chapter01.md

Modeling a single neuron

Overview

This directory is the repository's first executable modeling example. It is not a mirror of the book's vocabulary-only Chapter 1: it starts immediately with a conductance-based Hodgkin--Huxley (HH) simulation and a voltage trace.

Core ideas

A membrane stores charge, so its voltage changes only when the injected and ionic currents do not balance. Sodium activation provides rapid positive feedback; sodium inactivation and potassium activation provide delayed negative feedback. The three gating variables make those conductances depend on both voltage and recent history.

Essential model

The voltage balance implemented here is

$$ C\frac{dV}{dt}=I_{\mathrm{ext}}-g_{\mathrm{Na}}m^3h(V-E_{\mathrm{Na}}) -g_{\mathrm{K}}n^4(V-E_{\mathrm{K}})-g_{\mathrm{L}}(V-E_{\mathrm{L}}). $$

Here $V$ is membrane voltage, $t$ is time, $C$ is membrane capacitance, $I_{\mathrm{ext}}$ is applied current, $g_{\mathrm{Na}}$, $g_{\mathrm{K}}$, and $g_{\mathrm{L}}$ are maximal sodium, potassium, and leak conductances, and $E_{\mathrm{Na}}$, $E_{\mathrm{K}}$, and $E_{\mathrm{L}}$ are their reversal potentials. $m$, $h$, and $n$ are dimensionless sodium-activation, sodium-inactivation, and potassium-activation gates. Each gate $x\in{m,h,n}$ follows $dx/dt=\alpha_x(V)(1-x)-\beta_x(V)x$, where $\alpha_x$ and $\beta_x$ are voltage-dependent opening and closing rates.

Code examples

The example now lives in chapter01.ipynb: simulate_hh_voltage_trace integrates the four HH state variables from their voltage-dependent initial values, and an ipywidgets slider lets you adjust i_ext interactively.

What to look for

The trace is the compact consequence of current balance: a depolarizing input first recruits $m$, then $h$ falls and $n$ rises to end the spike. Try changing i_ext only after confirming how the same initial state is built from m_inf, h_inf, and n_inf.

Suggested order

1. Run HH_VOLTAGE_TRACE and identify the resting level, upstroke, and recovery.
2. Read the derivative function alongside the equation above.
3. Continue to the gating curves in Chapter 03 before changing rate laws.

Prerequisites and related chapters

Comfort with first-order ODEs, units of current and voltage, and Python arrays is enough. Chapter 03 separates the gates into steady-state and time-constant curves; Chapter 04 focuses on numerical trajectories.

Running the examples

Open chapter01.ipynb in Jupyter, or via the Colab badge at the top of the notebook. Run all cells top to bottom; the static figure reproduces the book's plot, and the interact(...) cell below it lets you adjust i_ext with a slider.