Michael R. Guevara

Phase locking, period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: a theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias.

A mathematical model for the perturbation of a biological oscillator by single and periodic impulses is analyzed. In response to a single stimulus the phase of the oscillator is changed. If the new phase following a stimulus is plotted against the old phase the resulting curve is called the phase transition curve or PTC (Pavlidis, 1973). There are two qualitatively different types of phase resetting. Using the terminology of Winfree (1977, 1980), large perturbations give a type 0 PTC (average slope of the PTC equals zero), whereas small perturbations give a type 1 PTC. The effects of periodic inputs can be analyzed by using the PTC to construct the Poincaré or phase advance map. Over a limited range of stimulation frequency and amplitude, the Poincaré map can be reduced to an interval map possessing a single maximum. Over this range there are period doubling bifurcations as well as chaotic dynamics. Numerical and analytical studies of the Poincaré map show that both phase locked and non-phase locked dynamics occur. We propose that cardiac dysrhythmias may arise from desynchronization of two or more spontaneously oscillating regions of the heart. This hypothesis serves to account for the various forms of atrioventricular (AV) block clinically observed. In particular 2∶2 and 4∶2 AV block can arise by period doubling bifurcations, and intermittent or variable AV block may be due to the complex irregular behavior associated with chaotic dynamics.

Bifurcation and chaos in a periodically stimulated cardiac oscillator

Abstract Periodic stimulation of an aggregate of spontaneously beating cultured cardiac cells displays phase locking, period-doubling bifurcations and aperiodic “chaotic” dynamics at different values of the stimulation parameters. This behavior is analyzed by considering an experimentally determined one-dimensional Poincare or first return map. A simplified version of the experimentally determined Poincare map is proposed, and several features of the bifurcations of this map are described.

Hysteresis and bistability in the direct transition from 1:1 to 2:1 rhythm in periodically driven single ventricular cells.

The transmembrane potential of a single quiescent cell isolated from rabbit ventricular muscle was recorded using a suction electrode in whole-cell recording mode. The cell was then driven with a periodic train of current pulses injected into the cell through the same recording electrode. When the interpulse interval or basic cycle length (BCL) was sufficiently long, 1:1 rhythm resulted, with each stimulus pulse producing an action potential. Gradual decrease in BCL invariably resulted in loss of 1:1 synchronization at some point. When the pulse amplitude was set to a fixed low level and BCL gradually decreased, N+1:N rhythms (N>/=2) reminiscent of clinically observed Wenckebach rhythms were seen. Further decrease in BCL then yielded a 2:1 rhythm. In contrast, when the pulse amplitude was set to a fixed high level, a period-doubled 2:2 rhythm resembling alternans rhythm was seen before a 2:1 rhythm occurred. With the pulse amplitude set to an intermediate level (i.e., to a level between those at which Wenckebach and alternans rhythms were seen), there was a direct transition from 1:1 to 2:1 rhythm as the BCL was decreased: Wenckebach and alternans rhythms were not seen. When at that point the BCL was increased, the transition back to 1:1 rhythm occurred at a longer BCL than that at which the {1:1-->2:1} transition had initially occurred, demonstrating hysteresis. With the BCL set to a value within the hysteresis range, injection of a single well-timed extrastimulus converted 1:1 rhythm into 2:1 rhythm or vice versa, providing incontrovertible evidence of bistability (the coexistence of two different periodic rhythms at a fixed set of stimulation parameters). Hysteresis between 1:1 and 2:1 rhythms was also seen when the stimulus amplitude, rather than the BCL, was changed. Simulations using numerical integration of an ionic model of a single ventricular cell formulated as a nonlinear system of differential equations provided results that were very similar to those found in the experiments. The steady-state action potential duration restitution curve, which is a plot of the duration of the action potential during 1:1 rhythm as a function of the recovery time or diastolic interval immediately preceding that action potential, was determined. Iteration of a finite-difference equation derived using the restitution curve predicted the direct {1:1<-->2:1} transition, as well as bistability, in both the experimental and modeling work. However, prediction of the action potential duration during 2:1 rhythm was not as accurate in the experiments as in the model. Finally, we point out a few implications of our findings for cardiac arrhythmias (e.g., Mobitz type II block, ischemic alternans). (c) 1999 American Institute of Physics.

