Canard Orbits Research Articles

Canards Underlie Both Electrical and Ca$^{2+}$-Induced Early Afterdepolarizations in a Model for Cardiac Myocytes

Early afterdepolarizations (EADs) are voltage oscillations that can occur during the plateau phase of a cardiac action potential. EADs at the cellular level have been linked to potentially deadly tissue-level arrhythmias, and the mechanisms for their arisal are not fully understood. There is ongoing debate as to which is the predominant biophysical mechanism of EAD production: imbalanced interactions between voltage-gated transmembrane currents or overactive Ca$^{2+}$-dependent transmembrane currents brought about by pathological intracellular Ca$^{2+}$ release dynamics. In this article, we address this issue using a foundational 10-dimensional biophysical ventricular action potential model which contains both electrical and intracellular Ca$^{2+}$ components. Surprisingly, we find that the model can produce EADs through both biophysical mechanisms, which hints at a more fundamental dynamical mechanism for EAD production. Fast-slow analysis reveals EADs, in both cases, to be canard-induced mixed-mode oscillations. While the voltage-driven EADs arise from a fast-slow problem with two slow variables, the Ca$^{2+}$-driven EADs arise from the addition of a third slow variable. Hence, we adapt existing computational methods in order to compute 2D slow manifolds and 1D canard orbits in the reduced 7D model from which voltage-driven EADs arise. Further, we extend these computational methods in order to compute, for the first time, 2D sets of maximal canards which partition the 3D slow manifolds of the 8D problem from which Ca$^{2+}$-driven EADs arise. The canard viewpoint provides a unifying alternative to the voltage- or Ca$^{2+}$-driven viewpoints while also providing explanatory and predictive insights that cannot be obtained through the use of the traditional fast-slow approach.

Canard Cycles and Relaxation Oscillations in a Singularly Perturbed Leslie–Gower Predator–Prey Model with Allee Effect

In this paper, we investigate the dynamics of a modified Leslie–Gower predator–prey model with Allee effect on prey. And the Holling type II functional response is considered in this model. When prey reproduces much faster than predator, by combining the normal form theory of slow–fast systems and the geometric singular perturbation theory, we observe much richer new dynamical phenomena than the existing ones. In the case of strong Allee effect, we prove the existence of canard cycles, homoclinic orbits and heteroclinic orbits. Furthermore, we focus on the case of weak Allee effect. In addition to the similar dynamics of that exhibited by the system with strong Allee effect, we also demonstrate the occurrence of relaxation oscillations created by entry-exit function. Moreover, the existence of canard explosion is further explained analytically and numerically with the help of sophisticated slow–fast techniques.

Classification of bursting patterns: A tale of two ducks.

Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s and is still ongoing. As a by-product, it establishes analytical and numerical foundations for studying complex temporal behaviors in multiple timescale models of cellular activity. In this review, we first present the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable systems. We recall a complementary mathematical classification approach by Bertram and colleagues, and then by Golubitsky and colleagues, which, together with the Rinzel-Izhikevich proposals, provide the state-of-the-art foundations to these classifications. Beyond classical approaches, we review a recent bursting example that falls outside the previous classification systems. Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. Namely, we provide a general framework for bursting systems with both subthreshold and superthreshold oscillations. A new class of bursters with at least 2 slow variables is then added, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. Key to this mechanism are so-called canard or duck orbits, organizing the underpinning excitability structure. We describe the 2 main families of folded-node bursters, depending upon the phase (active/spiking or silent/nonspiking) of the bursting cycle during which folded-node dynamics occurs. We classify both families and give examples of minimal systems displaying these novel bursting patterns. Finally, we provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast subsystem approach.

Amplitude-modulated spiking as a novel route to bursting: Coupling-induced mixed-mode oscillations by symmetry breaking.

