Fast–Slow Bursters in the Unfolding of a High Codimension Singularity and the Ultraslow Transitions of Classes
 Maria Luisa Saggio^{1},
 Andreas Spiegler^{1},
 Christophe Bernard^{1} and
 Viktor K. Jirsa^{1}Email author
https://doi.org/10.1186/s1340801700508
© The Author(s) 2017
Received: 21 July 2016
Accepted: 30 June 2017
Published: 25 July 2017
Abstract
Bursting is a phenomenon found in a variety of physical and biological systems. For example, in neuroscience, bursting is believed to play a key role in the way information is transferred in the nervous system. In this work, we propose a model that, appropriately tuned, can display several types of bursting behaviors. The model contains two subsystems acting at different time scales. For the fast subsystem we use the planar unfolding of a high codimension singularity. In its bifurcation diagram, we locate paths that underlie the right sequence of bifurcations necessary for bursting. The slow subsystem steers the fast one back and forth along these paths leading to bursting behavior. The model is able to produce almost all the classes of bursting predicted for systems with a planar fast subsystem. Transitions between classes can be obtained through an ultraslow modulation of the model’s parameters. A detailed exploration of the parameter space allows predicting possible transitions. This provides a single framework to understand the coexistence of diverse bursting patterns in physical and biological systems or in models.
Keywords
Bursting Timescale separation Degenerate and doubly degenerate Takens–Bogdanov singularity Ultraslow modulation Unfolding theory Minimal models1 Introduction
Modeling bursting behavior can help to uncover the mechanisms underlying the bursting dynamics in complex systems. Moreover, modeling gives the opportunity to perform in silico experiments to predict the outcome of manipulations of the system. For example, the Epileptor [15], which is a phenomenological model for the most common bursting behavior in epilepsy, has been used to predict seizure propagation and recruitment in highly personalized virtual epileptic brains [16]. Different treatment strategies can be tested in silico in these virtual epileptic patients, such as interventions on the network topology, stimulations and parameter changes, providing a tool throughout the presurgical evaluation.
Bursting activities, though, can present large differences, such as differences in amplitude and frequency. Different properties at the onset and offset of the burst (i.e. active phase; see the gray boxes in Fig. 1) have been linked to specific qualitative changes in the dynamics, which correspond to bifurcations occurring in a subsystem of the dynamical system [1, 17]. Izhikevich used the onset/offset bifurcations pair criterion to compile a taxonomy of possible bursting classes [17]. In the present study we provide a single autonomous model, comprising a minimal number of variables and parameters, able to produce many classes from this taxonomy. For this purpose, we make use of (i) the ‘dissection’ method applied by Rinzel [18] to the study of fast–slow bursters, namely bursters for which there is a timescale separation between the rhythm of oscillation within the active phase and the rhythm at which silent and active phases alternate; (ii) the unfolding theory approach proposed by Golubitsky et al. [19], based on the idea that the bifurcations involved in bursting activity can be ‘collapsed to a single local bifurcation, generally of higher codimension’.
Section 2 is a brief review of results in the literature upon which our work is built. In this section we will briefly recall both the dissection method [18] and the unfolding theory approach applied to bursting activity [19–21]. We will also describe a codimension 3 singularity, the degenerate Takens–Bogdanov (codim3 deg. TB) singularity [22–24], for which we recapitulate the results of the application of the unfolding approach for bursting activity [3, 19, 20, 25]. In Sect. 3 we will systematically extend the unfolding approach to the deg. TB singularity and show how this allows for a rich repertoire of bursting classes. The model in fact is able to display almost all types of bursting behavior that have been predicted for systems with timescale separation and a planar subsystem acting on the fast time scale [17]. We will explain in detail how to build the different classes of bursters. Furthermore, we will show how to obtain transitions among classes with an ultraslow modulation of the model parameters, as done for a conceptually similar model by Franci et al. [21]. In addition, we will show additional bursting classes obtained when varying a fourth parameter of the model, which correspond to exploring the codimension 4 doubly degenerate TB singularity [20, 26]. Finally, we will apply a measure for complexity based on codimensions [19] to the bursting classes found in the model. This can help to understand the occurrence of bursting phenomena in empirical data and models.
2 Modeling Fast–Slow Bursters
2.1 Dissection Method
At least two rhythms characterize a burster: the fast rhythm of the oscillations within the active phase, and the slower rhythm of the alternation between active and silent phases.
If the time scales of these two rhythms are sufficiently apart, we have a fast–slow burster. Rinzel [27] took advantage of this separation to analyze bursting in the Chay–Keyzer model for pancreatic β cells. He applied a powerful method, called ‘dissection’, that is at the base of our work. The idea behind this method is that we can distinguish two subsystems depending on the time scale at which they act, the slow and the fast ones, and that the variables of the slow subsystem enter the fast subsystem’s equation as parameters.
Both the fast and the slow subsystems can be analyzed in isolation. One can thus build a bifurcation diagram showing how the state space topology of the fast subsystem changes for different values of the slow variables z, here playing the role of bifurcation parameters. If the timescale separation holds, the coupling with the slowly changing z moves the fast subsystem x in this bifurcation diagram, without affecting the topology of the latter.
2.2 Classification of Bursters

Slowwave burster—The slow subsystem is a selfsustained oscillator, thus feedback from the fast to the slow subsystem is not required. In this case, the slow subsystem must be at least twodimensional, \(m\geqslant2\).

Hysteresisloop burster—The slow subsystem oscillates due to feedback from the fast subsystem. This can occur if the fast subsystem shows hysteresis between the silent and active states, which can be used to inform the slow subsystem about the state of the fast subsystem (e.g., by baseline). In this case one slow variable is enough, \(m\geqslant1\).
Bursters come in different flavors. They can differ, among other factors, in the amplitude–frequency pattern of the active phase and in the behavior of the baseline. In the first formal classification of bursters, proposed by Rinzel [1], the author used these features to determine the bifurcations responsible for oscillations onset and offset in the fast subsystem of bursters found in biological systems. This type of classification based on the onset/offset bifurcations pair has been later systematically extended by Izhikevich in his seminal paper [17], with the goal of including not only the known bursters but also all the possible fast–slow ones. His classification includes 120 different pairs of onset/offset bifurcations, of which 16 pertain to a planar fast subsystem with a fixed point like silent state (restingstate could also be modeled with a small amplitude limit cycle). Izhikevich proposed to label each burster by stating the dimensionality of the fast and slow subsystem (\(n+m\)), the onset/offset bifurcations pair and whether the burster is of slowwave or hysteresisloop type.
In this work, we focus on bursters with the smallest dimensionality, namely \(2+1\) for hysteresis loop and \(2+2\) for slow wave. In both cases we have a planar (\(n=2\)) fast subsystem. In general, planar systems can exhibit only four codim1 bifurcations (i.e. obtained by changing a single parameter) that allow the transition from a stable fixed point to a limit cycle, thus from the silent to the active phase. They are: saddlenode (SN), saddlenodeoninvariantcircle (SNIC), supercritical Hopf (supH) and subcritical Hopf (subH) bifurcations. Four bifurcations can be responsible for stopping the stable oscillation: SNIC, saddlehomoclinic (SH), supercritical Hopf and fold limit cycle (FLC). Considering all the pairs, we have sixteen different classes of planar bursters for slowwave and sixteen for hysteresis loop [17].
