Cardiac arrhythmias and circle maps-A classical problem

Suppression of quasiperiodicity in circle maps with quenched disorder

We show that introducing quenched disorder into a circle map leads to the suppression of quasiperiodic behavior in the limit of large system sizes. Specifically, for most parameters the fraction of disorder realizations showing quasiperiodicity decreases with the system size and eventually vanishes in the limit of infinite size, where almost all realizations show mode locking. Consequently, in this limit, and in strong contrast to standard circle maps, almost the whole parameter space corresponding to invertible dynamics consists of Arnold tongues.

Bifurcation, chaos, multistability, and organized structures in a predator-prey model with vigilance

There is not a single species that does not strive for survival. Every species has crafted specialized techniques to avoid possible dangers that mostly come from the side of their predators. Survival instincts in nature led prey populations to develop many anti-predator strategies. Vigilance is a well-observed effective antipredator strategy that influences predator-prey dynamics significantly. We consider a simple discrete-time predator-prey model assuming that vigilance affects the predation rate and the growth rate of the prey. We investigate the system dynamics by constructing isoperiodic and Lyapunov exponent diagrams with the simultaneous variation of the prey's growth rate and the strength of vigilance. We observe a series of different types of organized periodic structures with different kinds of period-adding phenomena. The usual period-bubbling phenomenon is shown near a shrimp-shaped periodic structure. We observe the presence of double and triple heterogeneous attractors. We also notice Wada basin boundaries in the system, which is quite rare in ecological systems. The complex dynamics of the system in biparameter space are explored through extensive numerical simulations.

Creation of discontinuities in circle maps

Circle maps frequently arise in mathematical models of physical or biological systems. Motivated by Cherry flows and ‘threshold’ systems such as integrate and fire neuronal models, models of cardiac arrhythmias, and models of sleep/wake regulation, we consider how structural transitions in circle maps occur. In particular, we describe how maps evolve near the creation of a discontinuity. We show that the natural way to create discontinuities in the maps associated with both threshold systems and Cherry flows results in a singularity in the derivative of the map as the discontinuity is approached from either one or both sides. For the threshold systems, the associated maps have square root singularities and we analyse the generic properties of such maps with gaps, showing how border collisions and saddle-node bifurcations are interspersed. This highlights how the Arnold tongue picture for tongues bordered by saddle-node bifurcations is amended once gaps are present. We also show that a loss of injectivity naturally results in the creation of multiple gaps giving rise to a novel codimension two bifurcation.

Order-indeterminant event-based maps for learning a beat

The process by which humans synchronize to a musical beat is believed to occur through error-correction where an individual's estimates of the period and phase of the beat time are iteratively adjusted to align with an external stimuli. Mathematically, error-correction can be described using a two-dimensional map where convergence to a fixed point corresponds to synchronizing to the beat. In this paper, we show how a neural system, called a beat generator, learns to adapt its oscillatory behavior through error-correction to synchronize to an external periodic signal. We construct a two-dimensional event-based map, which iteratively adjusts an internal parameter of the beat generator to speed up or slow down its oscillatory behavior to bring it into synchrony with the periodic stimulus. The map is novel in that the order of events defining the map are not a priori known. Instead, the type of error-correction adjustment made at each iterate of the map is determined by a sequence of expected events. The map possesses a rich repertoire of dynamics, including periodic solutions and chaotic orbits.

Nonequilibrium limit-cycle oscillators: Fluctuations in hair bundle dynamics

We develop a framework for the general interpretation of the stochastic dynamical system near a limit cycle. Such quasiperiodic dynamics are commonly found in a variety of nonequilibrium systems, including the spontaneous oscillations of hair cells of the inner ear. We demonstrate quite generally that in the presence of noise, the phase of the limit cycle oscillator will diffuse, while deviations in the directions locally orthogonal to that limit cycle will display the Lorentzian power spectrum of a damped oscillator. We identify two mechanisms by which these stochastic dynamics can acquire a complex frequency dependence and discuss the deformation of the mean limit cycle as a function of temperature. The theoretical ideas are applied to data obtained from spontaneously oscillating hair cells of the amphibian sacculus.