Mixed-mode oscillations consisting of alternating small- and large-amplitude oscillations are increasingly well understood and are often caused by folded singularities, canard orbits, or singular Hopf bifurcations. We show that coupling between identical nonlinear oscillators can cause mixed-mode oscillations because of symmetry breaking. This behavior is illustrated for diffusively coupled FitzHugh-Nagumo oscillators with negative coupling constant, and we show that it is caused by a singular Hopf bifurcation related to a folded saddle-node (FSN) singularity. Inspired by earlier work on models of pancreatic beta-cells [Sherman, Bull. Math. Biol. 56, 811 (1994)], we then identify a new type of bursting dynamics due to diffusive coupling of cells firing action potentials when isolated. In the presence of coupling, small-amplitude oscillations in the action potential height precede transitions to square-wave bursting. Confirming the hypothesis from the earlier work that this behavior is related to a pitchfork-of-limit-cycles bifurcation in the fast subsystem, we find that it is caused by symmetry breaking. Moreover, we show that it is organized by a FSN in the averaged system, which causes a singular Hopf bifurcation. Such behavior is related to the recently studied dynamics caused by the so-called torus canards.

Spike-Adding Canard Explosion in a Class of Square-Wave Bursters

This paper examines a spike-adding bifurcation phenomenon whereby small-amplitude canard cycles transition into large-amplitude bursting oscillations along a single continuous branch in parameter space. We consider a class of three-dimensional singularly perturbed ODEs with two fast variables and one slow variable and singular perturbation parameter $$\varepsilon \ll 1 $$ under general assumptions which guarantee such a transition occurs. The primary ingredients include a cubic critical manifold and a saddle homoclinic bifurcation within the associated layer problem. The continuous transition from canard cycles to N-spike bursting oscillations up to $$N\sim {\mathcal {O}}(1/\varepsilon )$$ spikes occurs upon varying a single bifurcation parameter on an exponentially thin interval. We construct this transition rigorously using geometric singular perturbation theory; critical to understanding this transition are the existence of canard orbits and slow passage through the saddle homoclinic bifurcation, which are analyzed in detail.

Spike-adding structure in fold/hom bursters

Square-wave or fold/hom bursting is typical of many excitable dynamical systems, such as pancreatic or other endocrine cells. Besides, it is also found in a great variety of fast-slow systems coming from other neural models, chemical reactions, laser dynamics, and so on. We focus on the spike-adding process and its connection with the homoclinic structure of the system. The creation of new fast spikes on a bursting neuron is an important phenomenon as it increases the duty cycle of the neuron. Here we mainly work with the Hindmarsh-Rose neuron model, a prototype of fold/hom bursting, but also with the pancreatic β-cell model, where, as already known from the literature, homoclinic bifurcations play an important role in bursting dynamics. Based on several numerical simulations, we present a theoretical scheme that provides a complete scenario of bifurcations involved in the spike-adding process and their connection with the homoclinic bifurcations on the parametric space. The global scheme explains the different phenomena of the spike-adding processes presented in literature (continuous and chaotic processes after Terman analysis) and moreover, it also indicates where each kind of spike-adding process occurs. Different elements are involved in the theoretical scheme, such as homoclinic isolas, canard orbits, inclination and orbit flip codimension-two bifurcation points and several pencils of period doubling and fold bifurcations, all of them illustrated with different numerical techniques. Some of these bifurcations needed in the process may be not visible on some numerical simulations because the organizing points are in different parametric planes due to the high dimension of the whole parameter space, but their effects are present. Therefore, we introduce a mechanism of the spike-adding process in fold/hom bursters in the whole space of parameters, even if apparently no role is played by the “far-away” homoclinic bifurcations. This fact is illustrated showing how the theoretical scheme provides a theoretical explanation to the different interspike-interval bifurcation diagrams (IBD) that have appeared in the literature for different models.

Early models of chemical oscillations failed to provide bounded solutions.

