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Two-dimensional bifurcation analysis

Overview

The historical folder title is retained, but these scripts correspond to the book's type-1 onset material. They reduce the RTM neuron to two variables and show its fixed points, invariant cycles, and saddle-node-on-invariant-circle (SNIC) onset geometry.

Core ideas

In a two-dimensional reduction, fixed points are intersections of nullclines and their Jacobian eigenvalues classify stability. A periodic orbit is an invariant cycle. At type-1 onset, a saddle and node meet on the cycle, so the orbit spends an increasingly long time near the collision and its firing rate approaches zero.

Essential model

The reduced RTM dynamics use instantaneous activation and the approximation $m=m_\infty(v)$, $h=1-n$:

$$ C\dot v=I_{\rm Na}(v,n)+I_{\rm K}(v,n)+I_L(v)+I,\qquad \dot n=\alpha_n(v)(1-n)-\beta_n(v)n. $$

Here $v$ is membrane voltage, $n$ is potassium activation, $I$ is applied current, and the sodium, potassium, and leak terms use RTM conductances and reversal potentials.

Code examples

Both examples now live in one notebook, chapter12.ipynb: simulate_rtm_2d_fp scans applied current, finds reduced RTM fixed points, and classifies their Jacobian eigenvalues; plot_rtm_2d_invariant_cycle plots trajectories and crossings around the RTM invariant cycle and its onset geometry.

What to look for

RTM_2D_FP separates branches by stability and eigenvalue type. In RTM_2D_INVARIANT_CYCLE, compare trajectories at and around the critical current: motion slows near the saddle-node region, which is the phase-plane signature behind the low-frequency type-1 onset.

Suggested order

  1. Run RTM_2D_FP to locate and classify equilibria.
  2. Run RTM_2D_INVARIANT_CYCLE to see how trajectories use those structures.
  3. Continue to Chapter 17 to measure the resulting frequency-current curve.

Prerequisites and related chapters

Chapter 10 introduces reductions and nullclines, and Chapter 11 introduces saddle-node collisions. Chapter 14 contrasts type-2 onset; Chapter 17 turns these onset mechanisms into firing-rate curves.

Running the examples

Open chapter12.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom.