(One in a series of six articles on Mathematics and Medicine being distributed by the Joint Policy Board for Mathematics in celebration of Mathematics Awareness Week 1994.) Magellan owed a lot to the compass, a simple device that helped the sailors of the Old World navigate safely across the vast oceans of the earth. What if scientists had a compass for exploring the mysterious river of human consciousness? University of California San Francisco researcher Dana Redington thinks that a branch of mathematics called nonlinear dynamics may provide scientists with such a tool. Redington is a psychophysiologist  a scientist who studies the nature of the connection between mind and body. He and research partner Steven Reidbord, a psychiatrist, are using the mathematical tools of nonlinear dynamics to analyze the way people's heart rates change when they talk to their therapists. Everyday experience tells us that the heart and mind are connected. In the classic "fightorflight" reaction, a sudden threat or flash of anger sends a squirt of adrenaline into the blood, causing the heart to pound. Redington and Reidbord are looking for something less obvious than that, however. They believe that because of the way the mind and body are wired together, what people think and feel can affect the complexity of the heart rate  how much it slows down or speeds up over a period of time. The scientists record this information using an electronic heart monitoring machine. The problem is that these heartrate changes are subtle and complex, and can be overlooked easily. This is why Redington and Reidbord turned to the mathematical area of nonlinear dynamics, which deals with very complex natural processes. "The thing that's so compelling about nonlinear dynamics is that extremely complicated forms of behavior can be described by very simple, almost elegant equations," Redington explains. Explaining what nonlinear dynamics is all about can be difficult. One half of the term is relatively simple, though: dynamics is the study of motion. Mathematically, it doesn't matter what kind of motion, whether a swinging pendulum or the way a person's heart rate slows down or speeds up. The nonlinear part is harder to define. Jerry Bona, head of the mathematics department at Pennsylvania State University, says that the best way to define nonlinear dynamics is by saying what it's not. A complicated linear process, he explains, can be described mathematically by breaking it down into simpler parts, analyzing each one separately, and then adding all the answers up. In principle, this is like the assemblyline system used to build automobiles. But to understand a nonlinear system, you can't do this. Mathematically, you have to look at the whole system all at once. Redington and many other scientists view the mind as a nonlinear system. Part of that nonlinearity stems from the fact that the brain is connected through the nervous system to the other organs of the body  and vice versa. The brain contains a number of interconnected circuits, or clocks, Redington explains. Each regulates a different bodily process or organ: breathing, heart rate, and digestion, for example. These clocks are connected to each other and to the wiring of higher brain functions such as thought and emotion. When everything is working right, these processes all work together to minimize wasted energy. For example, your heart beat changes slightly when you inhale to compensate for the greater pressure in your chest cavity. "Under healthy circumstances, all the organs of the body are working together in a nice, synchronized dance," Redington explains, "but when you get into a disease state, like stress, the dance becomes lopsided and unbalanced." Redington and Reidbord are trying to show that these shifts in one of the brain's internal clocks  heart rate  reflect our shifting mental states. They define these mental states as distinctive modes of thought, ranging from meditative calm to unmitigated panic. "We're trying to use physiology  the way patterns in the body occur in time  to give us more information about what's going on in a person's head," Redington explains.
This marriage of mathematics and psychology might someday better
explain the longsuspected links between mind and health. But
strictly in terms of science, says Redington, nonlinear dynamics is
just a good tool for exploring human consciousness. "When you look
through a telescope, you start to see in much finer detail, and
you're better able to describe things," Redington explains.
"That's exactly what I think nonlinear dynamics is all about."

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area. 