chapter08.md

Quadratic integrate-and-fire and theta neurons

Overview

Quadratic integrate-and-fire (QIF) neurons replace LIF's linear drift with a quadratic voltage flow. Their apparent divergence is converted into a regular spike cycle by a reset convention, or equivalently represented as smooth motion around the circle by the theta transformation.

Core ideas

A QIF trajectory can reach infinity in finite time; this is the model's spike event, not a claim that a physical membrane voltage is infinite. Resetting from the upper to lower branch makes a repeated trace. Under a voltage-to-phase change of variables, the two infinities meet at one point on a circle, so the same firing process becomes a continuous phase flow.

Essential model

The QIF examples use

$$ \frac{dV}{dt}=-\frac{V}{\tau_m}(1-V)+I, \qquad V=1-\cos\theta. $$

Here $V$ is normalized voltage, $t$ is time, $\tau_m$ is the membrane time constant, $I$ is constant input, and $\theta$ is circular phase. The QIF trace resets after passing the finite numerical threshold $V=1$, while the analytic infinite-threshold construction connects the positive and negative infinite branches. In the circle diagram, the transition occurs at $I=1/(4\tau_m)$, where $I$ and $\tau_m$ have the meanings above.

Code examples

All four examples now live in one notebook, chapter08.ipynb: simulate_qif_voltage_trace integrates a QIF trace with a midpoint step and reset convention; simulate_qif_infinite_threshold constructs the analytic finite-time blow-up branches and their continuation; simulate_theta_firing integrates phase $\theta$ and plots $1-\cos\theta$; plot_three_circles draws phase flows below, at, and above the firing transition on three circles (a schematic, no simulation). Each simulated example has an ipywidgets slider to explore its parameters interactively.

What to look for

The dashed segments in QIF_VOLTAGE_TRACE mark the reset convention. Compare that with QIF_INFINITE_THRESHOLD, where the finite-time divergence is shown explicitly. THETA_FIRING has no reset jump because phase wraps naturally; THREE_CIRCLES explains why the three input regimes have different geometry.

Suggested order

1. Run QIF_VOLTAGE_TRACE to see the practical event rule.
2. Run QIF_INFINITE_THRESHOLD to see what that rule abbreviates.
3. Run THETA_FIRING, then use THREE_CIRCLES to interpret the phase flow.

Prerequisites and related chapters

Chapter 07 supplies the linear integrate-and-fire baseline. Familiarity with one-dimensional ODEs, trigonometric functions, and phase portraits is useful. Later network examples use theta-neuron phase descriptions.

Running the examples

Open chapter08.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.