Phase locking, period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells

Chaos and peak-to-peak dynamics in a plankton-fish model

A plankton-fish model, comprising phosphorus, algae, zooplankton, and young fish, with light intensity and water temperature varying periodically with the seasons, is analyzed in this paper. For realistic values of the parameters the model behaves chaotically, but its dynamics within the strange attractor can be described by a few one-dimensional maps that allow one to forecast the next yearly peak of plankton or fish from the last peaks. This property is an unambiguous mark of a special form of chaos. Unfortunately, the estimate of such peak-to-peak maps from field data is possible only if plankton or young fish biomass has been sampled accurately and frequently for a paramount number of years. In conclusion, the analysis shows that it might be that plankton dynamics are characterized by an interesting and peculiar form of chaos, but that inferences from recorded data on the existence of these forms of chaos are premature.

Synchronous period-doubling in flicker vision of salamander and man

Periodic flashes of light have long served to probe the temporal properties of the visual system. Here we show that during rapid flicker of high contrast and intensity the eye reports to the brain only every other flash of light. In this regime, retinal ganglion cells of the salamander fire spikes on alternating flashes. Neurons across the entire retina are locked to the same flashes. The effect depends sharply on contrast and flash frequency. It results from a period-doubling bifurcation in retinal processing, and a simple model of nonlinear feedback reproduces the phenomenon. Pharmacological studies indicate that the critical feedback interactions require only cone photoreceptors and bipolar cells. Analogous period-doubling is observed in the human visual system. Under bright full-field flicker, the electroretinogram (ERG) shows a regime of period-doubling between 30 and 70 Hz. In visual evoked potentials from the occiput, the subharmonic component is even stronger. By analyzing the accompanying perceptual effects, we find that retinal period-doubling begins in the periphery of the visual field, and that it is the cause of a long mysterious illusory flicker pattern.

Heart rate dynamics during accentuated sympathovagal interaction

Concomitant sympathetic and vagal activation can occur in various physiological conditions, but there is limited information on heart rate (HR) behavior during the accentuated sympathovagal antagonism. Beat-to-beat HR and blood pressure were recorded during intravenous infusion of incremental doses of norepinephrine in 18 healthy male volunteers (mean age 23 +/- 5 yr). HR and blood pressure spectra and two-dimensional Poincaré plots were generated from the baseline recordings and from the recordings at different doses of norepinephrine. The mean blood pressure increased (from 90 +/- 7 to 120 +/- 9 mmHg, P < 0.001), HR decreased (from 60 +/- 9 to 48 +/- 7 beats/min, P < 0.001), and the high-frequency spectral component of HR variability increased (P < 0.001) during the norepinephrine infusion as evidence of accentuated sympathovagal interaction. Abrupt aperiodic changes in sinus intervals that were not related to respiratory cycles or changes in blood pressure occurred in 14 of 18 subjects during the norepinephrine infusions. These fluctuations in sinus intervals resulted in a complex or parabola-shaped structure of the Poincaré plots of successive R-R intervals and a widening of the high-frequency spectral peak. In four subjects, the abrupt fluctuations in sinus intervals were followed by a sudden onset of fixed R-R interval dynamics with a loss of respiratory modulation of HR, resulting in a torpedo-shaped structure of the Poincaré plots. These data show that HR behavior becomes remarkably unstable during accentuated sympathovagal interaction, resembling stochastic dynamics or deterministic chaotic behavior. These features of HR dynamics can be better identified by dynamic analysis of beat-to-beat behavior of R-R intervals than by traditional analysis techniques of HR variability.

Control of chaos in excitable physiological systems: A geometric analysis

Model-independent chaos control techniques are inherently well-suited for the control of physiological systems for which quantitative system models are unavailable. The proportional perturbation feedback (PPF) control paradigm, which uses electrical stimulation to perturb directly the controlled system variable (e.g., the interbeat or interspike interval), was developed for excitable physiological systems that do not have an easily accessible system parameter. We develop the stable manifold placement (SMP) technique, a PPF-type technique which is simpler and more robust than the original PPF control algorithm. We use the SMP technique to control a simple geometric model of a chaotic system in the neighborhood of an unstable periodic orbit (UPO). We show that while the SMP technique can control a chaotic system that has UPO dynamics which are characterized by one stable manifold and one unstable manifold, the success of the SMP technique is sensitive to UPO parameter estimation errors. (c) 1997 American Institute of Physics.

