New dynamics in cerebellar Purkinje cells: torus canards

We describe a transition from bursting to rapid spiking in a reduced mathematical model of a cerebellar Purkinje cell. We perform a slow-fast analysis of the system and find that-after a saddle node bifurcation of limit cycles-the full model dynamics temporarily follow a repelling branch of limit cycles. We propose that the system exhibits a dynamical phenomenon new to realistic, biophysical applications: torus canards.     

Embedding dynamics for round-off errors near a periodic orbit

We study the propagation of round-off errors near the periodic orbits of a linear map conjugate to a planar rotation with rational rotation number. We embed the two-dimensional discrete phase space (a lattice) in a higher-dimensional torus, where points sharing the same round-off error are uniformly distributed within finitely many convex polyhedra. The embedding dynamics is linear and discontinuous, with algebraic integer coefficients. This representation affords efficient algorithms for classifying and computing the orbits and their exact densities, which we apply to the case of rational rotation number with denominator 7, corresponding to certain algebraic integers of degree three. We provide evidence that the hierarchical arrangement of orbits previously detected in quadratic cases [Lowenstein et al., Chaos 7, 49-66 (1997)] disappears, and that the growth of the number of orbits with the period is algebraic.(c) 2000 American Institute of Physics.     

Complex dynamics in the Oregonator model with linear delayed feedback

The Belousov-Zhabotinsky (BZ) reaction can display a rich dynamics when a delayed feedback is applied. We used the Oregonator model of the oscillating BZ reaction to explore the dynamics brought about by a linear delayed feedback. The time-delayed feedback can generate a succession of complex dynamics: period-doubling bifurcation route to chaos; amplitude death; fat, wrinkled, fractal, and broken tori; and mixed-mode oscillations. We observed that this dynamics arises due to a delay-driven transition, or toggling of the system between large and small amplitude oscillations, through a canard bifurcation. We used a combination of numerical bifurcation continuation techniques and other numerical methods to explore the dynamics in the strength of feedback-delay space. We observed that the period-doubling and quasiperiodic route to chaos span a low-dimensional subspace, perhaps due to the trapping of the trajectories in the small amplitude regime near the canard; and the trapped chaotic trajectories get ejected from the small amplitude regime due to a crowding effect to generate chaotic-excitable spikes. We also qualitatively explained the observed dynamics by projecting a three-dimensional phase portrait of the delayed dynamics on the two-dimensional nullclines. This is the first instance in which it is shown that the interaction of delay and canard can bring about complex dynamics.     

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.     

Canards in a rheodynamic model of cardiac pressure pulsations

This paper reports on the canard phenomenon occurring in a rheodynamic model of cardiac pressure pulsations. By singular perturbation techniques the corresponding parameter value at which canards exist is obtained. The physiological significance of canards in this model is given.     

Food chain chaos with canard explosion

The "tea-cup" attractor of a classical prey-predator-superpredator food chain model is studied analytically. Under the assumption that each species has its own time scale, ranging from fast for the prey to intermediate for the predator and to slow for the superpredator, the model is transformed into a singular perturbed system. It is demonstrated that the singular limit of the attractor contains a canard singularity. Singular return maps are constructed for which some subdynamics are shown to be equivalent to chaotic shift maps. Parameter regions in which the described chaotic dynamics exist are explicitly given.     

Understanding anomalous delays in a model of intracellular calcium dynamics

In many cell types, oscillations in the concentration of free intracellular calcium ions are used to control a variety of cellular functions. It has been suggested [J. Sneyd et al., "A method for determining the dependence of calcium oscillations on inositol trisphosphate oscillations," Proc. Natl. Acad. Sci. U.S.A. 103, 1675-1680 (2006)] that the mechanisms underlying the generation and control of such oscillations can be determined by means of a simple experiment, whereby a single exogenous pulse of inositol trisphosphate (IP(3)) is applied to the cell. However, more detailed mathematical investigations [M. Domijan et al., "Dynamical probing of the mechanisms underlying calcium oscillations," J. Nonlinear Sci. 16, 483-506 (2006)] have shown that this is not necessarily always true, and that the experimental data are more difficult to interpret than first thought. Here, we use geometric singular perturbation techniques to study the dynamics of models that make different assumptions about the mechanisms underlying the calcium oscillations. In particular, we show how recently developed canard theory for singularly perturbed systems with three or more slow variables [M. Wechselberger, "A propos de canards (Apropos canards)," Preprint, 2010] applies to these calcium models and how the presence of a curve of folded singularities and corresponding canards can result in anomalous delays in the response of these models to a pulse of IP(3).     

