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brainpy.analysis.PhasePlane2D.plot_nullcline has two branches #718
Description
In my single compartment HH-like model, the nullcline of the voltage variable has two branches. When I use brainpy.analysis.PhasePlane2D.plot_nullcline() to plot the nullcline, the endpoints of the two branches are connected. Is there any way to avoid this?
Activity
chaoming0625 commentedon Feb 25, 2025
can you show the image?
Newtonpula commentedon Feb 25, 2025
chaoming0625 commentedon Feb 25, 2025
what does
the endpoints of the two branches are connectedmean?Newtonpula commentedon Feb 25, 2025
The middle part of the N-shape V nullcline should not exist, it just connects the endpoints of the left branch and the right branch of the V nullcline.
Attach is the same plot with fmt='o', but I want to use fmt='-'.
chaoming0625 commentedon Feb 25, 2025
Maybe you need to plot by yourself.
plot_nullcline(with_return=True)will return all points, and then you can draw what you want. Can this solve your question?Newtonpula commentedon Feb 25, 2025
Yes, it solves my question. Thanks very much.
github-actions commentedon Oct 7, 2025
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Han-Shen-math commentedon Oct 12, 2026
I investigated the branch-connectivity question behind this report and prepared a short research note and a reproducible test package. The existing
with_return=Trueworkaround helped the reporter; the attached work considers what guarantees an automatic replacement could offer.The general limitation: there is no algorithm that always terminates and correctly determines the complete connected-component structure for every smooth equation, even when the complete finite formula, all constituent functions, and all coefficients are visible. The precise formula language in the note allows rational constants, pi, sine, arithmetic, exponential, division on its domain, and explicitly written smooth piecewise definitions. Even deciding whether the zero set in the fixed square
[-1,1]^2has exactly two components is undecidable in this class.The proof reduces Wang's established real-zero existence theorem (1974) to that fixed-square question using an explicit smooth compactification. It permits degenerate zero sets and infinitely many components. It does not establish impossibility under additional regularity assumptions, or for rational polynomial curves. The note is an AI-assisted research exposition based on prior results, not a claim of a newly discovered foundational theorem.
A practical direction for BrainPy: specialize by the structure of the nullcline equation and preserve branch connectivity in the plotting representation. BrainPy accepts user-defined ODE functions, so the admissible inputs are not limited to a finite catalogue of built-in models. Nevertheless, common model families can be supported progressively:
The artificial connecting segments in this issue remain a concrete plotting problem that can be addressed without solving the unrestricted topology problem. A staged implementation would need additional model-specific work and regression cases covering disconnected branches, genuine connections, folds, small loops, and singularities. The original HH-like equations are not included in the issue, so these experiments do not reproduce that exact model.
Attached validation (standalone research code, Mac CPU):
The unresolved/unsupported cases are retained in the results. Tests exercise the implementations; the impossibility conclusion comes from the mathematical proof. These are not full BrainPy integration tests or evidence that #718 has already been fixed.
Files:
Start with
REPRODUCE.mdin the ZIP. The certificate checker is implemented separately from the generator, although both share the SymPy backend. The package is a research prototype with documented assumptions, not a proposed drop-in BrainPy patch.