Spirals, chaos, and new mechanisms of wave propagation

The chaos theory is based on the idea that phenomena that appear disordered and random may actually be produced by relatively simple deterministic mechanisms. The disordered (aperiodic) activation that characterizes a chaotic motion is reached through one of a few well-defined paths that are characteristic of nonlinear dynamical systems. Our group has been studying VF using computerized mapping techniques. We found that in electrically induced VF, reentrant wavefronts (spiral waves) are present both in the initial tachysystolic stage (resembling VT) and the later tremulous incoordination stage (true VF). The electrophysiological characteristics associated with the transition from VT to VF is compatible with the quasiperiodic route to chaos as described in the Ruelle-Takens theorem. We propose that specific restitution of action potential duration (APD) and conduction velocity properties can cause a spiral wave (the primary oscillator) to develop additional oscillatory modes that lead to spiral meander and breakup. When spiral waves begin to meander and are modulated by other oscillatory processes, the periodic activity is replaced by unstable quasiperiodic oscillation, which then undergoes transition to chaos, signaling the onset of VF. We conclude that VF is a form of deterministic chaos. The development of VF is compatible with quasiperiodic transition to chaos. These results indicate that both the prediction and the control of fibrillation are possible based on the chaos theory and with the advent of chaos control algorithms.

Quasiperiodicity and chaos in cardiac fibrillation

In cardiac fibrillation, disorganized waves of electrical activity meander through the heart, and coherent contractile function is lost. We studied fibrillation in three stationary forms: in human chronic atrial fibrillation, in a stabilized form of canine ventricular fibrillation, and in fibrillation-like activity in thin sheets of canine and human ventricular tissue in vitro. We also created a computer model of fibrillation. In all four studies, evidence indicated that fibrillation arose through a quasiperiodic stage of period and amplitude modulation, thus exemplifying the "quasiperiodic transition to chaos" first suggested by Ruelle and Takens. This suggests that fibrillation is a form of spatio-temporal chaos, a finding that implies new therapeutic approaches.

Nonlinear dynamics, chaos and complex cardiac arrhythmias

Periodic stimulation of a nonlinear cardiac oscillator in vitro gives rise to complex dynamics that is well described by one-dimensional finite difference equations. As stimulation parameters are varied, a large number of different phase locked and chaotic rhythms is observed. Similar rhythms can be observed in the intact human heart when there is interaction between two pacemaker sites. Simplified models are analysed, which show some correspondence to clinical observations.

Forced oscillations of a self-oscillating bimolecular surface reaction model

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-periodic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations.

The origins of aperiodicities in sensory neuron entrainment

Aperiodic entrainment to rhythmic sensory input was obtained with either a single neuron or an excitatory network model, without addition of a stochastic or "noisy" element. The entrainment properties of primary sensory neurons were well captured by the dynamics of the Hodgkin-Huxley ordinary differential equations with a quiescent resting state or threshold for spike output. The frequency-amplitude parameter space was compressed and aperiodic regimes were small in comparison to those of periodically activated pacemaker-like neurons. Transitions between phase-locked and aperiodic entrainment patterns were predictable and determined by the equation dynamics, supporting the contention that some aperiodicities observed in situ arise from the inherent membrane properties of neurons. When the rhythmically activated neuron was embedded in an excitatory network of Hodgkin-Huxley neurons with heterogeneous synaptic delays, aperiodic entrainment patterns were more frequently encountered and these were associated with asynchronous output from the network. Embedding the rhythmically activated neuron in a network with synaptic delays greatly reduced the range of entrained spike frequencies and increased the variability in the neuronal firing. The temporal coding of sensory stimuli may be dependent on these findings. Sensory stimuli are signaled in the periphery by a mixture of periodic and irregular interspike intervals. Most models of such temporal codes assume intrinsic rhythmicity arising from the ionic currents, with variations attributed to membrane or synaptic noise. In contrast, we demonstrate irregular neural codes that arise completely in the absence of noise. In the proposed model, the sources of these irregular sensory patterns are the extensive cross-connections and resultant interactions between neurons. The balance between the regular and irregular entrainment of a neuron in situ could uniquely identify a stimulus. Other biological mechanisms of modifying the entrainment properties and promoting aperiodic entrainment are discussed.

