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The classical HH ODEs

Overview

This chapter isolates the voltage-dependent gate functions used by the classical HH model. Rather than simulating a trace, it plots the equilibrium gate positions and their response speeds across voltage.

Core ideas

For a gate, an opening rate and a closing rate determine both where it settles and how quickly it gets there. Activation gates increase availability with depolarization in their relevant range; inactivation can move in the opposite direction. These curves explain why the same voltage equation can produce a brief spike instead of unbounded depolarization.

Essential model

For any gate $x$, the implementation can be written as

$$ \frac{dx}{dt}=\frac{x_\infty(V)-x}{\tau_x(V)},\qquad x_\infty(V)=\frac{\alpha_x(V)}{\alpha_x(V)+\beta_x(V)},\qquad \tau_x(V)=\frac{1}{\alpha_x(V)+\beta_x(V)}. $$

Here $x$ is one of the dimensionless gates $m$, $h$, or $n$; $V$ is membrane voltage; $x_\infty$ is its steady-state value; $\tau_x$ is its voltage-dependent time constant; and $\alpha_x$ and $\beta_x$ are the opening and closing rates, respectively.

Code examples

The example now lives in chapter03.ipynb: simulate_hh_gating_variables evaluates m_inf, h_inf, n_inf, and their time constants over a voltage grid, plotted as a six-panel figure.

What to look for

Compare the left-column steady-state curves with the right-column time constants. A high $m_\infty$ does not mean immediate sodium activation: the corresponding $\tau_m$ says how quickly it can approach that level.

Suggested order

  1. Inspect the rate functions alpha_* and beta_*.
  2. Relate each pair of rates to $x_\infty$ and $\tau_x$.
  3. Return to the voltage equation in Chapter 01 and identify each gate's conductance exponent.

Prerequisites and related chapters

Chapter 01 introduces the full HH balance equation. Chapter 04 uses the same rates during numerical integration. Familiarity with exponential functions and first-order relaxation is helpful.

Running the examples

Open chapter03.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom.