Mathematics has always responded to demands of applications, even as mathematics continued to develop its own internal structures. One need only look back to the mid-twentieth century to see the mathematics spawned by demands of the military needs of the time. Today we see a tremendous growth in applied mathematics related to biology and medicine.

As an example, consider SIAM’s final conference in the year of Mathematics of Planet Earth—the SIAM Conference on the Analysis of PDEs. Applied PDEs were at one time primarily (although not exclusively) driven by problems in fluid flow –- from dynamics of flows across airplane surfaces to large-scale motion of atmospheric systems. While these problems continue to have importance, more recent applications have grown as well, including problems in imaging and in biology. These trends are indicated in the invited talks at the conference.

Among the invited speakers are two talks related to biology that show the role of mathematics.

Benoit Perthame of the Université Peirre et Marie Curie, Paris VI, will talk about the role PDEs play in modeling neural networks. According to Perthame, “Neurons exchange information via discharges propagated by membrane potentials which trigger firing of the many connected neurons. How to describe large networks of such neurons? How can such a network generate a collective activity?” His talk will discuss how such questions can be tackled using nonlinear partial-integro-differential equations. Among this class of equations, the Wilson-Cowan equations describe globally brain spiking rates. Another classical model is the integrate-and-fire equation based on Fokker-Planck equations. Berthame will analyze these models and discuss synchronization phenomena.

Philip Maini of Oxford University will describe a completely different set of biological phenomena that are modeled by PDEs. He will describe “three different examples of collective cell movement which require different modeling approaches: movement of cells in epithelial sheets, with application to rosette formation in the mouse epidermis and monoclonal conversion in intestinal crypts; cranial neural crest cell migration which requires a hybrid discrete cell-based chemotaxis model; acid-mediated cancer cell invasion, modeled via a coupled system of nonlinear partial differential equations.” All these models can be expressed in the framework of nonlinear diffusion equations, which can be used to understand a range of biological phenomena.

Mathematics is playing an increasing important role in the understanding and analysis of biological phenomena.