A GEOMETRIC MECHANISM FOR MIXED-MODE BURSTING … · class of hybrid dynamical systems modeling neuronal activity. These dynamical systems, called non-linear adaptive integrate-and-

A GEOMETRIC MECHANISM FOR MIXED-MODE BURSTINGOSCILLATIONS IN A HYBRID NEURON MODEL

JUSTYNA SIGNERSKA-RYNKOWSKA ∗§¶, JONATHAN TOUBOUL ∗§ , AND

ALEXANDRE VIDAL‡§

Abstract. We exhibit and investigate a new type of mechanism for generating complex oscil-lations featuring an alternation of small oscillations with spikes (MMOs) or bursts (MMBOs) in aclass of hybrid dynamical systems modeling neuronal activity. These dynamical systems, called non-linear adaptive integrate-and-fire neurons, combine nonlinear dynamics modeling input integrationin a nerve cell with discrete resets modeling the emission of an action potential and the subsequentreturn to reversal potential. We show that presence of complex oscillations in these models relies ona fundamentally hybrid structure of the flow: invariant manifolds of the continuous dynamics governsmall oscillations, while discrete resets govern the emission of spikes or bursts. The decompositioninto these two mechanisms leads us to propose a purely geometrical interpretation of these complextrajectories, and this relative simplicity allows to finely characterize the MMO patterns through thestudy of iterates of the adaptation map associated with the hybrid system. This map is howeversingular: it is discontinuous and has unbounded left- and right-derivatives. We apply and developrotation theory of circle maps for this class of adaptation maps to precisely characterize the tra-jectories with respect to the parameters of the system. In contrast to more classical frameworks inwhich MM(B)Os were evidenced, the present geometric mechanism neither requires no more than twodimensions, does not necessitate to have separation of timescales nor complex return mechanisms.

Key words. hybrid dynamical systems, rotation theory, mixed-mode oscillations, bursting,chaos, nonlinear integrate-and-fire neuron model.

AMS subject classifications. 34K34, 37E45, 37E05, 37E10, 37N25, 92C20,

∗The Mathematical Neurosciences Team, Center for Interdisciplinary Research in Biology (CNRSUMR 7241, INSERM U1050, UPMC ED 158, MEMOLIFE PSL*) College de France, 11 placeMarcelin Berthelot, 75005 Paris, France§Project-Team MYCENAE, Inria Paris-Rocquencourt Research Centre, 78153 Le Chesnay,

France‡Laboratoire de Mathematiques et Modelisation d’Evry (LaMME), CNRS UMR 8071, Universite

d’Evry-Val-d’Essonne, 23 boulevard de France, 91037 Evry, France¶[email protected]

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1. Introduction. Neurons are electrically excitable cells that communicatethrough the emission of action potentials or spikes, that are stereotyped electricalimpulses. The neuronal information is transmitted through the timing of spikes andspecifically through the pattern of spike fired. Early electrophysiological recordingsrevealed that neurons in specific regions of the cortex could display complex rhythmicpatterns closely related to regular fluctuations of the underlying membrane potential.Rhythmic firing, both intrinsic or collective, is particularly important for specific func-tion and information coding. An essential property of cells encoding precise timingsis the presence of subthreshold oscillations. These are thought to subtend precisionand robustness of spike generation patterns, and are present in different cell typesand brain areas including inferior olive nucleus neurons [5, 38, 39], stellate cells ofthe entorhinal cortex [1, 2, 28], and dorsal root ganglia [4, 36, 37]. In response toconstant input, these cells display small oscillations of the voltage interspersed by oneor several spikes. Such trajectories alternating small voltage oscillations and spikesare referred to as Mixed-Mode (Bursting) Oscillations, MM(B)Os.

From the biological viewpoint, these electrical patterns rely on complex ionic andbiochemical mechanisms, that are accurately described by nonlinear dynamical sys-tems of relatively high complexity, such as the celebrated Hodgkin-Huxley model [22]and its numerous variants [11]. These models reproduce a broad range of excitabilitypatterns observed in neuronal cells, including regular spiking, bursting (periodic alter-nation of relatively fast spiking regimes and quiescence), MMOs [49, 50] and chaos [20].A number of simplified models were proposed, but the price to pay has been the versa-tility. For instance, planar dynamical systems, such as the popular FitzHugh-Nagumosystem [12], can reproduce accurately the excitable nature of nerve cells but cannotgenerate bursts of activity, MMOs or chaos.

First theoretical approaches accounting for the presence of MMOs in dynamicalsystems exhibited a possible relationship with complex structures of the dynamicalinvariants such as homoclinic tangencies [14, 15]. More recently, an important body oftheoretical works characterized MMOs in dynamical systems with multiple timescales,generally associated with canard explosions [8, 48]. Singular perturbation theory iscentral in this approach and allows characterizing the presence of funnels and mani-folds trapping the trajectories associated with MMOs in the limit of perfect separationof timescales [19, 46, 61]. In particular, a complete classification of the folded singu-larities and the associated MMO patterns emerging from that limit was achieved forthree dimensional systems with one fast and two slow variables [10, 18, 33]. Thesestudies characterize the properties of the MMO orbits generated by such systems,and showed in particular that the sequence of numbers of small oscillations separatedby large oscillations (signature) along a given orbit satisfied a few constraints. Itwas also shown that systems with more than three dimensions and/or more than twotimescales may display non chaotic changes in the signature of a MMO limit cyclewhen varying a parameter [32], as well as a wider range of possible signatures. But theinvestigation of such MMOs using multiple timescale analysis remains open since, todescribe the full dynamics, one is confronted to the major difficulty of understandingthe global return mechanism [34] which depends strongly on the interactions betweenthe different timescales. In this article, we propose a complementary approach by (i)considering a class of planar hybrid model that is able to generate a vast panel ofMM(B)Os, (ii) describing the underlying geometric mechanism for this generation,and (iii) introducing a general framework for studying the dynamical structure andthe associated orbits.

Geometric MMBOs in a hybrid neuron model 3

Contrasting with detailed biophysical models, integrate-and-fire models [7, 24,35, 57] are abstractions of the voltage dynamics that decouple the slow evolution ofthe voltage and the rapid firing of action potentials. Integrate-and-fire models arehybrid dynamical systems coupling a differential equation describing the dynamicsof the cell depolarization, with a discrete dynamics corresponding to the emissionof action potentials and the subsequent reset. It was shown that nonlinear bidimen-sional integrate-and-fire neuron models [7, 24, 52, 57] have a very rich phenomenology:they reproduce regular spiking, bursting and chaos, as well as the different types ofexcitability observed in nerve cells and also subthreshold oscillations under specifictechnical assumptions (see [57]). To date, MMOs were never reported in this class ofmodel. Mathematically, the study of the neural excitability and the presence of sub-threshold periodic attractors was performed in [57] and only involves the continuous-time dynamical system. The study of spike patterns dynamics is more complex. Itwas done through the study of a specific discrete dynamics [60], and was only in-vestigated in situations in which the subthreshold dynamics has no singular points.The stability of these patterns was investigated recently through the computation oftransverse Lyapunov exponents [13], and an explicit characterization of the orbits wasobtained in the linear bidimensional Mihalas and Niebur integrate-and-fire model [27].

Here, we undertake the rigorous study of the dynamics of the system in the pre-sence of singular points. As shown in [57], the subthreshold system in that situationhas generically two fixed points (except precisely at the saddle-node transition whereit has one), one of which being a saddle and the other one being either stable orunstable. We concentrate on the unstable case. For almost any initial condition(except precisely on the stable manifold of the saddle), the neuron fires a spike, andtherefore the adaptation map is defined everywhere except on a finite set of values.In this regime, we are led to study in depth the properties of discontinuous maps, andwe will show how the orbits are precisely related to MM(B)Os.

We will show that there exists a profound relationship between the type of MMOpattern and the rotation number of the adaptation map. This motivates us to char-acterize rotation numbers of the adaptation map. By doing so, we provide an over-arching framework for the analysis of the rotation number in a class of maps withdiscontinuities and infinite derivative naturally appearing in the present context, bycombining and extending a number of theoretical results [6, 17, 30, 41, 42, 44, 45]or applied to neuroscience [9, 16, 31, 56]. Recently developments on rotation theoryand orientation-preserving circle homeomorphisms and diffeomorphisms include thedetailed study of interspike intervals for periodically driven one-dimensional integrate-and-fire models [40, 53]. The study of the nonlinear integrate-and-fire neuron in theMM(B)O situation naturally leads to a complex problem, since the corresponding cir-cle maps are discontinuous with unbounded derivative, which prevents us from usingthe classical theories of Poincare and Denjoy.

Beyond its biological and mathematical interest for the characterization of orbitsof discontinuous maps, this study has important mathematical implications with re-gards to MMOs. It indeed provides a novel mechanism for studying rigorously andin great detail MM(B)O patterns. Alternatively to classical slow-fast or homoclinicmechanisms that generally require perturbative arguments or numerical continuation,the hybrid MM(B)Os exhibited here can be completely, explicitly and exactly charac-terized, do not require to have dynamics evolving on vastly different timescales, theirexistence is ensured in a relatively wide region of parameters and is not very sensitiveto perturbations, since they rely on a very resilient geometric mechanism that exists


even in elementary two-dimensional hybrid systems.The paper is organized as follows. In section 2, we introduce the equations we are

interested in and review a few results on their dynamics. We also describe formallythe regime in which MMOs and MMBOs appear, and their profound geometric origin.The type of MMO and MMBO fired is fundamentally related to the adaptation mapwhich is discontinuous with unbounded derivative, as we show in section 3. We providea precise description of the dynamics in the case where the adaptation map admitsone discontinuity in its invariant interval (section 4). Implications and perspectivesin dynamical systems and neurosciences, as well as for future analyses of cases withmore discontinuities are discussed in the conclusion (section 5).

2. Hybrid neuron model and the geometric MMO mechanism. We de-scribe and review the main properties of the class of neuron models studied in thismanuscript and describe its main properties.

2.1. The generalized nonlinear Integrate-and-Fire neuron model. Non-linear bidimensional integrate-and-fire neuron models are very popular models of neu-ronal dynamics widely used in applications. They are indeed simple yet very versatilerepresentations of neuronal dynamics that combine excitable nature of nerve cellsmembranes together with the discrete nature of spike emission. When the cell is notfiring an action potential, these models describe the dynamics of the membrane po-tential v together with an adaptation variable w as a nonlinear differential equation(subthreshold dynamics):

dvdt = F (v)− w + Idwdt = a(bv − w),

(2.1)

where a and b are real parameters accounting respectively for the time constant ratiobetween the adaptation variable and the membrane potential and for the couplingstrength between these two variables. The real parameter I represents the inputcurrent received by the neuron, and F is a real function accounting for the leak andspike initiation currents. Following [57, 60], we will assume that F is regular (at leastthree times continuously differentiable), strictly convex, and its derivative to have anegative limit at −∞ and an infinite limit at +∞. Moreover, we assume throughoutthe paper that the map F is superquadratic at infinity:

Assumption (A0). There exists ε > 0 such that F grows faster than v2+ε whenv → +∞ (i.e. there exists α > 0 such that F (v)/v2+ε ≥ α when v → +∞).

It was proved in [58] that, under this assumption, the membrane potential blowsup in finite time and, at this explosion time, say t∗, the adaptation variable convergesto a finite value w(t∗−). A spike is emitted at the time t∗ when the membrane potentialblows up1, i.e. limt→t∗−v(t) = +∞. At such an explosion time t∗, it is consideredthat the neuron has fired an action potential whereby the voltage is instantaneouslyreset and the adaptation variable is updated as follows:

v(t∗) = vr

w(t∗) = γw(t∗−) + d(2.2)

1If condition (A0) is not satisfied (e.g. when F is a quadratic polynomial [58]), an additionalcutoff parameter θ is introduced, and spikes are emitted at the time t∗ when the membrane potentialv reaches θ. This is the case of the classical Izhikevich model [24]. We expect most results to holdin these cases.


with γ ≤ 1 and d ≥ 0 corresponding to the effect on the adaptation variable ofthe emission of a spike. Note that in all models in the literature it is assumed thatγ = 1. However, going back to the biological problem, spikes are not Dirac masses butstereotypical electrical impulses s(t) = 1

δtS( tδt ) where S(t) is the typical spike shaperescaled on the dimensionless interval [0, 1], and δt the spike duration, assumed to bevery small 0 < δt 1/a. The adaptation variable integrates this sharp impulse:

w(t∗ + δt) = w(t∗−)e−aδt +

∫ δt

0

bs(t)e−a(δt−s) ds = γw(t∗−) + d

with γ = e−aδt < 1 and d = b∫ 1

0S(u)e−aδt(1−u) du. The classical nonlinear integrate

and fire neuron of [7, 24, 57] corresponds to the limit δt→ 0.By this reset mechanism, it is not hard to show that the system is globally well

posed, i.e. that one can define a unique forward solution for all times and initialconditions. This property requires to show that spikes do not accumulate in time,which can be done similarly as in [60].

