Hysteresis and Bistability in the Direct Transition from 1:1 to 2:1 Rhythm in Periodically Driven Single Ventricular Cells

When preparing the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added two homework problems (Problems 37 and 38) in Chapter 10 (Feedback and Control) about “cardiac restitution.” These problems contain a fascinating and elegantly simple example of restitution that provides insight into nonlinear dynamics and chaos. Problem 37 begins

Problem 37 The onset of ventricular fibrillation in the heart can be understood in part as a property of cardiac “restitution.” The action potential duration (APD) depends on the previous diastolic interval (DI): the time from the end of the last action potential until the start of the next one. The relationship between APD and DI is called the “restitution curve.” In cardiac muscle, a typical restitution curve has the form

APD_i+1 = 300 (1 – exp(–DI_i/100))

where all times are given in ms. Suppose we apply to the heart a series of stimuli, with period (or “cycle length”) CL. Since APD + DI = CL, we have DI_i+1 = CL – APD_i+1.

The problem then goes on to have the reader do some numerical calculations using various cycle lengths and initial diastolic intervals. Depending on the parameters, you can get (a) a simple 1:1 response between stimulation and action potential, (b) a 2:2 response in which every stimulus triggers an action potential but the APD alternates between long and short, a behavior called “alternans,” (c) a 2:1 response where an action potential is triggered by every second stimulus, with the tissue being refractory and not responding to the other stimuli, and (d) chaos. I have found this model is an excellent way to introduce students to chaotic behavior; even students with a weak mathematics background can understand it. When discussing this mathematical model with students, I often hand out a particularly clear paper to serve as background reading: J. N. Weiss, A. Garfinkel, H. S. Karagueuzian, Z. Qu, and P.-S. Chen (1999) “Chaos and the Transition to Ventricular Fibrillation: A New Approach to Antiarrhythmic Drug Evaluation,” Circulation, Volume 99, Pages 2819–2826.

Problem 38 explores how to understand this behavior by analyzing the slope of the restitution curve. If the slope is too steep, the behavior becomes more complex. Part (d) of Problem 38 says

Suppose you apply a drug to the heart that can change the restitution curve to

APD_i+1 = 300 (1 – b exp(–DI_i/100)) .

Plot APD as a function of DI for b = 0, 0.5, and 1. What value of b ensures that the slope of the restitution curve is always less than 1? Garfinkel et al. (2000) have suggested that one way to prevent ventricular fibrillation is to use drugs to “flatten” the restitution curve.

There is yet another type of behavior that is not discussed in Problems 37 or 38: a bistable response. Below is a new homework problem that discusses bistable behavior.

Problem 38 ½ Use the restitution curve from Problem 38, with b = 1/3 and CL = 250, to analyze the response of the system with initial diastolic intervals of 50, 60, 70, 80, and 90. You should find that the qualitative behavior depends on the initial condition. Which values of the initial diastolic interval give a 1:1 response, and which give 2:1? Determine the initial value of the DI, to three significant figures, for which the system makes a transition from one behavior to the other. When two qualitatively different behaviors can both occur, depending on the initial conditions, the system is “bistable.” To learn more about such behavior, see Yehia et al. (1999).

The full citation to the paper mentioned at the end of the problem is

Yehia, A. R., D. Jeandupeux, F. Alonso, and M. R. Guevara (1999) “Hysteresis and Bistability in the Direct Transition From 1:1 to 2:1 Rhythm in Periodically Driven Single Ventricular Cells,” Chaos, Volume 9, Pages 916–931.

The senior author on this article is Michael Guevara, of the Centre for Applied Mathematics in Bioscience and Medicine at McGill University. The introductory paragraph of their paper is reproduced below.

The majority of cells in the heart are not spontaneously active. Instead, these cells are excitable, being driven into activity by periodic stimulation originating in a specialized pacemaker region of the heart containing spontaneously active cells. This pacemaker region normally imposes a 1:1 rhythm on the intrinsically quiescent cells. However, the 1:1 response can be lost when the excitability of the paced cells is decreased, when there are problems in the conduction of electrical activity from cell to cell, or when the heart rate is raised. When 1:1 synchronization is lost in the intact heart, one of a variety of abnormal cardiac arrhythmias can arise. In single quiescent cells isolated from ventricular muscle, 1:1 rhythm can be replaced by a N+1:N rhythm (N≥2), a period-doubled 2:2 rhythm, or a 2:1 rhythm. We investigate below the direct transition from 1:1 to 2:1 rhythm in experiments on single cells and in numerical simulations of an ionic model of a single cell formulated as a nonlinear system of differential equations. We show that there is hysteresis associated with this transition in both model and experiment, and develop a theory for the bistability underlying this hysteresis that involves the coexistence of two stable fixed-points on a two-branched one-dimensional map.

For those interested in exploring the application of nonlinear dynamics to biology and medicine in more detail, two books Russ and I cite in Intermediate Physics for Medicine and Biology—and which I recommend highly—are From Clocks to Chaos by Leon Glass and Michael Mackey (both also at McGill) and Nonlinear Dynamics and Chaos by Steven Strogatz.