ultraslow adaptation of theane unfolding parameter in such a way that the global attractor of (9) will be determinedby a quasi-static modulation of (9a) through dierent persistent bifurcation diagrams.

Here, again, the role of singularity theory in distinguishing bifurcation and unfolding pa-rameters is crucial. The hierarchy between these parameters and the state variable, formalizedin the theory in [12, Denition III.1.1], is reected here in the hierarchy of time-scales.

The time-scale separation between (9a)(9b) and (9c) makes it possible once again toderive a global analysis of model (9) from the analysis of the steady state behavior of (7) as is varied. Such analysis can easily be derived geometrically in the singular limit 1 = 0. It issketched in Figure 4. For (SN , 0SH), the singularly perturbed model (7) exhibits rest-spike bistability, that is, the coexistence of a stable node Ns, a singular stable periodic orbit0, and a singular saddle separatrix W 0s . At =

0SH = 2(3 )3/2 the left and right branches

of the mirrored hysteresis bifurcation collide in a transcritical singularity that serves as aconnecting point for a singular homoclinic trajectory SH0. For > 0SH , the only (singular)attractor is the stable node Ns. At = SN , the saddle and the stable nodes merge in asaddle-node bifurcation SN . For < SN , the only attractor is the singular periodic orbit0. The dierent singular invariant sets in Figure 4A can be glued together to construct thethree-dimensional singular invariant set M0 in Figure 4B (left).

The singular invariant setM0 provides a skeleton for a threetime-scale bursting attractorthat shadows the branch L of stable xed points alternately with the branch P 0 of (singular)stable periodic orbits, as depicted in Figure 4B (right). To prove the existence of such anattractor, we need only understand how M0 perturbs for 1 > 0.

Near the singular limit, the branch of singular periodic orbits P 0 perturbs to a nearbybranch of exponentially stable periodic orbits P (see Figure 5), whereas the singular homo-clinic trajectory SH0 perturbs to an unstable homoclinic trajectory SH (at = SH). TheD

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V

n

LS

SH 0

UUUUU

P0

F

B)

A)

l

Ws

0

0

l 0SN

l 0

SH0

SH0

SN

Rest-spike bistable

M = L U S U S U U U P U SH0 0 0

NsNs

Ns

Figure 4. Singular steady-state behavior of (7) through a variation of the unfolding parameter . A.Singular phase portraits of (7) for = 0, = 1

3, and small negative . B. Gluing the dierent invariant sets

in A leads to the three-dimensional singular invariant set M0 (left), which provides a skeleton for a three-timescale bursting attractor (right) in the singularly perturbed system (9). The branch of stable xed points (resp.,saddle points) for < SN is drawn as the black solid curve L (resp., the black dashed curve S). The saddlenode bifurcation connecting them is denoted by F. The branch of unstable xed points is drawn as the blackdashed line U. The branch of stable singular periodic orbit for < 0SH is drawn as the blue cylindric surfaceP 0. The singular saddle homoclinic trajectory is drawn as the orange oriented curve SH0.

branch of unstable periodic orbits Q generated at SH eventually merges with P at a foldlimit cycle bifurcation FLC for some FLC (SH , 0SH). In the whole range (SN , FLC),model (7) exhibits the coexistence of a stable xed point and a stable spiking limit cycle. Thedetails of this analysis are contained in Lemma 6 in section B.2.

We follow [33, 32] to derive conditions on the bifurcation and unfolding parameters in (9a)(9b) and to place the hyperplane z = 0 (through a suitable choice of the parameters a, b, c R)such that an ultraslow variation of z can hysteretically modulate the slow-fast subsystem (9a)(9b) across its bistable range (SN ,

FLC) to obtain stable bursting oscillations. The existence

of such bursting oscillations is stated in the following theorem. Its proof is provided in sectionB.2.

Theorem 3. For all > wcusp, there exists an open set of bifurcation () and unfolding(, ) parameters near the pitchfork singularity at (, , ) = (PF (), PF (), PF ()) suchD

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F LS

U

x

Px max

x min

0SHSN SH FLC

QFLC

SH

Figure 5. Bifurcation diagram of (7) with respect the unfolding parameter for suciently small . Thebranch of stable xed points is depicted as the full thin line L, the branch of saddle points as the dashed thinline S, and the branch of unstable xed points as the dashed thin line U. The branch of stable periodic orbitsis depicted by the thick full lines P and the branch of unstable periodic orbits by the thick dashed lines Q.SH: Saddle-homoclinic bifurcation. FLC : Fold limit cycle bifurcation. F: Fold (saddle-node) bifurcation.The yellow strip between the saddle-node and fold limit cycle bifurcations denotes the rest-spike bistable range.

that, for all , , in those sets, there exist a, b, c, R such that, for suciently small 1 2 > 0, model (9) has a hyperbolic bursting attractor.

Theorem 3 uses the two regenerative phase portraits in Figure 3 left to construct a burst-ing attractor by modulating the unfolding parameter . The bursting attractor directly restsupon the bistability of those phase portraits. It should be noted that the same constructioncan be repeated on the restorative phase portraits in Figure 3 right. However, those phaseportraits are monostable, and their ultraslow modulation leads to a slow tonic spiking (i.e.,a single spike necessarily followed by a rest period). This attractor diers from a burstingattractor by the absence of a bistable range in the bifurcation diagrams of Figure 4. It canbe shown that the persistence of (rest-spike) bistability in the singular limit is a hallmark ofregenerative excitability (Figure 3 (left)) and that it cannot exist in restorative excitability(Figure 3 (right)). See [10] for a more detailed discussion. Modulation in (9) of the bifurca-tion parameter across the transcritical bifurcation of Figure 3 therefore provides a geometrictransition from the slow tonic spiking attractor to the bursting attractor. This transitionorganizes the geometric route into bursting discussed in the next section.

3. A physiological route to bursting.

3.1. A minimal threetime-scale bursting model. The recent paper [9] introduces theplanar neuron model

V = V V3

3 n2 + I,(10a)

n = (n(V V0) + n0 n).(10b)

Its phase portrait was shown to contain the pitchfork bifurcation of Figure 3 as an organizingcenter, leading to distinct types of excitability for distinct values of the unfolding parameters.The analysis of the previous section suggests that a bursting model is naturally obtained byD

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augmenting the planar model (10) with ultraslow adaptation:

V = kV V3

3 (n+ n0)2 + I z,(11a)

n = n(V ) (n(V V0) n) ,(11b)z = z(V )(z(V V1) z).(11c)

Model (10) is essentially model (11a)(11b) for k = 1 and z = 0, modulo a translationn n+n0. The dynamics (11b)(11c) mimic the kinetics of gating variables in conductance-based models, where the steady-state characteristics n() and z() are monotone increasing(typically sigmoidal) and the time-scaling n() and z() are Gaussian-like strictly positivefunctions. Details of model (11) for the numerical simulations of the paper are provided inAppendix C.