Entrainment Maps

Circadian oscillators found across a variety of species are subject to periodic external light-dark forcing. Entrainment to light-dark cycles enables the circadian system to align biological functions with appropriate times of day or night. Phase response curves (PRCs) have been used for decades to gain valuable insights into entrainment; however, PRCs may not accurately describe entrainment to photoperiods with substantial amounts of both light and dark due to their reliance on a single limit cycle attractor. We have developed a new tool, called an entrainment map, that overcomes this limitation of PRCs and can assess whether, and at what phase, a circadian oscillator entrains to external forcing with any photoperiod. This is a 1-dimensional map that we construct for 3 different mathematical models of circadian clocks. Using the map, we are able to determine conditions for existence and stability of phase-locked solutions. In addition, we consider the dependence on various parameters such as the photoperiod and intensity of the external light as well as the mismatch in intrinsic oscillator frequency with the light-dark cycle. We show that the entrainment map yields more accurate predictions for phase locking than methods based on the PRC. The map is also ideally suited to calculate the amount of time required to achieve entrainment as a function of initial conditions and the bifurcations of stable and unstable periodic solutions that lead to loss of entrainment.

The functions of atrial strands interdigitating with and penetrating into sinoatrial node: a theoretical study of the problem

The sinoatrial node (SAN)-atrium system is closely involved with the activity of heart beating. The impulse propagation and phase-locking behaviors of this system are of theoretical interest. Some experiments have revealed that atrial strands (ASs) interdigitate with and penetrate into the SAN, whereby the SAN-atrium system works as a complex network. In this study, the functions of ASs are numerically investigated using realistic cardiac models. The results indicate that the ASs penetrating into the central region of the SAN play a major role in propagating excitation into the atrium. This is because the threshold SAN-AS coupling for an AS to function as an alternative path for propagation is lower at the center than at the periphery. However, ASs penetrating into the peripheral region have a great effect in terms of enlarging the 1:1 entrainment range of the SAN because the automaticity of the SAN is evidently reduced by ASs. Moreover, an analytical formula for approximating the enlargement of the 1:1 range is derived.

Heart rate variability as determinism with jump stochastic parameters

We use measured heart rate information (RR intervals) to develop a one-dimensional nonlinear map that describes short term deterministic behavior in the data. Our study suggests that there is a stochastic parameter with persistence which causes the heart rate and rhythm system to wander about a bifurcation point. We propose a modified circle map with a jump process noise term as a model which can qualitatively capture such this behavior of low dimensional transient determinism with occasional (stochastically defined) jumps from one deterministic system to another within a one parameter family of deterministic systems.

Phase-locking behaviors in an ionic model of sinoatrial node cell and tissue

Phase-locking behaviors in sinoatrial node (SAN) are closely related to cardiac arrhythmias. An ionic model considering structural heterogeneity of SAN is numerically investigated. The bifurcations between phase-locking zones are interpreted by the map derived from the phase resetting curve. Furthermore, the validity of the circle map in describing phase locking of the actual SAN system is evaluated and explained. We reveal also how the phase-locking behaviors in heterogeneous tissue depend on the location of stimulating site and the coupling strength of the tissue. All these results may be of suggestive uses for understanding and controlling practical SAN dynamics.

Multi-frequency activation of neuronal networks by coordinated reset stimulation

We computationally study whether it is possible to stimulate a neuronal population in such a way that its mean firing rate increases without an increase of the population's net synchronization. For this, we use coordinated reset (CR) stimulation, which has previously been developed to desynchronize populations of oscillatory neurons. Intriguingly, delivered to a population of predominantly silent FitzHugh-Nagumo or Hindmarsh-Rose neurons at sufficient stimulation amplitudes, CR robustly causes a multi-frequency activation: different Arnold tongues such as 1 : 1 or n : m entrained neuronal clusters emerge, which consist of phase-shifted sub clusters. Owing to the clustering pattern the neurons' timing is well balanced, so that in total there is no synchronization. Our findings may contribute to the development of novel and safe stimulation treatments that specifically counteract cerebral hypo-activity without promoting pathological synchronization or inducing epileptic seizures.

Optimal waveform for the entrainment of a weakly forced oscillator

A theory for obtaining a waveform for the effective entrainment of a weakly forced oscillator is presented. Phase model analysis is combined with calculus of variation to derive a waveform with which entrainment of an oscillator is achieved with a minimum power forcing signal. Optimal waveforms are calculated from the phase response curve and a solution to a balancing condition. The theory is tested in chemical entrainment experiments in which oscillations close to and farther away from a Hopf bifurcation exhibited sinusoidal and higher harmonic nontrivial optimal waveforms, respectively.

A model for mechano-electrical feedback effects on atrial flutter interval variability

Atrial flutter is a supraventricular arrhythmia, based on a reentrant mechanism mainly confined to the right atrium. Although atrial flutter is considered a regular rhythm, the atrial flutter interval (i.e., the time interval between consecutive atrial activation times) presents a spontaneous beat-to-beat variability, which has been suggested to be related to ventricular contraction and respiration by mechano-electrical feedback. This paper introduces a model to predict atrial activity during atrial flutter, based on the assumption that atrial flutter variability is related to the phase of the reentrant activity in the ventricular and respiratory cycles. Thus, atrial intervals are given as a superimposition of phase-dependent ventricular and respiratory modulations. The model includes a simplified atrioventricular (AV) branch with constant refractoriness and conduction times, which allows the prediction of ventricular activations in a closed-loop with atrial activations. Model predictions are quantitatively compared with real activation series recorded in 12 patients with atrial flutter. The model predicts the time course of both atrial and ventricular time series with a high beat-to-beat agreement, reproducing 96+/-8% and 86+/-21% of atrial and ventricular variability, respectively. The model also predicts the existence of phase-locking of atrial flutter intervals during periodic ventricular pacing and such results are observed in patients. These results constitute evidence in favor of mechano-electrical feedback as a major source of cycle length variability during atrial flutter.

Phase response characteristics of sinoatrial node cells

In this work, the dynamic response of the sinoatrial node (SAN), the natural pacemaker of the heart, to short external stimuli is investigated using the Zhang et al. model. The model equations are solved twice for the central cell and for the peripheral cell. A short current pulse is applied to reset the spontaneous rhythmic activity of the single sinoatrial node cell. Depending on the stimulus timing either a delay or an advance in the occurrence of next action potential is produced. This resetting behavior is quantified in terms of phase transition curves (PTCs) for short electrical current pulses of varying amplitude which span the whole period. For low stimulus amplitudes the transition from advance to delay is smooth, while at higher amplitudes abrupt changes and discontinuities are observed in PTCs. Such discontinuities reveal critical stimuli, the application of which can result in annihilation of activity in central SAN cells. The detailed analysis of the ionic mechanisms involved in its resetting behavior of sinoatrial node cell models provides new insight into the dynamics and physiology of excitation of the sinoatrial node of the heart.

Calcium instabilities in mammalian cardiomyocyte networks

The degeneration of a regular heart rhythm into fibrillation (a chaotic or chaos-like sequence) can proceed via several classical routes described by nonlinear dynamics: period-doubling, quasiperiodicity, or intermittency. In this study, we experimentally examine one aspect of cardiac excitation dynamics, the long-term evolution of intracellular calcium signals in cultured cardiomyocyte networks subjected to increasingly faster pacing rates via field stimulation. In this spatially extended system, we observed alternans and higher-order periodicities, extra beats, and skipped beats or blocks. Calcium instabilities evolved nonmonotonically with the prevalence of phase-locking or Wenckebach rhythm, low-frequency magnitude modulations (signature of quasiperiodicity), and switches between patterns with occasional bursts (signature of intermittency), but period-doubling bifurcations were rare. Six ventricular-fibrillation-resembling episodes were pace-induced, for which significantly higher complexity was confirmed by approximate entropy calculations. The progressive destabilization of the heart rhythm by coexistent frequencies, seen in this study, can be related to theoretically predicted competition of control variables (voltage and calcium) at the single-cell level, or to competition of excitation and recovery at the cell network level. Optical maps of the response revealed multiple local spatiotemporal patterns, and the emergence of longer-period global rhythms as a result of wavebreak-induced reentries.

