chapter10.md

The slow-fast phase plane

Overview

These examples reduce conductance-based spiking to phase-plane geometry. They compare the visibly separated slow and fast variables in FitzHugh--Nagumo and Hodgkin--Huxley (HH) systems, then show how that separation organizes a periodic orbit.

Core ideas

When one state changes much faster than another, trajectories move quickly between branches of a slow manifold and linger near slow branches. Nullclines locate the directions with zero velocity, while a closed orbit describes repeated firing. A reduction with instantaneous sodium activation and $h+n=0.83$ makes this geometry visible in a two-dimensional HH system.

Essential model

The FitzHugh--Nagumo example uses

$$ \dot v=v-v^3/3-n+I,\qquad \dot n=(av-n)/\tau_n. $$

Here $v$ is the fast voltage-like variable, $n$ is the slow recovery variable, $I$ is applied current, $a$ sets the recovery nullcline, and $\tau_n$ makes recovery slow. The reduced HH examples take $m=m_\infty(v)$ and $h=0.83-n$, leaving $(v,n)$ as the phase plane.

Code examples

All five examples now live in one notebook, chapter10.ipynb: simulate_fn integrates FitzHugh-Nagumo and plots its nullclines, orbit, and voltage trace; simulate_hh_h_plus_n plots the HH combination $h+n$ against its 0.83 approximation; simulate_reduced_hh compares the reduced HH voltage and gates with the imposed $h=0.83-n$ relation; simulate_hh_nullclines_plus_solution draws the reduced HH nullclines and a trajectory with direction arrows; simulate_hh_cycle_speed colors portions of the HH orbit by their phase-plane speed. The nullcline construction shared by the last two lives in hh_v_nullcline. Each has an ipywidgets slider to explore its parameters interactively.

What to look for

In FN, find the long motion near the cubic nullcline and the rapid jumps between its outer branches. Compare HH_H_PLUS_N with REDUCED_HH before using the two-dimensional HH plots. In HH_CYCLE_SPEED, green points mark the slow portions of the orbit and blue points the faster traversal.

Suggested order

1. Run FN to connect nullclines to slow-fast motion.
2. Run HH_H_PLUS_N and REDUCED_HH to inspect the HH reduction.
3. Use HH_NULLCLINES_PLUS_SOLUTION and HH_CYCLE_SPEED to read its cycle.

Prerequisites and related chapters

This chapter builds on HH gates from Chapters 03--04 and the reduced models introduced in Chapter 05. Chapters 11--15 use the same phase-plane language to study bifurcations and canards.

Running the examples

Open chapter10.ipynb in Jupyter, or via the Colab badge at the top of the notebook. Run all cells top to bottom; each section's static figure reproduces the book's plot, and the interact(...) cell below it lets you adjust that example's parameters with sliders.