Teaching Dynamics to Biology Freshmen: a Modeling Approach
DateThursday, September 6, 2018 - 12:30pm
AbstractThere is a great need to reform how we introduce math to beginning students in Life Science. The usual “Calculus for Life Sciences” leaves students with two overwhelming impressions, as they have indicated in survey after survey: (1) “I hate math” and (2) “math has no application to biology”. Even worse, the math gateway courses into the life sciences serve as powerful filters keeping women and under-represented minorities out of the life sciences and medicine. Recently, there have been calls, from all the leading voices in US biology and medicine, for a new approach to mathematics for biology. We designed, and are currently teaching, a course like this. The course introduces students, on day 1, to the concept of modeling a system that has multiple interacting variables and nonlinear relations. The student quickly learns that models give rise to differential equations, and that differential equations can always be “solved” (that is, simulated numerically) using Euler’s method. They learn to program their own code for Euler’s method in a Python-like environment. We found that the key concept is the idea of a vector field, assigning “change arrows” to every point in state space. This 20th century concept has proven to be superior pedagogically to the 19th century math of differential equations as ‘expressions of the form. Students learn the typical sorts of behaviors that nonlinear differential equations exhibit, like equilibria, oscillations and even chaotic behavior. The major concepts of calculus, derivatives and integrals, are developed, as well as an introduction to matrices and eigenvalues and eigenvectors. Throughout, there is an emphasis on biological applications of these concepts, such as homeostatic (equilibrium) behavior in physiology and in ecological systems, multiple equilibria causing switch-like behavior, oscillation in insulin and glucose levels as well as in biological populations, etc.
Controlling Populations of Neural Oscillators
DateThursday, September 13, 2018 - 12:30pm
AbstractSome brain disorders are hypothesized to have a dynamical origin; in particular, it has been been hypothesized that some symptoms of Parkinson's disease are due to pathologically synchronized neural activity in the motor control region of the brain. This talk will describe several different approaches for desynchronizing the activity of a group of neurons, including maximizing the Lyapunov exponent associated with their phase dynamics, optimal phase resetting, controlling the phase density, and controlling the population to have clustered dynamics. It is hoped that this work will ultimately lead to improved treatment of Parkinson's disease via targeted electrical stimulation. Bio: Jeff Moehlis received a Ph.D. in Physics from UC Berkeley in 2000, and was a Postdoctoral Researcher in the Program in Applied and Computational Mathematics at Princeton University from 2000-2003. He joined the Department of Mechanical Engineering at the University of California, Santa Barbara in 2003. He has been a recipient of a Sloan Research Fellowship in Mathematics and a National Science Foundation CAREER Award, and was Program Director of the SIAM Activity Group in Dynamical Systems from 2008-2009. He received a Northrop Grumman Excellence in Teaching Award from UCSB in 2008. Jeff's current research includes the application of dynamical systems and control techniques to neuroscience, cardiac dynamics, and collective behavior.
Why is sampling in high dimensions difficult and when can we sample effectively in high dimensions?
DateThursday, September 27, 2018 - 12:30pm
AbstractSampling from posterior distributions which are implicitly defined by a computational model and noisy observations is required in many applications in science and engineering. I will give several examples of such sampling problems which I work on and which span the much of the physics of our planet: from the Earth's deep interior, to oceans and clouds. A characteristic of many of the sampling problems that occur in science is that their "dimension" is large. I will discuss what "dimension" can mean in this context and I will show, using simple examples, how that makes sampling difficult (impossible in many cases). I will then describe two situations in which sampling can be done efficiently even if the dimension is huge.