Arnold tongues in human cardiorespiratory systems

Arnold tongues are phase-locking regions in parameter space, originally studied in circle-map models of cardiac arrhythmias. They show where a periodic system responds by synchronizing to an external stimulus. Clinical studies of resting or anesthetized patients exhibit synchronization between heart-beats and respiration. Here we show that these results are successfully modeled by a circle-map, neatly combining the phenomena of respiratory sinus arrhythmia (RSA, where inspiration modulates heart-rate) and cardioventilatory coupling (CVC, where the heart is a pacemaker for respiration). Examination of the Arnold tongues reveals that while RSA can cause synchronization, the strongest mechanism for synchronization is CVC, so that the heart is acting as a pacemaker for respiration.

Periodic forcing of a model sensory neuron

We study the effects of sinusoidally modulating the current injected into a model sensory neuron from the weakly electric fish Apteronotus leptorhynchus. This neuron's behavior is known to switch from quiescence to periodic firing to bursting as the injected current is increased. The bifurcation separating periodic from bursting behavior is a saddle-node bifurcation of periodic orbits, and it has been shown previously that there is "type-I burst excitability" associated with this bifurcation, similar to the usual excitability associated with the transition from quiescence to periodic firing. Here we show numerically that sinusoidal modulation of the dc current injected into the model neuron can switch it from periodic to burst firing, or vice versa, depending on the frequency of modulation and the distance to the burst excitability threshold. This is explained by mapping resonance tongues in parameter space. We also show that such a model neuron can undergo stochastic resonance near the transition from periodic to burst firing, as a result of the burst excitability, regardless of the location (soma or dendrite) of the signal and noise. The novelty is that the "output event" is now a burst rather than a single action potential, and the neuron returns to almost periodic firing between bursts, rather than to the vicinity of a fixed point. Since the neuron under study is a sensory neuron that must encode signals with varying temporal structure in the presence of considerable intrinsic noise, these aspects are of potential importance to electrosensory processing and also to other bursting neurons that have periodic input.

Coupling patterns between spontaneous rhythms and respiration in cardiovascular variability signals

We performed a quantitative study of coupling patterns between respiration and spontaneous rhythms of heart rate and blood pressure variability signals by using the Recurrence Quantification Analysis (RQA). We applied RQA to both simulated and experimental data obtained in control breathing at three different frequencies (0.25, 0.20, and 0.13 Hz) from ten normal subjects. RQA succeeded in quantifying different degrees of non-linear coupling associated to several interference patterns. We found higher degrees of non-linear coupling when the respiratory frequency was close to the spontaneous Low Frequency (LF) rhythm (0.13 Hz), or almost twice the LF frequency (0.2 Hz), whereas weaker coupling was observed when the respiratory frequency was 0.25 Hz. Clinical applications of our approach should focus on new experimental protocols, featuring the stimulation of one of the two branches of the autonomic nervous system (ANS) or aimed at the analysis of pathologies linked to the ANS.

Phase relationships between two or more interacting processes from one-dimensional time series. I. Basic theory

A general approach is developed for the detection of phase relationships between two or more different oscillatory processes interacting within a single system, using one-dimensional time series only. It is based on the introduction of angles and radii of return times maps, and on studying the dynamics of the angles. An explicit unique relationship is derived between angles and the conventional phase difference introduced earlier for bivariate data. It is valid under conditions of weak forcing. This correspondence is confirmed numerically for a nonstationary process in a forced Van der Pol system. A model describing the angles' behavior for a dynamical system under weak quasiperiodic forcing with an arbitrary number of independent frequencies is derived.

Dynamical behavior of a pacemaker neuron model with fixed delay stimulation

In physiological and pathological conditions, many biological oscillators, such as pacemaker cells, operate under the influence of feedbacks. Fixed delay stimulation is a standard preparation to evaluate the effects of such influences. Through the study of the Hodgkin-Huxley model, we show that such recurrent excitation can lead to regular and irregular discharge trains with interdischarge intervals that are up to several multiples of the period of the oscillator. In other words, we show that recurrent excitation can considerably slow down the firings of the pacemaker. This result contrasts with previous studies of similar preparations that have reported that fixed delay stimulation leads to a bursting pattern in which regimes of high-frequency firing alternate with periods of quiescence. We elucidate the mechanisms underlying the behavior of the oscillator under fixed delay perturbation through the analysis of the dynamics of a well-known two-dimensional oscillator, namely, the Poincaré oscillator.

