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XPPAUT

XPPAUT. Differential Equations Tool B.Ermentrout & J.Rinzel. Preliminary Remarks. Nonlinear ODEs do not usually have closed form solutions Numerical solutions are needed Qualitative analysis: phase plane analysis, bifurcation analysis,stability of steady states

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XPPAUT

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  1. XPPAUT Differential Equations Tool B.Ermentrout & J.Rinzel

  2. Preliminary Remarks • Nonlinear ODEs do not usually have closed form solutions • Numerical solutions are needed • Qualitative analysis: phase plane analysis, bifurcation analysis,stability of steady states • XPPAUT can do all that for us! FOR FREE!

  3. Focus of this presentation: We will use XPPAUT for solving : -FitzHugh-Nagumo model of excitable membrane -Population growth model with time delay -Model of intracellular Calcium regulation

  4. Fitzhugh-Nagumo Neuron[2 & 3.p161-163 & 4.p422-431] • Simple model of an excitable membrane:

  5. Iapplied=0

  6. Iapplied=0.5

  7. Bifurcation Diagram:

  8. Population Growth Model[3.p2-9] • Simple model of growth:

  9. Solution:

  10. Sample Curve:

  11. Introduction of Time Delay • No closed-form solution available • Dynamic is more interesting

  12. Oscillatory Behavior in Model with Delay

  13. Calcium RegulationProc.Natl.Acad.Sci. U.S.A. (1990) 78,1461-1465

  14. Role of IP3( ) • Base parameter values are:

  15. [Ca] vs. Time(s)

  16. Bifurcation Diagram

  17. Calcium Entry From Extracellular Space

  18. [Ca] in ER

  19. Bifurcation Diagram

  20. Conclusion • XPPAUT is a powerful tool for: • Solving ordinary and delay differential equations • Understanding the solution through bifurcation analysis.

  21. References • [1] Goldbeter,A.,Dupont,G., and Berridge,M.(1990). Proc.Natl.Acad.Sci.U.S.A. 87 1461-1465. • [2] FitzHugh,R.(1961).Biophys J.1,445-466 • [3] Murray J.(1989) .Mathematical Biology,1st edition,Springer-Verlag,New York. • [4] Fall,C, et al,(2002) Computational Cell Biology,1st edition,Springer-Verlag,New York • [5] Bard Ermentrout XPPAUT5.41 Differential equations tool(August,2002) • www.math.pitt.edu/~bard/xpp/xpp.html

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