Abbreviations
Name  Alternative names  Abbreviation 

Codim1 Bifurcations  
SaddleNode  Fold, tangential, limit point  SN 
SaddleNode on Invariant Circle  Saddlenodeloop of codim1  SNIC 
Hopf  Poincaré–Andronov–Hopf  H 
Saddle Homoclinic  Saddle loop, homoclinic connection  SH 
Fold Limit Cycle  Double cycle, fold of cycles, saddlenode of limit cycles, saddle node of periodic orbits  FLC 
subcritical/supercritical  sub/sup  
Codim2 Bifurcations  
Cusp  C  
Bautin  Degenerate Hopf–Takens, generalized Hopf  B 
Takens–Bogdanov  TB  
Degenerate Loop  Neutralsaddlehomoclinic loop  DL 
SaddleNodeLoop  SNL  
Codim3 Bifurcations  
degenerate Takens–Bogdanov  deg. TB  
Regions in the bifurcation diagram  
Limit Cycle small—silent state outside the stable limit cycle  LCs  
Limit Cycle big—stable limit cycle surrounds the silent state  LCb  
Other relevant points in the bifurcation diagram (not bifurcation points)  
H and SN occur on two different fixed points  P_{1,2}  
FLC and SN occur for the same parameters’ values  P_{3} 
Planar bursters
Onset  Offset  

SNIC  SH  supH  FLC  
SN  c1  c2  c3  c4 
Triangular  Squarewave  Tapered  
Type I a, b  Type V  Type IV  
SNIC  c5  c6  c7  c8 
Parabolic  
Type II  
supH  c9  c10  c11  c12 
subH  c13  c14  c15  c16 
Elliptic  
Type III 
2.3 The Unfolding Theory Approach
One of the goals of the present work is to find a minimal descriptive model for bursters with a planar fast subsystem, for simplicity called planar bursters. We adopt a strategy developed by Golubitsky et al. [19], based on earlier work by Bertram et al. [3] (see also de Vries [25]).
Bertram et al. used as fast subsystem a model with a twoparameter bifurcation diagram, the Chay–Cook model for pancreatic β cells bursting [28]. They located, in this twoparameter bifurcation diagram, horizontal cuts crossing the codim1 bifurcation curves required for some of the bursting classes known at that time. Horizontal cuts are straight paths in the parameter plane along which only one parameter is changing. This parameter is then used as slow variable. Using the same model, they could produce different classes by changing the location of the cut in the twoparameter bifurcation diagram.
This strategy has been later formalized by Golubitsky et al. [19]. They realized that the codim1 bifurcations of the fast subsystem which are necessary for bursting can be collapsed to a single local singularity of higher codimension, that is, a singularity in a highdimensional parameter space, where the codim1 bifurcation curves coincide. A path for bursting activity can then be found in the socalled unfolding of the singularity.
The unfolding of a singularity of a dynamical system is a system that exhibits all possible bifurcations of that singularity [29]. This unfolding can be described by adding some terms containing extra parameters to the normal form of the singularity. The number of extra parameters necessary, called unfolding parameters, is the codimension of the singularity. In the unfolding parameter space there are manifolds (e.g. curves, surfaces) of lower codimension bifurcation points. These manifolds intersect at the origin, that is, where all the extra parameters are zero and the system is equal to the normal form of the singularity. In the unfolding, we can search for paths that cross the right sequence of codim1 bifurcations required by the burster, as done by Bertram et al. in the twoparameter bifurcation diagram of the Chay–Cook model.
Let us consider the subH/FLC burster, for instance. To have hysteresis for this class, no additional bifurcations, apart for those at onset and offset, are required. We can thus take the unfolding of the codim2 Bautin (also known as degenerate Hopf) singularity at which fold limit cycle and Hopf bifurcations occur together. In the unfolding, a curve of fold limit cycle bifurcations and a curve of Hopf (divided in a supercritical and a subcritical branch) stem from the Bautin point. We can thus locate a path for subH/FLC bursting. The path does not need to be horizontal, as long as it can be parametrized in terms of the slow variables. In this case, having hysteresis, one slow variable is enough.
The advantage of this approach is that we can use normal forms for the unfolding, if available, providing a minimal description for the fast subsystem.
Golubitsky and coworkers systematically investigated the unfoldings of codim1 and codim2 bifurcations, with respect to bursting paths. They also extended the work to some regions close to a codim3 singularity, but in a noncomplete fashion. With regard to bursters with a planar fast subsystem, they identified nine slowwave and three hysteresisloop bursters. The hysteresis loop can be harder to locate because, to exhibit hysteresis, they may require more bifurcations than their slowwave counterpart. For example, consider the supH/supH burster, two supercritical Hopf bifurcations alone are not enough to create hysteresis, but the slowwave burster can be built by going back and forth through a single supercritical Hopf point. On the other hand, hysteresisloop bursters have a simpler mechanism than slowwave bursters, with regard to the slow dynamics. In slowwave bursting the slow subsystem must be at least twodimensional and the path to follow in the unfolding must be completely specified. Hysteresisloop bursting can be obtained with just one slowly changing variable and it is enough to specify the curve on which the path has to lie and the orientation, while the points at which z inverts its direction over the course of time are determined by the crossing of the onset and offset bifurcation manifolds.
2.4 Codim3 Degenerate Takens–Bogdanov Singularity
The codim3 singularity used by Golubitsky et al. is called degenerate Takens–Bogdanov (deg. TB). Four topologically different unfoldings of this singularity have been identified by Dumortier et al. [22, 23]. These unfoldings are very rich, containing saddlenode, SNIC, saddlehomoclinic, supercritical Hopf, subcritical Hopf and fold limit cycle bifurcations [23]. It has been proposed that the unfolding of this singularity could provide a minimal model to understand neuronal excitability and its modulation [30, 31], or a qualitative model for a cortical mass [32]. In addition, the deg. TB singularity has appeared when investigating some of the models for neural bursting [3, 25] and its biological importance for bursting has been further underlined by Osinga et al. [20]. In one of the unfoldings of the deg. TB, the authors identified paths for many known bursters related to cell activity. They also implemented a slowwave bursting model.
In the present work we systematically extend the work by Golubitsky et al. to the deg. TB singularity and investigate its four unfoldings. We aim at uncovering the presence or absence not only of paths for bursters known from cell activity, but of all planar bursters present in Izhikevich’s classification. This would provide a general model ready to be applied in cell bursting and in any other field for which bursting classification is in progress, such as epileptic seizure modeling [15]. We give indications on how to build slowwave bursters, which is in line with the work of [20]. Furthermore we make use of hysteresis, when present, to build hysteresisloop bursters. This allows less constraints on the required path and make it simpler to implement transitions between different bursting classes (see also Franci et al. [21]).
