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XPPAUT

   From Scholarpedia
   Bard Ermentrout (2007), Scholarpedia, 2(1):1399. [3]doi:10.4249/scholarpedia.1399
    revision #136177 [[4]link to/cite this article]
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   Curator: [7]Bard Ermentrout
   Contributors:


   0.50 -

   [8]Eugene M. Izhikevich
   0.17 -

   [9]Warren Weckesser
   0.17 -

   [10]Nick Orbeck
   0.17 -

   [11]Martin Spacek

   [12]Richard Bertram

   [13]Benjamin Bronner
     * [14]Dr. Bard Ermentrout, Dept of Mathematics, Univ Pittsburgh, Pittsburgh PA

   Figure 1: XPPAUT interface with AUTO window

   XPPAUT is a general numerical tool for simulating, animating, and analyzing
   [15]dynamical systems. These can range from
     * discrete [16]finite state models (e.g., [17]McCulloch-Pitts neurons) to
     * [18]stochastic Markov models, to
     * discretization of [19]partial differential equations and
       [20]integro-differential equations.

   The program evolved from a DOS program that was originally written so that
   [21]John Rinzel and [22]Bard Ermentrout could easily illustrate the [23]dynamics
   of a simple model for an [24]excitable membrane. The DOS program, PHASEPLANE,
   became a commercial project and was used for many years by a number of patient
   folks.

Contents

     * [25]1 Current Status
     * [26]2 Features
     * [27]3 Examples
     * [28]4 References
     * [29]5 External Links
     * [30]6 See Also

Current Status

   The current version of XPPAUT runs under the X-window graphical environment and
   is distributed under the GNU public license. There are binary versions available
   for most [31]Linux distributions as well as for Mac OS X, iOS and various flavors
   of Windows.

Features

   XPPAUT contains many different integrators for
     * [32]stiff systems,
     * nonstiff systems,
     * [33]symplectic systems,
     * [34]discrete systems,
     * [35]Voltera systems,
     * [36]functional differential equations, and
     * stochastic systems.

   In addition, there are tools for analyzing the system such as the plotting of
   [37]Poincare sections, [38]delayed embeddings, [39]stability analysis,
   computation of one-dimensional [40]invariant manifolds, [41]nullclines, and
   [42]vector fields. Some statistical/data analysis is also incorporated, e.g.
   [43]spectral analysis, auto- and cross-[44]correlation, [45]histograms, mean, and
   variance. As XPPAUT was developed as a research tool for the author, it
   incorporates a number of specialized features such as the ability to compute the
   adjoint of a linearization about a [46]stable [47]limit cycle, a built in
   implementation of the [48]Gillespie algorithm, and built-in [49]averaging.

   XPPAUT also includes a frontend to [50]AUTO, a [51]continuation and
   [52]bifurcation package written by [53]Eusebius Doedel. This self-contained
   version of AUTO communicates seamlessly with XPP making it easy to continue the
   solutions to [54]boundary value problems as well as [55]equilibria, [56]fixed
   points and [57]limit cycles.

Examples

     * Ordinary differential equations.

   The [58]Morris-Lecar model has the form\[ C \frac{dV}{dt} = I-g_L(V-E_L)-g_K w
   (V-E_K)-g_{Ca} m_\infty(V)(V-E_{Ca}) \]

   \( \frac{dw}{dt} = \phi (w_\infty(V)-w)/\tau_w(V) \)

   The XPPAUT code for this set of equations is:
# [59]Morris-Lecar model Methods Chapter
dv/dt = ( I - gca*minf(V)*(V-Vca)-gk*w*(V-VK)-gl*(V-Vl))/c
dw/dt = phi*(winf(V)-w)/tauw(V)
# initial data
v(0)=-16
w(0)=0.014915
# functions
minf(v)=.5*(1+tanh((v-va)/vb))
winf(v)=.5*(1+tanh((v-vc)/vd))
tauw(v)=1/cosh((v-vc)/(2*vd))
# parameters
param vk=-84,vl=-60,vca=120
param i=0,gk=8,gl=2,c=20
param va=-1.2,vb=18
param vc=2,vd=30,phi=.04,gca=4.4
# parameter sets for different variants of the model
set hopf {vc=2,vd=30,phi=.04,gca=4.4}
set snic  {vc=12,vd=17.4,phi=.06666667,gca=4}
set homo {vc=12,vd=17.4,phi=.23,gca=4}
# some numerical settings
@ total=150,dt=.25,xlo=-75,xhi=75,ylo=-.25,yhi=.5,xp=v,yp=w
done

