Chaos in the Beeler-Reuter system for the action potential of ventricular myocardial fibres

Jensen, Johannes H.; Christiansen, P. L.; Scott, Alwyn C.; Skovgaard, Ove

doi:10.1016/0167-2789(84)90283-5

Cited by 11 publications

(4 citation statements)

References 18 publications

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“…In that case, one would also probably have to improve the accuracy of the numerical integration scheme. In an earlier study involving sinusoidal current stimulation of the space-clamped Beeler-Reuter membrane, only period-l, 2 and 4 rhythms were described; a period-8 rhythm was not seen before the onset of chaotic dynamics (Jensen et al, 1984). A similar finding occurs in the response to a pulsatile input (Vinet et al, 1990).…”

Section: Route To Chaossupporting

confidence: 49%

Chaotic dynamics in an ionic model of the propagated cardiac action potential

Lewis

Guevara

1990

Journal of Theoretical Biology

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We simulate the effect of periodic stimulation on a strand of ventricular muscle by numerically integrating the one-dimensional cable equation using the Beeler-Reuter model to represent the transmembrane currents. As stimulation frequency is increased, the rhythms of synchronization { 1 : 1 -* 2 : 2 -* 2 : 1 -* 4: 2 -* irregular-* 3 : 1 -* 6: 2 -* irregular-* 4: 1 -* 8 : 2 -*... --~ 1 : 0} are successively encountered. We show that this sequence of rhythms can be accounted for by considering the response of the strand to premature stimulation. This involves deriving a one-dimensional finiteditierence equation or "'map" from the response to premature stimulation, and then iterating this map to predict the response to periodic stimulation. There is good quantitative agreement between the results of iteration of the map and the results of the numerical integration of the cable equation. Calculation of the Lyapunov exponent of the map yields a positive value, indicating sensitive dependence on initial conditions ("chaos"), at stimulation frequencies where irregular rhythms are seen in the corresponding numerical cable simulations. The chaotic dynamics occurs via a previously undescribed route, following two period-doubling bifurcations. Bistability (the presence of two different synchronization rhythms at a fixed stimulation frequency) is present both in the simulations and the map. Thus, we have been able to directly reduce consideration of the dynamics of a partial differential equation (which is of infinite dimension) to that of a one-dimensional map, incidentally demonstrating that concepts from the field of non-linear dynamics--such as perioddoubling bifurcations, bistability, and chaotic dynamics--can account for the phenomena seen in numerical simulations of the cable equation. Finally, we sketch out how the one-dimensional description can be extended, and point out some implications of our work for the generation of malignant ventricular arrhythmias.

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Section: Route To Chaossupporting

confidence: 49%

Chaotic dynamics in an ionic model of the propagated cardiac action potential

Lewis

Guevara

1990

Journal of Theoretical Biology

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“…The Beeler-Reuter equations are a set of eight non-linear differential equations and like the Hodgkin-Huxley equations they are based on membrane ionic currents measured in voltage clamp experiments. The response of the Beeler-Reuter equations to sinusoidal stimulation was investigated by Jensen, Christiansen & Scott (1984). They found that for some stimulation frequencies the system exhibits phase locking whereas for other frequencies the response seems to be aperiodic.…”

Section: Chaos In Heart Beatmentioning

confidence: 99%

Chaos in biological systems

Olsen¹,

Degn²

1985

Quart. Rev. Biophys.

191

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Chaos is a widespread and easily recognizable phenomenon that hardly anybody took notice of until recently. The reason may be that chaos has something profoundly counterintuitive about it. It will not fit easily into any familiar cause–effect frame. The best introduction to chaos is by the way of an example. Consider a leaking faucet (Shaw, 1984). When the weight of the accumulating drop exceeds the surface tension the drop falls and a new drop begins to form. If the leak is small and the pressure in the faucet is constant, the time taken for the drop to reach the critical weight is constant. The dripping is perfectly periodic, the period depending on the leak rate. If the leak is slightly increased, the period of dripping will decrease slightly and vice versa. However, somewhere beyond this point the leaking faucet becomes a nuisance. When the leak is increased beyond a certain point the dripping looses its regularity. The time interval between the drops will first alternate periodically between a short and a long time interval. After a further increase of the leak this double periodic pattern will become unstable and change into a new pattern where four different time intervals between the drops alternate periodically. As the leak is further increased the period will double again and again and finally the dripping becomes completely irregular without any repeating pattern. When this occurs we are observing chaos. At the same time we are posed with the problem of understanding how such a ridiculously simple system can show random behaviour.

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“…The existence of period-2 and period-4 rhythms in ischemia, together with the experimental 42,45-47,88,109 and modeling 97,108,128,129 evidence for the period-doubling route to chaotic dynamics in cardiac tissue, leads very naturally to consideration of the possibility that the close temporal association of ventricular fibrillation with alternans might be an indication that ventricular fibrillation is a chaotic phenomenon somehow arising out of a cascade of period-doubling bifurcations. 9,32,37,38,52,53,111 Our results in the larger sheet ͑4 cmϫ4 cm, with a 2 cmϫ2 cm ischemic area͒ and the corresponding strand might be taken as providing some support for this hypothesis, in that we see the sequence ͕period-1→period-2→period-4→period-8→¯͖.…”

Section: Higher-order Period-doubled Rhythmsmentioning

confidence: 98%

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

Arce

González

et al. 2000

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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.

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Chaos in the Beeler-Reuter system for the action potential of ventricular myocardial fibres

Cited by 11 publications

References 18 publications

Chaotic dynamics in an ionic model of the propagated cardiac action potential

Chaotic dynamics in an ionic model of the propagated cardiac action potential

Chaos in biological systems

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

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