A description of the planar codim3 deg. TB singularity’s equations and unfoldings has been provided by Dumortier et al. [22, 23]. They identified four topologically different possibilities for this singularity and referred to them as codim3 deg. TB cases: cusp, saddle, focus and elliptic. We investigated the threeparameter unfoldings of all these cases looking for possible paths for bursting activity, considering both the time forward (\(t\rightarrow\infty\)) and the time reversed conditions (\(t\rightarrow\infty\)).
We found that the deg. TB singularity for the focus case in time reversed condition gives the largest amount of bursting paths. Exploring the other cases did not result in a description of new classes. For this reason, Sect. 2.5 is devoted to a detailed description of the focus case unfolding. In Sect. 3 we recall the presentation of the bursting classes identified in this unfolding by Bertram et al. [3] (see also the appendix in [20]) and we identify new classes. We then use this description to build a single model, which is able to display a vast repertoire of bursting activities.
Results for the focus case in time forward condition and for the cusp, saddle, and elliptic cases are briefly summarized in Sect. 3.6. More details are provided in Sects. 5.7 and 5.8.
2.5 Unfolding the Deg. TB Singularity of Focus Case
In the following three subsections we will discuss how changes in the unfolding parameters \(( \mu_{1},\mu_{2},\nu )\) affect the state space spanned by the variables \(( x,y )\). These results, including a complete analysis of the unfolding and its bifurcations, have been previously reported by Dumortier et al. [23] and we include them here for completeness.
2.5.1 Representation on a Sphere
The codim3 bifurcation occurs when the three unfolding parameters are equal to zero. Here saddlenode, Hopf, SNIC, saddlehomoclinic and fold limit cycle bifurcations coincide. From the origin of the parameter space, surfaces for codim1 bifurcations arise. At the intersection between surfaces of codim1 bifurcations we have curves of codim2 bifurcations.
2.5.2 Fixed Points and Local Bifurcations
The system is in a fixed point, or equilibrium, \((x_{0}, y_{0})\) when invariant with respect to time t, that is, \(\dot{x}=0\), \(\dot{y}=0\). The corresponding solution for Eq. (2) is \(y_{0}=0\), \(x_{0}^{3}+\mu_{2} x_{0}\mu_{1}=0\). Hence, the fixed points do not depend on ν. \(x_{0}\) is displayed in Fig. 3A as a function of \((\mu_{1}, \mu_{2})\). We can distinguish two regions in the space \((\mu_{1}\ \mu_{2})\): one in which a single fixed point, a focus, exists; the other in which we have a focus on an upper branch, a focus on a lower branch and a saddle on a middle branch. The saddle coalesces with the focus of the upper branch along the saddlenode bifurcation curve SN_{ r } and with the focus of the lower branch along SN_{ l }. Right and left (and later inferior, superior) refer to where the bifurcation occurs in the state space, as proposed in [23]. Figure 3B shows the saddlenode bifurcation in the complete parameter space of the unfolding and its intersection with the sphere: a closed curve which delimitates the region with three fixed points from the region with a single fixed point. The two saddlenode bifurcation surfaces, SN_{ l } and SN_{ r }, meet along a line of codimen2 cusp points. This line intersects with the sphere in two points, labeled in Fig. 4 as C_{ s } and C_{ i }.
The condition for the Hopf bifurcation can be found by equating the trace of the Jacobian (at the fixed point) to zero. The Hopf bifurcation takes place if \(x^{2}+x+\nu=0\) and \(x^{3}\mu_{2} x\mu_{1}=0\). The Hopf bifurcation in the unfolding is represented by green lines in Fig. 3C and Fig. 4, where a solid/dashed line is used for the supercritical/subcritical cases. On the sphere, we have two codim2 Takens–Bogdanov bifurcation points where the Hopf and saddlenode bifurcations, TB_{ l } on SN_{ l } and TB_{ r } on SN_{ r }, meet. Note that the other two intersections between the Hopf and saddlenode bifurcation curves in Fig. 4, P_{1} and P_{2}, are not Takens–Bogdanov points as the two bifurcations act on two different foci.
The saddlenode and the Hopf bifurcations just described are the local bifurcations in this unfolding, that is, the bifurcations changing the stability of fixed points.
2.5.3 Global Bifurcations
Results for the global bifurcations, affecting broader regions of the state space, are here obtained numerically and discussed analytically in [23]. At most one stable limit cycle exists in the system given by Eq. (2). To describe the unfolding, we can consider the stable limit cycle originating at the supH curve, between TB_{ l } and the Bautin point B, and we can follow its evolution and annihilation. Starting at the TB_{ l }, the limit cycle arises from the destabilization of the stable focus on the lower branch and grows until it meets the saddle in the middle branch. Here the limit cycle vanishes and we have a curve of saddlehomoclinic bifurcations SH_{ l }, which starts at TB_{ l } and terminates on SN_{ r } giving rise to the codim2 saddlenodeloop bifurcation SNLs_{ r }. ‘s’ denotes a small limit cycle, in the sense that it does not surround all the fixed points. From SNLs_{ r } to SNLb_{ r }, the limit cycle disappears through a SNIC bifurcation giving rise to a heteroclinic trajectory between the saddle and the stable focus appeared through SN_{ r }. SNLb_{ r } marks the point where the limit cycle has grown big enough to encircle all the fixed points. From here to the DL (degenerate loop) point, in fact, the limit cycle disappears through a ‘big saddlehomoclinic’ bifurcation SHb (a saddlehomoclinic bifurcation is said ‘big’ if the saddle’s unstable manifold returns to the saddle along the other direction of the saddle’s stable manifold [17, 33], this implies here that the limit cycle encompasses the stable fixed point). After DL, the limit cycle is not able to reach the saddle anymore and coalesces with an unstable limit cycle on the fold limit cycles curve FLC. This unstable limit cycle, which is always enclosed by the stable one, can originate in two ways: from the subcritical branch of the Hopf curve or from the subcritical branch of SHb. The unstable limit cycle can also disappear before reaching the FLC curve, via a SNIC bifurcation, from SNLs_{ l } to SNLb_{ l }.
3 Results
3.1 HysteresisLoop Bursting Classes
We investigated the twoparameter bifurcation topology (i.e. the unfolding on the sphere) to identify paths for bursting activity. Following [19, 20], the system given in Eq. (2) can be considered as the fast subsystem. We then used a onedimensional slow subsystem to slowly steer the fast subsystem in the parameter space so that bursting behavior can occur.
In the present work we are particularly interested in bursters driven by a single slow variable, which oscillates due to feedback from the fast subsystem. For this purpose, the state space of the fast subsystem must display hysteresis between the silent and the active states. The slow variable can be instructed, in the simplest form by linear feedback, to steer the path in a given direction when the system is close to a stable fixed point representing the silent phase, and in the opposite direction when the system has moved to another attractor and is thus far from the silent phase. If this second attractor is a limit cycle, the system is in the active phase.
A prerequisite of hysteresis is the existence of a regime in which at least two stable states coexist, that is, bistability. We find two regions on the sphere where bistability occurs (in yellow in Fig. 4). One region is in the lower portion of the bifurcation diagram, where the limit cycle surrounds the fixed point that acts as silent state, which we named LCb (Limit Cycle big) region. The other region is in the upper part of Fig. 4, here the silent state is outside the limit cycle. We named it LCs (Limit Cycle small) region. We added ‘b’ or ‘s’ to the labels of bursting classes to identify the region where they occur.
