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 Regulation Proc.Natl.Acad.Sci. U.S.A. (1990) 78,1461-1465

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

15. [Ca] vs. Time(s)

17. Bifurcation Diagram

18. Calcium Entry From Extracellular Space

20. [Ca] in ER

21. Bifurcation Diagram

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

23. 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,1 st edition,Springer-Verlag,New York. [4] Fall,C, et al,(2002) Computational Cell Biology,1 st edition,Springer-Verlag,New York [5] Bard Ermentrout XPPAUT5.41 Differential equations tool(August,2002) www.math.pitt.edu/~bard/xpp/xpp.html