The slow-fast subsystem (11a)(11b) shares the same geometric structure as (7). After atranslation V V + V0, the right-hand side of (11a) can easily be shown to be a universalunfolding of the winged cusp, and the slow dynamics (11b) modulate its bifurcation parameter.Plugging in the ultraslow dynamics (11c), one recovers the same structure as (9). Therefore,the conclusions of Theorems 2 and 3 apply to (11).

The dierence between (11) and (9) is that the model (11) has the physiological inter-pretation of a reduced conductance-based model, with V a fast variable that aggregates themembrane potential with all fast gating variables, n a slow recovery variable that aggregatesall the slow gating variables regulating neuronal excitability, and z an ultraslow adaptationvariable that aggregates the ultraslow gating variables that modulate the cellular rhythm overthe course of many action potentials. Finally, I models an external applied current.

3.2. Model parameters and their physiological interpretation.

The bifurcation parameter n0 models the balance between restorative and regenera-tive ion channels. The central role of the bifurcation parameter n0 in (11) was analyzed in[9, 10] and is illustrated in Figure 6. The transcritical bifurcation variety in Figure 3 cor-responds to the physiologically relevant transition from restorative excitability (large n0) toregenerative excitability (small n0). When the excitability is restorative, the recovery variablen provides negative feedback on membrane potential variations near the resting equilibrium,a physiological situation well captured by the FitzHughNagumo model (or the hysteresissingularity). In contrast, when excitability is regenerative, the recovery variable n providespositive feedback on membrane potential variations near the resting potential, a physiologicalsituation that requires the quadratic term in (11a) (or the winged cusp singularity).

The value of n0 in a conductance-based model reects the balance between restorative andregenerative ion channels that regulate neuronal excitability. How to determine the balancein an arbitrary conductance-based model is discussed in [10]. Note that the restorative orregenerative nature of a particular ion channel in the slow time-scale is an intrinsic propertyof the channel. A prominent example of a restorative channel is the slow potassium activationshared by (almost) all spiking neurons. A prominent example of a regenerative channel is theslow calcium activation encountered in most bursting neurons. The presence of regenerativechannels in neuronal bursters is well established in neurophysiology (see, e.g., [21, 1]).D

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n

V

n

V

Large n - restorative ( hysteresis)

0 Small n - regenerative0

TC

Figure 6. Transition from restorative excitability (tonic ring) to regenerative excitability (bursting) inmodel (11) by sole variation of the bifurcation parameter n0. The analytical expression of the steady statefunctions n() and z() and numerical parameter values are provided in Appendix C. The time scale is thesame in the left and right time series.

The ane unfolding parameter provides bursting by ultraslow modulation of the cur-rent across the membrane. For small n0, the modulation of the ultraslow variable z createsa hyperbolic bursting attractor through the hysteretic loop described in Figure 4. The bursterbecomes a single-spike limit cycle (tonic ring) for large n0 (restorative excitability), that is,in the absence of rest-spike bistability in the planar model.

The presence of ultraslow currents in neuronal bursters is well established in neurophysiol-ogy (see, e.g., [1]). A prominent example is provided by ultraslow calcium activated potassiumchannels.

Half activation potential aects the route to bursting. The role of the unfolding param-eter in (9) is illustrated in Figure 3: it provides two qualitatively distinct paths connectingthe restorative and regenerative phase portraits. This role is played by the parameter V0 in theplanar model (10) studied in [9], which has the physiological interpretation of a half activationpotential. The role of half activation potentials in neuronal excitability is well documented inneurophysiology (see, e.g., [27]). The role of this unfolding parameter in the route to burstingis discussed in the next subsection.

No spike without fast autocatalytic feedback. The role of the unfolding parameter kin (11) is to provide positive (autocatalytic) feedback in the fast dynamics. The prominentsource of this feedback in conductance-based models is the fast sodium activation. It is wellacknowledged in neurodynamics [19].

The reduced model (11) makes clear predictions about its dynamical behavior in theabsence of this feedback (i.e., k = 0). Those predictions are further discussed in section 5.2and are in close agreement with the experimental observation of small oscillatory potentialsD

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when sodium channels are shut down with pharmacological blockers [14, 35] or are poorlyexpressed during neuronal cell development [25].

3.3. A physiological route to bursting. A central insight of the reduced model (11) isthat it provides a route to bursting: xing all unfolding parameters and varying only thebifurcation parameter n0 leads to a smooth transition from tonic ring to bursting; see Figure7. Smooth and reversible transitions between those two rhythms have been observed in manyexperimental recordings [30, 34], making the route to bursting an important signaling mecha-nism. The fact that the modulation is achieved simply through the bifurcation parameter n0,i.e., the balance between restorative and regenerative channels, is of physiological importancebecause it is consistent with the physiology of experimental observations of routes to bursting[30, 34, 3].

n decreasesRestorative Regenerative

0

Figure 7. Route from tonic ring to bursting in model (11) via a smooth variation of the bifurcationparameter n0. Other parameters are as in Figure 6.

The analysis in the above sections shows that the transition from single spike to burstingis through the transcritical bifurcation variety in model (7). Looking at the singular limit = 0 of (7) near this transition variety provides further insight into the geometry of the routethat leads to the appearance of the saddle-homoclinic bifurcation organizing the bistablephase portrait. This route is organized by the path through the pitchfork bifurcation, whichprovides the most symmetric path across the transcritical variety. The generic transitions areunderstood by perturbing the degenerate path.

Figure 8A shows the qualitative projection of those paths onto the (V0, n0) parameter chartobtained in model (10) for I = 23 . The chart is reproduced from [9]. The same qualitativepicture is obtained for the (, ) parameter chart of the abstract model (7) at = TC(, )(see Appendix A). The chart associates dierent excitability types (as well as their restorativeor regenerative nature; see [10]) to distinct bifurcation mechanisms. Unfolding those pathsalong the I (or ) direction leads to the bifurcation diagrams in Figure 9B. They reveal (inthe singular limit) the onset of the bistable range organized by the singular saddle-homoclinicloop SH0 as paths cross the transcritical bifurcation variety.