Simulation of re-entry in a piece of myocardial tissue: strong sensitivity to spatial and temporal conditions

A simple model for the simulation of re-entrant excitation was created. The model consists of a matrix of 15x15 compartments, where each compartment has its own action potential that depends dynamically on four ion currents (INa, ICa, Ik and Ib) having time and voltage-dependent activation and inactivation kinetics. The compartments were combined with resistors to simulate electrotonic interaction. At short excitation intervals the action potential was shortened in duration, and at even shorter coupling intervals decremental propagation occurred. Re-entry around an obstacle could be elicited in response to a properly timed extra stimulus. A time dependent unidirectional block was made by making some of the action potentials longer in duration. An obstacle was not a necessary substrate for re-entry, but the timing of the extra stimulus was critical. In the presence of an obstacle, the induction of re-entry was critically dependent on the shape of the obstacle. The most important result of the simulations is that the system is highly sensitive to the initial spatial and temporal conditions. These sensitivities are generic features of dynamic systems that are described by non-linear differential equations and are typical for chaotic systems. The system studied shows features associated with deterministic chaos.

In vitro study of phase resetting and phase locking in a time-comparison circuit in the electric fish, Eigenmannia

The electric fish Eigenmannia generates on oscillating weak electric field. The amplitude and timing information of this electric field is perceived by electroreceptors distributed on its skin. The pathway of timing information, consisting of spherical cells and giant cells, was studied in an in vitro preparation. The giant cells were identified to be endogenous oscillators and thus have the functional advantage of phase locking more easily to a periodic stimulus with a frequency in the range of the intrinsic frequency. Their spontaneous rhythmic activity was perturbed by delivering excitatory single pulses or periodic pulses via their synaptic inputs. The regular and irregular dynamics produced by periodic stimulation were discussed in the context of a mathematical analysis of the response to single pulses. Ambiguous representations of the timing of the stimulus pulse were observed and could be related to this analysis. Some spontaneously firing cells could be silenced with periodic excitatory stimulation in a narrow frequency and amplitude range. Some irregularly firing cells continued to fire periodically for several seconds after phase locking to a periodic stimulus. This study is the first description of an endogenous oscillator in a system devoted to the precise timing of sensory events.

The D-1 dopamine receptor: past, present and future an idiosyncratic review

1. Key events in the experimental investigation of the D-1 dopamine receptor are reviewed. 2. The efficacy of D-1 receptor agonists in the treatment of experimental parkinsonism in MPTP-treated primates is demonstrated. The diminished dyskinetic liability of D-1 agonists is discussed. 3. The significance of the dopa-induced dyskinesias is discussed from the perspective of deterministic chaos. The unpredictibility and irreproducibility of dyskinetic movements is highlighted and compared with features of the logistic equation. 4. The authors propose that the dopa-induced dyskinesias should be considered to be a manifestation of a chaotic process within the basal ganglia. The loss of the dopaminergic innervation and the subsequent repeated exposure to dopamine (derived from the exogenous dopa administered to the subjects) alters the response properties of the basal ganglia circuitry so that stimulation of dopamine receptors now elicits the dyskinetic movements.

Age-structured and two-delay models for erythropoiesis

An age-structured model is developed for erythropoiesis and is reduced to a system of threshold-type differential delay equations using the method of characteristics. Under certain assumptions, this model can be reduced to a system of delay differential equations with two delays. The parameters in the system are estimated from experimental data, and the model is simulated for a normal human subject following a loss of blood. The characteristic equation of the two-delay equation is analyzed and shown to exhibit Hopf bifurcations when the destruction rate of erythrocytes is increased. A numerical study for a rabbit with autoimmune hemolytic anemia is performed and compared with experimental data.

A minimal single-channel model for the regularity of beating in the sinoatrial node

It has been suggested that the normal irregular beating of the heart is a manifestation of deterministically chaotic dynamics. Evidence proffered in support of this hypothesis includes a 1/f-like power spectrum, a small noninteger correlation dimension, and self-similarity of the time series. The major cause of the normal fluctuations in heart rate is the impingement of several neural and hormonal control systems upon the sinoatrial node, the natural pacemaker of the heart. However, intrinsic fluctuations of beat rate can be seen in the isolated node, devoid of all neural and hormonal inputs, and even in a single cell isolated from the node. The electrical activity in such a single cell is generated by ions flowing through discrete channels in the cell membrane.We decided to test the hypothesis that the fluctuations in beat rate in a single cell might be due to the fluctuations in the activity of this population of single channels. We thus assemble a model consisting of 6000 channels and probe its dynamics. Each channel has one or more gates, all of which must be open to allow current to flow through the channel. Since these gates are thought to open and close in a random manner, we model each gate by a Markov process, assigning a pseudorandom number to each gate every time that it changes state from open to closed or vice versa. This number, in conjunction with the classical voltage-dependent Hodgkin-Huxley-like rate constants that control the speed with which a gate will open or close, then determines when that gate will next change state. We also employ a second method that is much more efficient computationally, in which one computes the lifetime of the ensemble of 6000 channels. We show that the Monte Carlo model has behavior consistent with the hypothesis that the irregular beating seen experimentally in single nodal cells is due to the (pseudo)random opening and closing of single channels. However, since the pseudorandom number generator used in the simulations is deterministic, one cannot state that the activity in the model is random (or stochastic). Thus, it would be premature to claim that the irregularity of beating in a single nodal cell is accounted for by the stochastic behavior of a population of a few thousand single channels lying in the membrane of the cell. Finally, we consider some implications of our work for the naturally occurring in situ fluctuations in heart rate ("heart rate variability"). (c) 1995 American Institute of Physics.

Chaos and chaos control in biology

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Variance ECG detection of coronary artery disease--a comparison with exercise stress test and myocardial scintigraphy

Variance electrocardiogram (ECG) is a newly developed method by which resting ECG is registered with 24 leads during 220 beats. The temporal beat-to-beat QRS microamplitude variability is computed and a nondimensional diagnostic variance ECG coronary artery disease (CAD) index is derived from it. Consecutive outpatients (n = 160) were referred to myocardial scintigraphy (SPECT) investigation for the evaluation of angina pectoris. The variance ECG CAD index was compared with a symptom-limited exercise stress test and SPECT during and after the exercise test and with coronary angiography (n = 67). Discriminant accuracy was tested with receiver-operating characteristics (ROC). Relative to angiographic coronary pathology (prevalence 0.85), diagnostic information for the variance ECG CAD index and for SPECT were both p < 0.001, while the outcome of the exercise stress test was non-contributory. Prevalence of persistent or transient perfusion defects at SPECT was 0.59. The exercise stress test had a diagnostic capacity of p < 0.01 for transient perfusion defects and variance ECG CAD index showed a high diagnostic performance (p < 0.001) for persistent perfusion defects. Overall pathology at SPECT was better (p < 0.05) identified by variance ECG CAD index than by symptom-limited exercise stress test. It was concluded that in this high prevalence population the variance ECG CAD index has a diagnostic capacity at least as good as that of SPECT and better than that of the exercise stress test. The variance ECG CAD index was strongly diagnostic for persistent perfusion defects while exercise stress test was slightly diagnostic for transient perfusion defects. Therefore, the two tests provide complementary diagnostic information.

Generation of periodic and chaotic bursting in an excitable cell model

There are interesting oscillatory phenomena associated with excitable cells that require theoretical insight. Some of these phenomena are: the threshold low amplitude oscillations before bursting in neuronal cells, the damped burst observed in muscle cells, the period-adding bifurcations without chaos in pancreatic beta-cells, chaotic bursting and beating in neurons, and inverse period-doubling bifurcation in heart cells. The three variable model formulated by Chay provides a mathematical description of how excitable cells generate bursting action potentials. This model contains a slow dynamic variable which forms a basis for the underlying wave, a fast dynamic variable which causes spiking, and the membrane potential which is a dependent variable. In this paper, we use the Chay model to explain these oscillatory phenomena. The Poincaré return map approach is used to construct bifurcation diagrams with the 'slow' conductance (i.e., gK, C) as the bifurcation parameter. These diagrams show that the system makes a transition from repetitive spiking to chaotic bursting as parameter gK, C is varied. Depending on the time kinetic constant of the fast variable (lambda n), however, the transition between burstings via period-adding bifurcation can occur even without chaos. Damped bursting is present in the Chay model over a certain range of gK, C and lambda n. In addition, a threshold sinusoidal oscillation was observed at certain values of gK, C before triggering action potentials. Probably this explains why the neuronal cells exhibit low-amplitude oscillations before bursting.