2.2. Mathematical analysis of nonlinear integrate-and-fire models. Theexcitability properties of the system governed by the subthreshold system (2.1) wereinvestigated exhaustively in [57]. It was found that all models undergo a saddle-nodebifurcation and a Hopf bifurcation, organized around a Bogdanov-Takens bifurcation,along curves that can be expressed in closed form. The different curves are depicted

Saddle

-node

Hopf

V

w

Unstable focus

Saddle

Stable manifold

Unstable node

Saddle

Stable manifold

I

Saddle homoclinic

wA

B

C

BTSaddleStable

equilibrium

w

Stable manifold

Attraction bassinAttraction Basin

I

b

v

v

v

w

w

w

Figure 2.1. Bifurcations of the adaptive exponential model and its saddle-node (brown), Hopf(green), saddle homoclinic (purple) and Bogdanov-Takens (BT) bifurcations in the parameter plane(I, b). The curve given by (2.4) separating regions of unstable focus and unstable node is added indashed blue. Typical phase plane in the different regions of interest are depicted as smaller insets.They feature the nullclines (dashed black) and the stable manifold (red).

in the parameter space (I, b) in Fig. 2.1. They split the parameter space into a regionin which the system has no singular point (yellow region), a region in which theneuron has two singular points, one of which being a stable steady state (orange and


blue regions) and a region in which the system has two unstable singular points, oneof which being a saddle (magenta and pink regions). In the latter case, the stablemanifold of the saddle is made in part of a heteroclinic orbit connecting the saddleto the repulsive point. The heteroclinic orbit can either (i) wind around the unstablepoint when it is a focus (pink region B), or (ii) monotonically connect to the repulsivepoint (magenta region A), depending on whether the eigenvalues of the repulsive pointare real or not. The transition between these two regimes (dashed blue in Fig. 2.1)occurs when the linearized flow at the unstable singular point denoted (v−, F (v−)+I)has nonreal eigenvalues, i.e. when:

b >(a+ F ′(v−))2

4a. (2.3)

This condition that can be made explicit in all classical cases:• In the adaptive exponential model F (v) = ev − v, this curve reads:

I = (1 + b) log(1 + 2√ab− a)− (1 + 2

√ab− a). (2.4)

• In the quadratic model F (v) = v2 we find:

I = −a2

4− 3ab

2+ (a+ b)

√ab,

• and in the quartic model, the formula is made less transparent because of themore complex expression of the fixed points.

For producing the subsequent numerical simulations displayed in this article, we haveused the quartic model F (v) = v4 + 2av with the following parameter values2 :

a = 0.1, b = 1, I = 0.1175. (2.5)

In most simulations, value vr = 0.1158 has been used (unless otherwise stated), whilewe consider varying values of parameters d and γ.

Let us also briefly describe the case where the system has one stable and onesaddle singular points together with an unstable periodic orbit emerging from the Hopfbifurcation (orange region (C) in Fig. 2.1 between the Hopf and saddle-homocliniccurve). In that case, a simple application of Poincare-Bendixon’s Theorem ensuresthat the stable manifold of the saddle emerges from the unstable orbit and windsaround it infinitely many times. While we mostly concentrate on the case where bothfixed points are unstable (pink region B), the properties of the adaptation map andthe (MMBO-) shape of the orbits in this situation generally extend readily to the caseof region (C). The only difference is the existence of an attraction basin of the stablefixed point: if a trajectory resets within the area delineated by the unstable periodicorbit, it converges towards the stable fixed point and stops firing.

2.3. Spike Sequences and the geometric hybrid mechanism for mixedmode oscillations. In [60], the spike patterns fired were investigated theoreticallyin the case where the system has no singular point (yellow region of Fig. 2.1). Inthis parameter region, the spike dynamics appear relatively simple. Here, we extendthis analysis to the regions in which the system features two unstable fixed points.

2Note that in the quartic model, the map F uses the same parameter a in its linear term asthe time constant ratio of w relative to v, for simplicity: this choice allows avoiding consideringunessential parameters.


In region B (pink in Fig. 2.1), we will find mixed mode oscillations because of thegeometry of the vector field. Indeed, in that region of parameters, the stable manifoldof the saddle is made of a heteroclinic orbit spiraling around the unstable focus,constraining trajectories reset within the spiral to proceed to a prescribed number ofrotations around the unstable fixed point before firing (see Fig. 2.2). The rotationsaround the unstable fixed point are called small oscillations.

Definition 2.1. Mixed-Mode Oscillations (MMOs) for the system (2.1)-(2.2)are spiking orbits consisting of an alternation of small oscillations and spikes. ThoseMMOs featuring also bursts of two or more consecutive spikes are referred to Mixed-Mode Bursting Oscillations (MMBOs). MMO patterns are characterized by the MMOsignature Ls11 Ls22 Ls33 · · · , where Li denotes the number of consecutive spikes and si thenumber of subsequent small oscillations. Periodic signatures with period k are onlydenoted by finite sequence of length k, Ls11 Ls22 ...Lskk .

By bursts we mean consecutive spikes not separated by small oscillations andwith relatively short interspike-intervals between them.

We note that the periodicity of the MMO signature does not imply the peri-odicity of the corresponding voltage trajectory. In particular, there can be chaotic(nonperiodic) trajectories displaying MMOs with periodic signature.

When the reset line v = vr intersects the spiral, as in Fig. 2.2, the adaptationmap is undefined at each intersection with the stable manifold, since the orbit of(2.1) starting from such a point converges to the saddle. But for any initial condition(vr, w) between the points belonging to the stable manifold, the orbit will performa specific number of small oscillations before firing. We emphasize that the presentframework allows performing a very fine classification. First, the fact that the stablemanifold is bounded in the v variable implies that the amplitude of small oscillations,similarly to biological MMOs, have a bounded amplitude considerably smaller thanspikes amplitude. Moreover, small oscillations can be very finely characterized withthe precision of half a rotation.

When trajectories recur within the spiral of the stable manifold, the neuron dis-plays an alternation of spikes and small oscillations, hence MMOs or MMBOs. Thesignature of an MMBO is directly related to the sequence of values of the adaptationvariable after each reset, which can be seen as the iterations of the adaptation mapthat we now characterize in detail.

3. The adaptation map. Because of its direct relationship with firing patternsand MMOs, we now provide a fine characterization of the adaptation map, introducedin [60].

Definition 3.1. The definition domain D of the adaptation map is the setof adaptation values w ∈ R such that the point (vr, w) does not belong to the stablemanifold of the saddle. With any w ∈ D, the adaptation map associates the value Φ(w)of the adaption variable after reset for the orbit of the system with initial condition(vr, w), i.e.

Φ(w) := γW (t∗; vr, w) + d,

where (V (t; vr, w),W (t; vr, w)) is the solution of equation (2.1) with initial condition(vr, w) and t∗ satisfying limt→t∗− V (t; vr, w) =∞ is the time of the first spike for thissolution.

It is easy to see that this map is finite and uniquely defined under assump-tion (A0). The following sections characterize the properties of the adaptation mapand their relationship with spiking patterns.


Stablemanifold

Time18386080

7450275

Volta

geVo

ltage

-1 -0.5 0 0.5 1 1.5

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0

1

2

1.5

0.5

Unstablemanifold

33.52.5

0 321

3.5 0.51.52.5

Time

w w

v v

Figure 2.2. The geometry of Mixed-Mode Oscillations: (left) Phase plane with stable andunstable manifolds of the saddle: the stable manifold winds around the repulsive singular point.The reset line intersects the stable manifold separating regions corresponding to a specific number ofsmall oscillations (colored boxes, from 0 to 3 below the w-nullcline and from 3.5 to 0.5 above). Thesolution for one given initial condition in each case is provided in the corresponding panel below.Note that the time interval varies in the different plot (indicated on the x-axis). Simulations on theright had initial conditions vr = 0.012 and w chosen within the different intervals.

3.1. Properties of the adaptation map. By studying the orbits of the adap-tation map in the case where the system has no singular point (yellow region inFig. 2.1), it was shown in [60] that fixed points of the adaptation map correspondto regular spiking and periodic orbits to bursting trajectories. The assumption onno singular points therein was very convenient from the technical viewpoint, since inthat case the definition domain D is the whole real line, the map Φ is continuous onR and we can characterize exactly the convexity and monotonicity properties of themap. In the present case, we will see that Φ is no more continuous, it is undefined atspecific points, and has unbounded derivative.

We concentrate here on the case where the vector field (2.1) has two unstablesingular points. Several important structures are depicted in Fig. 3.1. The two sin-gular points of the vector field are denoted (v−, F (v−) + I) and (v+, F (v+) + I), withv− < v+. The former is repulsive, the latter is a saddle. We denote by Ws and Wu

the stable and unstable manifolds of this singular point. Each of these manifolds aremade of two branches, and we note Ws

− the branch of Ws pointing towards w < 0,and Wu

− and Wu+ the branches of Wu pointing towards v < 0 and v > 0 respectively.

The shape of the map Φ is organized around a few important points (see Fig. 3.1):• We denote by w∗ = F (vr) + I the intersection of the reset line v = vr with

the v-nullcline.• We denote by w∗∗ = bvr the intersection of the reset line with the w-nullcline.• We denote by (wi)

pi=1 the sequence of intersections of the reset line with Ws,


labeled in increasing order. Except for vr = v−, there exists a finite numberof such points or none depending on the parameter values: an even numberof intersections for v < v− and an odd number for v > v−. We denote by p1

the index such that (wi)i≤p1 are below the v-nullcline and (wi)i>p1 are above.When p is even, p1 = p/2, and otherwise, p1 = dp/2e is the smallest integerlarger than p/2. The points (wi) split the real line into p+1 intervals denoted(Ii)

pi=0. Remark that these intervals precisely correspond to those in which

the number of small oscillations occurring between two consecutive spikes isconstant except the interval Ip1 which is split into two subintervals by w∗ (seeFig. 2.2). The number of small oscillations for trajectories starting from Ii is

i if i < p1,

(p+ 1/2)− i if i > p1,

p1 if i = p1 and w < w∗,

p1 + 1/2 if i = p1 and w > w∗ and p is even

p1 − 1/2 if i = p1 and w > w∗ and p is odd.

(3.1)

• We denote by w−lim < w+lim < ∞ the limit of the adaptation variable when

v → +∞ along Wu− and Wu

+ respectively. In addition, we introduce thecorresponding values obtained through the reset mechanism:

β = γw−lim + d, α = γw+lim + d.

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

wlim

w+lim

1

0.5

0

-0.5

Wu

Wu+

Ws

Ws

vr

v

w

Figure 3.1. Geometry of the phase plane with indication of the points relevant in the charac-terization of the adaptation map Φ.

These points being defined, we are able to characterize the shape of the adaptationmap.

Theorem 3.2. The adaptation map Φ has the following properties.1. It is defined for all w ∈ D = R \ wipi=1.2. It is regular (at least C1) everywhere except at the points wipi=1.


3. In any given interval Ii with i ∈ 1 · · · p+1, the map is increasing for w < w∗

and decreasing for w > w∗.4. At the boundaries of the definition domain D, wipi=1, the map has well-

defined and distinct left and right limits:lim

w→w−iΦ(w) = α and lim

w→w+i

Φ(w) = β if i ≤ p1,

limw→w−j

Φ(w) = β and limw→w+

j

Φ(w) = α if j > p1.

5. The derivative Φ′(w) diverges at the discontinuity points:lim

w→w±iΦ′(w) = +∞ if i ≤ p1,

limw→w±i

Φ′(w) = −∞ if i > p1.

6. Φ has a horizontal plateau for w → +∞ provided that

limv→−∞

F ′(v) < −a(b+√

2). (3.2)

7. For w < min(

d1−γ , w1, w

∗∗)

, we have Φ(w) ≥ γw + d > w.

8. If vr < v+, Φ(w) ≤ α for all w ∈ D. Moreover, for any w taken between thetwo branches of the unstable manifold of the saddle, hence in particular forw ∈ (w1, wp), Φ(w) > β.

With a little abuse of terminology, we will refer to the points wi as the disconti-nuity points of Φ, although a priori Φ is not defined at wi.

In comparison to the case of the absence of singular points [60, Theorem 3.1], themap looses continuity, convexity, and uniqueness of fixed point, but the monotonicityproperty 3., the presence of a plateau 7. and the comparison with identity 8. remaintrue. However, the shape of Φ is much more singular and a number of nice propertiesare lost including continuity and concavity for w < w∗, which implies that the mapmay have now several fixed points.

The presence of discontinuities and divergence of the map derivative substantiallychange the nature of the dynamics as we will see below. It is worth noting that thisdivergence is a common property of correspondence maps in the vicinity of saddles,as we show in the following general result:3.

Lemma 3.3. Consider a two-dimensional smooth vector field (at least C2) withan hyperbolic saddle x0 associated with the eigenvalues −µ < 0 < ν of the linearizedflow. We denote by W s and Wu the stable and unstable manifolds of the saddle andconsider two transverse sections Ss and Su on each manifold intersecting W s andWu at xs and xu. There exists Ωs a one-side neighborhood of xs on Ss that mapsonto a one-side neighborhood Ωu of xu on Su. The correspondence map Ψ : Ωs 7→ Ωuis differentiable in Ωs and we denote by Ψ′ the one-side differential of Ψ at xs. Wehave:

Ψ′ =

∞ if ν − µ > 0

0 if ν − µ < 0(3.3)

3Note that this general result indicates that even if spikes are defined with a finite cutoff value,the adaptation map shows an infinite derivative at the discontinuities as well: the divergence of thederivative is independent of the divergence of the voltage.