The same qualitative picture persists for > 0. Figure 9 illustrates how the appearanceof the singular saddle-homoclinic loop is accompanied, for > 0, by a smooth transition froma monostable (SNIC - route (i)) or barely bistable (sub-Hopf - route (ii)) bifurcation diagramto the robustly bistable bifurcation diagram constructed in the sections above (Figure 5).Through ultraslow modulation of the unfolding parameter , this transition geometricallyD

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SN

TC

II

I

V

IV

i)ii)

(or n )0

(or V )0

(or z )TC (or n )0 TC

SH0SH0

(or n )0

(or z ) (or n )0

(or z )PF SH0

SNSN

SN

i) ii)

A)

B)

Bis

t.

Bis

t.

Bis

t.

Figure 8. Routes into bursting in the universal unfolding of the pitchfork bifurcation. A. Qualitativeprojection of routes into bursting onto the (V0, n0) (resp., (, )) of model (10) (resp., (7)) for I =

23(resp.,

= TC(, ); see Appendix A). Excitability is restorative in subregions I and II, mixed in subregion V, andregenerative in subregion IV. See [9] and [10] for details concerning the underlying bifurcation mechanisms. Thetransition path labeled with a star depicts the degenerate path across the pitchfork. The generic paths (i) and(ii) are distinguished by dierent half activation potentials V0 (resp., unfolding parameter ). B. Unfolding oftransition paths in A along the I (resp., ) direction. Black thick lines denote branches of saddle-node (SN)bifurcation. In paths (i) and (ii), the model undergoes a transcritical bifurcation (TC) as the path tangentiallytouches a branch of SN bifurcations. In the degenerate path, the model undergoes a pitchfork (PF) bifurcation asthe path enters the cusp tangentially to both branches of SN bifurcations. The singular saddle-homoclinic loop,geometrically constructed in Figures 4 and 9, is denoted by SH0 and determines the appearance of a singularbistable range persisting away from singular limit.

captures the transition from tonic spiking to bursting via the sole variation of the bifurcationparameter.

The strong agreement between the mathematical insight provided by singularity theoryand the known electrophysiology of bursting is a peculiar feature of the proposed approach.There is a direct correspondence between the bifurcation and unfolding parameters of thewinged cusp and the physiological minimal ingredients of a neuronal burster. In particular, ouranalysis predicts that any bursting neuron must possess at least one physiologically regulatedD

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Transcritical

Pitchfork

n

=0

>0

>0

0

n0

Figure 9. Geometry of the two generic routes into bursting in the unfolding of the pitchfork bifurcation inmodel (9) and model (11).

slow regenerative channel. This prediction needs to be tested systematically, but we haveDow

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found no counterexample in the bursting neurons we have analyzed to date.

4. Normal form reduction of conductance-based models.

4.1. A two-dimensional reduction. The winged cusp singularity emerges as an organizingcenter of rhythmicity in the reduced neuronal model (11), but a legitimate question is whetherthis singularity can be traced in arbitrary (high-dimensional) conductance-based models. Ourrecent paper [10] addresses a closely related question for the transcritical variety. It providesan analogue of the bifurcation parameter n0 in arbitrary conductance-based models of theform

CmV =

gma h

b (V E) + Iapp

=: Iion(V, xf , xs, xus) + Iapp,(12a)

xfj(V )xfj = xfj + xfj,(V ), j = 1, . . . , nf ,(12b)

xsj (V )xsj = xsj + xsj,(V ), j = 1, . . . , ns,(12c)

xusj (V )xusj = xusj + xusj,(V ), j = 1, . . . , nus,(12d)

where runs through all ionic currents, xf := [xfj ]j=1,...,nf denotes the nf -dimensional columnvector of fast gating variables, xs := [xsj ]j=1,...,ns denotes the ns-dimensional column vector ofslow gating variables, and xus := [xusj ]j=1,...,nus denotes the nus-dimensional column vector ofultraslow variables (see also [10] for more details on the adopted notation).

Following common analysis methods in neurodynamics, we want to reduce the (possibly)high-dimensional model (12) to a two-dimensional model of the form

V = F (V, n) + I,(13a)

(V )ns = n+ ns(V ),(13b)

where V is the fast voltage and n is a slow aggregate variable. We achieve this reductionby rst considering the singular limit of three time-scales leading to a quasisteady stateapproximation for fast gating variables, that is,

(14) xfj xfj,(V )

for all j = 1, . . . , nf , and freezing ultraslow variables, that is, setting

xusj xusjfor all j = 1, . . . , nus, where the values x

usj belong to the physiological range of the dierent

variables. The remaining dynamics read as

V = Iion(V, xf(V ), x

s, xus) + Iapp,

(V )xsj = (xsj + xsj,(V )), j = 1, . . . , ns,

which is a fast-slow system with V as fast variable and xs as slow variables.Dow

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The planar reduction proceeds from the change of variables

n = xs1,

ni = xsi xsi,(n1 (n)), i = 2, . . . , ns.

This change of variable is globally invertible by monotonicity of the (in)activation functionsxsi,. Under the additional simplifying assumption of identical time constants

(15) xsj (V ) = (V ) 1 1for all V R and all j = 1, . . . , ns, it is an easy calculation to show that

(V )ni = ni +O((n n)2, (n n)(V V ), (V V )2)around any equilibrium (V , (xsi )

) := (V , xsi,(V)). It follows that, locally around any

equilibrium, the two-dimensional manifold

Mred :={(V, xs) R [0, 1]ns : ni = 0, i = 2, . . . , ns

}={(V, xs) R [0, 1]ns : xsi = xsi,(n1 (n)), i = 2, . . . , ns

}is exponentially attractive.

It should be stressed that the (harsh) simplifying assumption (15) is necessary only aroundthe steady state value V and that the hyperbolic decomposition is robust to small perturba-tions [16]. It should also be observed that the proposed two-dimensional reduction is a straight-forward generalization of the classical two-dimensional reduction of the HodgkinHuxleymodel [8, 28] that rests on setting sodium activation to steady state (mNa mNa,(V ))and using an algebraic relationship between the sodium inactivation and the potassium acti-vation (usually in the form h 1 n).

4.2. The winged cusp planar model (7) is a local normal form of slow-fast conductance-based models. Given an equilibrium (V , n(V )) of (13), consider the (linear) change ofvariables

x = V V ,y =

n n(V )nV (V

).

