Impulse responses of automaticity in the Purkinje fiber

Nonlinear dynamics in cardiology

The dynamics of many cardiac arrhythmias, as well as the nature of transitions between different heart rhythms, have long been considered evidence of nonlinear phenomena playing a direct role in cardiac arrhythmogenesis. In most types of cardiac disease, the pathology develops slowly and gradually, often over many years. In contrast, arrhythmias often occur suddenly. In nonlinear systems, sudden changes in qualitative dynamics can, counterintuitively, result from a gradual change in a system parameter-this is known as a bifurcation. Here, we review how nonlinearities in cardiac electrophysiology influence normal and abnormal rhythms and how bifurcations change the dynamics. In particular, we focus on the many recent developments in computational modeling at the cellular level that are focused on intracellular calcium dynamics. We discuss two areas where recent experimental and modeling work has suggested the importance of nonlinearities in calcium dynamics: repolarization alternans and pacemaker cell automaticity.

Annihilation of single cell neural oscillations by feedforward and feedback control

Annihilation of neural oscillation by localized electrical stimulation has been shown to be a promising treatment modality in a number of neural diseases like Parkinson disease or epilepsy. The contributions presented in this manuscript comprise newly developed stimulation schemes to achieve annihilation of action potential generation and action potential propagation. The ability to achieve oscillation annihilation is demonstrated in computer simulations with a single compartment nerve cell model (annihilation of action potential generation), and with a multi-compartment nerve fiber model (annihilation of action potential propagation). Additionally, we show superiority of the new feedback based schemes over the feedforward schemes in terms of effectiveness, phase robustness, and reduced sensitivity to disturbances. At the end we propose a conditioned switched feedback control regime to be applied in actual applications, where oscillation annihilation is needed.

Bistability dynamics in simulations of neural activity in high-extracellular-potassium conditions

Modulation of extracellular potassium concentration ([K](o)) has a profound impact on the excitability of neurons and neuronal networks. In the CA3 region of the rat hippocampus synchronized epileptiform bursts occur in conditions of increased [K](o). The dynamic nature of spontaneous neuronal firing in high [K](o) is therefore of interest. One particular interest is the potential presence of bistable behaviors such as the coexistence of stable repetitive firing and fixed rest potential states generated in individual cells by the elevation of [K](o). The dynamics of repetitive activity generated by increased [K](o) is investigated in a 19-compartment hippocampal pyramidal cell (HPC) model and a related two-compartment reduced HPC model. Results are compared with those for the Hodgkin-Huxley equations in similar conditions. For neural models, [K](o) changes are simulated as a shift in the potassium reversal potential (E(K)). Using phase resetting and bifurcation analysis techniques, all three models are shown to have specific regions of E(K) that result in bistability. For activity in bistable parameter regions, stimulus parameters are identified that switch high-potassium model behavior from repetitive firing to a quiescent state. Bistability in the HPC models is limited to a very small parameter region. Consequently, our results suggest that it is likely some HPCs in networks exposed to high [K](o) continue to burst such that a stable, quiescent network state does not exist. In [K](o) ranges where HPCs are not bistable, the population may still exhibit bistable behaviors where synchronous population events are reversibly annihilated by phase resetting pulses, suggesting the existence of a nonsynchronous network attractor.

Effect of syncytium structure of receptor systems on stochastic resonance induced by chaotic potential fluctuation

To study a role of syncytium structure of sensory receptor systems in the detection of weak signals through stochastic resonance, we present a model of a receptor system with syncytium structure in which receptor cells are interconnected by gap junctions. The apical membrane of each cell includes two kinds of ion channels whose gating processes are described by the deterministic model. The membrane potential of each cell fluctuates chaotically or periodically, depending on the dynamical state of collective channel gating. The chaotic fluctuation of membrane potential acts as internal noise for the stochastic resonance. The detection ability of the system increases as the electric conductance between adjacent cells generated by the gap junction increases. This effect of gap junctions arises mainly from the fact that the synchronization of chaotic fluctuation of membrane potential between the receptor cells is strengthened as the density of gap junctions is increased.

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.

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.

Effects of gap junction conductance on dynamics of sinoatrial node cells: two-cell and large-scale network models

A computational model of single rabbit sinoatrial (SA) node cells has been revised to fit data on regional variation of rabbit SA node cell oscillation properties. The revised model simulates differences in oscillation frequency, maximum diastolic potential, overshoot potential, and peak upstroke velocity observed in cells from different regions of the node. Dynamic properties of electrically coupled cells, each with different intrinsic oscillation frequency, are studied as a function of coupling conductance. Simulation results demonstrate at least four distinct regimes of behavior as coupling conductance is varied: a) independent oscillation (Gc < 1 pS); b) complex oscillation (1 < or = Gc < 220 pS); c) frequency, but not waveform entrainment (Gc > or = 220 pS); and d) frequency and waveform entrainment (Gc > or = 50 nS). The conductance of single cardiac myocyte gap junction channels is about 50 pS. These simulations therefore show that very few gap junction channels between each cell are required for frequency entrainment. Analyses of large-scale SA node network models implemented on the Connection Machine CM-200 supercomputer indicate that frequency entrainment of large networks is also supported by a small number of gap junction channels between neighboring cells.