Temporal patterns of alpha 1-receptor stimulation regulate amplitude and frequency of calcium transients

The pulsatile release of neurotransmitters and many hormones might encode specific biological information according to temporal pattern. We tested this hypothesis by applying pulsed alpha 1-adrenoceptor stimulation to single aequorin-injected hepatocytes. The amplitude of free Ca2+ transients induced by rapid phenylephrine pulses (20-s interpulse interval) and continuous stimulation was similar (approximately 640 nM) but increased to approximately 1,000 nM as the interpulse interval was increased to 120 s. The same overall response was maintained despite a 13-fold reduction in average phenylephrine concentration. Some regimes of pulsed phenylephrine stimulation could give a higher frequency of pulsed phenylephrine stimulation could give a higher frequency of free calcium oscillations than continuous stimulation, or more rapid stimulation when some agonist pulses failed to elicit a free Ca2+ transient. For the same average phenylephrine concentration (0.3-0.6 microM), pulsed regimes could result in significantly higher frequencies and integrated responses than constant application. The lags between phenylephrine pulses and free Ca2+ transients reduced as the period between pulses increased. The amplitude and lag data are consistent with a refractory period of 18 s and a recovery phase with a time constant of approximately 100 s, perhaps corresponding to dephosphorylation of alpha 1-adrenoceptors phosphorylated by protein kinase C during each free Ca2+ transient.

A quantitative measurement of spatial order in ventricular fibrillation

INTRODUCTION

The degree of organization in ventricular fibrillation (VF) is not known. As an objective measurement of spatial order, spatial correlation functions and their characteristic lengths were estimated from epicardial electrograms of pigs in VF.

METHODS AND RESULTS

VF was induced by premature stimulation in five pigs. Electrograms were simultaneously recorded with a 22 x 23 array of unipolar electrodes spaced 1.12 mm apart. Data were obtained by sampling the signals at 2000 Hz for 20 minutes immediately after the initiation of FV. Correlations between all pairs of signals were computed at various times. Correlation lengths were estimated from the decay of average correlation as a function of electrode separation. The correlation length of the VF in pigs was found to be approximately 4 to 10 mm, varying as fibrillation progressed. The degree of correlation decreased in the first 4 seconds after fibrillation then increased over the next minute.

CONCLUSION

The correlation length is much smaller than the scale of the heart, suggesting that many independent regions of activity exist on the epicardium at any one time. On the other hand, the correlation length is 4 to 10 times the interelectrode spacing, indicating that some coherence is present. These results imply that the heart behaves during VF as a high dimensional, but not random, system involving many spatial degrees of freedom, which may explain the lack of convergence of fractal dimension estimates reported in the literature. Changes in the correlation length also suggest that VF reorganizes slightly in the first minute after an initial breakdown in structure.

Electrocardiogram signal variance analysis in the diagnosis of coronary artery disease--a comparison with exercise stress test in an angiographically documented high prevalence population