The y dynamics are particularly simple. Indeed, by simple Taylor expansion,

y = (x y) +O(x2), :=

1

(V ) 1.

In the new coordinates, (13) reads as

x = F

(x+ V , n(V ) +

nV

(V )y

)+ I,(16a)

y = (x y) +O(x2).(16b)Dow

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Simple computations show that (16a) satises

x

x(0, 0) =

IionV

+

nfi=1

Iion

xfi

xfi,V

,

x

y(0, 0) =

nsi=1

Iionxsi

xsi,V

,

where the right-hand sides are computed at V = V , xf = xf(V ), xs = xs(V ), and xus =xus.

We claim that the critical manifold x = 0 of (13) has a degenerate singularity providedthat

(i) the full slow-fast subsystem has a degenerate equilibrium; that is, the Jacobian ofthe slow-fast subsystem (12a)(12c) is singular;(ii) at such an equilibrium, the contributions of slow restorative and slow regenerativechannels [10] are perfectly balanced; that is,

nsi=1

Iionxsi

xsi,V

= 0.

To prove our claim we notice with computations similar to those in [10] that conditions (i)and (ii) imply that

IionV

+

nfi=1

Iion

xfi

xfi,V

= 0,

which is equivalent to the Jacobian of the fast subsystems (12a)(12b) being singular. Hence,when conditions (i) and (ii) are fullled,

(17)x

x(0, 0) =

x

y(0, 0) = 0.

Property (17) ensures that the critical manifold of (16) has a codimension > 0 singu-larity at the origin (where, as usual, the slow variable y plays the role of the bifurcationparameter). This singularity corresponds to the transcritical bifurcation detected in arbitraryconductance-based models in [10]. It is indeed proved in [10] that conditions (i) and (ii)enforce a transcritical bifurcation in the associated conductance-based model.

Algebraically, (17) ensures that, similarly to the bifurcation parameter in the wingedcusp universal unfolding (see section 2.3), y modulates the fast x dynamics nonmonotonically.Physiologically, it captures in the reduced model the nonmonotone modulation of membranepotential dynamics by slow restorative (providing negative feedback) and slow regenerative(providing positive feedback) ion channels.

We use the algorithm in [10] to detect the degenerate dynamics of (16) in arbitraryconductance-based models. This construction reveals that the transcritical bifurcation ispart of the transcritical transition variety in the universal unfolding of the winged cusp. Theresult is sketched in Figure 10 (left) and veried numerically in the HodgkinHuxley modelD

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augmented with a calcium current in Figure 10 (right). The model and its reduction arepresented and further discussed in section 4.3 below. The obtained phase plane is organizedby the mirrored hysteresis bifurcation diagram of the normal form (7) in Figure 2 in the lim-iting case in which the two hysteresis branches merge at the transcritical bifurcation. Thisprovides an indirect proof that the global phase plane is organized by the winged cusp. Thissingularity is indeed the only (codimension 3) singularity exhibiting the mirrored hysteresisin its universal unfolding (see [12, section IV.4]).

(V, x f , x s )

x

y

140 20

0

100

x

y

Figure 10. A transcritical bifurcation in the universal unfolding of the winged cusp organizes the dynamicsof the two-dimensional reduction of generic conductance-based models. Left: Sketch of the dynamics on thetwo-dimensional invariant manifold Mred. Right: Construction of the two-dimensional reduction (16) at thetranscritical bifurcation in the HodgkinHuxley model augmented with a calcium current (18)(19).

One can push forward the singularity analysis and derive an algorithm to enforce thedegenerate conditions of the winged cusp rather than the transcritical bifurcation by usingadditional model parameters as auxiliary parameters [12, section III.4]. This would lead tothe conclusion that the critical manifold of the reduced dynamics (16) is actually a versalunfolding of the winged cusp. Alternatively, one can modulate model parameters and showthat their variations recover all persistent bifurcation diagrams of the winged cusp. Suchcomputations, however, are lengthy and bring no new information to the picture presentedhere.

4.3. Application to the HodgkinHuxley model augmented with a regenerative chan-nel. The rst conductance-based model appears in the seminal paper of Hodgkin and Huxley[17]:

CV = gKn4(V VK) gNam3h(V VNa) gl(V Vl) + I,(18a)m(V )m = m+m(V ),(18b)n(V )n = n+ n(V ),(18c)h(V )h = h+ h(V ),(18d)

where the time constants x and the steady state characteristics x, x = m,n, h, are chosen inaccordance with the original model (see Appendix D). The model only accounts for two ioniccurrents: sodium, with its fast activation variable m and slow inactivation h, and potassium,with slow activation n. The classical phase portrait reduction [8, 28] is obtained with theD

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quasisteady state approximationm m(V ) and the empirical t h 1n. It is well knownthat in its physiological part (0 < n < 1) this phase portrait is qualitatively the FitzHughphase portrait in Figure 1. But we showed in [6, Figure 5] that the entire phase portrait(n R) indeed also contains the mirrored phase portrait of Figure 2. This observationsuggests that a winged cusp organizes the fast subsytem (18a)(18b) of HodkginHuxleydynamics. The singularity is found in a nonphysiological range of the phase space (n < 0),which is consistent with the absence of slow regenerative currents in the model.

The missing element in the HodgkinHuxley model to make the winged cusp physiologicalis a slow regenerative ion channel. Following [6], we add the calcium current

ICa,L = gCad(V VCa),(19a)d(V )d = d+ d(V ).(19b)

The algorithm in [10] detects a transcritical bifurcation for

V 61.2730, gCa 0.2520, I 30.7694.

Following the construction in section 4.2, in particular, (16), the associated reduced variationaldynamics at the transcritical bifurcation read

x = gK(n(V ) + y

nV

(V )

)4(V + x VK)

gNam(V + x)3(h(V ) + y

hV

(V ) +O(y2))(V VNa)

gCa(d(V ) + y

dV

(V ) +O(y2))(V VCa)

gl(V Vl) + Iy = (x y) +O(x2).

Its phase plane is drawn in Figure 10 (right).

We now apply the global two-dimensional reduction described in section 4.1, in particular,(13), to model (18)(19). To this aim, we express all variables in terms of potassium activationn. Since in the original model its activation function cannot be explicitly inverted, we use theexponential tting

n(V ) =1

1 + e0.06(11.6V ), n1 (n) = 11.6

1

0.06ln

(1

n 1

).