3.1.1 LCs Bursters
In the region LCs, oscillations can start through the SN bifurcation (SN_{ r } between SNLs_{ r } and P_{2}) or the supH. The limit cycle can vanish through the supHopf or the SH bifurcations. Consequently, we considered and verified the existence of four pairs of onset/offset bifurcations: c2s (SN/SH), c3s (SN/supH), c10s (supH/SH) and c11s (supH/supH). Among these classes, to the best of our knowledge, c11s was not previously identified in this unfolding (see references [3, 20, 25] and Sect. 4 for the other classes). This region contains in addition to these four cases a special case of burster in which no limit cycle exists and both active and silent phases are given by fixed points (pointpoint burster [17]). In this case both onset and offset are given by the SN bifurcation. When the stable focus, which represents the silent phase, destabilizes, the system spirals towards the other stable focus. This spiraling is the active phase. We attributed the number 0 to the SN/SN bursting class, which is not among the sixteen pointcycle classes.
3.1.2 LCb Bursters
In the LCb region, oscillations can be generated by the SN bifurcation (SN_{ r }) or the subHopf. Oscillations can be stopped through the SH or the FLC bifurcations. Consequently, we considered and verified the existence of classes given by four pairs of onset/offset bifurcations: c2b (SN/SH), c4b (SN/FLC), c14b (subH/SH) and c16b (subH/FLC). Class c14b was not identified before in this unfolding, while the others have been reported in [3, 20, 25].
Examples of paths are shown in Fig. 4, and bifurcation diagrams are shown in the bottom panel of Fig. 6, more details are in Sect. 5.3.
3.2 HysteresisLoop Bursters: A Unique Model
We use these equations to describe the fast subsystem. For bursting activity we need the fast subsystem to slowly move in the unfolding parameter space following a path to undergo the required bifurcations. We can parametrize this path in terms of a third variable z, which slowly changes in time. This variable steers the system through the parameter space and drives it into and out of oscillatory behavior. With reference to Eq. (1), this implies \(\mathbf {x}=(x,y)\) and \(\mathbf{z}=z\).
As described in [23] and summarized in Sect. 2.5, the unfolding parameters can be reduced to two if we restrict the movements to a spherical surface centered at the codim3 singularity. We can perform this reduction without loss of generality because the bifurcation curves on the sphere will be topologically equivalent to those on any other sphere, providing a small enough radius.
The simplest curve satisfying the requirements, considering that our twodimensional parameter space lies on a spherical surface, is the shortest arc on this surface between the initial and final point, known as great circle.
This parametrization (Eqs. (7)–(8)) and Eq. (6) provide a model able to reproduce all hysteresisloop bursting classes found in the unfolding of the deg. TB singularity, focus case.
To summarize the elements appearing in the model, we have a fast subsystem \(( x,y)\), which in some regions of the unfolding space presents bistability and hysteresis between a stable fixed point and a stable limit cycle; and a slow subsystem z, which depends on the feedback from the fast subsystem and moves the latter through the parameter space. When the fast subsystem shows hysteresis, the slow one can drive it in and out from the oscillatory behavior giving rise to bursting activity. The constant c determines the speed of the movement of the subsystem in the parameter space, as promoted by z. It should be small enough to guarantee timescale separation between the fast and slow subsystems. This constant affects the length of both silent and active phases: the slower the movement the bigger the number of oscillations in the active phase (under the condition that the other parameters are fixed). The role of the silent state is played by the upper branch of the equilibrium manifold, of state coordinates \((x,y)=(x_{s}(\theta(z),\phi(z)),0)\).
Once the slow dynamics is defined, what differentiates among classes is the location of the points A and B. We confined the movement to a sphere of radius R in the parameter space \((\mu_{2}, \mu_{1}, \nu)\) centered on the codim3 singularity at the origin. We fixed \(R=0.4\) in this work, unless otherwise specified. Points A and B determine the great circle on which the arc has to lie and the direction of movement (from A to B). The initial point of the path is A. The mechanism that forces z to change direction over the course of time will automatically set the final point of the path. In the following we will choose B so that it lies on the last bifurcation curve encountered by the system before z changes direction.
3.3 Transitions Between Classes
In general, transitions of classes are possible within the same region (LCs or LCb). To have a transition between classes in different regions, the system has to go through a simple oscillatory phase or a simple silent phase (as for example in the central part of Fig. 11A).
It is important to stress that A and B do not need to be, in general, the ending points of the path followed by the system, but determine the great circle it lies on, the direction of movement and the starting point. As z inverts its direction, those bifurcation points have the role of limiting the arc followed by the system. For this reason, even though in Fig. 11B we moved A and B downwards following the arcs connecting \(\mathbf{A}_{1}\) to \(\mathbf{A}_{2}\), and \(\mathbf{B}_{1}\) to \(\mathbf{B}_{2}\), respectively, the actual zigzag path followed by the fast subsystem is determined by the bifurcations that close the hysteresis loop.
This is not true for slowwave bursters, in which the whole trajectory in the unfolding must be specified. For this reason, and for the fact that path shapes are specific to each class, transitions between these kinds of bursters may be more difficult to implement.
3.4 SlowWave Bursters
3.4.1 SlowWave Bursting Classes
3.4.2 SlowWave Bursting Model
Slowwave bursting classes for which we have an hysteresisloop bursting path in the model can be simulated using Eq. (6) if z is substituted with a twodimensional selfoscillating slow subsystem, that is, \(\mathbf{z}\in\mathbb{R}^{m}\), \(m=2\). We did not produce these simulations. For the simulation of the other slowwave classes, we substituted Eqs. (7) and (8) with an appropriate parametrizations of the closed paths shown in Fig. 12 (see Sect. 5.5) and set the slow dynamics to \(\dot{z}=c\).
3.5 Summary for the Codim3 Deg. TB, Focus Case
3.6 Deg. TB Singularity—Elliptic, Saddle and Cusp Cases
The results described up to now pertain to the deg. TB focus case. We similarly investigated the unfoldings of the other three cases, the elliptic, saddle and cusp deg. TB singularities [22–24]. This did not add any new class with regard to those located in the focus case.
3.6.1 Elliptic
The elliptic case is described by Eq. (2) as well, but this time \(b>2\sqrt{2}\). Baer et al. [24] showed that the description of this unfolding is topologically equivalent to that of the focus case. We thus have the same bursting classes as in the latter. The authors pointed out that in the elliptic case the small limit cycle displays a ‘canardlike’ behavior close to the SN curves, with rapid changes in the amplitude. This renders the numerical continuation of the limit cycle harder than in the focus case. Another difference underlined by Baer et al. is that in the elliptic case, the orbit at SHb tends to the boundary of the elliptic sector rather than to the origin when approaching the codim3 singularity.
3.6.2 Saddle
3.6.3 Cusp
The equations for the cusp case are different from the equations for the other cases [22] (see Sect. 5.7). Unlike the other cases the cusp case only allows for two fixed points: one saddle and one focus. Bursting classes are the same as for the saddle case, but the time forward and time reversed behaviors are inverted. More details can be found in Sects. 5.7 and 5.8.