Linearized system

Smooth conjugacy

Ss

Su

Wu

W s

S0u

S0s

Figure 3.2. Typical topology of manifolds and sections in Lemma 3.3: we consider corre-spondence map between sections Ss (red) and Su (orange) transversal, respectively, to the stableand unstable manifolds (black lines) of the saddle (orange circle). Typical trajectories are plottedin blue. The key arguments are the characterization of correspondence maps associated with thelinearized system (upper left inset) between two transverse sections S′s and S′u, and the smoothconjugacy between the nonlinear flow and its linearization.

In the particular case µ = ν, the differential is finite and its value depends on theprecise location of the sections.

Proof. Let us start by considering the linearized system in the vicinity of thesaddle singular point. In the basis that diagonalizes the Jacobian, we can write thesystem in the simple form:

x = −µxy = νy

and considering a section S′s corresponding to y = y0 and a section S′u correspondingto x = x0 > 0, simple calculus leads to the formula that the correspondence map ϕof the linearized system between S′s and S′u is defined for x ≥ 0 by ϕ(x) = y0

xα0xα with

α = µ/ν. Hence, the differential of ϕ at 0+ is such that:• it diverges if α < 1, hence ν − µ > 0 (i.e. if the dilatation along the unstable

direction is stronger than the contraction along the stable direction)• it vanishes if α > 1, hence ν − µ < 0 (i.e. if the contraction along the stable

direction is stronger than the dilatation along the unstable direction)• when α = 1 (i.e. contraction and dilatation are of the same intensity), we

find ϕ′(0+) = y0/x0 which depends on the precise location of the sections.The lemma states that the same result holds for the nonlinear system. It is knownthat locally around the (hyperbolic) saddle, the Hartman-Grobman Theorem [21]ensures that the nonlinear system is conjugated to its linearization through an home-omorphism. For our purposes, we however need to ensure that the nonlinear andlinear flows are locally conjugated via smooth diffeomorphisms (at least C1). Thesmoothness of conjugacy is a subtle question for a general dynamical system thathas been the object of outstanding researches. In general, one needs to ensure thatthere are no resonances in the eigenvalues which may lead to relatively complex rela-tionship [51, 54]. Interestingly, in two dimensions, the problem is simpler and it was


proved by D. Stowe [55] that any C2 planar dynamical system in the neighbourhoodof a saddle is smoothly (with at least C1 regularity) conjugated with its linearization.In detail, there exists a neighborhood U of x0 and a C1 diffeomorphism h such thatthe trajectories x(t) of the linearized system and x(t) within u are related throughthe equality h x(t) = x(t) h. We also note that the differential of h is boundedaway from zero for U small enough (since it converges towards the identity as theneighborhood collapses). It is hence easy to see that the correspondence map of thenonlinear system between S′s and S′u, transversal sections to the stable and unstablemanifolds respectively, has the property (3.3). Completing the proof only amounts toshowing that the correspondence maps from a neighborhood of Ss to S′s and from S′uto a subset of Su are smooth with differential bounded away from zero and infinity.This is a classical consequence of the flow box theorem and regularity with respect tothe initial condition, that we will outline in more detail when needed in our specificsituation.

Now that this general result is proved, we proceed to the fine characterization ofthe adaptation map and the proof of Theorem 3.2.

Proof. Let us start by noting that the generalization of the reset mechanism doesnot substantially impact the shape of the adaptation map. Indeed, all propertiesrely on the map associating with a point on the reset line (vr, w) the value ϕ(w) :=W (t∗−; vR, w) of the adaptation variable at the time t∗ of the subsequent spike, sinceϕ(w) = (Φ(w)−d)/γ. In other words, the generalization of the reset mechanism doesnot introduce any new mathematical difficulty. Hence, the proofs for items 1. to 3.and 6. to 8. are straightforward extensions of the analogous proof in [57] or simplealgebra. Similarly, the proof of the property 4. uses akin argumentation to that of 3.,with left and right limits found by finely characterizing the shape of the trajectoriesas w approaches one of the discontinuity points wi. In all cases, the trajectory willinitially remain very close from the stable manifold, before leaving the vicinity of thestable manifold near the saddle and following the unstable manifold. Depending onwhether the trajectory approaches the saddle from the right or from the left, it willeither follow the left or right branch of the unstable manifold, hence either convergetowards w+

lim or w−lim.

We focus on the proof of property 5. that requires specific analysis. We useLemma 3.3 and prove that the conditions on the contraction and dilatation nearthe saddle are satisfied. We consider the specific sections that define Φ, namelySs = v = vr (which is valid as long as the stable manifold is not tangent to thereset line) and Su = v = +∞, which is slightly more subtle as we need to ensurethat the differential of the correspondence map does not vanish.

First, the dilatation always overcomes the contraction, since (v−, F (v−) + I) isan unstable focus, i.e. the linearized flow has two complex conjugate eigenvalues withpositive real part and therefore the trace of the Jacobian, given by F ′(v−) − a, isstrictly positive. Since F is convex, the trace of the Jacobian at the saddle equalsF ′(v+)− a ≥ F ′(v−)− a > 0. Hence the dilatation is stronger than the contraction,i.e. in the notation of (3.3) we have ν − µ > 0.

In order to show that the infinite derivative persists when one considers Su =v = ∞, we express the map Φ formally in the region below the stable manifoldof the saddle, that all spiking trajectories cross, and investigate its behavior. In thisregion, any trajectory has a monotonically increasing voltage (and blowing up in finite


time), and the orbit (v,W (v)) in the phase plane is solution of dW

dv=

a(bv −W

)F (v)−W + I

W (v0) = w0

(3.4)

The expression of the differential of W with respect to w0 at v is given by:

∂W

∂w0(v) = 1 +

∫ v

vr

(a(bu− F (u)− I))

(F (u)−W (u) + I)2

)∂W (u)

∂w0du, (3.5)

whose solution is given, as a function of the trajectory W , by4:

∂W

∂w0(v) = exp

(∫ v

vr

a(bu− F (u)− I)

(F (u)−W (u) + I)2du

). (3.6)

Hence, for any section Su = v = θ, the derivative of the map cannot vanish.Furthermore, for u large, we know that W (u) remains finite and thus the integrandin (3.6) behaves as −a/F (u) which is integrable at infinity under our assumption (A0).Consequently, the integral within the exponential term does not diverge towards −∞as v →∞ and the derivative (3.6) does not vanish at Su = v =∞. We further notethat all correspondence maps away from singularities (v−, F (v−)+I) and (v+, F (v+)+I) are well-defined and with finite derivative bounded away from zero for the samereason. The intervals (−∞, w1), Ii and (wp,∞) on the line v = vR are transversesections of the flow and correspondence maps from Ii to (−∞, w1) are non-decreasingfor i ≤ p1 (hence the left and right differentials of Φ at w±i for i ≤ p1 are equal to+∞) and non-increasing otherwise (hence the left and right differentials at w±i fori > p1 are equal to −∞).

3.2. Transient MM(B)O behaviors. We recall that at each discontinuitypoint wi, the right and left limits are always equal to either α or β. This prop-erty, related to the fact that all discontinuities correspond to intersections of the resetline with the same orbit (the orbit of the stable manifold), is a very important prop-erty that endows the system with a rich phenomenology, ensuring that the systemcan generate MMOs of any signature.

We start by treating the case where the adaptation map has an infinite numberof discontinuity points for simplicity of notations, before extending our results to thegeneral case. The situation in which the adaptation map has an infinite number ofdiscontinuities occurs in two cases:

1. vr = v− within the pink region in Fig. 2.12. the subthreshold dynamics has a stable fixed point with a circular attraction

basin bounded by the unstable limit cycle (parameters within the orangeregion in Fig. 2.1) and the reset line v = vR intersects this limit cycle.

We note (mi)i∈N the w value of the discontinuity points below the intersection w∗

of the reset line with the v-nullcline, with mi < mi+1 for any i. Similarly, we note(Mi)i∈N the w values of the discontinuity points satisfying Mi > w∗ and Mi > Mi+1,

4From this expression one can propose an alternative direct (but particular) proof of the diver-gence of the one-side derivative at the points wi. The interested Reader would indeed note that thestable manifold Ws(u) has, close to (v+)−, a linear expansion F (u) + I − Ws(u) ∼ −K(v+ − u)

with K = 12

(a+ F ′(v+) +

√(a+ F ′(v+))2 − 4ab

)> 0 and it is easy to deduce the divergence of

the integral term within the exponential when v → (v+)−.


i ∈ N. We notify that the left and right limits at mi (Mi) are well defined, equal toα and β, respectively (β and α, respectively).

Proposition 3.4. Assume that the reset line has an infinite number of intersec-tions with the stable manifold of the saddle. If moreover all the discontinuity points(mi)i∈N ∪ (Mi)i∈N belong to [β, α], then for every n ∈ N and every finite sequence(si)

ni=1, where si = ki + li/2, ki ∈ N, li ∈ 0, 1, there exists an interval J ⊂ [β, α]

such that for any w0 ∈ J , the orbit with initial condition (vr, w0) has a transientsignature

1s11s21s3 ...1sn ...

Proof. We recall that for w ∈ (mk,mk+1) (resp. w ∈ (Mk,Mk+1)), the orbit pass-ing through (vr, w) performs exactly k (resp. k+1/2) small oscillations before spiking.Thus, proving the proposition amounts to finding a set of initial conditions with a pre-scribed topological dynamics. In detail, given an MMO pattern 1s11s2 . . . 1sn , wheresi are as above, we are searching for sequences of Φ falling sequentially in the intervals

Isi :=

(mki ,mki+1) li = 0

(Mki ,Mki+1) li = 1.

The set of initial conditions corresponding to this prescribed signature is thereforeexactly Φ−1(Is1) ∩ Φ−2(Is2) ∩ ... ∩ Φ−n(Isn), and proving the theorem amounts toshowing that this set is not empty, which relies on the particular shape of the map Φ,and particularly on the fact that Φ(Isk) = (β, α) for any sk.

As we show now, this property implies that the considered set is a union of in-finitely (countably) many open non-empty intervals, with exactly one of these intervalscontained in Isi for every given si = ki + li/2, with ki ∈ N and li ∈ 0, 1. We willsay of such ensembles that they satisfy property (P).

The proof proceeds by induction with respect to n. First of all, we immediatelynote that this property is true for Φ−1(Is1) for arbitrary s1. This set clearly satisfiesthe property (P) since we already noted that

∀i ∈ N, Φ((mi,mi+1)) = Φ((Mi,Mi+1)) = (β, α).

It is immediate to note that this is also the case of Φ−1(I) for any open intervalI ⊂ (β, α).

We assume that the property is true at rank n, i.e. that for any sequence(s1, . . . sn), where numbers si are of the required form, we have Φ−1(Is1)∩Φ−2(Is2)∩... ∩ Φ−n(Isn) satisfying property (P). We show that this also holds for the set:

Φ−1(Is1) ∩ Φ−2(Is2) ∩ ... ∩ Φ−n(Isn) ∩ Φ−(n+1)(Isn+1)

= Φ−1(Is1 ∩ Φ−1(Is2) ∩ ... ∩ Φ−n(Isn) ∩ Φ−(n)(Isn+1)

),

for arbitrary admissible sequence (s1, ..., sn+1). By the recursion assumption, the setI := Is1∩Φ−1(Is2)∩...∩Φ−n(Isn)∩Φ−(n)(Isn+1

) is composed of a unique open interval,and we conclude that the set Φ−1(I), i.e. the set of initial conditions correspondingto the signature

1s11s21s3 ...1sn1sn+1 ...,


is a non-empty set satisfying assumption (P).Remark. We emphasize that Proposition 3.4 does not include the possibility of the presence

of bursts of activity (i.e. si ∈ 0, 1/2): this is due to the fact that Φ(β) and Φ(α) may be

strictly larger than β, thus Φ( (β,m1) ) = (Φ(β), α) and Φ( (M1, α) ) = (Φ(α), α) might be

strictly smaller than the whole interval (β, α) and the argument used in the proof no more

applies without additional technical assumptions.

If the reset map has a finite number of discontinuities, exactly the same proofapplies to show that any MMO pattern with accessible number of small oscillationsexists. This is rigorously summarized in the following:

Corollary 3.5. Suppose that the reset line v = vr has a finite number p ofintersections with Ws, ordered as w1 < w2 < ... < wl < wl+1 < ... < wl+k < .. < wp,with exactly k points in [β, α]: β ≤ wl+1, ..., wl+k ≤ α, k ≥ 2. Then for every N ∈ Nand a sequence (si)

Ni=1 with (k − 1) possible values for each element si, computed

for the intervals Il+j, 1 ≤ j ≤ k − 1 according to the formula (3.1), there exists aninterval J ⊂ [β, α] such that every initial condition w ∈ J fires MMO with pattern1s11s2 ...1sN ....

If the number of intersections wi of the reset line v = vr with Ws is infinite butonly k of them lie in the interval [β, α], then we have exactly two possibilities:

• all the points wi in [β, α] are not greater than w∗ and for every N ∈ N andevery sequence (si)

Ni=1 with si ∈ l + 1, l + 2, ..., l + k − 1 there exists an

interval J ⊂ [β, α] such that every initial condition w ∈ J fires MM(B)Owith pattern 1s11s2 ...1sN ..., where index l is obtained by ordering

w1 < w2 < ... < wl < β ≤ wl+1 < ... < wl+k ≤ α < wl+k+1 < ... ≤ w∗.

• all the points wi in [β, α] are greater than w∗ and for every N ∈ N andevery sequence (si)

Ni=1 with si ∈ l + 3/2, l + 5/2, ..., l + (k − 1) + 1/2 there

exists an interval J ⊂ [β, α] of initial conditions firing MM(B)O with pattern1s11s2 ...1sN ..., where index l is obtained from putting wi in the descendingorder:

w1 > ... > wl > α ≥ wl+1 > ... > wl+k ≥ β > wl+k+1 > ... ≥ w∗.

This corollary covers all cases studied in this paper, including finite or infinitenumber of intersections of the stable manifold with the reset line and applies in bothsituations when the system features an unstable focus or a stable fixed point sur-rounded by an unstable limit cycle: it is only the number of discontinuities in theinterval [β, α] that matters, limiting the possible MMO patterns.

4. Adaptation map with one discontinuity point in the invariant inter-val. The general description developed above does not allow to specify precisely thedynamics of the system. From now on, we shall assume that vR < v+ and we focuschiefly on the case where the adaptation map has exactly one discontinuity point w1

in the interval [β, α]. Our study distinguishes a few situations for the shape of mapΦ, represented in Fig. 4.1, and classified depending on the following inequalities:A1 There exists a unique discontinuity point w1 in the interval [β, α]:

β < w1 < α < w2 (4.1)

A2 The map is piecewise increasing on [β, α], i.e.

α < w∗ (4.2)


A2’ Alternative to (A2), we will consider the following condition:

w1 < w∗ ≤ α < w2 (4.3)

A3 The interval I := [β, α] is invariant, which is set by the condition:

Φ(β) ≥ β and Φ(α) ≥ β (4.4)

Non-transient regimes only depend on the properties of the map Φ in boundedinvariant interval. Indeed, we have seen in Theorem 3.2 that Φ is upperbounded andthat for w small enough Φ(w) > w, implying the existence of an invariant compact setI in which any trajectory is trapped after a finite number of iterations. This remarkopens the way to consider Φ as a circle map, and thus to use rotation theory in orderto rigorously discriminate (i) the nature of the firing (regular, bursting or chaotic),corresponding respectively to fixed points, periodic or chaotic orbits of Φ (see [60]),as well as (ii) the number of small oscillations before firing according to the partitionof Fig. 2.2, i.e. the signature of the MMO pattern fired.

Definition 4.1. Under assumptions (A1) and (A3), the lift Ψ of Φ : I → I isdefined on (β, α] as

Ψ := x ∈ (β, α] 7→

Φ(x) if β < x < w1

α if x = w1

Φ(x) + (α− β) if w1 < x ≤ α.(4.5)

For x ∈ R, we extend Ψ uniquely through the relationship:

∀w ∈ R, ∀k ∈ Z, Ψ(w + k(α− β)) = Ψ(w) + k(α− β).

We remark that any choice for extending Φ at the point w1 by β or α is equivalentfor constructing the map Ψ, since this map will be continuous in the interior of theinterval I. The only possible discontinuities of Ψ are the boundaries of I, which isgenerally the case since Φ(β) is usually distinct from Φ(α). We display an exampleof the lift in Fig. 4.5.

By convention we have chosen the left continuous version of Ψ. This choice doesnot have implications on rotation numbers as we will emphasize at relevant places.

We can then naturally define the rotation number at the point w ∈ R as:

%(Ψ, w) := limn→∞

Ψn(w)− wn(α− β)

(4.6)

provided that the limit exists. We finally emphasize that actually the maps Ψ and Φrestricted to [β, α] induce the same circle map ϕ : S|I| → S|I| on the circle of length|I| = α− β, and the orbits of Ψ coincide modulo |I| with the orbits of Φ.

The lift Ψ is continuous in the interior of I and makes a jump at the boundary.This jump is positive or negative depending on the respective values of Φ(α) andΦ(β), that distinguish between two important cases that are called non-overlappingand overlapping:A4 We say that the map is non-overlapping if (A1), (A2) and (A3) are satisfied,

and moreover:

Φ(α) ≤ Φ(β) (4.7)


When the inequality (4.7) does not hold, we identify another case:A4’ We say that the map is overlapping if conditions (A1), (A3) and either (A2)

or (A2’) are satisfied and:

Φ(α) > Φ(β) (4.8)

The terminology (chosen after [30]) refers to the fact that the map Φ is injectivein [β, α] under the condition (A4) and has an overlap in values in (A4’). Note thatin the non-overlapping case the map Ψ has positive jumps at discontinuity pointsα + k(α − β) (i.e. limw→α− Φ(w) ≤ limw→α+ Φ(w)), k ∈ Z, and in the overlappingcase the jumps are negative.

These conditions may seem complex to check theoretically since they involverelative values for the adaptation map Φ, the discontinuity points and the values of αand β. However, they are very easy to check numerically for a specific set of parame-ters. We illustrate these different situations for a particular choice of the subthresholdparameters and for a fixed value of the reset voltage vr, and identify the regions withrespect to the reset parameters γ and d where the above listed conditions are satisfiedin Fig. 4.1.

In the rest of the paper, we will work mainly with the lift Ψ of the adaptationmap, which is defined on the whole R, thus in particular at the point w1. However,as for every initial condition on the stable manifold, the system does not spike, theforthcoming results on the spike pattern fired are valid for trajectories with initialconditions outside the pre-images Φ−n(w1), n ∈ N ∪ 0. Nevertheless, as the pointsin these pre-images form a countable set, our analysis describes correctly the generaldynamical behavior of the spiking system in the cases considered.

In this article we focus on the case where the adaptation map has exactly onediscontinuity in the interval I, and provide an exhaustive description of the mapin terms of the MMOs patterns. One of the main limitation of this situation isthat MMOs have at most one small oscillation between spikes. Notwithstanding,the techniques developed in this situation already cover many difficulties one willencounter in cases with more intersections of the reset line and the stable manifoldof the saddle within the same invariant interval, yet the main advantage we find intreating this simpler case is that the number of possible configurations of the map andidentity line remains relatively limited, while it has a combinatorial explosion in caseswith more intersections. Therefore our results and methods can be extended to thesituation with more discontinuity points but under some further inevitable technicalassumptions.

We start by mentioning a simple remark relating orbits of the system in thepiecewise monotonic case to MMO patterns:

Proposition 4.2. Any periodic orbit of map Φ in I under conditions (A1),(A2) and (A3) corresponds to an MMBO of the system.

Proof. Under the current assumptions, map Φ is a strictly increasing on bothintervals [β,w1) and (w1, α], which implies that there is no periodic orbit of periodq > 1 fully contained inside [wmin, w1] or inside [w1, α]: every such orbit containspoints in both intervals. Hence, every point of an orbit of Φ falling in the interval(β,w1) fires a spike with no small oscillation, while any point of the orbit in theinterval (w1, α) induces one subthreshold oscillation before firing a spike. The orbitis thus necessarily an MMBO, which concludes the proof.

In our framework, we can classify MMOs with half-oscillation precision. However,in this section, we choose for sake of simplicity in the formulation of the results to


!

0 0.02 0.04 0.06 0.08 0.1 0.12

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

!

0 0.02 0.04 0.06 0.08 0.1 0.12

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

↵ < w1

w1 < < w1 < ↵ < w2

d

d

()(↵)

w1

↵

↵ w1 ↵ w1 ↵w1

↵ ↵ ↵

()

(↵)w1

()

w1

(↵)

↵ w1

↵

w

w↵

↵

↵

w2

↵

↵

w1

w2 2 (,↵)

A B C

D

Ew < ↵

↵ < w

Overlapping

Fixed point in [ , ] w1

D

E

CB

Non Overlapping

(↵) < w1 < ()

(↵) < () < w1

w1 < (↵) < ()

A

Figure 4.1. Partition of (d, γ) parameter space (for fixed values of the other parameters)according to the geometric properties of the Φ map for the quartic model: F = v4 + 2av, a = 0.1,b = 1, I = 0.1175 and vr = 0.1158 (case with only two intersections of the reset line with the stablemanifold).


consider integer numbers of small oscillations, i.e. the points in (β,w1) correspondto no small oscillations whereas the points in (w1, α) result in one small oscillation.Thus, referring to the signature of MMOs, in the discussed case we have si = 0 orsi = 1 and by grouping together in the signature consecutive spikes followed by nosmall oscillations, we can assume that si = 1 for any i.

When conditions (A1), (A2) and (A3) are satisfied, the singular case Φ(β) =Φ(α) can be treated using the classical Poincare theory of orientation preservingcircle homeomorphisms. In all other cases the corresponding lift is discontinuous andpossibly non-monotonic. Our study will develop upon previous works of Keener in [30],who proved the well-definition and gave characterizations of the rotation numberfor maps with bounded derivative (that we will extend to our case), and will makeimportant use of a number of more recent results on rotation numbers of maps thatdo not require boundedness of the map derivative [6, 41, 44].

4.1. Non-overlapping case. We start by investigating the non-overlappingcase (A4). In that situation, the lift Ψ is discontinuous (unless Φ(β) = Φ(α)) butconserves the orientation-preserving property since it only admits positive jumps.It is well-known that monotone circle maps conserve the properties of continuousorientation-preserving maps: they have a unique rotation number, and rational rota-tion numbers imply the existence of attractive periodic orbits.

In order to ensure convergence towards periodic orbits, one needs to take specialcare to the presence of discontinuities. Indeed, when Φ has a periodic orbit withperiod q, then necessarily there exists x0 ∈ R such that Ψq(x0) = x0 + p(α − β) forsome p ∈ N, p, q relatively prime, i.e. x0 is periodic mod (α − β) for the lift Ψ.However, since map Φ is discontinuous at w1, it might happen that, although therotation number is rational, no truly periodic orbit of Φ exists but point w1 acts asa periodic point. This means that one of the two following properties is necessarilyfulfilled, with x0 mod |I| = w1 (see [44]):

• for all t ∈ R, Ψq(t) > t+ p|I| and

∃x0 ∈ R, limt→x−0

Ψq(t) = x0 + p|I|; (4.9)

• for all t ∈ R, Ψq(t) < t+ p|I| and

∃x0 ∈ R, limt→x+

0

Ψq(t) = x0 + p|I|. (4.10)

Remark. By allowing the lift to be multivalued, Brette [6] and Alseda et al [17] avoid

the distinction of the three cases for rational rotation numbers (i.e. the existence of the

actual periodic orbit and the two cases listed above). This formalism indeed ensures that the

periodic orbit always exists (when the map is bivalued at w1), since the two situations above

can happen only if x0 mod |I| = w1, i.e. when the periodic orbit bifurcates.

For simplicity, with a little abuse of terminology, in both above cases, we willrefer to the orbit of x0( mod |I|) under Φ as the periodic orbit. Bearing that inmind we now relate the orbits of Φ to the dynamics of the neuron model and showthat the rotation number in the non-overlapping case fully characterizes the signatureof MM(B)O:

Theorem 4.3. We assume that the adaptation map Φ satisfies conditions (A4)and consider its lift Ψ : R → R. Then the rotation number %(Ψ, w) = % of Ψ existsand does not depend on w.


Moreover, the rotation number is rational % = p/q ∈ Q with p ∈ N∗ = N ∪ 0and q ∈ N relatively prime if and only if Φ has a periodic orbit, which is related tothe MM(B)O pattern fired in the following way:

i) If % = 0 the model generates tonic asymptotically regular spiking for everyinitial condition w0 ∈ [β, α] \ w1.

ii) If % = 1 the model generates asymptotically regular MMOs for every initialcondition w0 ∈ [β, α] \ w1, i.e. the signature is periodic: 111111... = (11).

iii) If % = p/q ∈ Q \ Z (p, q relatively prime, q > 1 and p < q), then the modelgenerates MMBOs for every initial condition w0 ∈ [β, α] \ w1. Defining0 ≤ l1 < · · · < lp ≤ q − 1 the integers such that lip/q mod 1 ≥ (q − p)/q andLi = li+1 − li for i = 1 · · · p (with the convention lp+1 = q + 1), the MMBOsignature is L1

1 · · · L1p.

iv) If % ∈ R\Q, then there is no fixed point and no periodic orbit, and the systemfires chaotic MMOs.

spike number

100 200 300 400 5000

1

2

3

4

0 10 20 30 40 50

0.08

0.09

0.1

0.11

0.12

- 1 - 0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

rese

t va

lue

100 200 300 400 500-1

0

1

2

3

4

0 5 10 15 20 25 300.09

0.1

0.11

-1 - 0.5 0 0.5 1 1.5- 0.2

- 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

spike number

w

w

w

v

v

t

t

vv

vr

vr

Ws

Ws

rese

t va

lue

w

Figure 4.2. Phase plane structure, v signal generated along attractive periodic orbits andsequence of w reset values for two sets of parameter values for which the map Φ is in the non-overlapping case (A4). In both cases, vr = 0.1 and γ = 0.05. The top case (d = 0.08) illustratesthe regular spiking behavior corresponding to the rotation number % = 0 mod 1. The bottom case(d = 0.08657) displays a complex MMBO periodic orbit with associated rational rotation number.