Figure 11 provides a comparison of the behavior of the original and reduced models. Despitequantitative dierences (in particular, as in the reduction of the original HodgkinHuxleymodel, treating fast variables as instantaneous increases spiking frequency), the reduced modelfaithfully captures the qualitative behavior of its high-dimensional counterpart, for instance,rest-spike bistability. Phase plane analysis of the associated normal form (7) provides a cleargeometrical interpretation of such dynamical behavior (Figure 2).D

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145

35V V

I app I app

-10 -10

-30

10

-30

10

t

t

t

t50 250 50 250

Full model Reduced model

Rest-spike bistability Rest-spike bistability145

35

Figure 11. Comparison of the full HodgkinHuxley model augmented with a calcium current (18)(19) andits two-dimensional reduction, obtained by applying the reduction procedure of section (4.1).

4.4. The role of ultraslow variables. Ultraslow variables appear in a variety of forms,such as ultraslow gating variables (e.g., inactivation of calcium channels), intracellular calcium(e.g., SK channels), metabotropic regulation of channel expression (e.g., regulation of calciumchannel expression by serotonin receptors), and homeostatic regulation of channel expression(e.g., calcium-dependent expression of ion channels). As such, they do not allow a systematicanalysis, unlike slow-gating variables. However, their eect on the model reduction (13) canbe understood in terms of modulation of the unfolding parameters of the associated normalform. The observation that the many (auxiliary) parameters of conductance-based modelsmight naturally provide a versal unfolding of the winged cusp organizing their fast criticalmanifold suggests that variations in ultraslow variables act as an ultraslow modulation ofthe unfolding parameters in the associated normal form. The eect of ultraslow variables isthus constrained to reshape the geometry of the slow-fast phase portrait. This might lead toultraslow adaptation mechanisms (similarly to the action of in Figure 4) or to even slowermodulation mechanisms (similarly to the action of k and V0 in Figure 12 below).

Clearly, this does not permit us to conclude precise results on the global dynamics of amultipletime-scale model but suggests that the low-dimensional bursting modulation mech-anism described here has a strong relevance for generic conductance-based models.

5. Modulation of bursting by unfolding parameters and its physiological interpretation.

5.1. Bursting modeling and unfolding theory. The rich literature on mathematical mod-eling of bursting calls for a few comparisons with the model proposed in the present paper.The geometry of our bursting attractor is the most classical one of a saddle-homoclinic burster(one out of the 16 bursting attractors in the recent classication of Izhikevich; see [19, page376]). Such an attractor is found, for instance, in the early bursting model of Hindmarshand Rose [15]. The two models exhibit an analogue geometry: the mirror of the classicalFitzHugh phase portrait, obtained here by mirroring the fast variable cubic nullcline, is ob-D

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tained there by mirroring the monotone activation function of the recovery variable. But theHindmarshRose model lacks the organization of some high-codimension singularity, makingit impractical for modulation studies (see, e.g., [31]) and for physiological interpretability.

The more recent literature on bursting has certainly exploited unfolding theory aroundhigh-codimension bifurcations to construct dierent types of bursting attractors. A nonex-haustive list is [29, 2, 11, 18] and the references discussed in [19, page 376]. The outcomeof those studies is a useful mathematical classication between dierent bursting attractorsorganized by dierent bifurcations, but it is not clear how to use this classication for mod-ulation studies. A possible reason is that most of those references construct bursting modelsfrom restorative phase portraits that retain the qualitative organization of FitzHugh modelby a hysteresis singularity. Such models lack the transcritical bifurcation that organizes thenormal form reduction of general bursting conductance-based models.

The approach of the present paper diers from earlier studies in starting from the cuspsingularity, inspired by our original observation that the mirrored hysteresis phase portraitorganizes the reduced HodgkinHuxley dynamics [6, Figure 5]. The direct link between themathematical unfolding of the cusp singularity and the local normal form of conductance-based models in the vicinity of their transcritical bifurcation is probably crucial in usingunfolding theory to understand the modulation of bursting in neuronal models.

5.2. A geometrical and physiological modulation of a burster across bursting types.The single geometric attractor of (11) contains a continuum of dierent bursting wave formsmodulated by the bifurcation and the unfolding parameters. Beyond the route to burstingstudied in section 3, Figure 12 illustrates a situation where the bifurcation parameter andthe ane unfolding parameters are xed but where the two remaining unfolding parametersare modulated in a quasi-static manner. The gure displays a variety of waveforms whichnevertheless share the same geometry of the bursting attractor as hysteretic paths in theuniversal unfolding of the winged cusp. For small autocatalytic feedback gain k, correspondingto low expression of fast sodium channels, the model emits small oscillatory potentials (SOPs)on the left. Increasing this gain, the waveform smoothly evolves toward a classical square-wave oscillation, on the right, after a transient tapered bursting activity, shown in theinset (see [19, page 376] and references therein for a discussion about the dierent burstingtypes). As in the case of the route from tonic spiking to bursting, the transition shown inFigure 12 has physiological relevance. For instance, a similar transition has been observedduring development of neuronal cells [25].

The geometry of the tapered-like bursting wave form in Figure 12 reveals another sub-tlety of the winged cusp unfolding. In addition to broad regions of restorative and regenerativeexcitability, Figure 8A shows a small parametric region of mixed excitability (type V in theterminology of [9]). Like regenerative phase portraits, phase portraits in this region have apersistent bistable range, but it is of fold/fold type, with a down-state that is a regenerativexed point and an up-state that is either a restorative xed point or a limit cycle (emergingfrom a Hopf bifurcation within or outside the bistable range). The bursting attractor observedin this region can be considered as a variant of the bursting attractor associated to regenera-tive excitability. Both bursting attractors share the same geometry of hysteretic paths in theunfolding of the winged cusp singularity but the fold/fold variant exhibits the peculiar waveD

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t

V

tk

0

tV0

Figure 12. Modulation of model (11) across dierent bursting wave forms. Increasing the fast positivefeedback gain k and decreasing the half-activation potential V0, the model smoothly evolves from (calcium driven)small oscillatory potentials SOP (on the left) to square-wave-like bursting (on the right) across tapered-likebursting (shown in the inset). Parameter values are provided in Appendix C.

form illustrated in Figure 13, usually studied under the name of tapered bursting in theliterature; see, e.g., [19, page 376].

SN

SN

HB (or z )TC

SH0(or n )0

SN

Bis

t.