3.7 Bursting Classes in the Partial Unfolding of the Doubly Degenerate TB Singularity
We have discussed in Sect. 2.5 the topology of the bifurcation diagram produced by the intersection of a sphere centered in the codim3 singularity and the unfolding of this singularity. Bifurcation diagrams obtained for different values of the radius of the sphere are topologically equivalent provided the radius is small enough [23]. Krauskopf and Osinga [26] showed that for increasing values of R one can observe a sequence of different topologies for the bifurcation curves. The authors explain how changing R corresponds to changing the parameter b in Eq. (2) and to exploring the partial unfolding of the codim4 doubly degenerate TB singularity. We numerically reconstructed the changes in topology identified in [26], with the goal of investigating how the paths for bursting activity are affected and whether new classes can be identified. It is important to stress that, while the paths for bursting discussed for small values of the radius (Sects. 3.1 and 3.4) can also be found in any arbitrary system exhibiting a deg. TB singularity, this is not necessarily true for the new paths we will describe in this section. These new paths, however, can be found in any system exhibiting a doubly degenerate TB singularity.
Bifurcation diagrams that correspond to layers of the unfolding of the deg. TB singularity have been identified in different models, for example the Bazykin predatorprey model [33, 36] or several neuron models (see Kirst et al. [31] and references therein). Other authors realized that the bifurcation diagrams identified in the models under investigation were deformations of this unfolding [37]. Studying the partial unfolding of the codim4 doubly degenrate TB singularity can help to understand some of these deformations. For example, the bifurcation diagram for the Morris–Lecar model produced by Govaerts et al. [37], which appears also in a model for a compression system [38], can be located in the stage (E) in Fig. 15. The presence of the other stages in the right ordering would be indicative for a codim4 doubly degenerate TB singularity in the model.
Do classes, which were found for small values of the radius, survive far from the codim3 singularity? We examined each of the bifurcation diagrams in Fig. 15 looking for paths for bursting activity. We observed that some of the classes persist through all the values of R analyzed: they are all the classes in LCb (c2b, c16b, c4b, c14b), plus c0 and c2s in LCs. The other classes found for a small radius (in A) disappear after C.
We were able to find paths that could be parametrized as arcs of great circles for c4s, c14s and c8b (the only one with SNIC bifurcation). For class c15s this was not possible. To simulate this class and produce its bifurcation diagram we chose as a path the circumference crossing both the supercritical and the subcritical branches of H, together with the SN_{ r } curve. This is not a great circle as it is not centered on the origin. To simulate this class, we modified Eq. (7) as described in Sect. 5.9. With regard to c8b, the only class with SNIC bifurcation here identified for which the path can be parametrized as an arc of great circle, it has an anomalous bifurcation diagram as compared to the other classes investigated up to now: in this case the lower branch of fixed points, and not the upper branch, plays the role of silent state and we had to modify Eqs. (7)–(8) and Eq. (6) accordingly (details can be found in Sect. 5.9). Bifurcation diagrams for c4s, c14s, c8b and c15s are obtained numerically together with the superimposed bursting trajectories.
For the other new classes (c14s, c12s, c1b, c5b, c6b and c13b) instead, paths are more complex than arcs of circumferences. We did not parametrize these paths, and the bifurcation diagrams in Figs. 16 and 17 are sketches. In all the classes with SNIC the role of silent state is played by the lower branch of the equilibrium manifold, as in c8b.
4 Discussion
4.1 A Unifying Framework for Fast–Slow Bursters
Here we have shown how to build fast–slow planar bursters by appplying the unfolding theory approach to the unfolding of the codim3 degenerate Takens–Bogdanov singularity [19]. We systematically explored the four possible unfoldings of this planar singularity: namely the focus, elliptic, saddle and cusp case [22–24]. For each of them we checked the time forward and time reversed behavior and located paths for slowwave and hysteresisloop bursters. We discovered that the focus case with time reversed covers the largest number of bursting classes. In fact, many bursting classes had already been identified in this unfolding by other authors [3, 19, 20, 25]. The other three cases did not bear new classes. In the unfolding of the focus case we found all the sixteen classes for slowwave bursting and seven of the hysteresisloop classes predicted as possible for planar bursting by Izhikevich [17]. We could identify seven additional hysteresisloop classes when exploring the equations far from the codim3 singularity, that is, the partial unfolding of the codim4 doubly degenerate TB singularity [20, 26]. The labels used for the bursting classes can be found in Table 2.
We built a model that can produce bursting activity for all the hysteresisloop classes found for which the path can be parametrized as an arc of great circle. These classes are all those found for small R (c2s, c3s, c10s, c11s, c2b, c4b, c14b and c16b) plus c4s and c8b. The model can be extended to c15s with some modifications. The equations of the fast subsystem of the model are those of the unfolding of the focus case of the deg. TB singularity. The unfolding parameters are parametrized in terms of one slow variable to follow the required path. The slow variable shows oscillations thanks to feedback from the fast subsystem. It can be observed from the phase flows in Fig. 4 that the fast subsystem is globally stable in each parameter configurations. This implies that, when the fast subsystem is in the bursting region, for all possible initial conditions the fast dynamics will converge to either the stable fixed point or stable limit cycle. The slow variable, on the other hand, is designed to ensure the alternation between these two states. This implies that all initial conditions will lead to the same bursting dynamics if the system is in the bursting region. The only parameters that determine the bursting class, if any, are the starting and ending points of the path. All the other parameters of the model can affect the shape of the orbit followed by the system without changing the class. For example, we showed that they can increase/decrease the number of oscillations in the active phase, alter their amplitude and frequency (within the constraints imposed by the bifurcations involved), or modify the silent/active phases lengths ratio.
This provides a unifying framework to investigate the underlying mechanisms of systems able to display different bursting behaviors. Furthermore, transitions between classes can easily be implemented through an ultraslow modulation of initial and final points of the path [21].
We could make some predictions about which transitions are easier to obtain and which instead require one to cross regions of the unfolding where the system is not bursting. We also clarified why transitions are easier to obtain for hysteresisloop bursters than for slowwave ones.
Submanifolds of the threeparameter unfolding of the deg. TB singularity can be identified in several neuron models and this unfolding has been proposed as a key element to understand neural excitability [30–32, 37, 39]. Our results allow one to extend the understanding of the dynamical repertoire hosted in this unfolding, by giving a complete description of the planar bursting activity that can be obtained thanks to a coupling of the planar unfolding with a slower system. This allows, once the codim3 deg. TB bifurcations of a model have been identified, to predict all possible planar bursting behaviors if the model is used as fast subsystem; whereas, if the system exhibits the codim4 doubly degenerate TB singularity, we can predict only some of the possible planar bursting classes, based on the partial unfolding of this codim4 singularity [20, 26].
