One place where Winfree’s work impacts our book is in Problems 39 and 40 in Chapter 10, discussing cellular automata. Winfree didn’t invent cellular automata, but his discussion of them in his wonderful book When Time Breaks Down is where I first learned about the topic.

“Box 5.A: A Cellular Excitable Medium

Take a pencil and a sheet of tracing paper and play with Figure 5.2 [a large hexagonal array of cells] according to the following game rules … Each little hexagon in this honeycomb is supposed to represent a cell that may be excited for the duration of one step (put a ‘0’ in the cell) or refractory (after the excited moment, replace the ‘0’ with a ‘1’) or quiescent (after that erase the ‘1’) until such time as any adjacent cell becomes excited: then pencil in a ‘0’ in the next step.

If you start with no ‘0’s,’ you’ll never get any, and this simulation will cost you little effort. If you start with a single ‘0’ somewhere, it will next turn to ‘1’ while a ring of 6 neighbors become infected with ‘0’. As the hexagonal ring of ‘0’s’ propagates, it is followed by a concentric ring of ‘1’ refractoriness, right to the edge of the honeycomb, where all vanish.

Now see what happens if you violate the rules just once by erasing a segment of that ring wave when it is about halfway to the edges: you will have created a pair of counter-rotating vortices (alias phase singularities), each of which turns out to be a source of radially propagating waves.

(Stop reading until you have played some.)

You may feel a bit foolish, since this is obviously supposed to mimic action potential propagation, and the caricature is embarrassingly crude. Which aspects of its behavior are realistic and which others are merely telling us ‘honeycombs are not heart muscle’? The way to find out is to undertake successively more refined caricatures until a point of diminishing returns is reached. For most purposes, it is reached surprisingly soon.”I consider cellular automata—whose three simple rules can be mastered by a child--to be among the best tools for illustrating cardiac reentry . In fact, I like this model so much that I generalized it to account for electrical simulation that produces adjacent regions of depolarization and hyperpolarization (Sepulveda et al., 1989; read more about that paper here). In “Virtual Electrodes Made Simple: A Cellular Excitable Medium Modified for Strong Electrical Stimuli,” published in the Online Journal of Cardiology, I added a fourth rule

“During a cathodal stimulus, the state of the cell directly under the electrode and its four nearest neighbors in the direction perpendicular to the fibers change to the excited state, and the two remaining nearest neighbors in the direction parallel to the fibers change to the quiescent state, regardless of their previous state.”Using this simple model, I was able to initiate “quatrefoil reentry” (Lin et al., 1999; read more here). I also could reproduce most of the results of a simulation of the “pinwheel experiment” (a point stimulus applied near the end of the refractory period of a previous planar wave front) predicted by Lindblom et al. (2000). I concluded

“This extremely simple cellular excitable medium--which is nothing more than a toy model, stripped down to contain only the essential features--can, with one simple modification for strong stimuli, predict many interesting and important phenomena. Much of what we have learned about virtual electrodes and deexcitation is predicted correctly by the model (Efimov et al., 2000; Trayanova, 2001). I am astounded that this simple model can reproduce the complex results obtained by Lindblom et al. (2000). The model provides valuable insight into the essential mechanisms of electrical stimulation without hiding the important features behind distracting details.”My online paper came out in 2002, the same year that Winfree died. In an obituary, Steven Strogatz wrote

“When Art Winfree died in Tucson on November 5, 2002, at the age of 60, the world lost one of its most creative scientists. I think he would have liked that simple description: scientist. After all, he made it nearly impossible to categorize him any more precisely than that. He started out as an engineering physics major at Cornell (1965), but then swerved into biology, receiving his PhD from Princeton in 1970. Later, he held faculty positions in theoretical biology (Chicago, 1969-72), in the biological sciences (Purdue, 1972-1986), and in ecology and evolutionary biology (University of Arizona, from 1986 until his death).

So the eventual consensus was that he was a theoretical biologist. That was how the MacArthur Foundation saw him when it awarded him one of its "genius" grants (1984), in recognition of his work on biological rhythms. But then the cardiologists also claimed him as one of their own, with the Einthoven Prize (1989) for his insights about the causes of ventricular fibrillation. And to further muddy the waters, our own community honored his achievements with the 2000 AMS-SIAM Norbert Wiener Prize in Applied Mathematics, which he shared with Alexandre Chorin.

Aside from his versatility, what made Winfree so special (and in this way he was reminiscent of Wiener himself) was the originality of the problems he tackled; the sparkling creativity of his methods and results; and his knack for uncovering deep connections among previously unrelated parts of science, often guided by geometrical arguments and analogies, and often resulting in new lines of mathematical inquiry.”

Art Winfree is right along side Richard Feynman in terms of modern geniuses and great teachers wrapped all in one. This book is a Must read for anyone interested in excitation and wavefront propagation in cells.

ReplyDeleteI've often wondered if 3-dimensional cellular automata would be a good way to model action potential propagation in a peripheral nerve fiber.

ReplyDeleteConsider Brad's 2-dimensional honeycomb of automata "cells" to represent a nerve fiber's cross-section--an array of axons. Project the honeycomb along a 3rd spatial dimension to define a 3-dimensional nerve fiber as a set of cross-sections.

Now let the contents of any "cellular" element of cross-section(N) at time=T be determined by the contents of its adjacent cellular elements in cross-section(N-1), cross-section(N), and cross-section(N+1)at time=T-1 according to some rules.

Of course in real nerve fibers, there are numerous axons bundled in spatial proximity at any cross-section. I've long wondered, do the adjacent axons' polarization states influence one another?