Boundary-equilibrium bifurcations in piecewise-smooth slow-fast systems

In this paper we study the qualitative dynamics of piecewise-smooth slow-fast systems (singularly perturbed systems) which are everywhere continuous. We consider phase space topology of systems with one-dimensional slow dynamics and one-dimensional fast dynamics. The slow manifold of the reduced system is formed by a piecewise-continuous curve, and the differentiability is lost across the switching surface. In the full system the slow manifold is no longer continuous, and there is an O(?) discontinuity across the switching manifold, but the discontinuity cannot qualitatively alter system dynamics. Revealed phase space topology is used to unfold qualitative dynamics of planar slow-fast systems with an equilibrium point on the switching surface. In this case the local dynamics corresponds to so-called boundary-equilibrium bifurcations, and four qualitative phase portraits are uncovered. Our results are then used to investigate the dynamics of a box model of a thermohaline circulation, and the presence of a boundary-equilibrium bifurcation of a fold type is shown.     

Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation

Numerical studies of higher-dimensional piecewise-smooth systems have recently shown how a torus can arise from a periodic cycle through a special type of border-collision bifurcation. The present article investigates this new route to quasiperiodicity in the two-dimensional piecewise-linear normal form map. We have obtained the chart of the dynamical modes for this map and showed that border-collision bifurcations can lead to the birth of a stable closed invariant curve associated with quasiperiodic or periodic dynamics. In the parameter regions leading to the existence of an invariant closed curve, there may be transitions between an ergodic torus and a resonance torus, but the mechanism of creation for the resonance tongues is distinctly different from that observed in smooth maps. The transition from a stable focus point to a resonance torus may lead directly to a new focus of higher periodicity, e.g., a period-5 focus. This article also contains a discussion of torus destruction via a homoclinic bifurcation in the piecewise-linear normal map. Using a dc-dc converter with two-level control as an example, we report the first experimental verification of the direct transition to quasiperiodicity through a border-collision bifurcation.     

Slow dynamics of embedded fluid in mesoscopic confining systems as probed by NMR relaxometry

Mesoscopic media such as porous materials or colloidal dispersions strongly influence the dynamics of the embedded fluid. In the strong-adsorption regime, it was recently proposed that the effective surface diffusion on flat surface is anomalous and exhibits long-time pathology, enlarging the time domain of the embedded-fluid dynamics towards the low-frequency regime. An interesting way to probe such a slow interfacial process is to use the field-cycling NMR relaxometry. This technique is used here to probe the fluid dynamics in two types of interfacial systems: i) a colloidal glass made of thin and flat particles; ii) a fully saturated porous media, the Vycor glass. Experimental results are critically compared to either a simple theoretical model of NMR dispersion involving elementary steps of the fluid dynamics near an interface (loops, trains, tails) or Brownian-dynamics simulations performed inside 3D reconstructions of these confined systems.Received: 1 January 2003, Published online: 14 October 2003PACS: 76.60.Es Relaxation effects - 61.43.Gt Powders, porous materials - 82.70.Dd Colloids     

Multirhythmic bursting

A complex modeled bursting neuron [C. C. Canavier, J. W. Clark, and J. H. Byrne, J. Neurophysiol. 66, 2107-2124 (1991)] has been shown to possess seven coexisting limit cycle solutions at a given parameter set [Canavier et al., J. Neurophysiol 69, 2252-2259 (1993); 72, 872-882 (1994)]. These solutions are unique in that the limit cycles are concentric in the space of the slow variables. We examine the origin of these solutions using a minimal 4-variable bursting cell model. Poincare maps are constructed using a saddle-node bifurcation of a fast subsystem such as our Poincare section. This bifurcation defines a threshold between the active and silent phases of the burst cycle in the space of the slow variables. The maps identify parameter spaces with single limit cycles, multiple limit cycles, and two types of chaotic bursting. To investigate the dynamical features which underlie the unique shape of the maps, the maps are further decomposed into two submaps which describe the solution trajectories during the active and silent phases of a single burst. From these findings we postulate several necessary criteria for a bursting model to possess multiple stable concentric limit cycles. These criteria are demonstrated in a generalized 3-variable model. Finally, using a less direct numerical procedure, similar return maps are calculated for the original complex model [C. C. Canavier, J. W. Clark, and J. H. Byrne, J. Neurophysiol. 66, 2107-2124 (1991)], with the resulting mappings appearing qualitatively similar to those of our 4-variable model. These multistable concentric bursting solutions cannot occur in a bursting model with one slow variable. This type of multistability arises when a bursting system has two or more slow variables and is viewed as an essentially second-order system which receives discrete perturbations in a state-dependent manner. (c) 1998 American Institute of Physics.     

