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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

XPPAUT

Differential Equations Tool

B.Ermentrout & J.Rinzel


Preliminary remarks
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!


Focus of this presentation
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


Fitzhugh nagumo neuron 2 3 p161 163 4 p422 431
Fitzhugh-Nagumo Neuron[2 & 3.p161-163 & 4.p422-431]

  • Simple model of an excitable membrane:





Population growth model 3 p2 9
Population Growth Model[3.p2-9]

  • Simple model of growth:




Introduction of time delay
Introduction of Time Delay

  • No closed-form solution available

  • Dynamic is more interesting



Calcium regulation proc natl acad sci u s a 1990 78 1461 1465
Calcium RegulationProc.Natl.Acad.Sci. U.S.A. (1990) 78,1461-1465


Role of ip3
Role of IP3( )

  • Base parameter values are:







Conclusion
Conclusion

  • XPPAUT is a powerful tool for:

  • Solving ordinary and delay differential equations

  • Understanding the solution through bifurcation analysis.


References
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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