Proof. Since the induced lift Ψ : R → R is strictly increasing, we can applythe monotone circle maps theory developed by Rhodes and Thompson [44, 45] andBrette [6]. The uniqueness of the rotation number is shown in [44, Theorem 1] and [6],


and the proof for orientation preserving homeomorphism applies5. The characteriza-tion of the orbits in the case of rational rotation numbers results from [44, Theorem2] and the fact that Ψ is strictly increasing.

Moreover, it can be shown that every non-periodic mod (α − β) orbit Oy :=Ψn(y)n∈N tends to some periodic mod (α − β) orbit, i.e. for every y ∈ R non-periodic mod (α− β), there exists a corresponding q-periodic point x0 such that

Ψnq+k(y) −−−−→n→∞

Ψk(x0) + np(α− β), k = 0, 1, ..., q − 1. (4.11)

This is a consequence of [6, Proposition 5] since the monotonicity of Ψ ensures thatthe underlying circle map is strictly orientation preserving. From the proof thereinit also follows that the asymptotic behavior is consistent for all the points of a givenorbit, i.e. that (4.11) holds and that any point of the orbit OΦ,y := Φn(y)n≥0 tendsunder Φq to its corresponding iterate x0 = Φq(x0),Φ(x0),Φ2(x0), ...., which providesthe classification of orbits for the adaptation map, analogous as the one for circlehomeomorphism with rational rotation number (cf. [29, Proposition 11.2.2]).

(i-ii)When %(Ψ) = 0 mod 1, the adaptation map admits a fixed point. Moreover,under the current assumptions we either have %(Ψ) = 0, if the fixed point belongs to(β,w1) (thus there is no (full) small oscillation between spikes), or %(Ψ) = 1, if thefixed point belongs to (w1, α) and the orbit displays one small oscillation between twoconsecutive spikes.

(iii) As mentioned in Proposition 4.2, periodic orbits necessarily correspond toMMBO. Moreover, it is not hard to show that orbits with rotation numbers p/qhave exactly p points to the right of w1. These points split the periodic orbitinto subintervals of firing of one spike or a burst, separated by a small oscillation.Since the lift preserves the orientation, the consecutive points of a periodic orbitw,Φ(w), . . . ,Φq−1(w) with rotation number p/q are ordered as the sequence ofnumbers (0, p/q, 2p/q, ..., (q−1)/q) in [0, 1] (up to the cyclic permutation, see e.g. [29,Proposition 11.2.1]). The signature of the MMBO is directly related to the indexesl ∈ 0, 1, ..., q−1 such that Φl(w) > w1, and hence such that lp/q ≥ (q−p)/q mod 1.We easily conclude that the signature of the MMBO indeed is L1

1L12 · · · L1

p.(iv) In the case of irrational rotation number, Φ admits no periodic orbit, and all

orbits under Φ have the same limit set Ω, which is either the circle or a Cantor-typeset as in the continuous case (Φ(β) = Φ(α)), as proved in [6, Proposition 6].

In cases where the point w1 acts as a periodic orbit, the forward attracting periodicorbit is unique. Otherwise, several attracting periodic orbits may exist with the samerational rotation number, i.e. necessarily with the same period and the same ordering.In [17], the authors have proved the uniqueness of the periodic orbit of maps suchas Φ in the non-overlapping case with the assumption that Φ is contractive on both(β,w1) and (w1, α) (see Remark 3.20 in [17]). Here, because of the divergence of thedifferential at the discontinuity points, we cannot use the contraction assumption.

We emphasize that since Ψ is a strictly increasing lift of a degree-one circle map,changing its value at a discontinuity point (while conserving monotonicity) does notchange the value of the rotation number (see e.g. [44]). The above remark means thatit does not matter for qualification of the dynamics of Φ how we define the lift Ψat its discontinuity points β ≡ α, neither that Φ is formally not defined at w1, since

limw→w−1

Φ(w) = α and limw→w+

1

Φ(w) = β.

5Continuity of the lift is not used in the classical proof of the uniqueness of the rotation numberfor orientation preserving circle homeomorphisms, see e.g. [29, Proposition 11.1.1].


Remark. It is important to note that, in the non-overlapping case, the qualitative shape ofthe orbit, in terms of MMBO signature, is equivalent to knowing the rotation number of theorbit. Moreover, we have provided in the proof an algorithm for constructing the signaturefrom a given rotation number. We illustrate this construction on two examples:

• Orbits of the adaptation map with rotation number % = 1/q have signature q1. When% = (q − 1)/q the signature is 2111 · · · 11 (with q − 2 repetitions of the pattern 11).

• For the rotation number % = 3/5, up to cyclic ordering, the orbits are equivalentto 0, 3

5, 2·3

5= 1

5mod 1, 3·3

5= 4

5mod 1, 4·3

5= 2

5mod 1. The three indices cor-

responding to values larger or equal to 2/5 are 1, 3, 4 ; hence L1 = 2, L2 = 1,L3 = 1 + 5− 4 = 2, and the signature is 211121.

We now provide a simple sufficient condition for the existence of 21 MMBOs:Proposition 4.4. Assume that Φ fulfills condition (A4) (non-overlapping case)

and moreover that Φ(α) < w1 < Φ(β). Then Φ admits a periodic orbit of period 2,thus the system has 21 MMBO.

50 100 150 200 250 300 350 400- 2

0

2

4

↵

(↵)

()

w1

w1

↵

0 450t

v

Adaptation Map

Generated signal

v

w

Phase Space

vr Ws

Figure 4.3. Phase plane and adaptation map fulfilling condition (A4) and additional conditionΦ(α) < w1 < Φ(β) and associated MMBO orbit of system (2.1). The rotation number is equal to0.5, hence the v signal along the orbit is a periodic alternation of a couple of spikes and one smalloscillation. The parameter values of the system corresponding to this simulation are vr = 0.1,γ = 0.05 and d = 0.087.

Proof. In this case, Φ2((β,w1)) ⊂ (β,w1), Φ2 is continuous on (β,w1) ∪ (w1, α),Φ2(β) > β and lim

w→w−1Φ2(w) = Φ(α) < w1. Hence, Φ2 admits a fixed point in (β,w1)

corresponding to a periodic point of period 2 for Φ. On the other hand, the secondpoint of this periodic orbit lies in (w1, α) since Φ((β,w1)) ⊂ (w1, α). Thus this orbitexhibits MMBO and necessarily %(Ψ) = 1/2.


This result is analogous to [30, Lemma 3.2] and does not necessitate the bound-edness assumption on the derivative of the map. A number of other results of [30]also extend to the case where Φ(α) < Φ(β) < w1. For instance, the criteria providedfor ensuring the existence of periodic orbits connected with the set of pre-imagesΦ−k(w1) of the discontinuity point w1 apply here. In detail, denoting by ∆ the setof pre-images of w1 lying in the interval (β,Φ(β)], it is shown that ∆ is not empty,and when ∆ is finite (i.e. either the sequence Φ−k(w1) terminates with some k or thepoint w1 is periodic), then Φ has a periodic orbit exhibiting MMBO (and a rationalrotation number).

When the second assumption of Proposition 4.4 is not valid and Φ(β) > Φ(α) >w1, the dynamics may generate complex orbits of higher period or even chaos. Differ-ent MMBO patterns may therefore be observed in the non-overlapping case, dependingsensitively on the parameters. We now focus on this dependence on the reset param-eters (γ, d). Basing ourselves on the corresponding result in [6], we can show that therotation number varies as a devil staircase, with the detailed proof in the Appendix.

Theorem 4.5. Assume that for any d ∈ [d1, d2], the adaptation map Φd remainsin the non-overlapping case (A4) and Φd(αd2) < Φd(βd1). Let %d be the uniquerotation number of Φd. Then:

• ρ : d 7→ %d is continuous and non-decreasing;• for all p/q ∈ Q ∩ Im(ρ), ρ−1(p/q) is an interval containing more than one

point, except, maybe, at the boundaries of the interval d1, d2;• ρ reaches every irrational number at most once;• ρ takes irrational values on a Cantor-type subset of [d1, d2], up to a countable

number of points.A similar result holds for the dependence of the rotation number on parameter

γ in the regime where we can ensure the suitable monotonicity of γ 7→ %γ . Fig. 4.4shows quantitatively an example of application of this theorem.

Beyond the continuity, the plateaus of rotation number observed in the devilstaircase situation are also a general property of our system, called locking. In detail,a non-decreasing degree-one map F is said to induce locking of the rotation number iffor every non-decreasing family (Fλ) of degree-one non-decreasing maps Fλ such that

Fλ0= F and Fλ

H−→ Fλ0(the convergence of the graphs of functions in the Hausdorff

metric) there is an interval of values of λ in which %(Fλ) = %(Fλ0). We can show,

using the theory of [45], that:Proposition 4.6. When the adaptation map Φ satisfies the assumption of

strictly non-overlapping case (A4) (i.e., Φ(α) < Φ(β)), then for parameters suchthat %(Ψ) is rational, Φ induces locking of the rotation number.Remark. We observe that the strictly non-overlapping case requires no condition to show

locking of the rational rotation number, which is not the case of continuous circle maps.

When Φ(α) = Φ(β), the lift Ψ would be in fact a lift of an orientation preserving circle

homeomorphism and thus locking of the rotation number at rational values requires that

there is no conjugacy with rational rotation for such a map (see e.g. Propositions 11.1.10

and 11.1.11 in [29]).

4.2. Overlapping case. Let now Φ satisfy the properties of the overlappingcase (A4’). In this case, the lift Ψ is no more increasing (it has in particular negativejumps at the points w = α + k(α − β) for k ∈ Z), and a number of importantproperties inherited from the well-behaved dynamics of orientation-preserving circlehomeomorphisms, persisting in the non-overlapping case [6, 44, 45], are now lost,leaving room for much richer dynamics.


Rota

tion

Num

ber

(a)

(b) (c)

(a) (b) (c)

0.08 0.082 0.084 0.086 0.088 0.09 0.0920

0.1

0.2

0.3

0.4

0.5

0.6

0.7

d

Adaptation map and attractive periodic orbit

Figure 4.4. Rotation number as a function of d. The parameter values vr = 0.1 and γ =0.05 have been chosen such that the adaptive map Φ fulfills conditions (A4) for any value of d ∈[0.08, 0.092]. Theorem 4.5 applies here, and the rotation number varies as a devil staircase.

In the current overlapping case, it is easy to see that our map restricted to itsinvariant interval [β, α] falls in the framework of the so-called old heavy maps [41],since it is a lift of a degree one circle map with only negative jumps. These mapshave interesting dynamics with non-unique rotation numbers. More precisely, we candefine a rotation interval [a(Ψ), b(Ψ)] with

a(Ψ) := infw∈R

lim infn→∞


, (4.12)

b(Ψ) := supw∈R

lim supn→∞


. (4.13)

As noted in [41], these two quantities are the (unique) rotation number of the orien-tation preserving maps:

Ψl(w) := infΨ(z) : z ≥ w, (4.14)

Ψr(w) := supΨ(z) : z ≤ w, (4.15)

i.e. a(Ψ) = %(Ψl) and b(Ψ) = %(Ψr). The maps Ψl and Ψr for the adaptation map ofthe hybrid neuron model are plotted in Fig. 4.5.

We can now conclude after [41]:Theorem 4.7. Under the current assumptions (A4’), we have:1. if Φ admits a q-periodic point w with rotation number %(Ψ, w) = p/q, then

a(Ψ) ≤ p/q ≤ b(Ψ);


↵

l

r

Figure 4.5. The orientation-preserving maps Ψl (green) and Ψr (blue) enveloping the adap-tation map Φ (blue dashed curve) in the overlapping case. The lift Ψ (red line) is non-monotonicand admits negative jumps.

2. if a(Ψ) < p/q < b(Ψ), then Φ admits a periodic point w of period q androtation number %(Ψ, w) = p/q.

Moreover, for any %1 and %2 such that a(Ψ) ≤ %1 ≤ %2 ≤ b(Ψ), there exists w0 suchthat

lim infn→∞

Ψn(w0)− w0

n(α− β)= %1, (4.16)

lim supn→∞

Ψn(w0)− w0

n(α− β)= %2. (4.17)

The second part of the theorem implies in particular that the rotation set in theoverlapping case is closed, and that every number % ∈ [a(Ψ), b(Ψ)] is the rotationnumber %(Ψ, w) of some w ∈ [β, α].

The property of having a non-trivial rotation interval implies coexistence of in-finitely many periodic orbits of distinct periods; this situation is sometimes referred toas ‘chaos’ (see [30]), a different notion than chaotic orbits with non-rational rotationnumbers.

Note that the fact that the number p/q (p and q relatively prime) belongs to theinterior of the rotation interval means that there exists a periodic trajectory having qspikes and p small oscillations in one cycle (as p points among q are located to the rightof w1). However, here from the rotation number we cannot directly deduce the MMOsignature since there can be different orderings of points on the orbit correspondingto the same rotation number p/q.

We also notify another important difference between the orientation-preserving(non-overlapping) maps and the other situations in which the rotation number is ingeneral not unique:Remark. The structure of the set of (minimal) periods of orbits for the non-invertible

(overlapping) case can be very complicated. For degree-one continuous non-injective maps,

this set can be fully characterized (see e.g. [3] or [42]). When we consider a map with dis-

continuities, then the situation is even more complicated and, to our knowledge, this problem

is solved only for lifts of monotonic modulo 1 transformations (see [23]), which corresponds

to the overlapping case with the additional monotonicity assumption (A2). This means in

particular that, even if the rotation set is reduced to singleton p/q (with p, q ∈ Z relatively

prime), we cannot assure that there cannot be periodic orbits with minimal period greater

than q. Although, in case they exist, each of them necessarily contributes only to the rotation


number p/q, thus its period can only be kq, for some k ∈ Z, and its non-reduced rotation

number is kp/kq.

Remark. If we additionally assume that the map Φ is piecewise increasing (A2), then

Proposition 4.2 remains valid and periodic orbits display MMBOs. In the case where as-

sumption (A2) is not valid, the periodic orbit of Φ can be fully contained in (w1, α), which

corresponds to the rotation number of the type q/q ≡ 1: such orbit displays MMO with signa-

ture 11, whereby a single small oscillation occurs between every pair of consecutive spikes.

An important property of rotation intervals, shown in [41, Theorem B], is thatthe boundaries a(Ψ) and b(Ψ) vary continuously with the parameters as soon as themaps Ψl and Ψr are also continuous with respect to these parameters (in the senseof uniform convergence in C0(R)). In our case, this property allows us to show thefollowing proposition, whose proof for the clarity of the article is presented in theAppendix.

Proposition 4.8. Consider fixed parameters vR, a, b, γ and I and vary d ∈[λ1, λ2] such that, for each d ∈ [λ1, λ2], the corresponding adaptation map Φd (theexponent d indicates the dependence in that parameter) satisfies the assumptions ofthe overlapping case. Then the maps d 7→ a(Ψd) and d 7→ b(Ψd), assigning to d theendpoints of the rotation interval of Φd, are continuous.

If we further assume that, for any pair (d1, d2) ∈ [λ1, λ2]2 with d2 < d1, we have

Φd2(βd1) ≤ Φd2(βd2) + d1 − d2, (4.18)

then the maps d 7→ Ψd, d 7→ Ψdr and d 7→ Ψd

l are increasing (Ψdr and Ψd

l denote,respectively, lower and upper enveloping maps of the lift of Φd). Consequently, themaps d 7→ a(Ψd) and d 7→ b(Ψd) behave like a devil staircase.Remark. The exemplary sufficient condition (4.18) for the devil staircase is equivalent to

Ψd2(βd1) ≤ Ψd2(β+d2

) + d1 − d2, (4.19)

where Ψd2(β+d2

) denotes the right limit of Ψd2 at β+d2

. Note that this is satisfied for instance

when, for every d ∈ [λ1, λ2], Φ′ < 1 in the whole interval [βd, βd + (λ2 − λ1)].

0.074 0.075 0.076 0.077 0.078 0.079 0.08 0.081 0.082 0.0830

0.1

0.2

0.3

0.4

0.5

0.6

!

r

l

Rota

tion

Num

ber

d

Figure 4.6. Rotation intervals according to d : maps d 7→ a(Ψd) = %(Ψdl ) and d 7→ b(Ψd) =

%(Ψdr) associated with the lifts Ψd of the adaptation maps Φd. Parameter values γ = 0.05 has been

chosen so that Φd remains in the overlapping case for any d ∈ [0.0745, 0.0825].


These general properties being proved, we now investigate special cases in whichthe behavior of spiking sequences is well characterized.

Proposition 4.9. Assume that Φ fulfills (A4’) and admits at least two fixedpoints wf ∈ [β,w1) and wf ∈ (w1, α). Then there exist periodic orbits of arbitraryperiod.

Proof. It suffices to prove that the rotation interval equals [0, 1]. Indeed, as thefixed point wf (which can be chosen arbitrarily among all the fixed points in (β,w1),if there are more than one fixed point) will be the fixed point of map Ψl, we obtainthat %(Ψl) = 0. Similarly, the fixed point wf can be chosen such that wf < w∗ and, inthis case, it satisfies Ψr(wf ) = wf +(α−β) for the upper envelope Ψr. Consequently,%(Ψr) = 1.

Proposition 4.10. Assume that Φ satisfies (A4’) and that Φ admits at least onefixed point in (w1, α). We denote by wf ∈ (w1, α) the lowest one. Assume moreoverthat there is no fixed point of Φ in [β,w1). Then

• if Φ(β) < wf , there exists q > 1 such that for all q > q, Φ admits a periodicorbit of period q;

• if Φ(β) ≥ wf , then Φ admits a trivial rotation interval [a(Ψ), b(Ψ)] = 1. Ifadditionally α ≤ w∗, then Φ admits no periodic orbit of period q > 1 and allorbits converge towards a fixed point.

Proof. We first assume that Φ(β) < wf . In this case, the lower envelope Ψl

intersects neither the identity (Id) line nor the Id + (α − β) line (and, obviously,none of the Id + k(α − β) for k ∈ Z). Thus the graph of Ψl is fully containedbetween the lines Id and Id + (α− β) and since the functions Ψl(w)− w + k(α− β),k ∈ Z, are continuous and α − β periodic they reach their suprema and infima.This means that the graph of Ψl is in fact away from Id − (α − β), i.e. there existsδ ∈ (0, 1) such that Ψl(w) < w + (α − β) − δ. Thus for every w and n ∈ N we haveΨnl (w)−Ψn−1

l (w) < α− β − δ and

a(Ψ) = %(Ψl) < 1− δ

(α− β).

On the other hand, Ψr(wf ) = wf + (α − β) and thus b(Ψ) = %(Ψr) = 1. This yieldsthat the rotation interval is not trivial and

[1− δ

(α− β), 1] ⊂ [a(Ψ), b(Ψ)].

For every q > 1 large enough, we have

a(Ψ) <q − 1

q< b(Ψ),

i.e. there exists a periodic orbit of Φ with period q and rotation number q−1q .

We now assume Φ(β) ≥ wf . Then wf is also a fixed point mod (α− β) of Ψl, i.e.

Ψl(wf ) = wf + (α− β),

a(Ψ) = %(Ψl) = %(Ψr) = b(Ψ) = 1.

Thus, if there were some periodic orbits of period q > 1, they would have the rotationnumber q

q = 1 and all the points of such orbits would lie in (w1, α). However, if


additionally α ≤ w∗, then the map Φ is increasing in (w1, α) and no periodic orbitcan be fully contained in this interval. Assuming that α ≤ w∗, we introduce the map

G(w) := Ψ(w)− (α− β)

on the interval [wf , wf + (α− β)]. Then

G(wf ) = wf and G(wf + (α− β)) = wf + (α− β),

i.e. G maps the interval [wf , wf + (α− β)] onto itself. Map G is continuous with theexception of one negative discontinuity jump at α. In the intervals of continuity it isalso increasing. We notice that every point w ∈ (α,wf + (α− β)) after at most a fewiterations of G enters the interval [wf , α] (the points wf +k(α−β) are always repellingfor orbits in its left neighborhood). The interval [wf , α] is invariant for G and G ismonotonically increasing in this interval. Hence, for every w ∈ [wf , α], there existsa fixed point wf ∈ [wf , α] of G such that lim

n→∞Gn(w) = wf . By previous argument,

the same result holds for every w ∈ [wf , wf + (α− β)]. However, this means that,

∀w ∈ R, Ψn(w)n→∞−−−−→ wf + n(α− β)

where wf is a fixed point mod (α− β) (that might depend on w). The statement forthe adaptation map Φ follows.

In the statement of Proposition 4.10, if Φ(β) ≥ wf but α > w∗ (i.e. map Φ isnon monotonic in (w1, α)), then we cannot exclude the existence of periodic orbits ofthe type q

q = 1 for some q ∈ N\1 (see Remark 4.2). However, since all the points ofsuch orbits would lie on the right of w1, then there will be exactly one small oscillationbetween any two consecutive spikes and thus the orbit will display MMOs of signature11.

Subsequent Theorem 4.13 considering a slightly more general situation completesthe above result for maps admitting fixed points in [β,w1) and no fixed point in(w1, α].

We can easily justify the following:Corollary 4.11. The existence of a fixed point wf of Φ (in the overlapping

case) with %(Ψ, wf ) = 0 and of a periodic orbit with period q > 1 implies existence ofperiodic orbits with all arbitrary periods greater than q and associated with MMBO.The same result holds if %(Ψ, wf ) = 1 provided that the q-periodic orbit is not of typeq/q (i.e. it admits points both on the left and the right side of w1).

In particular, if there exists a fixed point wf (%(Ψ, wf ) ∈ 0, 1) and a periodicorbit of rotation number 1/2, then there are periodic orbits of all periods, exhibitingMMBO.

In contrast to the non-overlapping case, in the overlapping case we have droppedthe assumption (A2) that the map was piecewise increasing. However, under thisassumption we can describe the chaotic behavior of the map’s iterates more precisely:

Theorem 4.12. Assume that Φ satisfies (A4’) with (A2), that Φ(α) < w1, thatΦ has at least two periodic orbits with periods q1 6= q2 and that exactly one point ofeach of these periodic orbits is greater than w1. Then the mapping w 7→ Φ(w) is ashift on a sequence space.

This is the immediate consequence of [30, Theorem 2.4], which extends to ourclass of discontinuous maps with unbounded derivative since the key element of theproof is a general description of how the intervals with endpoints being points ofperiodic orbits are permuted. Finally we make an important remark:


Remark. When we have multiple discontinuity points in the invariant interval, but the

map φ still falls within the domain of old heavy maps (i.e. suitably obtained lift Ψ has only

negative jumps), then one can use the same technics as for the overlapping case.

4.3. A general result for adaptation maps with one discontinuity inthe invariant interval [β, α]. In previous sections we have classified the dynamicsof the adaptation map and the associated spiking patterns in terms of rotation num-bers/rotation intervals. However, for particular values of the parameters, it is hard toanalytically determine whether the model has unique and rational or irrational rota-tion number since there is no explicit analytical expression for the rotation number.Below we get rid of this problem and only assume that [β, α] is an invariant interval(A3) and that the map has a unique discontinuity point within this interval (A1),regardless of whether the map is in the overlapping- or non-overlapping case or neitherof these (e.g. when the jumps at β + k(α− β) are positive but there is an overlap invalues of Φ|[β,w1) and Φ|(w1,α]).

Theorem 4.13. Under conditions (A1) and (A3) and assuming that there is afixed point in [β,w1), we have:

1. When Φ has exactly one fixed point wf in (β,w1) and• if maxΦ(w) : w ∈ (w1, α] ≤ wf , then

∀w ∈ R, limn→∞

Φn(w) = wf

and the rotation number %Φ(w) = 0 is unique.• if maxΦ(w) : w ∈ (w1, α] > wf and there is no fixed point in (w1, α],

then

∀w ∈ (−∞, wf ), limn→∞

Φn(w) = wf .

If additionally Φ(α) ≥ Φ(β), then Φ also admits periodic orbits in theinterval (wf , α). Moreover, there exists q ∈ N such that for every q ≥ qthere exists a periodic point w ∈ (wf , α) of period q.

2. When Φ has exactly two fixed points wf < wf,2 in (β,w1) and• if maxΦ(w) : w ∈ (w1, α] < wf,2, then

∀w ∈ R\wf,2, w1, w2, limn→∞

Φn(w) = wf

and the rotation number %Φ(w) = 0 is unique.• if maxΦ(w) : w ∈ (w1, α] ≥ wf,2 and Φ admits no fixed point in

(w1, α], then

∀w ∈ (−∞, wf,2), limn→∞

Φn(w) = wf .

If additionally Φ(α) ≥ Φ(β), then Φ also admits periodic orbits in theinterval (wf,2, α). Moreover, there exists q ∈ N such that for every q ≥ qthere exists a periodic point w ∈ (wf,2, α) of period q.

3. If Φ admits more than two fixed points in (β,w1), then a result analogous tostatement 2. holds, by replacing wf and wf,2 by the smallest and the largestfixed points in (β,w1) respectively. Yet, some points might be then attractednot by wf but by another (semi-) stable fixed points in (β,w1).