Before the development of the Brusselator, the Sel'kov and Turing models were considered as possible prototype models of chemical oscillations. We first analyse their Hopf bifurcation branches and show that they become vertical past a critical value of the control parameter. We explain this phenomenon by the emergence of canard orbits. Second, we analyse all solutions in the phase plane and show that some initial conditions lead to unbounded trajectories even if there exists a locally stable attractor. Our findings mathematically explain why these too simple two-variable models fail to account for the emergence of chemical oscillations. They support the conclusion that the Brusselator is the first minimal two-variable model explaining the onset of stable oscillations in a way fully compatible with thermodynamics and the law of mass action.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

Saddle Slow Manifolds and Canard Orbits in mathbb{R}^{4} and Application to the Full Hodgkin\u2013Huxley Model

Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron models typically involve observable differences in time scales. The Hodgkin–Huxley model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5–32, 2007) nondimensionalized this model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio ε. The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type.In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in mathbb{R}^{4}. We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin–Huxley model in organizing MMOs and determining the firing rates of action potentials. For ε sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the model.

Tangencies Between Global Invariant Manifolds and Slow Manifolds Near a Singular Hopf Bifurcation

Invariant manifolds of equilibria and periodic orbits are key objects that organize the behavior of a dynamical system both locally and globally. If multiple time scales are present in the dynamical system, there also exist so-called slow manifolds, that is, manifolds along which the flow is very slow compared with the rest of the dynamics. In particular, slow manifolds are known to organize the number of small oscillations of what are known as mixed-mode oscillations (MMOs). Slow manifolds are locally invariant objects that may interact with invariant manifolds, which are globally invariant objects; such interactions produce complicated dynamics about which only little is known from a few examples in the literature. We study the transition through a quadratic tangency between the unstable manifold of a saddle-focus equilibrium and a repelling slow manifold in a system where the corresponding equilibrium point undergoes a supercritical singular Hopf bifurcation. We compute the manifolds as families of orbit segments with a two-point boundary value problem setup and track their intersections, referred to as connecting canard orbits, as a parameter is varied. We describe the local and global properties of the manifolds, as well as the role of the interaction as an organizer of large-amplitude oscillations in the dynamics. We find and describe recurrent dynamics in the form of MMOs, which can be continued in parameters to Shilnikov homoclinic bifurcations. We detect and identify two such homoclinic orbits and describe their interactions with the MMOs.

A Lin's method approach for detecting all canard orbits arising from a folded node

Canard orbits are relevant objects in slow-fast dynamical systems that organize the spiraling of orbits nearby. In three-dimensional vector fields with two slow and one fast variables, canard orbits arise from the intersection between an attracting and a repelling two-dimensional slow manifold. Special points called folded nodes generate such intersections: in a suitable transverse two-dimensional section Σ, the attracting and repelling slow manifolds are counter-rotating spirals that intersect in a finite number of points. We present an implementation of Lin's method that is able to detect all of these intersection points and, hence, all of the canard orbits arising from a folded node. With a boundary-value-problem setup we compute orbit segments on each slow manifold up to Σ, where we require that the corresponding end points in Σ lie in a one-dimensional subspace known as the Lin space Z. The Lin space Z must be transverse to the slow manifolds and it remains fixed during the detection of canard orbits as zeros of the signed distance along Z. During the computation, a tangency of Z with one of the intersection curves in Σ may arise. To overcome this, we update the Lin space at an intermediate continuation step to detect a double tangency of Z to both curves in Σ, after which the canard detection is able to continue. Our method is demonstrated with the examples of the normal form for a folded node and of the Koper model.

