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PRACTICAL APPLICATIONS

Impact on the World

Utility—that critical, creative interface between theorists and experimentalists—is one important measure of mathematical accomplishment. That’s a point President Shirley Ann Jackson—herself, a theoretical physicist—emphasized in her inaugural address last September: “Because it embodies modeling, analysis, algorithmic development, even new mathematics, applied mathematics as a discipline can be viewed as a common core which undergirds all the sciences and engineering. It is inherently interdisciplinary.”

“What we’re doing is meaningful. It has an impact on the world,” says Mark Holmes, chairman of Rensselaer’s Department of Mathematical Sciences. “That’s why I like this department. It’s fun. You get to work with a lot of people outside your department in all sorts of fields.”   

The focus of Holmes’ research is an unlikely hybrid of disciplines— mathematical biology, with an emphasis on mechanopreception, which Holmes defined as “the study of how a living organism senses and then recognizes mechanical stimuli.”   

Holmes started his biology-oriented research with the sense of hearing. How do the auditory nerve and the brain turn sound waves into intelligible neural signals? Working with neurophysiologists, Holmes helped build a mathematical model of human auditory perception, showing how tiny, hairlike structures inside a snail-shaped organ called the cochlea convert sound waves into signals interpreted by the brain. Such a model is useful to computer engineers designing machines with voice-recognition capacity.   

Holmes has since moved on to what may be the most primitive and least understood of our senses—touch. By using mathematics, he’s attempting to fuse the tactile and the abstract. Holmes has developed equations that describe the functioning of the Pacinian corpuscles, microscopic structures embedded by the

 

 

Numerical simulation of the evolution to detonation of a combustible gas subject to a small initial gradient in temperature. The plot shows the variation in the pressure of the gas near the corner of a two-dimensional rectangular confinement at a time just after a curved ion wave (the high pressure ridge) has formed. (Contributed by D. Schwendeman.)

thousands beneath the surface of the skin. Holmes and his colleagues have published their findings in Progress in Neurobiology, making them probably the only mathematicians to appear in that journal.  

“The mechanisms through which the skin transduces tactile stimuli, changing them from mechanical to neural signals, are not understood. The sense of touch is like receiving 10,000 telephone calls at once, and yet being able to understand them all. But at the same time, the sense of touch is robust and primitive enough that you can lose some of it and not ruin the entire system. How does that happen?” Holmes says.   

Practical applications of a mathematical understanding of the tactile sense include the design and manufacture of prosthetic limbs, and sophisticated computer and virtual reality games, which Holmes described as “the big money area.”   

Working with neurophysiologists at Syracuse University, Holmes is constructing a mathematical model of human touch. In another project, Holmes is working with Rensselaer’s chairman of biomedical engineering, Robert Spilker, to develop a mathematical understanding of the cartilage in the knee. Together, they’re building a model of the soft tissue that acts as a cushion between the femur and the tibia, the major bones of the leg. Eventually, their work could lead to computer simulations that would aid in the diagnosis of various knee injuries.

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