Fig 1.
Ducks at the transition from rest to spikes in the FitzHugh–Nagumo model.
The dynamics are represented in the phase plane (upper panels) and as illustrative time courses (lower panels). When increasing the applied current I, the system’s dynamics transitions from a stable equilibrium (rest state, black dot in (a)), which is excitable, to a strongly attracting limit cycle (repetitive spiking state, in blue in (b)). The excitable structure in panel (a) is further illustrated with 2 trajectories whose initial conditions are at the stable equilibrium (rest state) and with a step current of slightly different amplitude: One trajectory (in red) stays below threshold, while the other one (in purple) crosses threshold, fires an action potential, and then returns back to the rest state. The transition from (a) to (b) is continuous but confined to a very small range of I values around IT. At I ∼ IT, special solutions called ducks emerge (c), which stay close to the unstable part of the V-nullcline, shown as a cubic curve whose attracting branches (resp. repulsive branch) are displayed as solid (resp. dashed) black lines. Two ducks shown in red stay below threshold and correspond to subthreshold oscillations, while one (in purple) crosses threshold and corresponds to a near-threshold spiking solution. Also shown in the top row is the w linear nullcline and the equilibrium point (filled circled when stable, open circle when unstable) at the intersection between the 2 nullclines. Single (double) arrows represent fast (slow) flow. Notice that the purple trajectory in panel (c) has the shape of a leftward-directed duck’s profile; see top-right inset. The bottom row shows the time courses of the membrane potential V for all trajectories displayed on the top row, keeping the same colors; for panel (c), only the time series of the largest red cycle and of the purple one are shown. Equations and parameter values are given in S1 Text.
Fig 2.
Example of electrophysiological recordings of bursting oscillations in 4 types of neurons: (a) parabolic-type bursting from the CeN neuron from the melibe (a sea slug) [26]; (b) square-wave-type bursting from a human β-cell [27]; (c) elliptic-type bursting from a dorsal-root-ganglia (DRG) neuron of a rat [28]; (d) pseudo-plateau-type bursting from a pituitary cell of a rat [29] (“Copyright 2011 Society for Neuroscience”).
In each case, we highlight with colors the 2 main phases of bursting oscillations: silent (quiescent) and active (burst). Figures adapted with permission.
Fig 3.
Electrophysiological recordings of the lateral thalamic nuclei neuron in cat from [31] show complex bursting oscillations.
Colored boxes highlight the active (burst) and silent (quiescent) phases of the bursting oscillations. The quiescent phase comprises 1 oscillation formed by a slow rise toward bursting threshold and a faster descent toward baseline. This complex bursting pattern is well captured by the “Folded-node/Homoclinic” bursting scenario proposed here. Figure adapted with permission.
Fig 4.
Rinzel classification of bursting patterns.
Square-wave bursting, here in the Hindmarsh–Rose model [40] (panels (a1)-(b1)); elliptic bursting, here in the FitzHugh–Rinzel model [6,41] (panels (a2)-(b2)); parabolic bursting, here in Plant’s model [42] (panels (a3)-(b3)). In each case, we show a phase-space projection of the bursting solution of the full system (orange) together with the bifurcation diagram of the fast subsystem (left panel) and the time course of membrane potential V (orange, right panel). Labels for bifurcations are as follows: Ho for homoclinic, HB for Hopf bifurcation, LP for saddle-node (limit point) bifurcation of equilibria, and SNP for saddle-node of periodic orbits. (a3) The critical manifold S0 (green) is the S-shaped (not fully rendered) surface of equilibria of the fast subsystem; this surface is folded along the fold curve . Equations and parameter values are given in S1 Text.
Fig 5.
Small homoclinic/big homoclinic bursting, corresponding to Fig 88 of [7]; shown is a new simulation with the same parameter values (available in [7]).
Panel (a) shows the slow-fast dissection in the (u, V) phase plane; the inset shows a zoom corresponding to the dashed rectangle, which better reveals the shape of the bursting cycle while the main plot better highlights the fast subsystem bifurcation structure. Labels HB, LP, and Ho refer to Hopf bifurcation, saddle-node bifurcation (fold or “limit point”), and homoclinic bifurcation of the fast subsystem, respectively. Panel (b) shows the V-time series of this bursting solution. Izhikevich’s classification allowed to characterize bursting patterns beyond square-wave, elliptic and parabolic, and already opened the door toward explaining more complex biological data. In particular, one can mention pathological brain activity related to, e.g., epileptic seizure [52] or spreading depolarization [16,53]. According to Izhikevich’s classification, bursting oscillations where the burst initiates via a fold bifurcation of the fast subsystem are termed fold-initiated bursting. In the present work, we will propose an extended classification based upon fold-initiated bursting cases.
Fig 6.
Folded-node bursting in a nutshell.
The top row shows the essentials of folded-node bursting: (a) A fold-initiated bursting system (f1, f2, s1) (f1,2 are fast and s1 is slow) with (b) an added slow variable s2 creating a folded node and corresponding to the main parameter of the 3D burster organizing spike-adding transitions gives (c) a 2 slow variables/2 fast variables folded-node burster. The bottom row is an extension of the top panel (b) and shows the essentials of folded-node dynamics (whose typical time course is shown in the top panel (b)): A canard point (ε = 0) in the (f1, f2, s1) bursting system with s2 as parameter (left panel) becomes a folded node (black dot, center panel, ε = 0) when the slow dynamics put on s2 is evolving, for ε = 0 along the attracting and repelling parts of the critical manifold; for small ε > 0, this folded node creates small-amplitude oscillations nearby, organized by attracting and repelling slow manifold
(perturbations of
) and responsible for the quiescent oscillations of the folded-node burster in the resulting 4D system. See S1 Text for a glossary of labels and technical terms.
