processes. Recent advances in neuroscience provide important foundations to begin
understanding how the brain works. Combined with experimental data, innovative mathematical
modeling provides an unparalleled opportunity to gain a revolutionary new understanding of
brain physiology, cognition (including sensory processing, attention, decision-making, etc.), and
neurological disease. With this new understanding, improved soldier performance, as well as
treatments for Post-Traumatic Stress Disorder, Traumatic Brain Injury, and other brain-related
disorders suffered by the warfighter will be able to be achieved more effectively, efficiently, and
ethically than via experimentation alone.
Thrust areas of the Biomathematics Program are as follows:
Fundamental Laws of Biology
The field of physics has long been "'mathematized" so that fundamental principles such as
Newton's Laws are not considered the application of mathematics to physics but physics itself.
The field of biology is far behind physics in this respect; a similar process of mathematization is
a basic and high-risk goal of the ARO Biomathematics Program. The identification and
mathematical formulation of the fundamental principles of biological structure, function, and
development applying across systems and scales will not only revolutionize the field of biology
but will motivate the creation of new mathematics that will contribute in as-yet-unforeseen ways
to biology and the field of mathematics itself. For example, heterogeneity/stochasticity is
ubiquitous in biological systems; is heterogeneity necessary for tipping points that result in
diseased individuals and epidemics and if so, what is its role? More generally, is heterogeneity in
biological systems necessary for their functioning or a problem to be overcome, or is the answer
system/function dependent?
Multiscale Modeling/Inverse Problems
Biological systems function through diversity, with large scale function emerging from the
collective behavior of smaller scale heterogeneous elements. This "'forward" problem includes
creating mechanistic mathematical models at different biological scales and synchronizing their
connections from one level of organization to another, as well as an important sub problem, how
to represent the heterogeneity of individual elements and how much heterogeneity to include in
the model. For example, the currently increasing ability to generate large volumes of molecular
data provides a significant opportunity for biomathematical modelers to develop advanced
analytical procedures to elucidate the fundamental principles by which genes, proteins, cells,
etc., are integrated and function as systems through the use of innovative mathematical and
statistical techniques. The task is complicated by the fact that data collection methods are noisy,
many biological mechanisms are not well understood, and, somewhat ironically, large volumes
of data tend to obscure meaningful relationships. However, traditionally "'pure" mathematical
fields such as differential geometry, algebra and topology, integration of Bayesian statistical
methods with mathematical methods, and the new field of topological data analysis, among
others, show promise in approaching these problems. Solutions to these types of multiscale
problems will elucidate the connection, for example, of stem cells to tissue and organ
development or of disease processes within the human body to the behavior of epidemics.
The "'inverse" problem is just as important as the forward problem. From an understanding of
the overall behavior of a system, is it possible to determine the nature of the individual elements?
For example, from knowledge of cell signaling, can we go back and retrieve information about
the cell? Although inverse problems have been studied for a long time, significant progress has