An S1 Gradient of Refractoriness is Not Essential for Reentry Induction by an S2 Stimulus

Sometimes the shortest papers are my favorites. Take, for example, an article that I published twenty years ago last month: a two-page communication in the IEEE Transactions on Biomedical Engineering titled “ An S1 Gradient of Refractoriness is Not Essential for Reentry Induction by an S2 Stimulus” (Volume 47, Pages 820–821, 2000). It analyzes the electrical stimulation of cardiac tissue, and focuses on the mechanism for inducing an arrhythmia.

The introduction is two short paragraphs (a mere hundred words). The first puts the work in context.

Successive stimulation (S1, then S2) of cardiac tissue can induce reentry. In many cases, an S1 stimulus triggers a propagating action potential that creates a gradient of refractoriness. The S2 stimulus then interacts with this S1 refractory gradient, causing reentry. Many theoretical and experimental studies of reentry induction are variations on this theme [1]-[9].

When I wrote this communication, the critical point hypothesis was a popular explanation for how to induce reentry in cardiac tissue. I cited nine papers discussing this hypothesis, but I associate it primarily with the books of Art Winfree and the experiments of Ray Ideker.

The critical point hypothesis.

The figure above illustrates the critical point hypothesis. A first (S1) stimulus is applied to the right edge of the tissue, launching a planar wavefront that propagates to the left (arrow). By the time of the upper snapshot, the tissue on the right (purple) has returned to rest and recovered excitability, while the tissue on the left (red) remains refractory. The green line represents the boundary between refractory and excitable regions: the line of critical refractoriness.

The lower snapshot is immediately after a second (S2) stimulus is applied through a central cathode (black dot). The tissue near the cathode experiences a strong stimulus above threshold (yellow), while the remaining tissue experiences a weak stimulus below threshold. The green curve represents the boundary between the above-threshold and below-threshold regions: the circle of critical stimulus. S2 only excites tissue that is excitable and has a stimulus above threshold (inside the circle on the right). It launches a wave front that propagates to the right, but cannot propagate to the left because of refractoriness. Only when the refractory tissue recovers excitability will the wave front begin to propagate leftward (curved arrow). Critical points (blue dots) are located where the line of critical refractoriness intersects the circle of critical stimulus. Two spiral waves -a type of cardiac arrhythmia where a wave front circles around a critical point, chasing its tail-rotate clockwise on the bottom and counterclockwise on the top.

A beautiful paper from Ideker’s lab provides evidence supporting the critical point hypothesis: N. Shibata, P.-S. Chen, E. G. Dixon, P. D. Wolf, N. D. Danieley, W. M. Smith, and R. E. Ideker (1988) “ Influence of Shock Strength and Timing on Induction of Ventricular Arrhythmias in Dogs,” American Journal of Physiology, Volume 255, Pages H891-H901.

The second paragraph of my communication begins with a question.

Is the S1 gradient of refractoriness essential for the induction of reentry? In this communication, my goal is to show by counterexample that the answer is no. In my numerical simulation, the transmembrane potential is uniform in space before the S2 stimulus. Nevertheless, the stimulus induces reentry.

The critical point hypothesis implies the answer is yes; without a refractory gradient there is no line of critical refractoriness, no critical point, no spiral wave, no reentry. Yet I claimed that the gradient of refractoriness is not essential. To explain why, we must consider what happens following the second stimulus.

Cathode break excitation.

The tissue is depolarized (D, yellow) under the cathode but is hyperpolarized (H, purple) in adjacent regions along the fiber direction on each side of the cathode, often called virtual anodes. Hyperpolarization lowers the membrane potential toward rest, shortening the refractory period (deexcitation) and carving out an excitable path. When S2 ends, the depolarization under the cathode diffuses into the newly excitable tissue (dashed arrows), launching a wave front that propagates initially in the fiber direction (solid arrows): break excitation. Only after the surrounding tissue recovers excitability does the wave front begin to rotate back, as if there were four critical points: quatrefoil reentry.