Variance electrocardiography (variance ECG) is a new resting procedure for detection of coronary artery disease (CAD). The method measures variability in the electrical expression of the depolarization phase induced by this disease. The time-domain analysis is performed on 220 cardiac cycles using high-fidelity ECG signals from 24 leads, and the phase-locked temporal electrical heterogeneity is expressed as a nondimensional CAD index (CAD-I) with the values of 0-150. This study compares the diagnostic efficiency of variance ECG and exercise stress test in a high prevalence population. A total of 199 symptomatic patients evaluated with coronary angiography was subjected to variance ECG and exercise test on a bicycle ergometer as a continuous ramp. The discriminant accuracy of the two methods was assessed employing the receiver operating characteristic curves constructed by successive consideration of several CAD-I cutpoint values and various threshold criteria based on ST-segment depression exclusively or in combination with exertional chest pain. Of these patients, 175 with CAD (> or = 50% luminal stenosis in 1 + major epicardial arteries) presented a mean CAD-I of 88 +/- 22, compared with 70 +/- 21 in 24 nonaffected patients (p < 0.01). Variance ECG provided a stochastically significant discrimination (p < 0.01) which was matched by exercise test only when chest pain variable was added to ST-segment depression as a discriminating criterion. Even then, the exercise test diagnosed single-vessel disease with a significantly lower sensitivity. At a cutpoint of CAD-I > or = 70, compared with ST-segment depression > or = 1 mm combined with exertional chest pain, the overall sensitivity of variance ECG was significantly higher (p < 0.01) than that of exercise test (79 vs. 48%). When combined, the two methods identified 93% of coronary angiography positive cases. Variance ECG is an efficient diagnostic method which compares favorably with exercise test for detection of CAD in high prevalence population.

Oscillators in the human body and circular-muscle gymnastics

There is a growing body of literature about the role of oscillators in the living body, and about the interactions between different oscillators. Considering the importance of endogenous oscillators in regulating the body's functions, and the existence of 'dynamical diseases', diseases of control systems which involve oscillators in the body, a way to mend dysfunctioning oscillators seems to be needed. Circular-muscle gymnastics, a method of physical activity which has been developed in Israel, reveals some phenomena which may point in a promising direction. Some of these phenomena call to mind known facts and theories about oscillators and their effects.

Periodicity and chaos in a photosynthetic system

In this paper experimental results of investigation of the oscillations in a photosynthetic system are presented and a model for their interpretation is suggested. Periodicities in photosynthetic systems detected in earlier studies by physical chemical methods can be also detected by means of recording the potential difference between two point electrodes. The observed dependences demonstrate a wide range of various types of behaviour of the system, working, e.g. in periodic, quasiperiodic, chaotic or 'pulse' regimes. Since the until-now-used 2-dimensional theoretical model, based on the existence of two dominant autocatalytical processes, appeared not to be sufficient for explaining such types of the regimes, a generalized 3-dimensional autocatalytical model is suggested, which is able to explain all the above mentioned photosynthetic regimes.

Feedback and delays in neurological diseases: a modeling study using dynamical systems

Numerous regulatory mechanisms in motor control involve the presence of time delays in the controlled behavior of the system. Experimentally, we have shown that an increase of the time delay in visual feedback induces different oscillations in control subjects and in patients with neurological diseases during the performance of a simple compensatory tracking task. A preliminary model is proposed to describe the oscillations observed in control subjects and in patients with neurological diseases. The influence of delays in two feedback loops are the main components of the motor control circuitry involved in this task and are studied from an analytical and physiological perspective. We analytically determine the influence in the model of each of these delays on the stability of the finger position. In addition, the influence of stochastic elements ("noise") in the modeling equation is seen to contribute qualitatively to a more accurate reproduction of experimental traces in patients with Parkinson's disease but not in patients with cerebellar disease.

Model of chemically excitable membranes generating autonomous chaotic oscillations

A simple mathematical model of the chemically excitable membranes leading to autonomous chaotic oscillations is presented. The model assumes two kinds of autocatalytic ion channels, one is for cations and the other is for anions. Self-consistency between the ion distributions and the electric potentials is taken into account by including the counter ions explicitly. Cations and anions pass through their own channels with their permeabilities changing nonlinearly with the densities of ions at the surfaces of the membrane. Cation and anion transport systems then form two subsystems that oscillate and interact with each other through the membrane potential. When the coupling strength between the two ion systems and adsorption rate of ions to channels are varied, various types of chaotic oscillations are generated autonomously, i.e., without a stimulating periodic force. Experimental evidence to the present model is discussed. It is suggested that endogenous chaos in biological systems may appear from the electric coupling among different kinds of ion transport systems.

