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Unformatted text preview: Computational Methods in Biology (Spring 2011) Assignment 4 (due in class on April 15) This assignment focuses on bursting oscillations and uses geometric singular per- turbation analysis and 2-parameter bifurcation diagrams. The assignment is to be done using XPPAUT. 1. [20 points] This problem examines the rst model for the bursting electrical activity of pancreatic -cells. These cells are located in cell clusters called islets of Langerhans and secrete the hormone insulin when they spike. Thus, bursts of electrical activity induce pulses of insulin secretion into the capillaries that penetrate the islets. The rst mathematical model for bursting in -cells was developed by Chay and Keizer in 1983. The model we will look at is a hybrid of the Chay-Keizer model and the Morris-Lecar model. The code ( CK.ode ) can be downloaded from my web site. The di erential equations are: dV dt =- ( I K + I Ca + I K ( Ca ) + I K ( ATP ) ) /C m dn dt = ( n ( V )- n ) / n dc dt = autoc ( cknot- c ) + (1- autoc ) f J mem where I K ( ATP ) is K + current that is inactivated by ATP (just think of it as a leak current), J mem is the Ca 2+ ux through the plasma membrane, and autoc and cknot are used to clamp c at the value cknot . To clamp c (i.e., make it a parameter), set autoc = 1. If autoc = 0 (the default value), c will change with time (i.e., it will be a variable). (a) Start the CK.ode code with XPP. You should see a bursting pattern. The goal now is to perform a fast/slow analysis of this system by treating calcium concentra- tion ( c ) as a slowly-changing parameter of the system. The rst step is to construct a) as a slowly-changing parameter of the system....
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