Slow dynamics in colloidal glasses and porous media as probed by NMR relaxometry: assessment of solvent levy statistics in the strong adsorption regime

Mesoscopic media such as porous materials or colloidal pastes develop large specific surface area which strongly influence the dynamics of the embedded fluid. This fluid confinement can be used either to probe the interfacial geometry (frozen porous media) or the particle dynamics (paste and colloidal glass). In the strong adsorption regime, it was recently proposed that the effective surface diffusion on flat surface is anomalous and exhibits long time pathology (Lévy walks). This phenomena is directly related to the time and space properties of loop trajectories appearing in the bulk between a desorption and a readsorption step. The Lévy statistics extends the time domain of the embedded fluid dynamics toward the low frequency regime. An interesting way to probe such a slow interfacial process is to use field cycling NMR relaxometry. In the first part of this paper, we propose a simple theoretical model of NMR dispersion which only involves elementary time steps of the solvent dynamics near an interface (loops, trains, tails in relation with the confining geometry). In the second part, field cycling NMR relaxometry is used to probe the slow solvent dynamics in two type of interfacial systems: (i) a colloidal glass made of thin and flat particles (ii) two fully saturated porous media, the Vycor glass and MCM48 respectively. Experimental results are critically compared to closed-form analytical expressions and numerical simulations.     

兴奋性作用诱发神经簇放电个数不增反降的分岔机制

非线性动力学在识别神经放电的复杂现象、机制和功能方面发挥了重要作用.不同于传统观念,本文提出了兴奋性作用可以降低而不是增加簇内放电个数的新观点.在簇放电模式休止期的适合相位施加强度合适的脉冲或自突触电流,能诱发簇内放电个数降低;电流的施加相位越早,所需的强度阈值越大,簇内放电个数越少.进一步,利用快慢变量分离获得的簇放电的动力学性质进行了理论解释.簇放电模式表现出低电位的休止期和高电位的放电的交替,存在于快子系统的鞍结分岔点和同宿轨分岔点之间;放电起始于鞍结分岔、结束于同宿轨分岔;越靠近同宿轨分岔从休止期跨越到放电所需的电流强度越大.因此,电流在休止期上的作用相位越早,就越靠近同宿轨分岔,因而从休止期跨越到放电需要的电流强度阈值越大,放电起始相位到同宿轨分岔之间的区间变小导致放电个数变少.研究结果丰富了非线性现象及机制,对兴奋性作用提出了新看法,给出了调控簇放电模式的新途径.     

Chaotically spiking canards in an excitable system with 2D inertial fast manifolds

We introduce a new class of excitable systems with two-dimensional fast dynamics that includes inertia. A novel transition from excitability to relaxation oscillations is discovered where the usual Hopf bifurcation is followed by a cascade of period doubled and chaotic small excitable attractors and, as they grow, by a new type of canard explosion where a small chaotic background erratically but deterministically triggers excitable spikes. This scenario is also found in a model for a nonlinear Fabry-Perot cavity with one pendular mirror.     

Novel routes to chaos through torus breakdown in non-invertible maps

The paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non-invertible maps. By means of detailed, numerically calculated phase portraits we first describe how three- and five-layered tori arise through period-doubling and/or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We then describe several different mechanisms for the destruction of five-layered tori in a system of two linearly coupled logistic maps. One of these scenarios involves the destruction of the two intermediate layers of the five-layered torus through the transformation of two unstable node cycles into unstable focus cycles, followed by a saddle-node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set. Other scenarios involve different combinations of local and global bifurcations, including bifurcations that lead to various forms of homoclinic and heteroclinic tangles. We finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two-dimensional non-invertible maps that have previously been used to study the properties of invariant sets.     