4.2 Finding New Paths for Bursting
Bertram et al. [3] searched for bursting paths in the twoparameter bifurcation diagram of the Chay–Cook model. This bifurcation diagram, as pointed out by the authors, is a layer of the unfolding of the deg. TB singularity, focus case. It can be obtained by keeping \(\mu_{2}\) fixed at a negative value. Such a layer describes a bifurcation diagram that excludes some of the points that we find on the sphere: the two cusp points C_{ s }, C_{ i } (they require \(\mu _{2}=0\)), and the Bautin point (requires a positive \(\mu_{2}\)). In this layer, Bertram et al. identified c2s, c2b, c4b and c16b with hysteresis, and c5 without hysteresis. De Vries [25] added the path for c3s with hysteresis. The complete unfolding on the sphere has been investigated by Osinga et al. [20]. The authors located paths for the bursters known to Bertram et al. and de Vries and they added the path for c10s. By exploring the unfolding far from the codim3 singularity they also added a new class. This extra class is the pointpoint SN/subH burster. Izhikevich [17] and Golubitsky et al. [19] identified additional slowwave classes close to two codim2 bifurcations that exists also in the unfolding of the deg. TB singularity: the SNL point (c1, c6) and the Bautin point (c11, c12, c15). In this article we have provided a systematic framework of bursting activity, located paths for bursting in parameter space and identified novel paths for c4s, c8b, c11s, c14b, c14s, c12s, c15s, c1b, c5b, c6b and c13b for hysteresisloop and all the missing classes for slowwave (excluding those for which there is already an hysteresisloop path, they are c7, c8, c9 and c13).
4.3 Complexity of Bursting Classes
Golubitsky et al. [19] introduced a notion of complexity in the characterization of bursters. They defined the complexity of a bursting class as the codimension of the lowest codimension singularity in which unfolding that class can be found. Onset and offset bifurcations, in fact, are not enough to describe the sequence of bifurcations required by a bursting class. Other bifurcations may be required to obtain a class and even a greater number of them may be needed for the hysteresisloop types. If more bifurcations are required for a class, then more parameters must be tuned to obtain a given sequence. This increases the complexity of the class. The number of parameters to be tuned is reflected in the codimension of the singularity in which unfolding the class first appears. Thus, Golubitsky et al. argued that the more complex a class is, the less likely it is to encounter that class, in empirical data or in models. They proposed to complement Izhikevich’s classification by providing information on this measure of complexity.
With regard to hysteresisloop bursters, the class of smallest complexity is subH/FLC (c16), which exists close to the codim2 Bautin bifurcation as outlined by Golubitsky et al. Class SN/SH (c2) lives close to the codim2 SaddleNodeLoop bifurcation [17], but this bifurcation is not local and the lowest codimension singularity for this class is the codim3 deg. TB one [19]. Figure 19 reports the complexity of the classes identified in the present work. We classified the classes appearing far from the codim3 singularity, thus in the partial unfolding of the doubly degenerate TB singularity, as codim4 classes. With regard to these classes, it is an open question whether they could be located in the proximity of a codim3 singularity other than the deg. TB. As pointed out by Osinga et al. [20], not all the unfoldings of codim3 singularities are available. Nonetheless, the authors noted that the unfolding of the deg. TB is the only one to present both a codim3 cusp point and a codim2 TB bifurcation (at which Hopf and saddlehomoclinic bifurcations coincide). From this we can conclude that classes that require all these conditions, such as c14s, cannot be found in the unfolding of other codim3 singularities. Thus, their complexity is four. Further work is required to determine the codimension of classes c8b, c15s, c13b, c12s, c15s, c5b, c1b and c6b.
An additional indicator of the complexity of a class is given by the number of bifurcations that its path has to cross to produce the desired bursting behavior and to close the hysteresis loop. Among the classes of codim3, for example, paths for c2 need only to cross saddlenode and saddlehomoclinic bifurcation curves, while, on the other extreme, c11s requires one to go through four bifurcation curves. This implies that a more refined tuning of the parameters of the path is required for c11s to obey all constraints. For this reason, we added in Fig. 19 a column stating the minimum number of bifurcations encountered by the paths we identified. When the number of bifurcations to cross is high, it may become harder to cross them using a path as simple as an arc of great circle. In this work, this was not possible for all the classes that needed six or five bifurcations, and for some of those needing four bifurcations.
In the present work, we focused on pointcycle bursters, that is, bursters with a fixed point like silent phase and with a limit cycle for the active phase. We also provided two examples of a pointpoint burster, in which both phases are given by fixed points. Other possibilities exist that we did not address, as for example cyclecycle bursters, in which the silent phase is characterized by small amplitude oscillations, requiring the coexistence of two stable limit cycles.
4.4 Modeling Approaches
In the present work we have considered bursting that can be obtained in systems with two fast variables and one or two slow variables. However, more than two time scales can interact to produce more complex bursting patterns not possible with only two time scales [40]. The oscillatory activity of the fast subsystem itself can arise from the interplay of multiple time scales [39]. Franci et al. [21] applied the unfolding theory approach to a codim3 singularity to build a threevariable model for bursting activity with three time scales, motivated by the spikelike oscillations of neuronal bursters. They used the unfolding of the codim3 winged cusp singularity, which is described by a single variable x. The unfolding presents regions with only one fixed point and regions with coexistence of three fixed points. No limit cycle can live in one dimension. To generate oscillations Franci and coworkers expressed the unfolding parameters in terms of a second variable y acting on a slower time scale. The second variable receives feedback from the fast one and exploits the hysteresis present in the fast variable to create a limit cycle in the plane \(( x,y )\) (relaxation oscillations). This second variable plays the same role as our z in class c0 in Fig. 4: no limit cycle is present in the fast subsystem but, thanks to hysteresis, one can be created in the \(( x,y )\) plane (note that the spiraling towards the fixed point in c0 is due to the presence of the second fast variable and is thus absent in the limit cycle generated by Franci et al. in the \(( x,y )\) plane). The oscillations created by this limit cycle have a spikelike shape. The authors introduced a third variable, z, acting on a slower time scale than y, that allows for the alternation between active and silent phases with a similar mechanism of that used here. By changing the parameters they provided a model for c2, c3, c4 and c16 hysteresisloop bursters. They also showed an example of how an ultraslow modulation can lead to transitions between classes.
Our approach, based upon the planar unfolding of the codim3 deg. TB singularity, allowed us to identify a richer repertoire of bursting activity using the same number of variables. This is due to the fact that the planar unfolding of the codim3 deg. TB is richer, in terms of limit cycles and bifurcations, than the onedimensional unfolding of the codim3 winged cusp coupled with a slow variable. While the extra classes found with this method may not play a role in neuronal bursting [21], they could be relevant for other types of bursting system, for which oscillations are not necessarily spikelike, for example epileptic seizures. Another possibility to implement timescale separation is to introduce it within the planar system of the deg. TB singularity. The slowfast TB singularity has already been studied in [41] and appears in what has been proposed as a minimal model for neural excitability (displaying also canard explosions) [39]. The bifurcation diagram of this model, in fact, resembles a layer of the unfolding of the deg. TB singularity (focus or elliptic) which passes through the origin. It would be interesting to explore what is added, in terms of bursting dynamics, by the introduction of this third time scale within the deg. TB singularity.
The unfolding theory approach proves to be a valuable tool to build bursters when timescale separation holds. When this does not happen, then different phenomena can occur, including chaos [17].