Modelling the dynamics of angles of human R-R intervals

Heart rate variability (HRV) data from young healthy humans is expanded into two components, namely, the angles and radii of a map of R-R intervals. It is shown that. for most subjects at rest breathing spontaneously, the map of successive angles reveals a highly deterministic structure after the frequency range below approximately 0.05 Hz has been filtered out. However, no obvious low-dimensional structure is found in the map of successive radii. A recently proposed model describing the map of angles for a periodic self-oscillator under external periodic and quasiperiodic forcing is successfully applied to model the dynamics of such angles.

Characteristics of a piecewise smooth area-preserving map

We are reporting a study carried out in a system concatenated by two area-preserving maps. The system can be viewed as a model of an electronic relaxation oscillator with over-voltage protection. We found that a border-collision bifurcation may interrupt a period-doubling bifurcation cascade, and that some special features, such as "quasicoexisting periodic orbits crossing border" as well as the transition between "quasitransience" and chaotic orbits, accompany the process. These features belong to the so-called "quasidissipative" properties. Here "quasitransience" denotes the behavior of iterations outside elliptic islands. They are "attracted" to the islands. As soon as it reaches the islands, the iteration follows the conservative regulations exactly. This induces a kind of escaping from strange sets. The scaling behavior of the escaping rate is obtained numerically.

On the firing maps of a general class of forced integrate and fire neurons

Integrate and fire processes are fundamental mechanisms causing excitable and oscillatory behavior. Van der Pol [Philos. Mag. (7) 2 (11) (1926) 978] studied oscillations caused by these processes, which he called 'relaxation oscillations' and pointed out their relevance, not only to engineering, but also to the understanding of biological phenomena [Acta Med. Scand. Suppl. CVIII (108) (1940) 76], like cardiac rhythms and arrhythmias. The complex behavior of externally stimulated integrate and fire oscillators has motivated the study of simplified models whose dynamics are determined by iterations of 'firing circle maps' that can be studied in terms of Poincaré's rotation theory [Chaos 1 (1991) 20; Chaos 1 (1991) 13; SIAM J. Appl. Math. 41 (3) (1981) 503]. In order to apply this theory to understand the responses and bifurcation patterns of forced systems, it is fundamental to determine the regions in parameter space where the different regularity properties (e.g., continuity and injectivity) of the firing maps are satisfied. Methods for carrying out this regularity analysis for linear systems, have been devised and the response of integrate and fire neurons (with linear accumulation) to a cyclic input has been analyzed [SIAM J. Appl. Math. 41 (3) (1981) 503]. In this paper we are concerned with the most general class of forced integrate and fire systems, modelled by one first-order differential equation. Using qualitative analysis we prove theorems on which we base a new method of regularity analysis of the firing map, that, contrasting with methods previously reported in the literature, does not requires analytic knowledge of the solutions of the differential equation and therefore it is also applicable to non-linear integrate and fire systems. To illustrate this new methodology, we apply it to determine the regularity regions of a non-linear example whose firing maps undergo bifurcations that were unknown for the previously studied linear systems.

Synchronization and rhythmic processes in physiology

Complex bodily rhythms are ubiquitous in living organisms. These rhythms arise from stochastic, nonlinear biological mechanisms interacting with a fluctuating environment. Disease often leads to alterations from normal to pathological rhythm. Fundamental questions concerning the dynamics of these rhythmic processes abound. For example, what is the origin of physiological rhythms? How do the rhythms interact with each other and the external environment? Can we decode the fluctuations in physiological rhythms to better diagnose human disease? And can we develop better methods to control pathological rhythms? Mathematical and physical techniques combined with physiological and medical studies are addressing these questions and are transforming our understanding of the rhythms of life.

Phase synchronization between several interacting processes from univariate data

A novel approach is suggested for detecting the presence or absence of synchronization between two or three interacting processes with different time scales in univariate data. It is based on an angle-of-return-time map. A model is derived to describe analytically the behavior of angles for a periodic oscillator under weak periodic and quasiperiodic forcing. An explicit connection is demonstrated between the return angle and the phase of the external periodic forcing. The technique is tested on simulated nonstationary data and applied to human heart rate variability data.

Transient phase locking patterns among respiration, heart rate and blood pressure during cardiorespiratory synchronisation in humans

The interactions between respiration, heart rate and blood pressure variability (HRV, BPV), are considered to be of paramount importance for the study of the functional organisation of the autonomic nervous system (ANS). The aim of the reported study is to detect and classify the intermittent phase locking (PL) phenomena between respiration, HRV and BPV during cardiorespiratory synchronisation experiments, by using the following time-domain techniques: Poincaré maps, recurrence plots, time-space separation plots and frequency tracking locus. The experimental protocol consists of three stages, with normal subjects in paced breathing at 15, 12 and 8 breaths min-1. Transient phenomena of coordination between respiration and the major rhythms of HRV and BPV (low and high frequency, LF and HF) have been detected and classified: no interaction between LF and HF rhythms at 15 breaths min-1; short time intervals of stable 1:2 frequency and phase synchronisation during the 12 breaths min-1 stage; 1:1 PL during the 8 breaths min-1 stage. 1:1 and 1:2 PL phenomena occurred when the respiration frequency was quite close to the LF frequency or when it was about twice the LF frequency, respectively. The complex organisation of the ANS seems to provoke transient rather than permanent PL phenomena between the co-ordinating components of respiration and cardiovascular variability series.

Piecewise linear models for the quasiperiodic transition to chaos

We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode locking and the quasiperiodic transition to chaos. For instance, for these families, we obtain complete solutions to several questions still largely unanswered for families of smooth circle maps. Our main results describe (1) the sets of maps in these families having some prescribed rotation interval; (2) the boundaries between zero and positive topological entropy and between zero length and non-zero length rotation interval; and (3) the structure and bifurcations of the attractors in one of these families. We discuss the interpretation of these maps as low-order spline approximations to the classic "sine-circle" map and examine more generally the implications of our results for the case of smooth circle maps. We also mention a possible connection to recent experiments on models of a driven Josephson junction. (c) 1996 American Institute of Physics.

Controlling the dynamical behavior of a circle map model of the human heart

One-dimensional circle maps are good models for describing the nonlinear dynamical behavior of two interacting oscillators. They have been employed to characterize the interaction between a periodic external forcing stimulus and an in vitro preparation of chick embryonic cardiac cells. They have also been used to model some human cardiac arrythmias such as modulated ventricular parasystole. In this paper, we describe several techniques involving engineering feedback control theory applied to a circle map model of human heart parasystole. Through simulations of the mathematical model, we demonstrate that a desired target phase relationship between the normal sinus rhythm and an abnormal ectopic pacemaker can be achieved rapidly with low-level external stimulation applied to the system. Specifically, we elucidate the linear, self-tuning, and nonlinear feedback approaches to control. The nonlinear methods are the fastest and most accurate, yet the most complex and computationally expensive to implement of the three types. The linear approach is the easiest to implement but may not be accurate enough in real applications, and the self-tuning methods are a compromise between the other two. The latter was successful in tracking a variety of period-1, period-2, and period-3 target phase trajectories of the heart model.

A decoding problem in dynamics and in number theory

Given a homeomorphism f of the circle, any splitting of this circle in two semiopen arcs induces a coding process for the orbits of f, which can be determined by recording the successive arcs visited by the orbit. The problem of describing these codes has a two hundred year history (that we briefly recall) in the particular case when the arcs are limited by a point and its image; in modern language, it is the kneading theory of such maps, and as such is relevant for our understanding of dynamical problems involving oscillations. This paper deals with questions attached to the general case, a problem considered by many mathematicians in the 50's and 60's in the case where f is a rotation, and which has recently found some applications in physiology. We show that, except for trivial cases, any code determines the rotation number, up to the orientation, of the homeomorphism which generates it. In the case the code is periodic, we can also determine whether or not it can be generated in this way. An equivalent problem in arithmetic consists of finding +/-p, knowing a collection of classes in Z/qZ of the form {m,m+p,.,m+(k-1)p}, where 2</=k</=q-2 and p and q are relatively prime. We describe this equivalence, and give simple solutions of the decoding problem both in the dynamical context and in the number theoretic context.