Chaos and Coexistence Theory

Because most functions in the human body, including the heartbeat, depend on other bodily events, such as hormone and chemical levels, the body is a dynamic system -- very slight changes in startup conditions can result in drastically different outcomes, so that the system is virtually unpredictable. Recognizing this, Chay's research involves applying the insights of a branch of modern mathematical physics known as nonlinear dynamics or, more popularly, chaos theory. This approach involves examining the mathematical description of a physical phenomenon through the full range of its variables, to discern what patterns may exist within the apparent unpredictability. This field of study has come into existence only with the advent of powerful computers, because it requires a tremendous number of iterations of complex differential equations, and Chay's investigations can be accomplished only with supercomputing.

Chay first studied a simple system -- just one cell and how it behaves as it gets sicker. In its healthy state, the cell is perfectly still and ready to receive an impulse from the sinus node. When that occurs, the cell sends the impulse to a neighboring cell and returns to a quiescent state. When the cell is sick, however, it holds too many positive charges, and under these conditions, Chay's calculations show that when an extra impulse arrives at the wrong moment -- known as the vulnerable period -- the cell begins sending its own signals across the heart, which interferes with pulses from the sinus node. As a result, arrhythmia kicks in.

Even for a sick cell, however, when a pulse arrives outside the vulnerable period, nothing happens. Arrived at using an approach from chaos theory called "bifurcation analysis," these results indicate that the beating and quiescent states of a single cell coexist; both possibilities are present, and which of the two occurs depends on the timing and magnitude of the triggering impulse. It is similar to the coexistence of water and ice when temperature is at the melting point. "At a certain level, you have two phases coexisting," Chay says, "so I was able to explain that complex dynamics exist even in a single cardiac cell." Her studies showed this coexistence of states -- dubbed the "coexistence theory" -- for the first time.

Multiple Cell Models

To make her model more realistic -- for studying reentrant arrhythmia, Chay added a second cell. In this model, both cells begin at rest, then begin beating when an electrical pulse arrives from the sinus node. But a mistimed pulse causes signals to zing back and forth between the two cells, similar to reentrant arrhythmia -- though it's not continuous and eventually stops. Since the two cells either beat independently or coupled for a short time, these results lend further support to the coexistence theory.

Calculations on an even more realistic, six-cell model, showed the coexistence of several states -- reentrant arrhythmia, self beating and quiescent state -- all of which can be triggered by electrical impulses given at the precise time and magnitude. "When a cell has several states coexisting, it becomes very susceptible to small perturbations," Chay says. "When cardiac tissues are in a healthy state, there's no coexistence. And even if you keep shocking it, nothing happens."

Now that Chay has developed a theoretical model to explain arrhythmia, she has extended the model to larger systems and plans to examine anti-arrhythmic drugs. Essentially, the drugs block channels that allow ions to flow into and out of cells. These moving ions, which are molecules with a positive or negative charge, create an electrical imbalance that in turn alters the passage of electrical pulses. "By blocking channels," Chay says, "I can see how to prevent arrhythmia, and I'd like to explain everything, instead of doing an experiment in the laboratory. I'd like to see precisely how the bifurcating phases change."

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