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:
dV
 B.V .(V   ).(  V )  C.w  I applied
dt
dw
  (V   .w)
dt
V  Membrane potential
w  Gating variable
I applied  Current injected into the cell
Iapplied=0
Iapplied=0.5
Bifurcation Diagram:
Population Growth Model[3.p2-9]
• Simple model of
growth:
dN
• dt  rN (1  N / k )
r  Rate constant of growth
k  Environmen tal capacity
Solution:
rt
N (0)k .e
N (t ) 
rt
[k  N (0).(e  1)]
N (t )  K as t  
Sample Curve:
Introduction of Time Delay
• No closed-form solution available
• Dynamic is more interesting
dN
 rN (1  N (t  T ) / k )
dt
T  Time delay for propagatio n of inhibitory
signal.
Oscillatory Behavior in Model with
Delay
Calcium Regulation
Proc.Natl.Acad.Sci. U.S.A. (1990) 78,1461-1465
d [Ca]
 v0  v1.[ IP3]  pump  JRyR  k 4 [Ca]  k5 [Ca]ER
dt
d [Ca]ER
 pump  JRyR  k5 [Ca]ER
dt
2
[Ca]2
[Ca]ER
[Ca]4
pump  v 2 2
, JRyR  v3
. 4
2
2
2
K 2  [Ca]
K R  [Ca]ER K A  [Ca]4
v0  Calcium leak into cytosol from extracellu lar space
v1  IP3 induced calcium release from ER.
pump  ATP dependent calcium pump .
JRyR  Calcium induced Calcium release from ER
k 4  Calcium eliminatio n through plasma membrane
k5  Calcium leak from ER vesicles into cytosol
Role of IP3()
• Base parameter values are:
v0  1M / s
v3  500 M / s
v1  7.3M / s
K R  2M
  0 M
K A  0.9M
v2  65M / s
K 2  1M
k4  10s 1
k5  1s 1
[Ca] vs. Time(s)
 0
  0.5M
Bifurcation Diagram
Calcium Entry From Extracellular
Space
v0  1M / s
v0  3.2M / s
[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