Cross Section

This section demonstrates how to use COPASI’s cross section task. We start by importing an oscillating model from the BioModels Database, explore its dynamics with a time course simulation, and then extract characteristic events using the cross section task.

The Model

First, download BioModel 239 from the BioModels Database. Import this model into COPASI. Then, navigate to the Tasks section, and select Time Course. Set the duration to 100 seconds and use the automatic interval size.

To visualize the simulation run, use the output assistant to create a plot for “Concentrations, Volumes, and Global Quantity Values.” Click “Hide All,” then select “[Calcium]” to show just its trace. The output will be similar to:

To understand how changing a parameter affects the simulation, use COPASI’s slider feature. Select Tools > Show Sliders or click the slider icon in the toolbar. Click New Sliders and select the “constant” parameter of reaction R2.

The initial value is 2.9. Try changing it from 1.5 to 3 while observing the plot. You’ll see the oscillatory dynamics switch from basic oscillations to bursting:

Cross Section

The cross section task lets you detect when the system crosses a specified value of a variable (a “surface” in phase space), in a specified direction. For this tutorial, we identify the maxima (peaks) of the previously observed calcium oscillations. Select the rate of calcium as the variable in the cross section task, detect crossings of the value zero in the negative direction, and stop detection after 100 seconds.

To visualize results, modify the plot so that values are shown as circles rather than lines:

Now, run the cross section task and you will obtain:

These points correspond to the maxima observed earlier.

Creating a Bifurcation Diagram

The oscillation behavior changes when adjusting reaction R2’s parameter. We can systematically illustrate this using a scan task in combination with the cross section task to generate a bifurcation diagram.

1. Define a scan over R2.constant, varying it in 100 intervals from 1.5 to 3.
2. Set the subtask to “Cross Section.”
3. Enable “output during subtask execution.”

For the plot, set R2.constant as the x-axis and calcium concentration as the y-axis. Use circles for points. (Alternatively, the output assistant’s “Scan of Concentrations, Volumes and Global Quantity Values” is suitable.)

This plot shows all maxima, not just those from the limit cycle. To filter for relevant crossings, configure the cross section task to only collect crossings after a certain condition, e.g., when “time > 50”.

Re-running the parameter scan now yields only those crossings of interest:

Cross Section Task Options

The cross section task has options to efficiently find crossing points. They fall into two groups:

First Group: When to Stop Collecting Crossings

• If crossings (#) larger: Stop after a set number of crossing points.
• If a crossing is repeated within tolerance: Stop if crossing points start repeating (within a set relative accuracy), typically indicating a limit cycle.

Note: You must set a maximum simulation time to ensure simulation ends even if crossings are absent or not repeating.

Second Group: When to Start Reporting Crossings

• If crossing (#) larger: Begin reporting after a set number of crossings.
• If a crossing is repeated within tolerance: Start only once crossings reoccur within the given tolerance. This helps when looking for a limit cycle, but isn’t useful for chaotic dynamics.

Complete Example Download:
Download BIOMD0000000329_scan.cps

Run in COPASI

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Example 2: Route to Chaos via Period Doubling

As another example, we illustrate the route to chaos through period doubling bifurcations using a chemical-kinetics analog of Rössler’s chaotic attractor [1],[2].

As before, set up the cross section task to detect maxima of a variable (X2). Recommended settings:

• Start plotting after at most 50 oscillations, or sooner if a limit cycle is detected (accuracy: 1e-5).
• Stop plotting after 1000 time units or after 50 maxima are detected, whichever comes first. Also stop earlier if a limit cycle is found.

The scan task scans the autocatalytic expansion parameter from 0.05 to 0.2. Define a plot with the scan parameter on x-axis and X2 concentration on the y-axis.

This yields the classic Feigenbaum (period doubling) diagram:

Complete Example Download:
Download chaos3v2.cps Run in COPASI

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References

[1] O. E. Rössler: An Equation for Continuous Chaos. Physics Letters Vol. 57A no 5, pp 397-398, 1976.

[2] Baier, G., & Sahle, S. (1995). Design of hyperchaotic flows. Physical Review E, 51(4), R2712–R2714. doi:10.1103/PhysRevE.51.R2712