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)

Phase resetting of embryonic chick atrial heart cell aggregates. Experiment and theory

The influence of brief duration current pulses on the spontaneous electrical activity of embryonic chick atrial heart cell aggregates was investigated experimentally and theoretically. A pulse could either delay or advance the time of the action potential subsequent to the pulse depending upon the time in the control cycle at which it was applied. The perturbed cycle length throughout the transition from delay to advance was a continuous function of the time of the pulse for small pulse amplitudes, but was discontinuous for larger pulse amplitudes. Similar results were obtained using a model of the ionic currents which underlie spontaneous activity in these preparations. The primary ion current components which contribute to phase resetting are the fast inward sodium ion current, INa, and the primary, potassium ion repolarization current, IX1. The origin of the discontinuity in phase resetting of the model can be elucidated by a detailed examination of the current-voltage trajectories in the region of the phase response curve where the discontinuity occurs.

Bistabilities and annihilation phenomena in electrophysiological cardiac models

We have investigated the oscillatory behavior of cardiac cellular elements simulated by two electrophysiological models: the van Capelle and Durrer (VCD) model and the sinoatrial node cell model of Yanagihara, Noma, and Irisawa (YNI). The VCD model behavior was examined systematically by using continuation-bifurcation analysis. Bifurcation diagrams were constructed as a function of Qit1, an intrinsic parameter of the model, which sets both maximum diastolic potential and depolarization threshold of the cell. The existence of stable high amplitude oscillations was evidenced between two Hopf bifurcation points (HB). Near each HB, a zone of bistability was detected. Close to the HB that corresponded to high values of Qit1, a high amplitude periodic stable state coexisted with a stable steady state. Close to the other HB, in a narrow range of lower Qit1 values, a relatively high amplitude periodic stable state coexisted with a low amplitude periodic stable state. There was no stable steady state in the latter bistability zone. Through the use of phase-plane representations and the determination of separatrices between the different attractor basins, we could deduce the conditions of timing, polarity, and strength needed for a pulse perturbation to send the system from one state to another and vice versa. The YNI model was analyzed by numerical simulation, and the oscillatory behavior of the sinoatrial node cell was explored while applying a depolarizing bias current of various strengths. Results were similar to those obtained from the VCD model in that there were two bistability regions for two different ranges of applied bias current. Depending on current intensity, annihilation of pacemaker activity could be achieved in both zones. However, the coexistence of two oscillatory stable states was never observed in the YNI model. From the behavioral similarities of these different models, we can conclude that bistabilities and annihilation phenomena can be found in transitional zones between quiescence and rhythmic activity.

Nonlinear dynamic behavior of cardiac oscillators

By studying a mathematical model of the effects of brief current impulses on the oscillation of cardiac cells, we have simulated the phase transition curves of the oscillators. This has permitted discussions of the interactions of two coupled sinus oscillators and the periodic stimulation of one ventricular oscillator. The results have been compared with findings from the normal electrocardiogram and applied to explain the mechanisms of some arrhythmias.

Phase resetting in a model of cardiac Purkinje fiber

The phase-resetting response of a model of spontaneously active cardiac Purkinje fiber is investigated. The effect on the interbeat interval of injecting a 20-ms duration depolarizing current pulse is studied as a function of the phase in the cycle at which the pulse is delivered. At low current amplitudes, a triphasic response is recorded as the pulse is advanced through the cycle. At intermediate current amplitudes, the response becomes quinquephasic, due to the presence of supernormal excitability. At high current amplitudes, a triphasic response is seen once more. At low stimulus amplitudes, type 1 phase resetting occurs; at medium amplitudes, a type could not be ascribed to the phase resetting because of the presence of effectively all-or-none depolarization; at high amplitudes, type 0 phase resetting occurs. The modeling results closely correspond with published experimental data; in particular type 1 and type 0 phase resetting are seen. Implications for the induction of ventricular arrhythmias are considered.

Phase resetting and bifurcation in the ventricular myocardium

With the dynamic differential equations of Beeler, G. W., and H. Reuter (1977, J. Physiol. [Lond.]. 268:177-210), we have studied the oscillatory behavior of the ventricular muscle fiber stimulated by a depolarizing applied current I app. The dynamic solutions of BR equations revealed that as I app increases, a periodic repetitive spiking mode appears above the subthreshold I app, which transforms to a periodic spiking-bursting mode of oscillations, and finally to chaos near the suprathreshold I app (i.e., near the termination of the periodic state). Phase resetting and annihilation of repetitive firing in the ventricular myocardium were demonstrated by a brief current pulse of the proper magnitude applied at the proper phase. These phenomena were further examined by a bifurcation analysis. A bifurcation diagram constructed as a function of I app revealed the existence of a stable periodic solution for a certain range of current values. Two Hopf bifurcation points exist in the solution, one just above the lower periodic limit point and the other substantially below the upper periodic limit point. Between each periodic limit point and the Hopf bifurcation, the cell exhibited the coexistence of two different stable modes of operation; the oscillatory repetitive firing state and the time-independent steady state. As in the Hodgkin-Huxley case, there was a low amplitude unstable periodic state, which separates the domain of the stable periodic state from the stable steady state. Thus, in support of the dynamic perturbation methods, the bifurcation diagram of the BR equation predicts the region where instantaneous perturbations, such as brief current pulses, can send the stable repetitive rhythmic state into the stable steady state.