A) B)

Figure 13. Variant of the saddle-homoclinic bursting attractor in model (11). A. When the fast-slow sub-system (11a)(11b) exhibits Type V excitability [9], the bistable range is of fold/fold type leading to a taperedbursting waveform. Parameter values are provided in section C. B. The hysteretic path associated to this typeappears along path (ii) of Figure 8. At the two ends of the bistable range, the up and down attractors are stableequilibria losing stability in a saddle-node bifurcation. Depending on the excitability subtype, the burst onsetcan exhibit either damped spiking oscillations ending in a Hopf bifurcation within the bistable range (a situationcaptured by the bifurcation diagram in [9, Figure 5.2]) or a single action potential (a situation captured by thebifurcation diagram in [9, Figure 5.3]).D

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It is remarkable that the four dierent wave forms shown in Figures 12 and 13 can bemodeled by the same geometric attractor. A companion paper in preparation further investi-gates the physiological mechanisms that modulate the bursting wave within the unfolding ofthe winged cusp singularity.

6. Conclusions. The paper proposes that conductance-based models exhibiting burstingattractors are organized by a winged cusp singularity. The geometry of the resulting attractoris classical (a hysteretic modulation of a slow-fast portrait over a rest-spike bistable range),but singularity theory is used to identify key parameters for the modulation of the burstingattractor.

The cusp singularity organizes the slow-fast phase portrait around the mirror hysteresisof section 2.3 in contrast to the standard hysteresis of classical phase portrait reductions ofthe HodgkinHuxley model.

The bifurcation parameter has the convenient physiological interpretation of an ionic bal-ance recently studied in [10]. Its modulation through the transcritical variety of the cuspunfolding governs a geometric transition from tonic spiking to bursting in the threetime-scale normal form (9): it provides a physiologically relevant route to bursting.

The ane unfolding parameter has the physiological interpretation of an ultraslow ioniccurrent, typically driven by the intracellular calcium concentration. Its modulation providesthe classical adaptation variable of the threetime-scale bursting attractor.

The two remaining unfolding parameters have the physiological interpretation of a fastautocatalytic gain (the maximal sodium conductance) and of an average half activation poten-tial, respectively. Their quasi-static modulation evolves the bursting attractor across dierentbursting wave forms, consistently with what is observed experimentally in neuronal develop-ment, for instance.

In spite of the vast diversity of ion channels encountered in dierent neurons and theresulting vast diversity of regulation pathways, singularity theory and time-scale separationsuggest an apparent simplicity and universality in the underlying modulation mechanismsas paths in the universal unfolding of the winged cusp. Those features are appealing as weaddress system theoretic questions such as sensitivity, robustness, and homeostasis issues.

Appendix A. Codimension-one and -two bifurcation varieties in (7). The xed pointequation of (7) is organized by a winged cusp at xwcusp :=

13 , wcusp := 0, wcusp := 127 ,

wcusp := 13 , wcusp := 2. Codimension-one transcritical and hysteresis bifurcation transi-tion varieties in its unfolding are dened by

(20) TC(, ) = x3TC (TC + xTC)2 + xTC xTC(TC + xTC)with

xTC(, ) =2 (4 + 48)1/2

12,(21a)

TC(, ) = xTC(2 + )2

(21b)

and

(22) HY (, ) = x3HY (HY + xHY )2 + xHY xHY (HY + xHY )Dow

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with

xHY () = 1 + 3

,(23a)

HY (, ) = 3x2HY xHY (2 + 2)

2 + ,(23b)

respectively.The codimension-two pitchfork variety is dened by

PF () =

(3 +

(92 +

1

27

)1/2)1/3 1(

3 +(92 + 127

)1/2)1/3 1,(24)PF () = x3PF (PF + xPF )2 + xPF xPF (PF + xPF )(25)

with

xPF () = 1 + PF3

,(26a)

PF () = 3x2PF xPF (2 + 2PF )

2 + PF.(26b)

Appendix B. Proofs.

B.1. Proof of Theorem 2. We rely on geometric singular perturbation arguments [7, 20,24, 22, 23]. The reduced dynamics associated to (7), evolving on the slow time-scale = t,are given by

0 = Gswcusp(x, + y; , , ),(27a)

y = x y,(27b)whereas the associated layer dynamics, evolving on the fast time-scale t, are given by

x = Gswcusp(x, + y; , , ),(28a)

y = 0.(28b)

We construct the singular bistable phase portrait starting from the degenerate situation inFigure 3 center, corresponding to a pitchfork bifurcation. The same qualitative phase portraitis obtained on the pitchfork variety (24) for all > wcusp. Perturbing out of the pitchforkvariety, but remaining on the transcritical variety dened by (20), the phase portrait perturbsto one of the two qualitative situations in Figure 3 (center top or bottom). Finally, for belowand suciently near TC(, ) and below and suciently near TC(, ), one obtains thequalitative slow-fast dynamics in Figure 14A, which leads to the singular phase portrait inFigure 14B. The following lemma summarizes this construction.

Lemma 4. For all > wcusp, there exists > 0 such that, for all (PF () , PF () + ), there exists > 0 such that, for all (TC(, ) , TC(, )),there exists > 0 such that, for all (TC(, ) , TC(, )), the following hold(refer to Figure 14 (left) for the notation):D

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xA) B)

ySa

+Sr_

_

Saup

Sr

+Sa

F1

F2F3

F4

F2(y , x )1 1

(y ,x )2 2(y ,x )3 3

Ws0l 0

Figure 14. Slow fast dynamics associated to (27)(28).

(i) The critical manifold of the slow-fast dynamics (27)(28) has a mirrored hysteresisshape. In particular, it is composed of the attractive branches Sa , S+a , and S

upa , the

repelling branches Sr and S+r , and the four folds Fi, i = 1, . . . , 4, connecting them.(ii) There are exactly three nullcline intersections (yi, xi), i = 1, . . . , 3, belonging to S

a ,

Sr , and S+r , respectively.A direct geometric inspection reveals the presence of a singular periodic orbit 0 and a

singular saddle separatrix W 0s . These objects persist for > 0, as proved in the followinglemma, which proves Theorem 2.

Lemma 5. Let (yi, xi), i = 1, . . . , 3, be dened as in the statement of Lemma 4 (ii). For all, , , satisfying the conditions of Lemma 4, there exists such that, for all (0, ), thefollowing hold.

(i) (y1, x1) is locally exponentially stable, (y2, x2) is a hyperbolic saddle, and (y3, x3) islocally exponentially unstable.

(ii) There exists an exponentially stable relaxation oscillation limit cycle surrounding(y3, x3).

(iii) The stable manifold W s of (y2, x2) separates the basin of attraction of (y1, x1) and .