Mixed-Mode Oscillations and Twin Canard Orbits in an Autocatalytic Chemical Reaction

A mixed-mode oscillation (MMO) is a complex waveform with a pattern of alternating small-amplitude oscillations (SAOs) and large-amplitude oscillations (LAOs). MMOs have been observed experimentally in many physical and biological applications, but most notably in chemical reactions. We are interested in MMOs of an autocatalytic chemical reaction that can be modeled by a system of three ordinary differential equations with one fast and two slow variables. This difference in time scales provides a mechanism for generating small and large oscillations. Provided the time-scale ratio $\varepsilon$ is sufficiently small, geometric singular perturbation theory predicts the existence of two-dimensional locally invariant manifolds called slow manifolds. Slow manifolds and their intersections, which occur along so-called canard orbits, give great insight into the mechanisms for generating SAOs. The mechanisms for LAOs are less well understood and involve analysis of the global dynamics. We study the autocatalytic ...

Planar Canards with Transcritical Intersections

We review essential techniques in the study of families of periodic orbits of slow-fast systems in the plane. The techniques are demonstrated by treating orbits passing through unfoldings of transcritical intersections of curves of singular points in the most generic setting. We show that such transcritical intersections can generate canard type orbits. The stability of limit cycles of canard type containing that pass near transcritical intersections is examined by means of the slow divergence integral.

Canard trajectories in 3D piecewise linear systems

We present some results on singularly perturbed piecewise linear systems, similar to those obtained by the Geometric Singular Perturbation Theory. Unlike the differentiable case, in the piecewise linear case we obtain the global expression of the slow manifold ${\mathcal S}_{\varepsilon}$. As a result, we characterize the existence of canard orbits in such systems. Finally, we apply the above theory to a specific case where we show numerical evidences of the existence of a canard cycle.

An elementary model of torus canards

We study the recently observed phenomena of torus canards. These are a higher-dimensional generalization of the classical canard orbits familiar from planar systems and arise in fast-slow systems of ordinary differential equations in which the fast subsystem contains a saddle-node bifurcation of limit cycles. Torus canards are trajectories that pass near the saddle-node and subsequently spend long times near a repelling branch of slowly varying limit cycles. In this article, we carry out a study of torus canards in an elementary third-order system that consists of a rotated planar system of van der Pol type in which the rotational symmetry is broken by including a phase-dependent term in the slow component of the vector field. In the regime of fast rotation, the torus canards behave much like their planar counterparts. In the regime of slow rotation, the phase dependence creates rich torus canard dynamics and dynamics of mixed mode type. The results of this elementary model provide insight into the torus canards observed in a higher-dimensional neuroscience model.

Numerical continuation of canard orbits in slow–fast dynamical systems

A trajectory of a system with two clearly separated time scales generally consists of fast segments (or jumps) followed by slow segments where the trajectory follows an attracting part of a slow manifold. The switch back to fast dynamics typically occurs when the trajectory passes a fold with respect to a fast direction. A special role is played by trajectories known as canard orbits, which do not jump at a fold but, instead, follow a repelling slow manifold for some time. We concentrate here on the case of a slow–fast system with two slow and one fast variable, where canard orbits arise geometrically as intersection curves of two-dimensional attracting and repelling slow manifolds. Canard orbits are intimately related to the dynamics near special points known as folded singularities, which in turn have been shown to explain small-amplitude oscillations that can be found as part of so-called mixed-mode oscillations.In this paper we present a numerical method to detect and then follow branches of canard orbits in a system parameter. More specifically, we define well-posed two-point boundary value problems (BVPs) that represent orbit segments on the slow manifolds, and we continue their solution families with the package AUTO. In this way, we are able to deal effectively with the numerical challenge of strong attraction to and strong repulsion from the slow manifolds. Canard orbits are detected as the transverse intersection points of the curves along which attracting and repelling slow manifolds intersect a suitable section (near a folded node). These intersection points correspond to a unique pair of orbits segments, one on the attracting and one on the repelling slow manifold. After concatenation of the respective pairs of orbit segments, all detected canard orbits are represented as solutions of a single BVP, which allows us to continue them in system parameters. We demonstrate with two examples—the self-coupled FitzHugh–Nagumo system and a three-dimensional reduced Hodgkin–Huxley model—that branches of canard orbits can be computed reliably. Furthermore, our computations illustrate that the continuation of canard orbits allows one to find and investigate new types of dynamics, such as the interaction between canard orbits and a saddle periodic orbit that is generated in a singular Hopf bifurcation.