Although Theorem 4.13 might seem a bit technical, it allows to determine thestructure of the firing pattern and possible MMOs for given initial condition (vR, w0)


since the fixed points and periodic orbits correspond, respectively, to regular firingand bursting.Remark. Since lim

w→w−1

Φ′(w) = ∞ and Φ′ is C1 in (−∞, w1), most of the time Φ′

is increasing in (β,w1) and thus Φ is strictly convex. In that case, there can be at most

two fixed points wf and wf,1 in (β,w1). However, in general, we cannot exclude the case

where Φ admits an inflection point in this interval and thus more fixed points are possible.

Theorem 4.13 is stated independently of the number of fixed points in order to incorporate

all possible cases.

Proof. First, we note that for every point w ∈ R, there exists a non-negativeinteger N = N(w) such that, for all n ≥ N , Φn(w) ∈ [β, α], i.e. the successiveiterations by Φ of every point eventually enter the invariant interval [β, α] after atransient. It is thus sufficient to investigate the dynamics of the map within theinvariant interval [β, α]. The lift Ψ associated with Φ is discontinuous at β+k(α−β),k ∈ Z, with negative jumps if Φ(α) ≥ Φ(β) and positive jumps if Φ(α) < Φ(β). Wealso use Ψl and Ψr, the enveloping maps of Ψ introduced in equations (4.14)-(4.15).

We first assume that map Φ admits exactly one fixed point wf ∈ (β,w1) and thatmaxΦ(w) : w ∈ (w1, α] ≤ wf . Then every point w ∈ [β, α] \ w1 is mapped byΦ, after a few iterations, into (β,wf ), which is the domain of attraction of wf andthus lim

n→∞Φn(w) = wf . The uniqueness of the rotation number %(Ψ, w) = 0 follows

immediately from this asymptotic behavior.Now we assume that map Φ admits exactly one fixed point wf ∈ (β,w1), that

maxΦ(w) : w ∈ (w1, α] > wf , and that Φ admits no fixed points in (w1, α].Obviously, any orbit starting from [β,wf ) tends to the fixed point wf . If additionallyΦ(α) ≥ Φ(β) then we are in the overlapping case and the fixed point wf (and wf +k(α − β), k ∈ Z) is still a fixed point of the lower-map Ψl but it is no longer a fixedpoint for the upper-map Ψr (which has no fixed points). We recall that maps Ψl

and Ψr are continuous, non-decreasing, and admit unique rotation numbers. Thusa(Ψ) = %(Ψl) = 0. In order to calculate the second endpoint b(Ψ) of the rotationinterval, we need to calculate the rotation number of map Ψr. We notice that as Ψr

admits no fixed point, its graph lies above the identity line and:

∃δ > 0, ∀w ∈ R, ∀i ∈ N, Ψir(w)−Ψi−1

r (w) > δ.

Consequently,

∀w, ∀n ∈ N,Ψnr (w)− wn(α− β)

=1

n(α− β)

n−1∑i=0

(Ψi+1r (w)−Ψi

r(w))≥ δ

α− β > 0.

Thus the rotation interval [a(Ψ), b(Ψ)] of Ψ is not trivial since it contains the non-trivial interval [0, δ/(α− β)]. Consider

g = min

k ∈ N ; 0 <

1

k<

δ

α− β

.

Then 1/q ∈ (0, δ/(α− β)) and

∀q ≥ q, 1

q∈(

0,δ

α− β

)⊂ (a(Ψ), b(Ψ)) ,

which means that adaptation map Φ admits a periodic point wq ∈ (wf , α) (its wholeorbit lies in (wf , α)). This completes the proof of the first statement.


The second statement of the theorem is concerned with the situation in which Φhas exactly two fixed points wf and wf,2 in (β,w1). It is proved in a similar way asthe first statement by establishing whether the rotation interval is trivial or not.

Statement 3 also follows almost immediately.

4.4. Evolution of the rotation number along a segment of (d, γ) values.In sections 4.1 to 4.5, we have investigated the rotation number or the rotation intervalin various subcases existing under general assumption (A1), i.e. the adaptation mapfeatures a unique discontinuity point in the interval [β, α]. In this section, we illustratenumerically the dependence of the rotation number (thus also the MMBO patternfired) and its possible uniqueness on the values of parameters d and γ.

!

0 0.02 0.04 0.06 0.1 0.12

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.1

0.2

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0.088 0.089 0.09 0.091 0.092 0.093 0.0940

0.1

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0.6

0.7

0.8

0.9

1

0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065

d

r

l

d

0.08d

A

B C

D

E

A

BC

DE

A

A B C D E

Rotation Number of w = 0

!

0 0.02 0.04 0.06 0.1 0.12

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.1

0.2

0.3

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0.088 0.089 0.09 0.091 0.092 0.093 0.0940

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065

d

Rota

tion

Num

ber

r

l

d

0.08d

A

B C

DE

A

BC

DE

A

A B C D E

Rotation Number of w = 0Ro

tatio

n Nu

mbe

r

Figure 4.7. Rotation numbers according to (d, γ). Left panel : rotation number of w = 0together with the boundaries of the regions A to E corresponding to the different subcases when w1

is the unique discontinuity of the adaptation map lying in the [β, α]. Right panel : rotation numbersof the left and right lifts Ψl and Ψr associated with Φ for (d, γ) varying along the blue segmentdrawn in the insert.

The left panel of Fig. 4.7 shows the rotation number of the point w = 0 associatedwith the adaptation map for (d, γ) in [0, 0.12] × [0.01, 0.15]. The various regions inthe (d, γ)-plane corresponding to the different subcases studied above and alreadyshown in Fig. 4.1 are superimposed on the colormap. On one hand, regions A, B andC compose the non-overlapping case, i.e. assumption (A4) is fulfilled, and generalTheorem 4.3 applies for (d, γ) values in these regions. In particular, the rotationnumber of Φ is unique, i.e. does not depend on the initial condition.

• In region A, Φ(α) < Φ(β) < w1. Along certain paths in this region, Theorem4.5 applies and the rotation number varies as illustrated in Fig. 4.4.

• In region B, Φ(α) < w1 < Φ(β), hence Proposition 4.4 applies and ensuresthe existence of a period 2 orbit of Φ.

• In region C, w1 < Φ(α) < Φ(β). This region may feature very rich dynamicsincluding all types of behavior arising in the other regions.

On the other hand, regions D and E compose the overlapping case and Φ may admitdifferent rotation numbers depending on the initial condition. For (d, γ) in theseregions, the lift Ψ associated with the adaptation map exhibits only negative jumps.The general Theorem 4.7 applies, which ensures the existence of a rotation interval.Using the left and right lifts Ψl and Ψr associated with Φ, one computes the endpointsof the rotation interval and their evolution according to parameter d (Proposition 4.8


and Fig. 4.6). For (d, γ) in region D, α < w∗ and Φ is piecewise increasing, whilew∗ < α in region E.

Several points are worth noticing. First, the global Theorem 4.13 applies in allregions A to E. One may track the appearance and disappearance of the fixed pointsaccording to the value of d and γ together with the evolution of the rotation numberor rotation interval. Second, the well-definition of the rotation number or rotationinterval has been shown in regions A to E, i.e. when considering that the adaptationmap features a single discontinuity in the invariant interval. Outside these regions, theconstruction of the lift is more complex due to the presence of additional discontinuitypoints. Yet, the numerical calculation of the rotation number can be performed for agiven initial condition (e.g. w = 0 as for producing the left panel of Fig. 4.7).

The second panel of Fig. 4.7 illustrates the evolution of the rotation number (orrotation interval) along a segment of (d, γ) values crossing all regions from A to E.We have computed the (unique) rotation number of Ψl and Ψr along this segment.Hence, we show

• In region A, the rotation number associated with Φ is uniquely defined (forany initial condition) and varies along the path as in Fig. 4.4.

• In region B, the rotation number is uniquely defined and constant equal to1/2.

• In region C, the rotation number is uniquely defined. Note that the constantvalue 1/2 obtained in the simulation only depends on the choice of the segmentfor (d, γ) values. As shown in the top panel, the rotation number can differfrom this value for other values of (d, γ) in region C (e.g. in the right part ofregion C).

• In region D, the rotation number is not uniquely defined in the general case.Nevertheless, along the particular chosen path in the parameter space (d, γ),Ψl and Ψr present the same rotation number 1/2 and the rotation numberof Φ does not depend on the initial condition. This particular simulationillustrates a way to evidence that the rotation number of the adaptation mapis unique by showing that the rotation interval is reduced to a singleton.

• In region E, the rotation numbers of Ψl and Ψr differ and the rotation intervalof the adaptation map evolves according to (d, γ) in the tunnel bounded bythe black and red lines.

5. Discussion. Nonlinear bidimensional neuron models are easily defined andshow an astonishingly rich mathematical phenomenology. A number of studies hadalready revealed their common subthreshold dynamical properties [57], investigatedtheir spike patterns in the absence of any equilibrium state [60], and all showed theirversatility [7, 24, 52] and capability to reproduce neuronal dynamics [25, 26, 43, 59].Completing this body of works by the study of spiking patterns of bidimensionalspiking neuron models in the presence of multiple unstable fixed points, we havebeen led to investigate in depth in the present manuscript the properties of iteratesof a particular class of interval maps that present both discontinuities and diver-gence of the derivative. Interestingly, in the presence of an unstable focus of thesubthreshold dynamics, we have shown that the spike patterns fired may correspondto complex oscillations that combine action potentials (or bursts of action potentials)and subthreshold oscillations, trajectories known as MMOs or MMBOs in continuousdynamical system.

Contrasting with classical mechanisms for the generation of MMOs, the hybridnature of the system allows for these complex oscillations to occur in system with


only two variables. Moreover, the mechanism of generation of these trajectories isnovel: it is a purely geometric feature of the model that depends on the topologyof the invariant manifolds of the continuous-time dynamics. As such, these trajec-tories occur in systems that do not necessarily display separation of timescales, andmore elementary dynamical systems methods allows to rigorously exhibit their exis-tence and characterize them, for the first time, up to half-small oscillation precision,without relying on complex numerical computations of funnels and manifolds in high-dimensional systems. One may however wonder whether if there exists a relationshipbetween the two systems, and particularly it is tempting to interpret the hybrid sys-tem as the reduction of a differentiable multiple timescale system in a certain singularlimit. Yet, the wide panel of MMOs associated with the versatile geometric structurepresented in this article indicates that such system is at least four-dimensional andinvolves more than three timescales. The construction of a suitable return mechanismfor such system remains a challenging problem from the dynamical viewpoint. Thisstudy thus constitutes a showcase for MMBOs that is both rigorous and relativelystraightforward. Additionally, it can be directly used for the analyses of the reducedsystems emerging from highly complex dynamics involving multiple timescales thatmay be encountered in future developments of canard-induced MMO analysis.

Beyond the mathematical interest of this new mechanism for the generation ofMM(B)Os, this study came back to the biophysical origin of the model. By doingso, we have found important to take into account the consequence of the durationof a spike fired in the reset mechanism, which has resulted in introducing a slightlymodified reset mechanism (featuring an additional parameter, γ, accounting for anattenuation of the adaptation variable) and showed that the introduction of this newparameter allows the quartic model (or, presumably, all other models of the class,including the Izhikevich model [24] or the adaptive exponential [7]) to be in any of thecases identified. This property completes nicely the repertoire of possible behaviors ofthis class of models: these simple dynamical systems have the mathematical advantageto present a very rich phenomenology, and the computational advantage to reproduceall trajectories displayed by real neurons. It is important to note that the partitionof the parameter space in Fig. 4.1 is obtained from explicit conditions involving thedynamical invariants of the continuous dynamics. Therefore, it gives a useful tool forsolving the problem of parameter estimation for the model in order to reproduce modeloutputs fulfilling a list of qualitative and quantitative specifications. In particular,the ability to reproduce fine trajectories of MMBOs is very useful in situations inwhich synchronization is essential: in the presence of noise, the presence of smallsubthreshold oscillations supports the generation of precise and robust rhythmic spikegeneration patterns, as recorded in specific rhythmic pattern generators as the inferiorolive nucleus [5, 38, 39], in the stellate cells of the entorhinal cortex [1, 2, 28], and inthe dorsal root ganglia [4, 36, 37].

From the computational neuroscience viewpoint, it is important to understandthe role of the different biophysical parameters on the qualitative responses of neu-rons. This raises at least two perspective works. First, in the flavor of [59], we couldgo deeper into the analysis of the shape of the adaptation map in the particular caseof the adaptive exponential integrate-and-fire system, and thus relate the presenceand possible signature of MMBOs to variations in biophysical parameters. Second,understanding the structural stability of trajectories and their possible bifurcations,as a function of the parameters, in hybrid systems is still a complex issue that wouldbe very informative from the application viewpoint. First works in that direction have


been developed in [13]: taking into account the infinite contraction of the trajectoriesin the voltage variable associated with the reset, the authors proposed to computeexpansion or contraction exponents along transverse directions, providing a notionof stability of hybrid orbits which is more explicit than criteria on the shape of theadaptation map. It would be interesting to develop these methods in the cases ofnon-monotonic spiraling trajectories associated with the presence of MMBOs. Al-ternatively, using models with simpler subthreshold dynamics, for instance linear orpiecewise linear [27, 47], may allow for a derivation of an explicit expression of thereset maps, thus for fine characterization of the stability of the orbits.