Proof of Lemma 5. (i) From Lemma 4, the xed point (y1, x1) belongs to the attractivebranch Sa of the critical manifold S. Moreover, it is an exponentially stable xed point of thereduced dynamics (27). From standard persistence arguments [7], there exists 1 such that,for all (0, 1], (y1, x1) is an exponentially stable xed point of (7). The xed point (y2, x2)belongs to the repelling branch Sr of the critical manifold S. Moreover, it is an exponentiallystable xed point of the reduced dynamics (27). Again from [7], there exists 2 such that,for all (0, 2], there exists an exponentially unstable local invariant manifold W s,loc suchthat all trajectories starting in W s,loc approach (y2, x2) exponentially fast. W

s,loc is the local

stable manifold of (y2, x2). Its unstable manifold is given by the ber of the unstable manifoldof W s,loc passing through (y2, x2). The xed point (y3, x3) belongs to the repelling branch

S+r of the critical manifold S; moreover, it is an exponentially unstable xed point of thereduced dynamics (27). By [7], there exists 3 > 0 such that, for all (0, 3], (y3, x3) is anexponentially unstable xed point of (7).

(ii) The slow fast dynamics possesses a singular periodic orbit 0 (see Figure 14). Following[24], there exists 4 such that, for all (0, 4], there exists an exponentially stable relaxationoscillation limit cycle surrounding (y3, x3).

(iii) In backward time, trajectories of the reduced dynamics (27) starting on Sr in aDow

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neighborhood of (y2, x2) approach either the fold F1 or the fold F2. Following [22], thereexists 5 such that, for all (0, 5], all trajectories starting in the local stable manifoldW s,loc approach (in backward time) either the fold F1 or the fold F2 along an invariantmanifold W s , which continues after the fold singularities roughly parallel to trajectories of thelayer problem. Therefore, the branch that continues after F1 extends to x = , whereas thebranch that continues after F2 extends to x = +. The invariant manifold W s is the saddlestable manifold and separates the plane in two disconnected regions that contain, respectively,the two attractors (y1, x1) and

.

Items (i), (ii), and (iii) are proved by picking = mini=1,...,5 i.

B.2. Proof of Theorem 3. Starting from a set of parameters satisfying the conditionsof Lemma 4 and increasing to = TC(, ), the two folds F1 and F4 in Figure 14Aapproach each other and eventually collide in a transcritical singularity TC, as in the slow-fast dynamics in Figure 15A. A direct geometrical inspection reveals the presence of a singularsaddle-homoclinic trajectory SH0 (Figure 15B) for which the transcritical singularity servesas the connecting point. This homoclinic orbit persists for > 0, as sketched in Figure 16A.On the contrary, decreasing , the two folds move away from each other until the left branchof the mirrored hysteresis is tangent to the y nullcline at a saddle-node bifurcation SN andeventually remains on its left, as in Figure 16B. The following lemma summarizes this analysis.For its statement, we refer to Figures 14 and 15.

xA) B)

ySa

+Sr_

_

Saup

Sr

+Sa

F2F3F2

(y , x )1 1

(y ,x )2 2(y ,x )3 3

TCSH0

Figure 15. Slow fast dynamics associated to (27)(28).

A) B)

SN

Figure 16. Phase portrait of (7), with parameters and the function y satisfying the conditions of Lemma6. A. Singularly perturbed saddle-homoclinic trajectory for = 0SH + c(

). B. Saddle-node bifurcation.

Lemma 6. For all > wcusp, there exists > 0 such that, for all (PF () , PF () + ), there exists > 0 such that, for all (TC(, ) , TC(, )),there exists > 0, such that, for in(0, ], the following hold.D

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(i) Let 0SH := TC(, ). There exists a smooth function c() dened on[0,], and

satisfying c(0) = 0 and c() < 0, for all (0, ], such that, for = SH := 0SH +

c(), (7) has an unstable saddle-homoclinic orbit SH.

(ii) For all > 0, there exists SN < 0SH such that (7) has a nondegenerate saddle-node

bifurcation for = SN at which the node (y1, x1) and the saddle (y2, x2) merge.(iii) For all [SN , 0SH), the nullcline intersection (y3, x3) belongs to the repelling

branch S+r (where S+r is dened as in Figure 14A).

(iv) For all [SN , SH ], there exists an exponentially stable relaxation oscillation limitcycle surrounding (y3, x3).

(v) There exists FLC (SH , 0SH) such that, for = FLC, the family P of stableperiodic orbits merges at a fold limit cycle bifurcation with the family of unstable periodicorbits Q emerging from the unstable saddle-homoclinic bifurcation.

Proof of Lemma 6. (i) For in a neighborhood of PF (), = TC(, ) and smallerthan and suciently near to TC(, ), there are exactly three nullcline intersections (yi, xi),i = 1, 2, 3, belonging to the attractive branch Sa , the repelling branch Sr , and the repellingbranch S+r , respectively. Relying on the results in [23] and following exactly the same stepsas [9, section 6.1], we can nd 1, such that the existence part of the point (i) holds with = 1. The resulting saddle-homoclinic trajectory is sketched in Figure 16A. To prove thatsuch homoclinic trajectory is unstable, recall that the stability of a saddle-homoclinic orbitis determined by the saddle quantity , that is, the trace of the Jacobian computed at thesaddle and at the saddle-homoclinic bifurcation: if > 0 (resp., < 0), the homoclinicorbit is unstable (resp., stable). The Jacobian JSH of (7) computed at (y2, x2) at the saddle-homoclinic bifurcation has the form

JSH =

(a bc d

), a, d > 0, b, c R.

Therefore, the saddle quantity = a d > 0 for all 0 < < a/d.(ii) For in a neighborhood of PF (), = TC(, ), and smaller than and suciently

near to TC(, ), the (cubic) xed point equation Gwcusp(x, + x;, , ) has three roots,corresponding to the three xed points (yi, xi), i = 1, 2, 3, of point (i). Decreasing , thetwo smaller roots (corresponding to the xed points (y1, x1) and (y2, x2)) approach each otherand eventually merge in a quadratic zero for = SN corresponding to a nondegeneratesaddle-node bifurcation.

(iii) We prove the statement for = PF () since, by continuity, the same will hold in aneighborhood. When = PF (), = PF (), and is smaller than and suciently near toPF (), the nullcline intersection (y3, x3) lies on S

+r . By continuity, the same is true for all

close to PF (). Since the value SN PF () continuously as PF (), one can pick suciently close to PF () such that (y3, x3) lies on S

+r for all [SN , PF ()).