Universal Bifurcations and the Classification of Cardiac Arrhythmias

An appreciation of the extraordinarily rich dynamics of cardiac arrhythmias can be obtained from examination of any text in electrocardiography. Typically, there are a large number (100 or more) of rhythms that are identified on the basis of changes outside of normal limits of rhythm and wave morphology in the electrocardiogram.’*’ The point of this paper is to show that mathematics can provide a framework for studying several cardiac arrhythmias. It is possible to theoretically predict dynamics in some biological models and clinical situations using appropriate mathematical models. The main theoretical technique used is to represent cardiac systems by a finitedifference equation of the form,

Alternans and higher-order rhythms in an ionic model of a sheet of ischemic ventricular muscle

Life-threatening arrhythmias such as ventricular tachycardia and fibrillation often occur during acute myocardial ischemia. During the first few minutes following coronary occlusion, there is a gradual rise in the extracellular concentration of potassium ions ([K(+)](0)) within ischemic tissue. This elevation of [K(+)](0) is one of the main causes of the electrophysiological changes produced by ischemia, and has been implicated in inducing arrhythmias. We investigate an ionic model of a 3 cmx3 cm sheet of normal ventricular myocardium containing an ischemic zone, simulated by elevating [K(+)](0) within a centrally-placed 1 cmx1 cm area of the sheet. As [K(+)](0) is gradually raised within the ischemic zone from the normal value of 5.4 mM, conduction first slows within the ischemic zone and then, at higher [K(+)](0), an arc of block develops within that area. The area distal to the arc of block is activated in a delayed fashion by a retrogradely moving wavefront originating from the distal edge of the ischemic zone. With a further increase in [K(+)](0), the point eventually comes where a very small increase in [K(+)](0) (0.01 mM) results in the abrupt transition from a global period-1 rhythm to a global period-2 rhythm in the sheet. In the peripheral part of the ischemic zone and in the normal area surrounding it, there is an alternation of action potential duration, producing a 2:2 response. Within the core of the ischemic zone, there is an alternation between an action potential and a maintained small-amplitude response ( approximately 30 mV in height). With a further increase of [K(+)](0), the maintained small-amplitude response turns into a decrementing subthreshold response, so that there is 2:1 block in the central part of the ischemic zone. A still further increase of [K(+)](0) leads to a transition in the sheet from a global period-2 to a period-4 rhythm, and then to period-6 and period-8 rhythms, and finally to a complete block of propagation within the ischemic core. When the size of the sheet is increased to 4 cmx4 cm (with a 2 cmx2 cm ischemic area), one observes essentially the same sequence of rhythms, except that the period-6 rhythm is not seen. Very similar sequences of rhythms are seen as [K(+)](0) is increased in the central region (1 or 2 cm long) of a thin strand of tissue (3 or 4 cm long) in which propagation is essentially one-dimensional and in which retrograde propagation does not occur. While reentrant rhythms resembling tachycardia and fibrillation were not encountered in the above simulations, well-known precursors to such rhythms (e.g., delayed activation, arcs of block, two-component upstrokes, retrograde activation, nascent spiral tips, alternans) were seen. We outline how additional modifications to the ischemic model might result in the emergence of reentrant rhythms following alternans. (c) 2000 American Institute of Physics.

Phase resetting of the rhythmic activity of embryonic heart cell aggregates. Experiment and theory

Injection of a current pulse of brief duration into an aggregate of spontaneously beating chick embryonic heart cells resets the phase of the activity by either advancing or delaying the time of occurrence of the spontaneous beat subsequent to current injection. This effect depends upon the polarity, amplitude, and duration of the current pulse, as well as on the time of injection of the pulse. The transition from prolongation to shortening of the interbeat interval appears experimentally to be discontinuous for some stimulus conditions. These observations are analyzed by numerical investigation of a model of the ionic currents that underlie spontaneous activity in these preparations. The model consists of: Ix, which underlies the repolarization phase of the action potential, IK2, a time-dependent potassium ion pacemaker current, Ibg, a background or time-independent current, and INa, an inward sodium ion current that underlies the upstroke of the action potential. The steady state amplitude of the sum of these currents is an N-shaped function of potential. Slight shifts in the position of this current-voltage relation along the current axis can produce either one, two, or three intersections with the voltage axis. The number of these equilibrium points and the voltage dependence of INa contribute to apparent discontinuities of phase resetting. A current-voltage relation with three equilibrium points has a saddle point in the pacemaker voltage range. Certain combinations of current-pulse parameters and timing of injection can shift the state point near this saddle point and lead to an interbeat interval that is unbounded . Activation of INa is steeply voltage dependent. This results in apparently discontinuous phase resetting behavior for sufficiently large pulse amplitudes regardless of the number of equilibrium points. However, phase resetting is fundamentally a continuous function of the time of pulse injection for these conditions. These results demonstrate the ionic basis of phase resetting and provide a framework for topological analysis of this phenomenon in chick embryonic heart cell aggregates.