Canard cycles in the presence of slow dynamics with singularities

We study the cyclicity of limit periodic sets that occur in families of vector fields of slow–fast type. The limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point. At this turning point a stability change takes place: on one side of the turning point the dynamics point strongly towards the curve of singularities; on the other side the dynamics point away from the curve of singularities. The presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point, i.e. orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve. All existing results deal with a non-zero slow movement, permitting a good estimate of the cyclicity by considering the slow-divergence integral along the curve of singularities. In this paper we study what happens when the slow dynamics exhibit singularities. In particular, our study includes the cyclicity of the slow–fast two-saddle cycle, formed by a regular saddle connection (the fast part) and a part of the curve of singularities (the slow part). We see that the relevant information is no longer merely contained in the slow-divergence integral.

Feedback control of canards

We present a control mechanism for tuning a fast-slow dynamical system undergoing a supercritical Hopf bifurcation to be in the canard regime, the tiny parameter window between small and large periodic behavior. Our control strategy uses continuous feedback control via a slow control variable to cause the system to drift on average toward canard orbits. We apply this to tune the FitzHugh-Nagumo model to produce maximal canard orbits. When the controller is improperly configured, periodic or chaotic mixed-mode oscillations are found. We also investigate the effects of noise on this control mechanism. Finally, we demonstrate that a sensor tuned in this way to operate near the canard regime can detect tiny changes in system parameters.

Smoothness of transition maps in singular perturbation problems with one fast variable

This paper deals with the smoothness of the transition map between two sections transverse to the fast flow of a singularly perturbed vector field (one fast, multiple slow directions). Orbits connecting both sections are canard orbits, i.e. they first move rapidly towards the attracting part of a critical surface, then travel a distance near this critical surface, even beyond the point where the orbit enters the repelling part of the critical surface, and finally repel away from the surface. We prove that the transition map is smooth. In a transcritical situation however, where orbits from an attracting part of one critical manifold follow the repelling part of another critical manifold, the smoothness of the transition map may be limited, due to resonance phenomena that are revealed by blowing up the turning point! We present a polynomial example in R 3 .

CONTROL-BASED CONTINUATION OF PERIODIC ORBITS WITH A TIME-DELAYED DIFFERENCE SCHEME

This paper presents a method that is able to continue periodic orbits in systems where only output of the evolution over a given time period is available, which is the typical situation in an experiment. The starting point of our paper is an analysis of time-delayed feedback control, a method to stabilize periodic orbits experimentally that is popular among physicists. We show that the well-known topological limitations of this method can be overcome by an embedding into a pseudo-arclength continuation and prove that embedded time-delayed feedback control is able to stabilize periodic orbits that have at most one unstable Floquet multiplier sufficiently close to the unit circle. In the second part we introduce preconditioning into the time-delayed feedback control. In this way, we extract a nonlinear system of equations from time profiles, which we solve using Newton iterations. We demonstrate the feasibility of our method by continuing periodic orbits in a laser model through folds, and by computing the family of canard orbits of the classical stiff Van der Pol system with constant forcing.

Canard orbits in Fabry-Perot cavities induced by radiation pressure and photothermal effects

A theoretical study of a high-finesse Fabry-Perot cavity considering radiation pressure and photothermal displacement is reported. We show that the competition between these two effects induces a different kind of dynamic behavior in such a system, consisting of canard orbits and excitability. The transition between the excitable regime and the canard oscillations, occurring through a supercritical Hopf bifurcation, appears in an order compatible with the van der Pol-FitzHugh-Nagumo equations. Besides its interest as a study of general nonlinear dynamics, the characterization of the effects described is critical for high sensitivity interferometric displacement measurements as those employed for gravitational waves detection.