(iv) By points (ii) and (iii) above and the same arguments as the proof of point (ii) inLemma 6, we can nd 2 > 0 such that, for all (0,min(1, 2), where 1 is dened as in theproof of point (i) above, and all [SN , SH), there exists a periodic orbit surrounding(y3, x3), and, moreover, this periodic orbit is exponentially stable. For =

SH , the stable

periodic orbit coexists with the unstable homoclinic orbit, since by [5, Theorem 3.5], a branchof stable periodic orbits cannot end at an unstable homoclinic bifurcation.D

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(v) The existence of FLC (SH , 0SH) satisfying the statement follows by two mainobservations. First, again by [5, Theorem 3.5] there exists a family Q of unstable periodicorbits emerging at = SH from the unstable homoclinic bifurcation. Second, simple geo-metric arguments show that for = 0SH (and suciently small) no periodic orbit can exist.The existence of the fold limit cycle bifurcation then follows by noticing that the fold limitcycle is the only planar bifurcation of periodic orbits not involving a Hopf point and thatboth the unstable homoclinic bifurcation and the fold limit cycle bifurcation are genericallyfound in the unfolding of the degenerate situation in which the saddle quantity is zero,corresponding to a resonant homoclinic orbit. The unfolding of this bifurcation, also calledresonant side-switching, is detailed in [4, Theorem A].

Figure 17 summarizes the results in Lemma 6.

FF

LL

S

U

x

z+

P

0SH

SN

SH FLC

QFLC

My

Figure 17. Bifurcation diagram of (9a)(9b) with respect to z and parameters satisfying the conditions ofLemma 6. See the main text describing Figure 5 for the notation.

We now follow [33, 32] to derive suitable conditions on the four parameters a, b, c in (9c)such that z hysteretically modulates (9a)(9b) along its rest-spike bistable range. To thisaim, note that the minimum value of y along the family of singular periodic orbits 0 and thesingular homoclinic trajectory SH0 for [SN , SH0] (see Figure 4) is necessarily strictlylarger then the maximum value of y along the branch of stable xed points L and at thesaddle-node bifurcation F . By persistence arguments, the same holds true in the nonsingularcase for suciently small. It follows that there exists a plane in the three-dimensionalspace x, y, z that, for [SN , 0SH ], never intersects the family of stable periodic orbits P and the branch of stable xed points L, and that intersects once the branch of saddle pointsS, say, for = M (see Figure 17). Clearly, splits R3 into two open half spaces. Let c bethe half space containing the family of singular periodic orbits. Then we pick a, b, c such that = {(x, y, z) : z + ax + by + c = 0} and z + ax + by + c > 0 for (x, y, z) c. UnderDo

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these conditions on a, b, c, Theorem 3 follows the same reasoning as the proofs in [33] (for theanalysis near the branch of the stable steady states and the jump up at the fold bifurcation)and [32] (for the analysis near the branch of periodic orbits and the jump down at the foldlimit cycle bifurcation).

Appendix C. Parameter for numerical simulations in Figures 6, 7, 12, 13. For the sakeof an easy numerical implementation and the reproduction of nice time series, we suggestthe following piecewise linear approximation of (11):

V = kV V3

3 (n+ n0)2 + I z,(29a)

n = n (n(V V0) n) ,(29b)z = z(z(V V1) z),(29c)

where

n(V V0) :={kn(V V0) if V < V0,kn+(V V0) if V V0

with 0 kn < 1 and kn+ > 1, and

z(V V1) :={kz(V V1) if V < V1,kz+(V V1) if V V1

with 0 kz < 1 and kz+ > 1.Parameters used in Figures 6 and 7 are k = 1, I = 11/3, n = 0.02, z = 0.0005, V0 = 0.5,

kn = 0.4, kn+ = 7, V1 = 1, kz = 0, kz+ = 50. The bifurcation parameter is n0 = 0.3 in Figure6 (left) and n0 = 1.1 in Figure 6 (right). In Figure 7 n0 is linearly (in time) decreased from0.3 to 1.1.

Parameters used in Figures 12A are n0 = 1.1, I = 11/3, n = 0.02, z = 0.0001,kn = 0.9, kn+ = 7, V1 = 1.2, kz = 0, kz+ = 100. The time-varying parameters k and V0evolve as k(t) = 0.5 + 2.5t/T and V0(t) = 0.5 0.75 min(1, 1.3t/T ).

Parameters used in Figures 13A are n0 = 0.2, k = 0.7, I = 11/3, n = 0.02, z = 0.001,V0 = 1.25, kn = 0.9, kn+ = 7, V1 = 1.2, kz = 0, kz+ = 50.

Appendix D. Parameter for numerical simulations of the HodgkinHuxley model insection 4.3. All the parameters and activation and inactivation rates are taken from theoriginal paper [17]. The time constants x(V ) and steady state functions x(V ) are relatedto the activation and inactivation rates x(V ) and x(V ), x = m,n, h, as follows:

x(V ) =1

x(V ) + x(V ), x(V ) =

x(V )

x(V ) + x(V ).

Acknowledgment. Prof. M. Golubitsky is gratefully acknowledged for insightful com-ments and suggestions during the visit of the rst author at the Mathematical BioscienceInstitute (Ohio State University).

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[2] R. Bertram, M. J. Butte, T. Kiemel, and A. Sherman, Topological and phenomenological classi-cation of bursting oscillations, Bull. Math. Biol., 57 (1995), pp. 413439.

[3] C Beurrier, P Congar, B Bioulac, and C Hammond, Subthalamic nucleus neurons switch fromsingle-spike activity to burst-ring mode, J. Neurosci., 19 (1999), pp. 599609.

[4] S.-N. Chow, B. Deng, and B. Fiedler, Homoclinic bifurcation at resonant eigenvalues, J. Dynam.Dierential Equations, 2 (1990), pp. 177244.

[5] S.-N. Chow, C. Li, and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, CambridgeUniversity Press, Cambridge, UK, 1994.

[6] G. Drion, A. Franci, V. Seutin, and R. Sepulchre, A novel phase portrait for neuronal excitability,PLoS ONE, 7 (2012), e41806.

[7] N. Fenichel, Geometric singular perturbation theory for ordinary dierential equations, J. DierentialEquations, 31 (1979), pp. 5398.

[8] R. FitzHugh, Impulses and physiological states

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Modeling the modulation of neuronal bursting: a singularity theory approach