In our study bursting behavior was engineered by describing slow changing parameters. However, this is not the only way of eliciting bursts. Both slow and fast interventions can provoke bursts. For parameter changes the changes are usually slow (as discussed throughout the paper). However, the state variables are sensitive to perturbations, for example, because of multistability. In this study, the limit cycle that encircles all fixed points is less sensitive to perturbations than the limit cycle that does not encircle all fixed points (see Fig. 5). The effect of perturbations is mostly reversible in the latter case, where in the former case the fast event needs to be coordinated (e.g. temporally) to reverse an effect of perturbations (see Spiegler et al., [42], for example Figs. 10–12). The existence of a ‘small’ limit cycle next to others or fixed points can be directly used to describe bursting behavior by fast events, for instance, in repetitive spiking sequences. Other dynamics associated with bursting behavior are quasiperiodicity, deterministic chaos and intermittency [43].
5 Methods
5.1 Reproducing the Unfolding
To reproduce the unfoldings of the deg. TB singularity focus case (Fig. 3C and Fig. 4), we used the Matlabbased software Matcont and CL_Matcont [44]. We described the parameter space with spherical coordinates, fixed the radius \(R=0.4\) and explored the \((\theta,\phi)\) parameter space. We computed the equilibrium manifold for a fixed value of θ, identified the SN points and performed their twoparameter continuations. From the Takens–Bogdanov points located on the SN curves, we started the continuation of the H and SH curves. From the Bautin point on H we computed the FLC curve and from the last point of the FLC curve we started the continuation of the SHb curve.
5.2 Investigation of the State Space
To produce the phase flows in Fig. 5, we choose a point \((\theta,\phi)\) in parameter space for each region of the unfolding labeled with a Roman numeral. We computed x and ynullclines analytically, while the orbits where simulated with Matcont.
To investigate the amplitude and frequency of the stable limit cycle, we discretized the parameter space with 720 points for ϕ and 360 for θ. For each point we integrated the fast subsystem (using Matlab function ‘ode45’) using as initial conditions: a big value for \((x,y)\) (\((x,y)=(10,10)\)) so that the system stabilizes on the limit cycle in regions V to X; a point close to the fixed point on the lower branch (we added \(\epsilon=0.01\) to the y coordinate of the fixed point) to reach the limit cycle in region III. We simulated 3000 s, removed the first 500 s to avoid the transient behavior and used the last 2500 s to compute the amplitude and frequency. For the amplitude we took the difference between the maximum and minimum of the timeseries; for the frequency we used Hann window and then applied Discrete Fourier Transform.
To obtain the flat representation we applied the Lambert Equal Area Azimuthal Projection, the Matlab identifier for this projection is ‘equaazim’.
5.3 HysteresisLoop Bursting Classes
Bursting classes in the unfolding of the deg. TB singularity, focus case
Class  Label  Bifurcations sequence  Point A  Point B 

LCs region  
SN/SH  c2s  SH_{ l }, SN_{ r }  SH_{ l }  SN_{ r } ∈ [P_{2}, SNLs_{ r }] 
SN/supH  c3s  SN_{ l }, H, SN_{ r }  SN_{ l } ∈ [TB_{ l }, C_{ s }]  SN_{ r } ∈ [P_{2}, SNLs_{ r }] 
supH/SH  c10s  SH_{ l }, H, SN_{ r }  SH_{ l }  SN_{ r } ∈ [C_{ s }, P_{2}] 
supH/supH  c11s  SN_{ l }, H, H, SN_{ r }  SN_{ l } ∈ [TB_{ l }, C_{ s }]  SN_{ r } ∈ [C_{ s }, P_{2}] 
LCb region  
SN/SH  c2b  SHb, SN_{ r } ^{a}  SHb ∈ [DL, SNLb_{ r }]  SN_{ r } ∈ [SNLb_{ r },TBr] 
SN/FLC  c4b  FLC, SN_{ r } ^{a}  FLC  SN_{ r } ∈ [SNLb_{ r }, TBr] 
subH/SH  c14b  SHb, H^{a}  SHb ∈ [DL, SNLb_{ r }]  H ∈ [B, TB_{ r }] 
subH/FLC  c16b  FLC, H^{a}  FLC  H ∈ [B, TB_{ r }] 
5.4 Transitions
5.5 SlowWave Bursting Classes
5.6 Focus Case Time Forward Behavior
5.6.1 HysteresisLoop Bursting
There are two bistability regions: one in the central part of the bifurcation diagram and one in the lower part (in yellow in Fig. 23, left panel). In the first, the only bifurcation for the onset of the oscillations is the SN bifurcation, for the offset there are SH and FLC bifurcations. We identified paths for all the possible pairs: c2b and c4b. In this region the limit cycle surrounds the silent state (lower branch of the equilibrium manifold). In the lower part of the bifurcation diagram two possible onsets, SN and supH, and two possible offsets, SH and supH, give four possible pairs for which we located paths: c2s, c3s, c10s and c11s. The limit cycle surrounds the upper branch of the equilibrium manifold, while the role of silent state is played by the lower branch.
5.6.2 SlowWave Bursting
A requisite for bursting is that there should be at least one attractor at each point along the path. The area which satisfies this condition is colored in light gray in Fig. 23, right panel. The bifurcation curves within this area are: SN, SNIC and supH for onset; SN, SH, supH and FLC for offset. Bifurcations at the border of the region cannot be considered because there would be no attractor once the curve is crossed. We identified closed paths for all the possible pairs: 12 out of 16 classes are present (see Fig. 13). The four missing are those with subH.
5.7 Saddle Case Time Forward and Cusp Case Time Reversed
5.7.1 HysteresisLoop Bursting
The bifurcation in the bistability region are: subH for onset; SH and FLC for offset. Paths for the two possible pairs, giving c14b and c16b bursting, can be located in the unfolding.
5.7.2 SlowWave Bursting
The bifurcation curves within the stability area are: supH and subH for onset; SH, supH and FLC for offset. Paths for all the possible pairs, for which an hysteresis loop has not been identified, can be found in the unfolding (c10, c11, c12 and c15).
5.8 Saddle Case Time Reversed and Cusp Case Time Forward
The saddle case with time reversed has the same bursting classes of the cusp case in the time forward condition. The unfolding diagrams are shown in Fig. 24C, D.
5.8.1 HysteresisLoop Bursting
There is no bistability region, so hysteresisloop bursting is not possible.
5.8.2 SlowWave Bursting
The only bifurcation within the stability region that can be used for seizure onset and/or offset is the supH. We located a path for supH/supH (c11) bursting.