Russ Hobbie and I discuss break excitation in a homework problem in Chapter 7 of Intermediate Physics for Medicine and Biology.

Problem 48. During stimulation of cardiac tissue through a small anode, the tissue under the electrode and in the direction perpendicular to the myocardial fibers is hyperpolarized, and adjacent tissue on each side of the anode parallel to the fiber direction is depolarized. Imagine that just before this stimulus pulse is turned on the tissue is refractory. The hyperpolarization during the stimulus causes the tissue to become excitable. Following the end of the stimulus pulse, the depolarization along the fiber direction interactselectrotonically with the excitable tissue, initiating an action potential (break excitation). (This type of break excitation is very different than the break excitation analyzed on page 181.)

(a) Sketch pictures of the transmembrane potential distribution during the stimulus. Be sure to indicate the fiber direction, the location of the anode, the regions that are depolarized and hyperpolarized by the stimulus, and the direction of propagation of the resulting action potential.

(b) Repeat the analysis for break excitation caused by a cathode instead of an anode. For a hint, see Wikswo and Roth (2009).

Now we come to the main point of the communication; the reason I wrote it. Look at the first snapshot in the illustration above, the one labeled S1 that occurs just before the S2 stimulus. The tissue is all red. It is uniformly refractory. The S1 action potential has no gradient of refractoriness, yet reentry occurs. This is the counterexample that proves the point: a gradient of refractoriness is not essential.

The communication contains one figure, showing the results of a calculation based on the bidomain model. The time in milliseconds after S1 is in the upper right corner of each panel. S1 was applied uniformly to the entire tissue, so at 70 ms the refractoriness is uniform. The 80 ms frame is during S2. Subsequent frames show break excitation the development of reentry.

An illustration based on Fig. 1 in “An S1 Gradient of Refractoriness is Not Essential for Reentry Induction by an S2 Stimulus” (IEEE TBME, 47:820–821, 2000). It is the same as the figure in the communication, except the color and quality are improved.

The communication concludes:

My results support the growing realization that virtual electrodes, hyperpolarization,deexcitation, and break stimulation may be important during reentry induction [8], [9], [14], [15], [21]-[24]. An S1 gradient of refractoriness may underlie reentry induction in many cases [1]-[6], but this communication provides a counterexample demonstrating that an S1 gradient of refractoriness is not necessary in every case.

This is a nice calculation, but is it consistent with experiment? Look at Y. Cheng, V. Nikolski, and I. R. Efimov (2000) “ Reversal of Repolarization Gradient Does Not Reverse the Chirality of Shock-Induced Reentry in the Rabbit Heart,” Journal of Cardiovascular Electrophysiology, Volume 11, Pages 998–1007. These researchers couldn’t produce uniform refractoriness, so they did the next best thing: repeated the experiment using S1 wave fronts propagating in different directions. They always obtained the same result, independent of the location and timing of the critical line of refractoriness.

Does this calculation mean the critical point hypothesis is wrong? No. See my paper with Natalia Trayanova and her student Annette Lindblom (“ The Role of Virtual Electrodes in Arrhythmogenesis: Pinwheel Experiment Revisited,” Journal of Cardiovascular Electrophysiology, Volume 11, Pages 274–285, 2000) to examine how this view of reentry can be reconciled with the critical point hypothesis.

One of the best things about this calculation is that you don’t need a fancy computer to demonstrate that the S1 gradient of refracoriness is not essential; A simple cellular automata will do. The figure below sums it up (look here if you don’t understand).

A cellular automata demonstrating that an S1 gradient of refractoriness is not essential for reentry induction by an S2 stimulus.

Originally published at



Professor of Physics at Oakland University and coauthor of the textbook Intermediate Physics for Medicine and Biology.

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Brad Roth

Professor of Physics at Oakland University and coauthor of the textbook Intermediate Physics for Medicine and Biology.