Oscillatory thermal-diffusive instability of combustion waves in a model with chain-branching reaction and heat loss

In this paper we investigate the properties and the linear stability of premixed combustion waves in a non-adiabatic thermal-diffusive model with a two-step chain-branching reaction mechanism. Here we focus only on the emergence of the pulsating instabilities, and the stability analysis is carried out for Lewis numbers for fuel greater than one, and various values of Lewis number for radicals. We consider the problem in two spatial dimensions to allow perturbations of a multidimensional nature. It is demonstrated that the flame speed as a function of the parameters is a double-valued C-shaped function, i.e. for a given set of parameter values there are either two solutions, fast and slow solution branches, propagating with different speed, or the combustion wave does not exist. The extinction of combustion waves occurs at finite values of the parameters and non-zero flame speed. The flame structure demonstrates a slow recombination regime behaviour with negligible fuel leakage for the fast solution branch away from the extinction condition. For parameter values close to the extinction condition and on the slow solution branch, the fuel leakage is significant and a fast recombination regime is observed. It is demonstrated that two types of instabilities emerge in the model: the uniform planar and the travelling instability. The slow solution branch is always unstable due to the uniform perturbations. The fast solution branch is either stable or loses stability due to the travelling or uniform perturbations. The switching between the onset of various regimes of instability is due to the bifurcation of co-dimension two. In the adiabatic limit this bifurcation is found for Lewis number for fuel equal to one, whereas in the non-adiabatic case it moves towards values above unity. The properties of the travelling instability are studied in detail.     

Molecular rotations studied by nuclear resonant scattering

Nuclear resonant scattering techniques can be used to study both fast and slow dynamics of Mössbauer nuclei. The influence of rotational dynamics in molecular systems is studied applying three types of scattering techniques: (1) Synchrotron radiation perturbed angular correlation (SRPAC) yields direct and quantitative evidence for rotational dynamics in the μs-ns regime. (2) Nuclear inelastic scattering (NIS) monitors the relative influence of intra- and intermolecular forces via the vibrational density of states, which can be influenced by the onset of molecular rotation. (3) In nuclear forward scattering (NFS), information both on rotational and on translational dynamics can be extracted. Results using SRPAC and NIS on a plastic crystal and NFS on ferrocene confined in a molecular sieve are presented.     

New Canards Bursting and Canards Periodic-Chaotic Sequence

A trajectory following the repelling branch of an equilibrium or a periodic orbit is called a canards solution. Using a continuation method, we find a new type of canards bursting which manifests itseff in an alternation between the oscillation phase following attracting the limit cycle branch and resting phase following a repelling fixed point branch in a reduced leech neuron model Via periodic-chaotic alternating of infinite times, the number of windings within a canards bursting can approach infinity at a Gavrilov-Shilnikov homoclinic tangency bifurcation of a simple saddle limit cycle.     

Bifurcations of mixed-mode oscillations in a stellate cell model

Experimental recordings of the membrane potential of stellate cells within the entorhinal cortex show a transition from subthreshold oscillations (STOs) via mixed-mode oscillations (MMOs) to relaxation oscillations under increased injection of depolarizing current. Acker et al. introduced a 7D conductance based model which reproduces many features of the oscillatory patterns observed in these experiments. For the first time, we present a comprehensive bifurcation analysis of this model by using the software package AUTO. In particular, we calculate the stable MMO branches within the bifurcation diagram of this model, as well as other MMO patterns which are unstable. We then use geometric singular perturbation theory to demonstrate how the bifurcations are governed by a 3D reduced model introduced by Rotstein et al. We extend their analysis to explain all observed MMO patterns within the bifurcation diagram. A key role in this bifurcation analysis is played by a novel homoclinic bifurcation structure connecting to a saddle equilibrium on the unstable branch of the corresponding critical manifold. This type of homoclinic connection is possible due to canards of folded node (folded saddle-node) type.     

Compressibility effects on double tearing mode interlocking in differentially rotating plasmas

Effects of compressibility on the double tearing modes (DTMs) in rotating plasmas are numerically investigated by using a compressible magnetohydrodynamics (MHD) model. It is found that due to the compressibility effects, the threshold of the interlocking magnetic island width in the slow and intermediate rotation regimes is larger than the counterpart in the incompressible plasmas. In the fast rotation regime, the compressible effect makes the DTM islands interlock more easily and faster. Moreover, in the very fast rotation regime, the plasma rotation can more effectively suppress the DTM islands. The scalings of the interlocking threshold in the different rotation regimes are obtained. Effects of plasma viscosity and beta on the DTM interlocking in the compressible plasmas are also discussed.