5.9 HysteresisLoop Bursting Classes in the Partial Unfolding of the Doubly Degenerate TB Singularity
Bursting classes far from the deg. TB singularity, focus case
Class  Label  Bifurcations sequence  Point A  Point B 

LCs region  
SN/FLC  c4s  SN_{ l }, FLC_{ B }, SN_{ r }; SN_{ l }, FLC_{ B }, subH, SN_{ r }  SN_{ l } ∈ [TB_{ l }, C_{ s }]  SN_{ r } ∈ [P_{3}, SNLs_{ r }] 
supH/FLC  c12s  SN_{ l }, FLC_{ B }, subH, supH, SN_{ r }  SN_{ l } ∈ [TB_{ l }, C_{ s }]  SN_{ r } ∈ [C_{ s }, P_{3}] 
subH/SH  c14s  SH_{ l }, subH, FLC_{ B }, SN_{ r }  SH_{ l }  SN_{ r } ∈ [C_{ s }, P_{3}] 
subH/supH  c15s  SN_{ l }, supH, subH, FLC_{ B }, SN_{ r }  SN_{ l } ∈ [TB_{ l }, C_{ s }]  SN_{ r } ∈ [C_{ s }, P_{3}] 
SNIC/FLC  c8b  H, SNIC_{ l }, FLC_{ B }, SN_{ r }, FLC_{ B }, SH_{ l }, SNIC, FLC_{ B }; SN_{ r }, SH_{ l }, SNIC, FLC_{ B }  H ∈ [TB_{ l }, P_{2}]  FLC_{ B } ∈ [P_{3}, B_{2}] 
SN/SNIC  c1b  SN_{ r }, SH_{ l }, SNIC, SN_{ r }, SH_{ l }, H, SN_{ r }, FLC, SH_{ l }, SNIC, SN_{ r }, SH_{ l }, H  SN_{ r } ∈ [C_{ s }, P_{3}]  H ∈ [TB_{ l }, P_{2}] 
SNIC/SNIC  c5b  SN_{ r }, SH_{ l }, SNIC, SNIC, H, SN_{ r }, FLC, SH_{ l }, SNIC, SNIC, H  SN_{ r } ∈ [C_{ s }, P_{3}]  H ∈ [TB_{ l }, P_{2}] 
SNIC/SH  c6b  SN_{ r }, SH_{ l }, SH_{ l }, SN_{ r }, SNIC, H  SN_{ r } ∈ [C_{ s }, P_{3}]  H ∈ [TB_{ l }, P_{2}] 
subH/SNIC  c13b  SN_{ r }, SH_{ l }, SNIC, H, SN_{ r }, FLC, SH_{ l }, SNIC, H,  SN_{ r } ∈ [C_{ s }, P_{3}]  H ∈ [P_{2}, B_{2}] 
The LCb region is not affected by changes in R (except in stage E; see below for details) and the classes leaving in that region persist through all the stages. Classes in the LCs region are, instead, influenced.
5.9.1 Stage B
A new separated region of bistability, XI, is created by the appearance of the FLC_{ B } curve. The only onset bifurcation curve in the region is subH and the only possible offset bifurcation is FLC. We verified the existence of a sketched bifurcation diagram compatible with the existence of subH/FLC (c16b) bursting.
5.9.2 Stage C
The new region of bistability created by FLC_{ B } merges with the LCs region (to distinguish it from LCb, we will continue to identify this region in the upper part of the unfolding with the label ‘LCs’, even though in the region XI the limit cycle surrounds the silent state). We now have in the bistability region three bifurcations for the onset (SN, supH and subH) and three for the offset (SH, supH and FLC). This leads to nine possible pairs (c2, c3, c4, c10, c11, c12, c14, c15, c16). We found paths for all of them and verified that the sketched bifurcation diagrams along these paths were consistent with the existence of bursting trajectories of the desired types. Some of these classes were not present in previous stages: c4s, c12s, c14s and c15s. We could use arcs of great circles as paths for c4s and c14s, while we used a different parametrization (see details below) for c15s. Bifurcation diagrams are shown in Fig. 16. Those for c4s, c14s and c15s are obtained numerically, that for c12s is a sketch.
For the bifurcation diagrams in Fig. 16 we used these values: for c4s \(R=0.6\), \(\mathbf{A}=[0.5225,0.1454,0.2566]\), \(\mathbf{B}=[0.5657,0.1638,0.1149]\), \(c=0.00005\), \(d^{*}=0.3\); for c14s \(R=0.75\), \(\mathbf{A}=[0.7148,0.2262,0.02091]\), \(\mathbf {B}=[0.7117,0.2311,0.05016]\), \(c=0.0000001\), \(d^{*}=0.2\); and for c15s \(R=0.6\), \(\mathbf{A}=[0.4935,0.1334,0.3142]\), \(\mathbf {B}=[0.5775,0.0195,0.168]\), \(\mathbf{D}=[0.5583,0.1606,0.1499]\), \(c=0.00002\), \(d^{*}=0.8\).
To obtain c12s we need to cross SN_{ l }, FLC_{ B }, the subcritical and the supercritical branches of H and SN_{ r } in this order. This requires a path with a knot as shown in Fig. 15.
5.9.3 Stage D
At this stage, part of the H curve passes below SH_{ l } and becomes entirely subcritical. This implies that all the classes present in the previous stage and having the supH bifurcation as onset (c10s, c11s, c12s) or offset (c3s, c11s, c15s) are not present anymore. There are not new bifurcations in the bistability region with regard to the previous stage, so new classes are not possible. We verified the existence of sketched bifurcation diagrams consistent with c2s, c4s, c10s and c16s bursting.
5.9.4 Stage E
The H curve is now completely below SH_{ l } and part of the SNIC_{ r } curve becomes part of the bistability region. We have three possible onset bifurcations (SN, subH and SNIC) and three possible offsets (SH, FLC and SNIC), which gives nine classes to verify. We found paths for all of them (c2s, c4s, c14b, c16b, c5b, c6b, c8b, c13b, c1b). All the classes not containing a SNIC bifurcation were already present in previous stages. For class c14b in LCs we could not represent the path with a straight line in the cartoon bifurcation diagram in Fig. 15 and we did not verify the required shape on the numerical bifurcation diagram. For this class it is the lower branch of the equilibrium manifold that plays the role of silent state. After the offset, at SH_{ l }, if the fast subsystem goes back to the silent state z changes direction and moves towards the onset. In this case the path will start on SH_{ l }, enter the XII region, then region XI to end on H. If the fast subsystem is attracted by the upper branch of the equilibrium manifold, z continues to decreases until SN_{ r } is crossed and the fast subsystem settles in the silent state. In the latter case, the path followed will be longer and cross SN_{ r }, FLC_{ B }, SH_{ l }, SH_{ l }, SN_{ r }, subH as shown in Fig. 16.
Only one of the classes involving a SNIC bifurcation, SNIC/FLC (c8b) bursting, had a path that could be parametrized as an arc of great circle. In this class as well the role of the silent state is played by the lower branch of equilibrium. To simulate the timeseries we used the x coordinate of this lower fixed point for \(x_{s}\) in Eq. (6). The values of the parameters used for the simulation are \(R=1\), \(\mathbf{A}=[0.8207,0.5572,0.1263]\), \(\mathbf {B}=[0.9455,0.3043,0.116]\) and \(c=0.001\). The numerical bifurcation diagram can be found in Fig. 16, the other bifurcation diagrams for SNIC bursting are sketches.
5.9.5 Stage F
The subcritical branch of the H curve now crosses also SHb and LCs merges with LCb. The bifurcations present in the bistability region are the same as in the previous stage, as well as the bursting classes.
Declarations
Acknowledgements
The research reported herein was supported by the Brain Network Recovery Group through the James S. McDonnell Foundation and funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 720270. The authors thank Silvan Siep for helpful discussion.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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