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.

The saline oscillator consists of an inner vessel containing salt water partially immersed in an outer vessel of fresh water, with a small orifice in the center of the bottom of the inner vessel. There is a cyclic alternation between salt water flowing downwards out of the inner vessel into the outer vessel through the orifice and fresh water flowing upwards into the inner vessel from the outer vessel through that same orifice. We develop a very stable (i.e., stationary) version of this saline oscillator. We first investigate the response of the oscillator to periodic forcing with a train of stimuli (period=Tp) of large amplitude. Each stimulus is the quick injection of a fixed volume of fresh water into the outer vessel followed immediately by withdrawal of that very same volume. For Tp sufficiently close to the intrinsic period of the oscillator (T0) , there is 1:1 synchronization or phase locking between the stimulus train and the oscillator. As Tp is decreased below T0 , one finds the succession of phase-locking rhythms: 1:1, 2:2, 2:1, 2:2, and 1:1. As Tp is increased beyond T0 , one encounters successively 1:1, 1:2, 2:4, 2:3, 2:4, and 1:2 phase-locking rhythms. We next investigate the phase-resetting response, in which injection of a single stimulus transiently changes the period of the oscillation. By systematically changing the phase of the cycle at which the stimulus is delivered (the old phase), we construct the new-phase--old-phase curve (the phase transition curve), from which we then develop a one-dimensional finite-difference equation ("map") that predicts the response to periodic stimulation. These predicted phase-locking rhythms are close to the experimental findings. In addition, iteration of the map predicts the existence of bistability between two different 1:1 rhythms, which was then searched for and found experimentally. Bistability between 1:1 and 2:2 rhythms is also encountered. Finally, with one exception, numerical modeling with a phenomenologically derived Rayleigh oscillator reproduces all of the experimental behavior.

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.

No abstract

The cardiac restitution curve describes functional relationships between diastolic intervals and their corresponding action potential durations. Although the simplest relationship is that restitution curves are monotonic, empirical studies have suggested that cardiac patients present a more complex dynamical process characterized, for instance, by a non-monotonic restitution curve. The purpose of this chapter is to analyze the dynamical properties of a non-monotonic cardiac restitution curve model derived from previously published clinical data. To achieve this goal, we use Recurrence Quantitative Analysis combined with Lyapunov exponents and Supertrack Functions in order to describe the complex dynamics underlying non-monotonic restitution curves. We conclude by highlighting that a consequence of the advanced complex dynamics that emerges from the aforementioned non-monotonicity, is the increasing risk of alternant rhythms.