Graduate Student Seminar

Speaker 1: Jon Graf

Title

Implementation of the SDIRK method for the solution of the ODE system within a method of lines discretization for parabolic PDE

Abstract

Singly Diagonally Implicit Runge-Kutta (SDIRK) are multi-stage methods appropriate for the solution of stiff ODE problems. Implicit methods are necessary to solve stiff problems, but these methods require the computational cost of solving a non-linear system that explicit methods do not. We first verify our method implementation in Matlab with convergence studies for two scalar test problems with both implicit and explicit Runge-Kutta methods. Next, two time-dependent PDE problems are used to present results comparing a third order SDIRK method to Matlab's built in ode15s method. The method of lines is used for the numerical solution of the PDEs. The result of the spatial discretization is a large system of ODEs that must be solved at each time step thus at each time step we apply the ODE method. We use adaptive time stepping in which the high order of the method leads to larger time steps and thus fewer non-linear systems that need to be solved. Finally, we establish the context of this work within the three dimensional Calcium Induced Calcium Release (CICR) model, a three species non-linear reaction diffusion coupled PDE system.

Speaker 2: Nicole Massarelli

Title

Effects of Perturbations on Sensory Feedback During Lamprey Locomotion

Abstract

Lamprey locomotion is a complex closed-loop system involving muscles which move the body, neural activity in the spinal cord which activates the muscles, sensory feedback which regulates the neural signal, and fluid interactions which exert forces back on the body.  It is hard to describe how perturbations affect swimming because each subsystem has an input which is really the output of another subsystem.  For this reason we study open-loop maps, specifically how bending affects the sensory feedback in the lamprey which then acts as an input to the spinal cord.  To characterize sensory feedback properties, we perform bending experiments on the isolated spinal cord and record from edge cells which detect stretch in the body.  Frequency domain analysis allows us to characterize the response across a wide range of frequencies with only one type of bending signal.  This saves time and money by reducing the number of experiments needed to fully understand how the edge cells respond to perturbations.  I will describe these techniques and show the results for the edge cell experiments for the lamprey.