Fig 7.
(Top part) Main idea of the Rinzel/Izhikevich, Bertram/Golubitsky and colleagues and folded-node bursting classifications. ((ai)-(di), i = 1,2,3) Exemplary “folded-node/homoclinic” bursting, presented in the full 4D system and in its 2D fast and slow subsystems (resp.), showing that both subsystems are required to fully understand this bursting profile; all equations are given in the left column (a1)-(a3). Top row (b1)-(d1), full system bursting solution in 2 different 3D phase-space projections: 2 slow/1 fast in (b1)-(c1) and 1 slow/2 fast in (d1), also showing the critical manifold (fast subsystem’s set of equilibria, in green), the fast subsystem’s limit cycles envelope (blue), as well as relevant bifurcations. In (c1), the trajectory is zoomed near its small oscillations, which follow attracting (red) and repelling (blue) slow manifolds , perturbations of the attracting and repelling parts
of the critical manifold, and pass near the folded node (dot). Middle row (b2)-(d2), fast subsystem information: the bifurcation diagram with respect to 1 slow variable (s1) in (b2), which we can assume persists as such for a small interval of values of the other slow variable (s2); this allows to superimpose the projection of the full system bursting orbit (c2), as done in the Rinzel/Izhikevich classification, and to compute loci of bifurcation points of this diagram in the 2-parameter plane (s1, p), as done in the Bertram/Golubitsky and colleagues classification. However, both approaches classify this bursting pattern as fold/homoclinic (square-wave), hence failing to capture the reason for its small oscillations during quiescence, which can only be unraveled by studying the slow subsystem’s information in the bottom row (b3)-(c3) and find the existence of a folded node in the slow singular limit; details on labels in S1 Text.
Fig 8.
Folded-node/Homoclinic bursting.
Panels (a-b) show the spike-adding transition in system (1): (a) in the (z, x) plane where we show several limit cycles for β-values exponentially close to −1.656996 superimposed onto the fast subsystem bifurcation diagram; (b) bifurcation diagram of the associated 3D bursting system (1) with respect to parameter β, showing the sharp rise of the amplitude of the limit cycle branch (orange), corresponding to spike-adding transitions. Panels (c-d) show a folded-node/homoclinic bursting orbit in the extended 4D system (6): (c) in the (β, z, x) space (single/double arrows indicate slow/fast motion); (d) time course of the fast variable x. The bottom panels show a comparison between this folded-node bursting orbit from (6) and experimental data from [31]. Equations and parameter values are given in S1 Text.
Fig 9.
Panels (a-b) show the spike-adding transition in system (1): (a) in the (z, x) plane where we show several limit cycles for β-values exponentially close to −1.391279 superimposed onto the fast subsystem bifurcation diagram; (b) bifurcation diagram of the associated 3D bursting system (1) with respect to parameter β, showing the sharp rise of the amplitude of the limit cycle branch (orange), corresponding to spike-adding transitions. Panels (c-d) show a folded-node/Hopf bursting orbit in the extended 4D system (6): (c) in the (β, z, x) space (single/double arrows indicate slow/fast motion); (d) time course of the fast variable x. Equations and parameter values are given in S1 Text.
Fig 10.
Folded-node/fold-of-cycles bursting.
Panels (a-b) show the spike-adding transition in system (1): (a) in the (z, x) plane where we show several limit cycles for β-values exponentially close to 0.320207 superimposed onto the fast subsystem bifurcation diagram; (b) bifurcation diagram of the associated 3D bursting system (1) with respect to parameter β, showing the sharp rise of the amplitude of the limit cycle branch (orange), corresponding to spike-adding transitions. Panels (c-d) show a folded-node/fold-of-cycles bursting orbit in the extended 4D system (6): (c) in the (β, z, x) space (single/double arrows indicate slow/fast motion); (d) time course of the fast variable x. Equations and parameter values are given in S1 Text.
Fig 11.
Cyclic folded-node bursting cases.
We use polar coordinates in order to construct idealized models. The top panels show the slow-fast dissection for the amplitude variable r of the underlying bursting model, with 3 different torus canard scenarios (a), (b), and (c). Adding a slow dynamics on a parameter β controlling the slow nullcline then yields associated cyclic folded-node bursting scenarios for which we show both the slow-fast dissection in the (a, r) plane and the x time series: (a) initiated by a subcritical Hopf bifurcation; (b) terminated by a fold of cycles; (c) initiated by a fold of cycles. Equations and parameter values are given in S1 Text.
Fig 12.
A folded-node/cyclic folded-node bursting example.
The x-times series of the folded-node/cyclic folded-node bursting solution is shown in panel (a), where the upper envelope of the burst phase has been traced in black in order to better show the small-amplitude oscillations of this envelope due to the presence of a cyclic folded node; panel (b) is a zoom of panel (a) near the classical folded node highlighting small-amplitude oscillations throughout the silent phase. Equations and parameter values are given in S1 Text.
Fig 13.
A conductance-based episodic bursting example [46].
Left panels: Folded-node bursting orbits shown in the (s1, s2, V)-space projection together with the 2D critical manifold S0, the lower fold curve , the folded-node singularity fn; we also show the location of the Hopf bifurcation point (HB) of the 3D fast subsystem assuming only s2 as a slow variable. Right panels: V time series. The top panels show a bursting orbit for the original parameter values from [30], whereas the bottom ones show a similar bursting solution for a different parameter set where only the kinetics of the 2 slow currents have been modified. In the second parameter set, the HB point moves out of the subthreshold oscillation region and, hence, the one-slow-variable scenario does not fully explain the bursting pattern, which is better cast as folded-node bursting. The parameters of the slow currents that we modify to obtain the new set are as follows:
. All equations and parameter values are given in S1 Text.