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Numerical solution of HH ODEs

Overview

These examples use numerical integration to turn the HH equations into time-series and state-space plots. Together they show a transient approaching a repeating orbit and how the evolving gates produce refractoriness.

Core ideas

An ODE solver approximates a continuous trajectory at a finite set of times. After a transient, a periodically spiking HH state can approach a limit cycle. During and after a spike, sodium inactivation and potassium activation leave the state temporarily less excitable; a pulse therefore has a different effect at different phases.

Essential model

The solver advances the state vector

$$ \mathbf{x}=(V,m,n,h),\qquad \frac{d\mathbf{x}}{dt}=\mathbf{F}(\mathbf{x};I_{\mathrm{ext}}). $$

Here $V$ is membrane voltage; $m$, $n$, and $h$ are the dimensionless HH gates; $t$ is time; $I_{\mathrm{ext}}$ is applied current; and $\mathbf{F}$ is the HH voltage-and-gate right-hand side defined in each derivative function. A limit cycle is a closed trajectory in this state space that repeats after one period.

Code examples

All three examples now live in one notebook, chapter04.ipynb: simulate_hh_solution integrates voltage and all three gates at a constant input; simulate_hh_limit_cycle starts away from equilibrium and plots $n$ against $V$; simulate_hh_refractoriness adds a short current pulse at several onset times. Each has an ipywidgets slider to explore i_ext interactively.

What to look for

HH_SOLUTION shows the initial transient before the repeating regime. HH_LIMIT_CYCLE removes time from view so the returning state is visible. In HH_REFRACTORINESS, compare the response to each red pulse marker: the same perturbation can fail, delay, or advance a spike depending on gate state.

Suggested order

  1. Run HH_SOLUTION and identify its transient and later periodic segment.
  2. Run HH_LIMIT_CYCLE to see that periodic segment as a closed projection.
  3. Run HH_REFRACTORINESS to connect trajectory phase with excitability.

Prerequisites and related chapters

Chapter 01 supplies the HH current balance and Chapter 03 explains its gate functions. The LIF comparison in Chapter 07 offers a reset-based alternative with no explicit ionic gates.

Running the examples

Open chapter04.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 i_ext with a slider.