A reduction in the correlation dimension of heartbeat intervals precedes imminent ventricular fibrillation in human subjects

Reduced reflexive control of heartbeat intervals occurs with advanced heart disease and is an independent risk factor for mortality. Based on a previous study of experimental myocardial infarction in pigs, we hypothesized that a deterministic measure of heartbeat dynamics, the correlation dimension of R-R intervals (D2), may be a better predictor of risk than a stochastic measure, such as the standard deviation (SD). We determined the point estimates of the heartbeat D2 (i.e., PD2s) in Holter electrocardiographic recordings from 11 high-risk patients who manifested ventricular fibrillation (VF) during the recording and in high-risk controls having only nonsustained ventricular tachycardia (14 patients) or premature ventricular complexes (13 patients). We found that PD2 reduction (i.e., PD2s < 1.2) precedes lethal arrhythmias by hours, but is not reduced in high-risk controls (p < 0.001; sensitivity, 91%; specificity, 85%). Heartbeat SD did not discriminate among the patients. Thus PD2 of heartbeat intervals may provide an important diagnostic test and early warning sign of VF.

Controlling cardiac chaos

The extreme sensitivity to initial conditions that chaotic systems display makes them unstable and unpredictable. Yet that same sensitivity also makes them highly susceptible to control, provided that the developing chaos can be analyzed in real time and that analysis is then used to make small control interventions. This strategy has been used here to stabilize cardiac arrhythmias induced by the drug ouabain in rabbit ventricle. By administering electrical stimuli to the heart at irregular times determined by chaos theory, the arrhythmia was converted to periodic beating.

Studies on re-entrant arrhythmias and ectopic beats in excitable tissues by bifurcation analyses

A phase-plane bifurcation analysis is a useful way to theoretically understand how various types of arrhythmias may arise from excitable tissues. In this paper, we have performed phase-plane bifurcation analysis to characterize arrhythmogenic states in excitable tissues. To achieve this, we have first formulated a model which is simple enough to be mathematically tractable, yet captures the non-linear features of cardiac excitation and conduction. In this model, single cells are connected in a circular fashion by gap conductances. Each cell carries the following two types of currents: a passive outward current and an inward "excitable" current which contains an activation and an inactivation gate. The activation gate is responsible for the upstroke of action potential and inactivation gate is responsible for the termination of the plateau potential. With this model, we have constructed bifurcation diagrams as a function of a bifurcation parameter. The parameter chosen as the bifurcation parameter has the property of raising maximum diastolic potential while shorting the refractory period. Our analysis revealed the existence of three distinct multi-stable phases in certain ranges of the bifurcation parameter: (1) bistability between a rotor and a quiescent state, (2) bistability between rotor and ectopic beats, and (3) three stable states co-existing among quiescent state, rotor, and ectopic beats. In these three regions, external impulses exert very distinct effects: In region 1, a brief current pulse can annihilate a re-entrant arrhythmia to quiescence. To initiate re-entry from a quiescent tissue, however, it takes two pulses (a primary pulse followed by a premature pulse at a site different from the "primary" site). In region 2, a brief pulse can convert a re-entrant arrhythmia to ectopic beats. To convert the ectopic beats back to circus movement, these beats have to be suppressed by a few brief current pulses to initiate one-way propagation. Depending on the frequency and strength of impulses in region 3, the tissue can switch back and forth among quiescence, circus movement, and ectopic beats. For comparison, we have also included a more complete Beeler-Reuter cardiac cell model in our analysis and obtained essentially the same results. From the behavioral similarities of these models, we conclude that re-entrant and ectopic arrhythmias must be intrinsic properties of excitable tissues and external stimuli can convert one mode of arrhythmia to another in the multistability regions.(ABSTRACT TRUNCATED AT 400 WORDS)

Microbial predation in a periodically operated chemostat: a global study of the interaction between natural and externally imposed frequencies

Predator-prey systems in continuously operated chemostats exhibit sustained oscillations over a wide range of operating conditions. When the chemostat is operated periodically, the interaction of the natural oscillation frequency with the external forcing gives rise to a wealth of dynamic behavior patterns. Using numerical bifurcation techniques, we perform a detailed computational study of these patterns and the transitions (local and especially global) between them as the amplitude and frequency of the forcing vary. The transition from low-forcing-amplitude quasiperiodicity to entrainment of the chemostat behavior by strong forcing (involving the concerted closing of resonance horns) is analyzed. We concentrate on certain strong resonance phenomena between the two frequencies and provide an extensive atlas of computed phase portraits for our model system. Our observations corroborate recent mathematical results and case studies of periodically forced chemical oscillators. In particular, the existence and relative succession of several distinct types of global bifurcations resulting in chaotic transients and multistability are studied in detail. The location in the operating diagram of several key codimension 2 local bifurcations of periodic solutions is computed, and their interaction with an interesting feature we name "real-eigenvalues horns" is examined.