     * Delay equations

   The [60]Mackey-Glass equation has the form\[ \frac{dx}{dt} = -x(t) +
   \frac{A}{1+x(t-\tau)^p} \]

   and the XPPAUT code for this is:
x'=-x+a/(1+delay(x,tau)^p)
par a=5,p=2,tau=3
@ delay=10,total=200,xlo=0,xhi=200,ylo=0,yhi=5
done

   The penultimate line in the code contains delay=10 which tells XPPAUT how much
   storage to allocate for the delay. It must be larger than the largest delay in
   your model.
     * Networks

   Figure 2: Synaptic activity in a network of 100 cells

   A network of [61]quadratic integrate and fire models with synaptic coupling and
   adaptation can be written as\[ x_i' = x_i^2+a + \sum_{j=1}^N w(i-j) s_j(t) - g
   z_j \]

   \( s_i' = -s_i/\tau_s + \sum_m \delta(t-t_i^m) \)

   \( z_i' = -z_i/\tau_z + \sum_m \delta(t-t_i^m) \)

   Each time \( x_j \) reaches the "spiking" value, it is reset and the synapses, \(
   s_j \) and adaptation \( z_j \) are incremented. The XPPAUT code for this network
   is
# chain of QIF with adaptation
# weight matrix
table w % 25 -12 12 gsyn*exp(-abs(t)/sigma)/(2*sigma)
par sigma=4
# define the discrete convolution structure
special k=conv(even,100,12,w,s0)
# ODEs
x[0..99]'=x[j]^2+a+k([j])-g*z[j]
s[0..99]'=-s[j]/taus
z[0..99]'=-z[j]/tauz
# look for events
global 1 x[0..99]-xspike {x[j]=xreset;s[j]=1;z[j]=z[j]+1}
# parameters
par a=-.04,gsyn=1,taus=5,tauz=30,g=.1,xreset=-2,xspike=20
# start them at rest and set a few off
init x[4..99]=-.2
init x[0..3]=1
# set up plots
@ total=200,xhi=200,yp=x50,ylo=-5,yhi=20
done

   A sample simulation is shown at the right. Time is vertical and space is
   horizontal. A wave propagates producing several spikes before terminating. The
   synaptic variables are shown in color code. An animation is also shown in figure
   3. The code for this is
fcircle [0..99]/100;(x[j]+8)/30;.015;z[j]/4
end

   Figure 3: Animation of the potential colored according to the degree of
   adaptation

References

   Bard Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide
   to XPPAUT for Researchers and Students, SIAM 2002, Philadelphia, USA.

   Internal references
     * John W. Milnor (2006) [62]Attractor. [63]Scholarpedia, 1(11):1815.
     * Timothy D. Sauer (2006) [64]Attractor reconstruction. Scholarpedia,
       1(10):1727.
     * Jan A. Sanders (2006) [65]Averaging. Scholarpedia, 1(11):1760.
     * John Guckenheimer (2007) [66]Bifurcation. Scholarpedia, 2(6):1517.
     * James Meiss (2007) [67]Dynamical systems. Scholarpedia, 2(2):1629.
     * Willy Govaerts, Yuri A. Kuznetsov, Bart Sautois (2006) [68]MATCONT.
       Scholarpedia, 1(9):1375.
     * Harold Lecar (2007) [69]Morris-Lecar model. Scholarpedia, 2(10):1333.
     * Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) [70]Periodic orbit.
       Scholarpedia, 1(7):1358.
     * Carmen C. Canavier (2006) [71]Phase response curve. Scholarpedia, 1(12):1332.
     * Philip Holmes and Eric T. Shea-Brown (2006) [72]Stability. Scholarpedia,
       1(10):1838.
     * Lawrence F. Shampine and Skip Thompson (2007) [73]Stiff systems.
       Scholarpedia, 2(3):2855.

External Links

   [74]Author's webpage

   [75]XPPAUT page

   [76]XPP for Ios

See Also

   [77]Attractor, [78]AUTO97, [79]Bifurcation, [80]Dynamical Systems, [81]MATCONT,
   [82]Numerical Continuation, [83]Periodic Orbit, [84]Phase Model, [85]Phase
   Response Curve, [86]Stability

   [87]PyDSTool - XPPAUT alternative in Python
   Sponsored by: [88]Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the
   peer-reviewed open-access encyclopedia
   [89]Reviewed by: [90]Richard Bertram, Department of Mathematics, Florida State
   University, Tallahassee, Florida
   [91]Reviewed by: [92]Anonymous
   Accepted on: [93]2006-10-26 16:32:12 GMT
   Retrieved from
   "[94]http://www.scholarpedia.org/w/index.php?title=XPPAUT&oldid=136177"
   [95]Categories:
     * [96]Dynamical Systems
     * [97]Bifurcations
     * [98]Simulation Environment

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