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

• XPPAUT can do all that for us! FOR FREE!

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]

• Simple model of an excitable membrane:

cell theinto injectedCurrent I

variableGatingw

potential MembraneV

).(

.)).(.(.

applied

wVdt

dw

IwCVVVBdt

dVapplied

Iapplied=0

Iapplied=0.5

Bifurcation Diagram:

Population Growth Model[3.p2-9]

• Simple model of growth:

•

capacity talEnvironmenk

growth ofconstant Rate

)/1(

r

kNrNdt

dN

Solution:

tKtN

eNk

ekNtN

rt

rt

as )(

)]1).(0([

.)0()(

Sample Curve:

Introduction of Time Delay

• No closed-form solution available • Dynamic is more interesting

signal.

inhibitory ofn propagatiofor delay Time

)/)(1(

T

kTtNrNdt

dN

Oscillatory Behavior in Model with Delay

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

cytosol into vesiclesER fromleak Calcium

membrane plasman through eliminatio Calcium

ER from release Calcium induced CalciumJRyR

. pump calciumdependent ATP

ER. from release calcium induced 3

spacelar extracellu from cytosol intoleak Calcium0

][

][.

][

][,

][

][2

][][

][][]3.[0][

5

4

1

44

4

22

2

3222

2

5

541

k

k

pump

IPv

v

CaK

Ca

CaK

CavJRyR

CaK

Cavpump

CakJRyRpumpdt

Cad

CakCakJRyRpumpIPvvdt

Cad

AERR

ER

ERER

ER

Role of IP3( )

• Base parameter values are:

sMv /10

sMv /3.71

M 0

sMv /652

MK 12

sMv / 5003

MKR 2

MK A 9.0

14 10 sk

15 1 sk

[Ca] vs. Time(s) 0

M 5.0

Bifurcation Diagram

Calcium Entry From Extracellular Space

sMv /10

sMv /2.30

[Ca] in ER

Bifurcation Diagram

Conclusion

• XPPAUT is a powerful tool for:

• Solving ordinary and delay differential equations

• Understanding the solution through bifurcation analysis.

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