Application of chaos theory to biology and medicine

The application of "chaos theory" to the physical and chemical sciences has resolved some long-standing problems, such as how to calculate a turbulent event in fluid dynamics or how to quantify the pathway of a molecule during Brownian motion. Biology and medicine also have unresolved problems, such as how to predict the occurrence of lethal arrhythmias or epileptic seizures. The quantification of a chaotic system, such as the nervous system, can occur by calculating the correlation dimension (D2) of a sample of the data that the system generates. For biological systems, the point correlation dimension (PD2) has an advantage in that it does not presume stationarity of the data, as the D2 algorithm must, and thus can track the transient non-stationarities that occur when the systems changes state. Such non-stationarities arise during normal functioning (e.g., during an event-related potential) or in pathology (e.g., in epilepsy or cardiac arrhythmogenesis). When stochastic analyses, such as the standard deviation or power spectrum, are performed on the same data they often have a reduced sensitivity and specificity compared to the dimensional measures. For example, a reduced standard deviation of heartbeat intervals can predict increased mortality in a group of cardiac subjects, each of which has a reduced standard deviation, but it cannot specify which individuals will or will not manifest lethal arrhythmogenesis; in contrast, the PD2 of the very same data can specify which patients will manifest sudden death. The explanation for the greater sensitivity and specificity of the dimensional measures is that they are deterministic, and thus are more accurate in quantifying the time-series. This accuracy appears to be significant in detecting pathology in biological systems, and thus the use of deterministic measures may lead to breakthroughs in the diagnosis and treatment of some medical disorders.

Evolution of rhythms during periodic stimulation of embryonic chick heart cell aggregates

During periodic stimulation of spontaneously beating chick heart cell aggregates, there is often an evolution of coupling patterns between the stimulator and the aggregate action potential. For example, at rapid stimulation frequencies, a rhythm that is initially 1:1 (stimulus frequency:aggregate frequency) can evolve to other rhythms such as 5:4 and 4:3. Time-dependent effects generated during periodic stimulation are characterized by three types of experiments to determine 1) the effect of periodic stimulation on the intrinsic cardiac beat rate (overdrive suppression), 2) the effect of periodic stimulation on the phase resetting properties of the aggregate, and 3) the time-dependent changes in the coupling patterns between the stimulator and the aggregate during periodic stimulation. The protocols involved variations in the duration and rate of periodic stimulation. A mathematical model is developed in the form of a two-dimensional finite difference equation based on the data from experiments 1 and 2. The model is used to predict the data generated by experiment 3. There is good correspondence with the experiments in that the theory reproduces complex transitions between various rhythms and displays irregular rhythms similar to those observed experimentally. These results have implications for the evolution of cardiac arrhythmias such as atrioventricular heart block and modulated parasystole.

Changes in heart rate variability during fainting

Studies of heart rate variability in people who faint may yield insights into normal physiologic mechanisms which probably are dynamic. These insights might be gained because fainting appears to be due to a breakdown of these mechanisms. Tilt table testing reliably induces fainting in patients with a history of fainting and can be used to study these mechanisms. During tilt tests ending in fainting heart rate changes markedly, with a loss of high-frequency components on power spectral analysis and a progressive slowing of overall sinus node discharge. These changes appear to be due to changes in efferent vagal nerve traffic. Several possible mechanisms of these changes in heart rate variability are discussed.

The estimation of the Kolmogorov entropy from a time series and its limitations when performed on EEG

A method to estimate a lower bound of the Kolmogorov entropy-the so called K2-entropy-from a time series is presented which avoids use of the generalized correlation integral. The influence of the norm is studied. The method is demonstrated on some standard examples. The entropy of the attractor apparent in the EEG of the foetal sheep is estimated and the results are compared with results obtained from synthesized data featuring some basic properties of EEG. This gives an insight into the limitations of the procedure.