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.

Alternans in periodically stimulated isolated ventricular myocytes: Experiment and model

Publisher Summary This chapter describes the alternans in periodically stimulated isolated ventricular myocytes. Single ventricular cells are isolated from rabbit hearts using standard techniques. For the pulse amplitude sufficiently low, only a subthreshold response to each stimulus of the pulse train is obtained. For a sufficiently high pulse amplitude at a given time between stimuli, the stimulus is suprathreshold, and so action potentials can be obtained. In the case when 1:1, 2:2, and 2:1 rhythms are seen in different areas of the ventricle, it is possible that the 2:1 pattern is because of block of propagation of the cardiac impulse, with the 2:2 pattern being seen in the region of block. The 2:2 patterns seen in this circumstance is because of detrimental conduction and can be seen when 2:1 block occurs in tissue taken from virtually all areas of the heart. The possibility then exists that the 2:2 pattern in the region of block is partly because of electrotonic injection of current from the distal 2:1 region, in a manner demonstrated in modeling work on a one-dimensional strand of Purkinje fiber.

Triggered alternans in an ionic model of ischemic cardiac ventricular muscle

It has been known for several decades that electrical alternans occurs during myocardial ischemia in both clinical and experimental work. There are a few reports showing that this alternans can be triggered into existence by a premature ventricular contraction. Detriggering of alternans by a premature ventricular contraction, as well as pause-induced triggering and detriggering, have also been reported. We conduct a search for triggered alternans in an ionic model of ischemic ventricular muscle in which alternans has been described recently: a one-dimensional cable of length 3 cm, containing a central ischemic zone 1 cm long, with 1 cm segments of normal (i.e., nonischemic) tissue at each end. We use a modified form of the Luo-Rudy [Circ. Res. 68, 1501-1526 (1991)] ionic model to represent the ventricular tissue, modeling the effect of ischemia by raising the external potassium ion concentration ([K(+)](o)) in the central ischemic zone. As [K(+)](o) is increased at a fixed pacing cycle length of 400 ms, there is first a transition from 1:1 rhythm to alternans or 2:2 rhythm, and then a transition from 2:2 rhythm to 2:1 block. There is a range of [K(+)](o) over which there is coexistence of 1:1 and 2:2 rhythms, so that dropping a stimulus from the periodic drive train during 1:1 rhythm can result in the conversion of 1:1 to 2:2 rhythm. Within the bistable range, the reverse transition from 2:2 to 1:1 rhythm can be produced by injection of a well-timed extrastimulus. Using a stimulation protocol involving delivery of pre- and post-mature stimuli, we derive a one-dimensional map that captures the salient features of the results of the cable simulations, i.e., the {1:1-->2:2-->2:1} transitions with {1:1<-->2:2} bistability. This map uses a new index of the global activity in the cable, the normalized voltage integral. Finally, we put forth a simple piecewise linear map that replicates the {1:1<-->2:2} bistability observed in the cable simulations and in the normalized voltage integral map. (c) 2002 American Institute of Physics.

Iteration of the Human Atrioventricular (AV) Nodal Recovery Curve Predicts Many Rhythms of AV Block

The atrioventricular nodal recovery curve provides a quantitative description of how the conduction time through the atrioventricular node of a prematurely elicited atrial beat increases as the recovery time since the immediately preceding activation of the bundle of His decreases. This curve can be well approximated in human beings with normal atrioventricular nodal function by a single exponential function (the “standard” curve). Assuming that the response of the atrioventricular node to any atrial stimulus with a given recovery time during periodic pacing of the atrium is independent of the atrial rate, a simple equation (“map”) can be derived, using the recovery curve. The rhythm of atrioventricular conduction expected at any atrial rate is then obtained by numerically iterating this map on a digital computer. As the atrial rate is increased, rhythms resembling normal sinus rhythm, first-degree atrioventricular block, millisecond Wenckebach, Wenckebach periodicity, reverse Wenckebach periodicity, alternating Wenckebach periodicity, and higher grades of block are successively encountered. A mathematical theorem about the map is invoked to show that these are the only rhythms of conduction permitted. In addition, the order in which the various rhythms will appear as the atrial rate is increased as well as the ordering of blocked atrial beats within a given rhythm of block are also derived. Iteration using recovery curves other than the standard one leads to rhythms in which there is atypical Wenckebach, alternation of the conduction time, or coexistence of two different conduction times. Since this work puts into a common framework many different rhythms of atrioventricular conduction, it forms the beginnings of a “unified theory” of atrioventricular block.