In these studies, we have made a crucial use of the planar nature of the system.MMBOs will of course exist in higher dimensional hybrid dynamical systems, and willrequire fine characterization of the invariant manifolds. The extension of the theory tohigher dimensional systems would be particularly interesting from the computationalneuroscience viewpoint for understanding the behavior of neuron networks in whichseveral neurons driven by this dynamics are coupled and communicate at the timesof the spikes.

One difficulty of the study is related to the fact that the map under scrutiny, theadaptation map, is not known analytically. Our mathematical analysis have coveredin detail the cases of overlapping and non-overlapping maps with one discontinuity.These situations do not cover all possible shapes of adaptation maps which can in-duce lifts with more discontinuity points, thus presumably with positive and negativejumps, or lifts with only positive jumps but non-monotonous. A direct mathematicalperspective of this work is thus to develop rotation theory for this kind discontinuousdegree-one circle maps. While in these cases it is still possible to obtain an upperand lower over-bounds for the rotation set by computing the rotation numbers of thenon-decreasing maps Ψl and Ψr, defined in the same way as in the overlapping case, itremains an open question to whether if any value within this interval corresponds tothe rotation number of a given orbit, and it is not hard to find elementary examplesin which this is false6. Thus the general, complete and precise characterization ofthe dynamics of the system is a complex and rich mathematical problem that raisesseveral deep questions of iterates of interval maps with discontinuities. The otherquestion, still open even in the more classical overlapping case, is to characterize thestability of the orbits in the case where the system has multiple possible rotation num-bers of points. A possible approach would be to use and develop symbolic dynamicsand kneading theory for such discontinuous interval maps. However, we emphasizethat in our purpose to characterize the orbits and the pattern of complex oscillationfired, the rotation theory is the most efficient since we have a univocal bidirectionallink between the rotation number and the signature of the MMBO (Theorem 4.3),which allowed us to characterize situations in which the neuron shows regular spiking,MMO, bursting, MMBO or chaotic behavior.

The aim of this work was first of all to demonstrate the new geometric andintuitive mechanism for generating MM(B)Os in one of the most popular hybridneurone models. However, all the above deep mathematical questions open excitingperspectives for future work.

Acknowledgements: The authors warmly thank Jonathan Rubin for interestingdiscussions and suggestions. Justyna Signerska-Rynkowska was partly supported byPolish National Science Centre grant 2014/15/B/ST1/01710.

6We thank Micha l Misiurewicz for interesting discussion on this topic.


Appendix A. Proofs of Theorems 4.5 and 4.8.

Proof of Theorem 4.5 A general theorem for continuous orientation-preserving circlemaps is shown in [29], and is extended to the case of non-continuous orientation-preserving maps in [6] and in [45]. This theory is valid under non-degeneracy con-ditions on the dependence of the maps on the parameters. In particular, a generalresult on the monotone family of increasing lifts Ψs, s ∈ [λ1, λ2], can be shown underthe assumption that the map s 7→ Ψs is increasing and continuous with respect to theHausdorff topology of H-convergence, which is equivalent to the uniform convergenceat the continuity points (see [45]), i.e. under the condition

∀s0∈[λ1,λ2]

w 6=αs0+k(αs0−βs0 )

ε>0

∃δ>0ξ>0

∀s∈[λ1,λ2]

w∈R

|s−s0| < ξ∧|w−w| < δ =⇒ |Ψs(w)−Ψs0(w)| < ε

(A.1)where αs0 + k(αs0 − βs0), k ∈ Z, denotes the discontinuity point of the lift Ψs0 .

As the reset parameter d is increased, the map Φ is rigidly increased by thesame amount. This particularly simple dependence of the map on d allows to controlprecisely how the dynamical features of the map vary with d. In particular, we notethat the boundaries of the invariant interval αd and βd are also simply translated asd varies, and in particular the amplitude θ := αd − βd of the invariant interval isconstant. Moreover, we also observe that for any d ∈ [d1, d2], the maps Φd have thesame discontinuity point wd1 = w1, the lifts Ψd are continuous at points w1+k(αd−βd),have positive jumps at αd + k(αd−βd) and satisfy Ψd(w+ θ) = Ψd(w) + θ. So in factall these lifts Ψd can be seen as lifts of non-continuous invertible circle maps underthe same projection p : t 7→ exp( 2πıt

θ ).However, even if the map Φd is increasing with d, this is not necessarily the case

of Ψd, because of the simultaneous fluctuation of the invariant interval. Indeed, wheneach lift Ψd is obtained from Φd

∣∣[βd,αd]

the relation Ψd1(w) < Ψd2(w) for d1 < d2

might be violated in the intervals [βd1 , βd2 ], when we glue the right part of the graphof Φd2 to its left part (in [βd2 , w1)). But noticing that under the additional conditionΦd(αd2) < Φd(βd1) for any d ∈ [d1, d2], the interval [βd1 , αd2 ] constitutes a particularinvariant interval in which the adaptation map Φd is piecewise increasing and non-overlapping, we can build well-behaved lifts Ψd : R → R based on the shape of themap Φd on this bigger invariant interval [βd1 , αd2 ]. To the difference of Ψd, thesenew lifts are discontinuous at the points w1 + k(αd2 − βd1) (where they have positivejumps of amplitude d2 − d1), in addition to their discontinuity at αd2 + k(αd2 − βd1),k ∈ Z. The latter jump remains also positive under our assumption that Φd(βd1) isstrictly greater than Φd(αd2). While this construction did not substantially modifiedthe dynamics, it has the advantage of ensuring that the mapping (w, d) 7→ Ψd(w) isincreasing in both variables. Moreover, we note that enlarging the invariant intervalhas no effect on the orbits, since any orbit of Φd with an initial condition in [βd1 , αd2 ]enters after a few iterations in the interval [βd, αd]. Since the orbits Ψn

d (w) project

mod (αd2 − βd1) to the orbits Φnd (w), we therefore have %(Ψd) = %(Ψd).

Concluding the proof therefore only amounts to showing that the map d 7→ Ψd iscontinuous in the Hausdorff topology, which is very simple once noted, as mentionedabove, that this property is equivalent to the uniform convergence at all points in theinterior of [βd1 , αd2 ] \ w1 and that Ψd − Ψd′ = d − d′ on this interval. Thus themapping ρ : d 7→ %(Ψd) has the properties listed in the theorem and consequently, thesame holds for ρ : d 7→ %(Ψd). We have noticed that while continuity of the lifts under


the Hausdorff topology was always satisfied in our case, an additional assumption isnecessary to ensure that the mapping (s, w) 7→ Ψs(w) (where s denotes a parameter,here d or γ) is increasing in both variables, which otherwise is not always true. Weemphasize that even in situations in which this mapping is not increasing in bothvariables, the rotation number remains continuous under the H-convergence providedthat the limit function Ψs0 is strictly increasing, see [45, Proposition 5.7]. 2

Proof of Theorem 4.8 The first part of the proof amounts to showing that the inducedmaps d 7→ Ψd

l and d 7→ Ψdr are continuous (as mappings from the subset [λ1, λ2] of

(R, | · |) into the functional space C0(R)).This regularity readily stems from the fact that Φd and Φd0 are simply shifted by

the amount d−d0. But as in the proof of Theorem 4.5, one needs to be careful to thevariation of the invariant intervals at points βd and αd. These also have an additiverelationship in d (i.e. βd − βd0 = d − d0 and similarly for αd). Thus close from thediscontinuity, we do not have an additive relationship in Ψd in general, but for themaps Ψd

l and Ψdr , we can prove even the uniform continuity in d ∈ [λ1, λ2]:

∀ε > 0,∃ξ > 0,∀(d1, d2) ∈ [λ1, λ2]2, |d1 − d2| < ξ =⇒ dC0(R)(Ψd1l ,Ψ

d2l ) < ε. (A.2)

We now fix ε, ξ > 0 and (d1, d2) ∈ [λ1, λ2]2 with d1 − d2 < ξ, and analyse themaps Ψd1

l and Ψd2l in the interval [βd2 , αd2 ] without loss of generality, since it suffices

to consider the maps on an arbitrary interval of length θ := αd − βd.For w ∈ [βd1 , αd2 ], basing on the shape of Φd, we immediately obtain

Ψd1l (w)−Ψd2

l (w) = d1 − d2 < ξ.

For w ∈ [βd2 , βd1 ], we find

Ψd1l (w) = minΦd1(w + θ),Φd1(βd1),

Ψd2l (w) = Φd2(w) ≤ Φd2(βd1) = Φd1(βd1)− (d1 − d2) < Φd1(βd1).

We now distinguish between two cases depending on whether Ψd1l (w) ≥ Ψd2

l (w) ornot. When this inequality is true, we find

|Ψd1l (w)−Ψd2

l (w)| = Ψd1l (w)−Ψd2

l (w)

≤ Ψd1(βd1)−Ψd2(βd2)

≤ Ψd2(βd1)−Ψd2(βd2) + d1 − d2 ≤ (1 + C)ξ,

where C := max(Φd)′(w) : w ∈ [βλ1 , βλ2 ] is actually a constant independent of d.If, on the contrary, Ψd1

l (w) < Ψd2l (w), then we have

Ψd2l (w) ≤ Φd2(βd1) = Φd1(βd1)− (d1 − d2) < Φd1(αd1)− (d1 − d2)

using the overlapping condition. Similarly, Ψd1l (w) ≥ Ψd1

l (βd2) = Φd1(αd2). Equipped

with these estimates, we can compute that |Ψd1l (w) − Ψd2

l (w)| ≤ (1 + C)ξ, where

C := maxΦ′d(w) : w ∈ [αλ1, αλ2

] is independent of d, which proves (A.2) for Ψdl .

Similar methods will work for proving the property for upper-enveloping maps Ψdr

concluding the proof of the continuity.Note that, in contrast to the proof of Theorem 4.5, we did not consider here the

maps Φd1 and Φd2 on a common bigger invariant interval, e.g. [βd2 , αd1 ] for d1 > d2,


because such lifts would have positive jumps at w1 and, consequently, would no morecorrespond to heavy maps.

For proving the second statement, we consider again (d1, d2) ∈ [λ1, λ2]2 such thatd1 > d2. For d ∈ [d1, d2], we build the maps Ψd, Ψd

r and Ψdl on the interval [βd2 , αd2 ].

Note that Ψd1(w)−Ψd2(w) = d1−d2 > 0 for w ∈ [βd1 , αd2 ] ⊂ [βd2 , αd2 ]. The relationΨd1(w)−Ψd2(w) > 0 can only be violated in [βd2 , βd1 ]. However, Ψd2(w) ≤ Ψd2(βd1)for w ∈ [βd2 , βd1 ] since Ψd2 is monotonically increasing on this interval. On the otherhand, depending on whether w∗(d1) ∈ [βd2 + θ, αd1 ] or not, Ψd1 in [βd2 , βd1 ] is eithermonotonous (non-decreasing or non-increasing) or has exactly one local extremumbeing w∗(d1). This yields

Ψd1(w) ≥ minΨd1(βd2),Ψd1(β−d1)

for every w ∈ [βd2 , βd1 ]. Additionally, since Ψd1 fulfills the overlapping condition,

Ψd1(β−d1) > Ψd1(β+d1

) = Ψd2(βd1) + d1 − d2 > Ψd2(βd1)

and Ψd1(β−d1) > Ψd2(w) for every w ∈ [βd2 , βd1 ]. Using overlapping argument for Ψd2 ,we obtain

Ψd1(βd2) = Ψd2(β−d2) + d1 − d2 > Ψd2(β+d2

) + d1 − d2 ≥ Ψd2(βd1)

due to (4.18). Thus Ψd1(βd2) > Ψd2(w) for every w ∈ [βd2 , βd1 ]. It follows thatΨd1 > Ψd2 also in [βd2 , βd1 ] and the mapping d 7→ Ψd is increasing. Now, by definitionof enveloping maps Ψd

l and Ψdr , the fact that Ψd2 < Ψd1 on R for d2 < d1 implies that

Ψd2r < Ψd1

r and Ψd2l < Ψd1

l on R. Thus the maps d 7→ Ψdr and d 7→ Ψd

l are increasingas well. Since we already know that these maps are continuous, the statement aboutthe devil staircase follows. 2Remark. To ensure that the mapping t 7→ %(Ft) behaves as a devil staircase for the

continuous increasing family Ftt∈[T1,T2] of continuous non-decreasing degree-one maps Ft,

we also need to make sure that there exists a dense set S ⊂ Q such that, for s ∈ S, no map

Ft is conjugated to the rotation Rs by s and that the map t 7→ %(Ft) is not constant (see

Proposition 11.1.11 in [29]). However, in practice, these two specific cases do not occur for

the adaptation map.

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A GEOMETRIC MECHANISM FOR MIXED-MODE BURSTING … · class of hybrid dynamical systems modeling neuronal activity. These dynamical systems, called non-linear adaptive integrate-and-