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Data analysis

The Analysis window (toolbar button Analysis, or Run > Analyse the run) holds XPPAUT's tools from Numerics > stocHastic and Numerics > Averaging. Each tool analyses the last run, exactly as XPPAUT's menus do, with XPPAUT's own code. Pick a tool on the left, set its options, and press Compute; the result is plotted on the right. Drag to pan, scroll to zoom, and right-click the plot for more options.

Every tool also has a command, listed in each section below.

Spectra

Spectrum estimates the power spectral density by Welch's method: the run is cut into windows that overlap by half, each window is tapered (Hamming by default) and transformed, and the results are averaged. Longer windows resolve frequencies more finely; shorter ones average more windows and give a smoother estimate. Cross spectrum and coherence compare two columns: coherence near 1 at a frequency means the two signals move in step there.

Frequencies are in cycles per unit of the model's time. A neuron firing every 18 ms shows a peak near 0.055 and its harmonics.

Fourier series treats the whole run as one period and gives the cosine and sine coefficients of each mode, or with Amplitude and phase their size and phase (XPPAUT's Power).

Commands: spectrum V 4096, spectrum V W coherence 1024, fourier V, power V.

Histograms and correlations

Histogram counts the points of a column in each bin, over the column's range or one you set. A condition such as W>0.2 counts only the points where it holds. 2-D histogram gives the fraction of points in each cell of a grid over two columns, drawn as a colour map.

Correlation is the cross-correlation of two columns, or the autocorrelation of one, with their means subtracted. Lags are in time units, so the first peak after zero of an autocorrelation is the period of an oscillation.

Commands: hist V 50, hist V 50 -0.5 0.5, hist2d V W 20, xcorr V, xcorr V W 200.

Statistics and the Liapunov exponent

Statistics reports the mean and standard deviation of a column, and the largest Liapunov exponent of the run. XPPAUT computes it by carrying a small perturbation along the run one output step at a time, renormalising it each step and averaging the logarithms of its growth. It is negative near a stable equilibrium, close to zero on a limit cycle and positive for chaos (about 0.9 for the Lorenz attractor). Use a long run that has settled.

Command: mean V.

Phase response curves

Phase response works on a model with a stable limit cycle. It does what XPPAUT users do by hand before Averaging > New adjoint:

  1. integrate past the transient onto the cycle (Transient; 0 uses the total time, and longer runs are tried if needed);
  2. measure the period from the upward crossings of the first variable;
  3. integrate exactly one period, which becomes the run;
  4. compute XPPAUT's adjoint along it.

Each column of the result is the infinitesimal phase response curve to small kicks in that variable: how far a kick at each phase moves the cycle forward, per unit of kick, in time units. Choose the variable with Show.

Try it: open Morris-Lecar, type use snic and set iapp=0.09, then compute the phase response (prc). Near this saddle-node bifurcation every kick advances the neuron (a type I curve). With use hopf and set iapp=0.26 early kicks delay it instead (type II).

Command: prc, or prc 500 to settle for 500 time units first.

Weakly coupled oscillators: the H function

H function computes the interaction function H(φ) of two copies of the oscillator coupled weakly: the coupling averaged over one cycle with the adjoint. Give one formula per variable; primed names are the other copy's variables. For gap-junction coupling through V, write V'-V for V and 0 for the others. XPP Lab finds the cycle and its adjoint first.

For a symmetric pair the phase difference φ changes at a rate proportional to −Hodd(φ), so the stable phase-locked states are the zeros of Hodd where it slopes upward. Pick Hodd or Heven with Show.

Command: hfun V'-V; 0.

Fitting parameters to data

Fit to data adjusts parameters, and initial conditions, so that the model matches measured data, by XPPAUT's Levenberg-Marquardt method. The data file holds rows of numbers: time first, then the measured columns.

  • Variables: the model variables to match, separated by commas.
  • Columns: their columns in the file, counting time as column 1.
  • Vary: the parameters and initial conditions to fit. They start from their current values.

The model's initial conditions apply at the first time in the file. When the fit converges, the fitted values replace the parameters and the model is run again to show the fitted solution.

Command: fitdata recording.dat V 2 gca,gk.